Effects of Kinematic Hardening of Mucus Polymers in an Airway Closure Model

Abstract

The formation of a liquid plug inside a human airway, known as airway closure, is computationally studied by considering the elastoviscoplastic (EVP) properties of the pulmonary mucus covering the airway walls for a range of liquid film thicknesses and Laplace numbers. The airway is modeled as a rigid tube lined with a single layer of an EVP liquid. The Saramito-Herschel-Bulkley (Saramito-HB) model is coupled with an Isotropic Kinematic Hardening model (Saramito-HB-IKH) to allow energy dissipation at low strain rates. The rheological model is fitted to the experimental data under healthy and cystic fibrosis (CF) conditions. Yielded/unyielded regions and stresses on the airway wall are examined throughout the closure process. Yielding is found to begin near the closure in the Saramito-HB model, whereas it occurs noticeably earlier in the Saramito-HB-IKH model. The kinematic hardening is seen to have a notable effect on the closure time, especially for the CF case, with the effect being more pronounced at low Laplace numbers and initial film thicknesses. Finally, standalone effects of rheological properties on wall stresses are examined considering their physiological values as baseline.

1. Introduction

Inner walls of airways are coated with a liquid film called airway surface liquid (ASL). In a healthy human lung, thickness of liquid film is approximately 2 – 4% of the airway diameter [1]. For the first 15-16 generations of the pulmonary airways, ASL is comprised of two liquid layers. The radially inward layer is mucus and the radially outward layer, also known as periciliary liquid (PCL), is serous [2]. For the remaining generations, the liquid film becomes comprised of only a single layer, with attributes similar to those of water.

Airway closure could occur when thickness of liquid film, compared to the diameter of the airway, exceeds a critical level as discussed by Gauglitz and Radke [3]. In such a case, a very slight perturbation of the liquid film triggers the Plateau-Rayleigh instability driven by the surface tension at the liquid-air interface. The instability leads to formation of a liquid plug that blocks the airway and all distal parts. This phenomenon is known as the airway closure that can occur in healthy and pathological conditions and it is most usually observed in small airways during exhalation [4, 5].

Several studies have been conducted [6, 7, 8] to examine the rheological properties of the pulmonary mucus. These studies confirm viscoelastic, viscoplastic, and shear thinning characteristics of the pulmonary mucus suggesting that it can be classified as an elastoviscoplastic (EVP) fluid. EVP materials exhibit complex, non-Newtonian behavior, and can be encountered frequently in everyday life. There have been many constitutive models developed to describe behavior of such complex fluids. Saramito[9] integrated both the viscoplastic Bingham and the viscoelastic Oldroyd-B models in a thermodynamic framework. The model has a capability to treat fluid as a viscoelastic solid below the yield stress and as a viscoelastic fluid above the yield stress. An improvement to this model was later proposed by Saramito[10], which combines the viscoplastic Herschel-Bulkley model with the viscoelastic Oldroyd-B model. Saramito’s models and the model introduced by Park and Liu [11] based on the works of Papanastasiou[12] and Belblidia et al. [13] could be regarded as the same family of models that are capable of incorporating EVP properties in physically-sound constitutive equations.

In recent years, an experimental procedure called Large Amplitude Oscillatory Shear (LAOS) test [14] has been utilized for determining rheological properties of yield-stress fluids. Through this procedure, it is possible to extract information about fluid, such as the storage, $G^{\prime }$, and the loss, $G^{″}$, moduli with respect to strain amplitude or stress. When data acquired from LAOS tests are considered, the previously mentioned EVP models fail to correctly predict loss-modulus of certain EVP fluids, especially in regions of low strain amplitude in strain amplitude sweep testing [15]. This is due to the fact that, below yield stress, aforementioned EVP models treat fluid as an elastic solid in the absence of solvent viscosity and therefore, they fail to properly account for energy dissipation in this region. To tackle this problem, Dimitriou et al. [16] introduced the concept of kinematic hardening, which allows energy dissipation before yielding. With utilization of an isotropic kinematic hardening mechanism, the loss modulus of the fluid is captured more accurately in low-strain amplitude regions of the LAOS tests. Fraggedakis et al. [15] incorporated the isotropic kinematic hardening model of Dimitriou et al. [16] into the Saramito EVP model, and demonstrated that the new model results in much better agreement with the experimental data than the original EVP model for the loss modulus. In this paper, the isotropic kinematic hardening model is integrated into the Saramito-Herschel-Bulkley (Saramito-HB) EVP model to accurately capture behavior of pulmonary mucus.

Studies show that airway closure may have an important impact on overall health of pulmonary epithelial cells. Formation, propagation, and finally rupture of a liquid plug that blocks an airway may lead to very high mechanical stresses on the epithelial cells forming the airway wall and can cause serious damage [17, 18, 19, 20]. Numerical and experimental investigations [21, 22] show that, in a rigid pipe of microscopic scale, formation of a liquid plug can cause large enough stresses on pipe walls suggesting that stresses on the epithelial cells during plug formation in airways may exceed the critical limits for cell viability [19, 17]. Recently, a numerical study by Romano et al. [23] considered the whole airway closure process using a single-layer Newtonian liquid model. Their results show that the highest values of stresses on the walls occur just after the coalescence. They later considered the viscoelastic Oldroyd-B and FENE-CR models for the liquid film [24] and demonstrated that, in addition to the first Newtonian stress peaks occurring right after the coalescence, a curvature-induced viscoelastic instability can lead to secondary stress peaks that are as large as the first ones, especially in the cases of high Weissenberg and Laplace numbers. Romano et al. [25] also examined impact of a surfactant under a single-layer setting with a Newtonian model for the ASL. They showed that the closure is slowed and magnitudes of wall stresses are reduced significantly as surfactant concentration is increased. Erken et al. [26] proposed a two-layer all-Newtonian model for the ASL and performed numerical simulations to examine effects of serous layer on the airway closure phenomenon. They showed that the existence of the serous layer leads to an earlier closure and damping of mechanical stresses on airway wall compared to the single-layer model. In the recent study by Erken et al. [27], a single-layer Saramito-HB EVP fluid model is used to examine effects of elastoviscoplasticity on the airway closure process considering the rheological properties of mucus in healthy, asthma, chronic obstructive pulmonary disease (COPD) and cystic fibrosis (CF) conditions.

In the present paper, an isotropic kinematic hardening model is incorporated into the Saramito-HB EVP model [10] to investigate effects of kinematic hardening on the airway closure, and report on the shortcomings of the Saramito-HB model via a direct comparison with the findings of Erken et al. [27], which constitutes the main novelty of the present study. For this purpose, extensive numerical simulations are performed to simulate airway closure using the Saramito-HB model without and with the isotropic kinematic hardening model, and the results are compared in terms of airway closure time and mechanical stresses on airway walls. A single-layer model is used for the ASL to isolate the sole effects of the EVP properties of the pulmonary mucus on airway closure. The baseline simulations are performed for the rheological properties of mucus in healthy and CF conditions using the experimental data of Patarin et al. [7]. To the best knowledge of the authors, this is the first study in the literature where the effects of kinematic hardening are examined for the pulmonary mucus using an EVP fluid model. In addition, individual effects of material properties of pulmonary mucus such as elastic modulus, yield stress, polymeric viscosity, and yield strain indicator are examined systematically in terms of airway closure time and mechanical stresses on the airway wall.

2. Formulation and numerical method

As shown in Fig. 1, a segment of an airway with a length of $L^{\ast }$ and a radius of $a^{\ast }$ is considered. Flow is assumed to be incompressible and axisymmetric, and the airway wall is rigid. The liquid layer adjacent to the wall is modeled as an EVP fluid while air is Newtonian. Following Romanò et al. [23, 24], the properties of the mucus and the serous layers are homogenized into a single layer.

Fig. 1.

(Left) Illustration of an airway segment. (Right) Schematic of the airway closure model. The domain is axisymmetric with a length of $L^{\ast }$ and a radius of $a^{\ast }$. Periodic and no-slip boundary conditions are applied in the axial direction and on the wall, respectively. The EVP liquid layer is adjacent to the wall and is surrounded by air. Surface tension at the air-liquid interface is ${\sigma }^{\ast }$. The undisturbed initial thickness of the liquid layer is $h^{\ast }$ and the initial radial distance from the centerline to the liquid layer is $R_I^{\ast }$. The mechanical analog distinguishing the kinematic hardening model from the common viscoelastic and elastoviscoplastic models is highlighted on the right side of the sketch.

The flow equations are described in the framework of the finite-difference/front-tracking method using a one-field formulation [28, 29, 27]. All the equations are solved in their dimensional forms, indicated by ${\text{superscript}}^{\ast }$, however, results are presented in the non-dimensional forms. In this formulation, the momentum and mass conservation equations can be written for the whole computational domain as

$$\begin{gathered} \frac{\partial {\rho }^{\ast }{\mathit{u}}^{\ast }}{\partial t^{\ast }}+{\nabla }^{\ast }\cdot ({\rho }^{\ast }{\mathit{u}}^{\ast }{\mathit{u}}^{\ast })=-{\nabla }^{\ast }{p}^{\ast }+{\nabla }^{\ast }\cdot {\mu }_{L,S}^{\ast }\left({\nabla }^{\ast }{\mathit{u}}^{\ast }+{\nabla }^{\ast }{\mathit{u}}^{\ast T}\right)+{\nabla }^{\ast }\cdot {\tau }^{\ast } \\ +{\int }_{S^{\ast }}{\sigma }^{\ast }{\kappa }^{\ast }\mathit{n}\delta ({\mathit{x}}^{\ast }-{\mathit{x}}_f^{\ast })d{S}^{\ast }, \end{gathered}$$ (1)

$${\nabla }^{\ast }\cdot {\mathit{u}}^{\ast }=0,$$ (2)

where $t^{\ast }$, ${\mathit{u}}^{\ast }$, $p^{\ast }$, ${\tau }^{\ast }$, ${\rho }^{\ast }$ and ${\mu }_{L,S}^{\ast }$ are time, velocity vector, pressure field, extra stress tensor, discontinuous density and solvent viscosity, respectively. The term on the right-hand side of Eq. (1) represents a body force due to surface tension where ${\sigma }^{\ast }$ is surface tension coefficient, ${\kappa }^{\ast }$ is twice the mean curvature of the interface, $\mathit{n}$ is a unit normal vector to the interface and $S^{\ast }$ is the surface area. The surface tension applies only on the interface between air and liquid as represented by the Dirac delta function $\delta$. The point ${\mathit{x}}^{\ast }$ identifies where the equation is evaluated and ${\mathit{x}}_f^{\ast }$ denotes the coordinates of the interface.

The extra stress tensor, which is solved only in the EVP region, is modeled using the Saramito-HB model [10] augmented with the isotropic kinematic hardening model of Dimitriou and McKinley [30], Dimitriou et al. [16] and Fraggedakis et al. [15] to account for energy dissipation before yielding. The model can be written in the following generic form of a transport equation

$$\frac{\partial \mathit{B}}{\partial t^{\ast }}+{\mathit{u}}^{\ast }\cdot \nabla \mathit{B}-\mathit{B}\cdot \nabla {\mathit{u}}^{\ast }-\nabla {\mathit{u}}^{\ast T}\cdot \mathit{B}=-\frac{F}{{\Lambda }^{\ast }}(\mathit{B}-\mathit{I}),$$ (3)

where $\mathit{B}$ is the conformation tensor, $\mathit{I}$ is an identity tensor, $F$ is the stretch function and ${\Lambda }^{\ast }$ is the polymer relaxation time [31]. In the Saramito-HB model, the relaxation time can be expressed in terms of the other material properties as ${\Lambda }^{\ast }={\mu }_{L,P}^{\ast }∕G^{\ast }$ where $G^{\ast }$ is the elastic modulus of the liquid layer and ${\mu }_{L,P}^{\ast }$ is the polymeric viscosity, which can be evaluated as ${\mu }_{L,P}^{\ast }=K^{\ast }{\left(\frac{a^{\ast }{\mu }_{L,S}^{\ast }}{{\sigma }^{\ast }}\right)}^{1-n}$. The total viscosity of the liquid layer is then defined as ${\mu }_L^{\ast }={\mu }_{L,S}^{\ast }+{\mu }_{L,P}^{\ast }$. The $F∕{\Lambda }^{\ast }$ term on the right-hand side of Eq. (3) can be expressed as $F∕{\Lambda }^{\ast }=G^{\ast }\max {\left(0,\frac{\mid {\tau }^{\ast d}\mid -{\tau }_y^{\ast }}{K^{\ast }{\mid {\tau }^{\ast d}\mid }^n}\right)}^{\frac{1}{n}}$ where ${\tau }_y^{\ast }$ is the yield stress of the liquid layer, ${\tau }^{\ast d}$ is the deviatoric part of the extra stress tensor, $K^{\ast }$ is the consistency parameter and $n$ is the power law index used to quantify the shear thinning property of the fluid [31, 29], The magnitude of ${\tau }^{\ast d}$ is computed as

$$\mid {\tau }^{\ast d}\mid ={\left(\frac{1}{2}{\tau }_{ij}^{\ast d}{\tau }_{ij}^{\ast d}\right)}^{\frac{1}{2}}.$$ (4)

After Eq. (3) is solved for the conformation tensor $\mathit{B}$, the extra stress tensor is then evaluated as ${\tau }^{\ast d}=\frac{{\mu }_{L,P}^{\ast }}{{\Lambda }^{\ast }}(\mathit{B}-\mathit{I})=G^{\ast }(\mathit{B}-\mathit{I})$.

The difference between the Saramito-HB model and the Saramito-HB model with the isotropic kinematic hardening (Saramito-HB-IKH) is embedded in the term $F∕{\Lambda }^{\ast }$ in Eq. (3). In the Saramito-HB-IKH model, $F∕{\Lambda }^{\ast }$ is expressed as $F∕{\Lambda }^{\ast }=G^{\ast }\max {\left(0,\frac{\mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }}{K^{\ast }{\mid {\tau }^{\ast d}\mid }^n}\right)}^{\frac{1}{n}}$ where ${\tau }_{y,e}^{\ast }$ is effective yield stress that evolves dynamically with the flow [30, 16, 15], which is in contrast with the constant yield stress ${\tau }_y^{\ast }$ used in the Saramito-HB model. The relation between ${\tau }_{y,e}^{\ast }$ and ${\tau }_y^{\ast }$ can be written as [32]

$${\tau }_{y,e}^{\ast }={\tau }_y^{\ast }qA,$$ (5)

where $q$ is an auxiliary material constant and $A$ is a dimensionless evolution parameter that governs evolution of ${\tau }_{y,e}^{\ast }$ in the flow, and is solved only in the EVP region. Following Dimitriou et al. [16], Dimitriou and McKinley [30] and Fraggedakis et al. [15], the evolution equation for the kinematic hardening parameter, $A$, can be written as

$$\frac{DA}{D{t}^{\ast }}=\frac{\mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }}{\mid \mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }\mid }{\left(\frac{\mid \mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }\mid }{K^{\ast }}\right)}^{\frac{1}{n}}-qA{\left(\frac{\mid \mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }\mid }{K^{\ast }}\right)}^{\frac{1}{n}},$$ (6)

where $D∕D{t}^{\ast }=(\partial ∕\partial t^{\ast })+{\mathit{u}}^{\ast }\cdot {\nabla }^{\ast }$ is the material derivative.

In the present simulations, $A$ is initialized such that ${0\le {\tau }_{y,e}^{\ast }\mid }_{t^{\ast }=0}\le {\tau }_y^{\ast }$. The two extreme cases of initial conditions, namely ${{\tau }_{y,e}^{\ast }\mid }_{t^{\ast }=0}^{\ast }=0$ and ${{\tau }_{y,e}^{\ast }\mid }_{t^{\ast }=0}={\tau }_y^{\ast }$ are referred to as the no-pre-energy-defect and the high-pre-energy-defect conditions, respectively. The no-preenergy-defect condition corresponds to a state of the material with zero defect energy at the beginning of deformation [16]. Therefore, this initial condition considers no accumulation of ${\tau }_{y,e}^{\ast }$ prior to the simulations and ${\tau }_{y,e}^{\ast }$ is initiated as zero. The other initial conditions tested in this study are selected in the range ${0<{\tau }_{y,e}^{\ast }\mid }_{t^{\ast }=0}\le {\tau }_y^{\ast }$ to mimic the possible residuals lingering from the previous breathing cycles, where ${{\tau }_{y,e}^{\ast }\mid }_{t^{\ast }=0}={\tau }_y^{\ast }$ corresponds to a high-level of such residuals.

While the initial condition for ${\tau }_{y,e}^{\ast }$ is varied in the Saramito-HB-IKH model to account for the residual stresses remaining from the previous breathing cycles, it should be noted that such residuals can also be found in the Saramito-HB model, in the form of stresses trapped in unyielded material. Depending on these initial residual stresses, the flow may reach different steady states as discussed by Cheddadi et al. [33], Syrakos et al. [34]. In the present study, these type of residual stresses are not considered, i.e., the stresses are initially set to be zero.

It is further assumed that, following a fluid particle, the material properties remain constant, i.e.,

$$\begin{gathered} \frac{D{\rho }^{\ast }}{D{t}^{\ast }}=0,\frac{D{\mu }_{L,S}^{\ast }}{D{t}^{\ast }}=0,\frac{{DG}^{\ast }}{{Dt}^{\ast }}=0, \\ \frac{D{\tau }_y^{\ast }}{D{t}^{\ast }}=0,\frac{Dn}{D{t}^{\ast }}=0,\frac{D{K}^{\ast }}{D{t}^{\ast }}=0,\frac{Dq}{D{t}^{\ast }}=0. \end{gathered}$$ (7)

Across the liquid-air interface, the material properties given in Eq. (7) vary discontinuously, and their values are evaluated using an indicator function, $\phi$, defined as $\phi =0$ in the liquid region and $\phi =1$ in the gas region. For instance, once $\phi$ is determined, ${\rho }^{\ast }$ is evaluated in the entire computational domain as

$${\rho }^{\ast }={\rho }_G^{\ast }\phi (r,z,t)+{\rho }_L^{\ast }[1-\phi (r,z,t)],$$ (8)

where the subscripts ‘$G$’ and ‘$L$’ denote the properties of gas and liquid regions, respectively. All the other material properties in Eq. (7) are computed similarly as done for the density in Eq. 8.

In the front-tracking method used here, the field equations are solved using a finite-difference method on a staggered stationary Eulerian grid while a Lagrangian grid is used to track the interface between the liquid and air regions [28, 35]. The numerical method is essentially the same as that used in the previous publications and readers are referred to [35, 29, 31, 36, 26, 27] for details.

The topological changes such as breakup and coalescence are not automatically handled and must be done explicitly in the front-tracking method [35]. A simple algorithm proposed by Olgac et al. [37] is utilized here for the rupture of air core to create a liquid plug. In this method, the air-liquid interface is monitored at every time step and the interface is ruptured when the smallest distance between the centerline and the interface gets smaller than a prespecified threshold value of $l_{\text{th}}$ by simply deleting the front element closest to the centerline and connecting the interface to the symmetry axis [37, 26]. Following Erken et al. [26], the threshold is set to $l_{\text{th}}=1.5\Delta r$ where $\Delta r$ is the Eulerian grid size in the radial direction at the centerline.

The present front-tracking method has been extensively validated for the airway closure [23, 25, 26, 27] and reopening [36] in similar settings as considered here. Therefore, a similar study is not repeated here.

3. Problem statement

A schematic of the airway segment and the computational domain are shown in Fig. 1. Periodic boundary conditions are used for flow at $z^{\ast }=0$ and $z^{\ast }=L^{\ast }$. Flow is assumed to be axisymmetric with respect to the centerline, and the no-slip boundary conditions are imposed at the rigid wall. In order to initiate the Plateau-Rayleigh instability, the liquid layer is slightly perturbed from its initial thickness of $h^{\ast }$, and the interface is initialized as

$$r^{\ast }=R_I^{\ast }=a^{\ast }-h^{\ast }[1-0.1\times \cos (2\pi z^{\ast }∕L^{\ast })],$$ (9)

where $z^{\ast }$ and $r^{\ast }$ are the axial and the radial coordinates, respectively, and $R_I^{\ast }$ is the initial radial location of the liquid-air interface. Following Romano et al. [23] and Erken et al. [27], the initial perturbation is set to 10% of the initial thickness of the liquid layer. Note that an instability can be initiated by a smaller perturbation but this value is used to reduce the computational time and hence facilitate extensive simulations.

As stated before, while the equations are solved in their dimensional forms, results are non-dimensionalized before they are presented. The flow quantities are non-dimensionalized using the length scale $a^{\ast }$, the time scale ${\mu }_{L,S}^{\ast }{a}^{\ast }∕{\sigma }^{\ast }$, the velocity scale ${\sigma }^{\ast }∕{\mu }_{L,S}^{\ast }$ and the stress scale ${\sigma }^{\ast }∕a^{\ast }$. The resulting non-dimensional numbers relevant for the study can be summarized as

$$\begin{gathered} \rho =\frac{{\rho }_G^{\ast }}{{\rho }_L^{\ast }},\mu =\frac{{\mu }_G^{\ast }}{{\mu }_{L,S}^{\ast }},\epsilon =\frac{h^{\ast }}{a^{\ast }}, \\ \lambda =\frac{L^{\ast }}{a^{\ast }},La=\frac{{\rho }_L^{\ast }{\sigma }^{\ast }{a}^{\ast }}{{{\mu }_{L,S}^{\ast }}^2},Bi=\frac{{\tau }_y^{\ast }{a}^{\ast }}{{\sigma }^{\ast }},Wi=\frac{{\Lambda }^{\ast }{\sigma }^{\ast }}{a^{\ast }{\mu }_{L,S}^{\ast }}, \end{gathered}$$ (10)

where $\rho$ is gas-to-liquid density ratio, $\mu$ is gas-to-liquid dynamic viscosity ratio, $\epsilon$ is non-dimensional initial undisturbed film thickness, $\lambda$ is length-to-radius ratio of airway tube, $La$ is the Laplace number, $Bi$ is the Bingham number and $Wi$ is the Weissenberg number.

As previously stated, the airways in the lungs form a branched structure, and with each new branch signifying a new generation, the airway radius becomes smaller. Airway closure can only be observed in airways where radius of an airway is sufficiently small. In this study, the airway radius is taken to be $a^{\ast }=0.065\text{cm}$ [38], which corresponds approximately to a radius of ninth or tenth generation, where airways are small enough for closure to occur [39]. It should also be noted that this radius is small enough so that the gravitational effects on flow become negligible [36], hence they are not taken into consideration.

For a one-layer Newtonian liquid covering inside of a rigid tube, Gauglitz and Radke [3] demonstrated that the Plateau–Rayleigh instability occurs when $h_c^{\ast }∕a^{\ast }\ge 0.12$. In this study, for the non-dimensional initial film thickness, a range of $0.25\le \epsilon \le 0.35$ is used to mimic different mucus hypersecretion intensities. In addition, simulations are repeated for three different surface tension values to mimic the normal and surfactant-deficient conditions. These values are ${\sigma }^{\ast }\in [0.026,0.052,0.078]$ N/m [40, 41] which correspond to $La\in [100,200,300]$, respectively. The solvent viscosity of the liquid layer is taken as ${\mu }_{L,S}^{\ast }=0.013\text{Pa}\cdot \mathrm{s}$ [21], and the density of the liquid layer is assumed to be ${\rho }_L^{\ast }=1000$ kg/m³ [23]. The length-to-radius ratio of the airway is set to $\lambda =6$ [42], which is representative of an airway of an adult lung in the 9th-10th generation.

Simulations are performed using the Saramito-HB-IKH model and the material properties are selected according to the experimental data obtained for healthy and pathological condition of CF in human airways. The experimental data for storage $G^{\prime }$ and loss $G^{″}$ moduli for the mucus are taken from Patarin et al. [7]. Then a non-linear regression is used to obtain material properties of liquid layer including elastic modulus $G^{\ast }$, yield stress ${\tau }_y^{\ast }$, consistency parameter $K^{\ast }$, power law index $n$ and auxiliary material constant $q$. The fitting procedure is explained in detail in Appendix A and an additional validation case is presented in Appendix B.

4. Results and Discussion

Effects of kinematic hardening on an airway closure are computationally examined in terms of closure time and wall stresses considering the healthy as well as the CF conditions. Simulations are performed for a range of $La$ and $\epsilon$ values to mimic the surfactant deficiency and mucus hypersecretion conditions.

In the simulations, a tensor-product structured grid is used. While the grid is uniform in the axial direction, it is stretched radially to better resolve the thin liquid layer region near the wall. As a result of stretching, the radial grid size, $\Delta r$, near the wall is three times smaller than that at the centerline. The computation domain has a radial length of $a^{\ast }$ and an axial length of $L^{\ast }=6a^{\ast }$. After checking grid convergence (see Appendix C), for all the relevant flow quantities it is found that a grid containing 96 × 576 grid cells in the radial and axial directions is sufficient to reduce the spatial errors below the highest value of 1.55% in critical regions for certain cases, and much lower in general. Therefore, this grid resolution is used for all the simulations presented in this study.

The present numerical method is explicit, so time step is restricted to maintain numerical stability. Following Muradoglu and Tryggvason [43], the restrictions due to momentum diffusion (viscosity), convection (CFL condition) and surface tension (capillary wave) are considered. The total viscosity is used in determining the diffusive time scale. The time step is then multiplied by a safety factor to account for the nonlinear effects. The safety factor is taken as 0.85 in the present simulations. We note that temporal numerical error is generally found to be negligibly small compared to spatial error mainly due to a small time step imposed by the numerical stability requirements in the present explicit flow solver [36].

4.1. Rheological Fitting

In their experimental study, Patarin et al. [7] examined mucus samples collected from patients of asthma, COPD and CF as well as from people with no known lung disease. By performing a strain sweep under a fixed frequency of 0.6 Hz, they tracked the rheological response of the samples under strain amplitude variation. The response is reported in terms of evolution of the storage modulus $G^{\prime }$ and the loss modulus $G^{″}$ for the healthy samples and each pathological condition. Our study utilizes the Saramito-HB-IKH model to obtain rheological fits to the experimental data of $G^{\prime }$ and $G^{″}$ for the healthy and the CF cases by using a similar procedure as described in Fraggedakis et al. [15]. Through the fitting procedure, the model parameters are determined and are subsequently used in the airway closure simulations. For a detailed explanation of the fitting procedure, readers are referred to Appendix A.

The acquired mucus properties from the fitting procedure are summarized in Table 1. These results indicate an increase in the yield stress in the case of the pathological conditions, which is in good agreement with the yield stress ranges provided in Patarin et al. [7] as well as the other experimental studies [44]. It can also be seen that the elastic modulus is lower in the healthy case while it increases significantly for the CF case. The differences in Bingham number for both cases indicate variations in the yield stress of the mucus. Relatively lower values of Weissenberg number for the CF case are mainly due to the higher elastic modulus of the mucus for that condition.

Table 1.

The properties of mucus for the Saramito-HB-IKH model determined through rheological fitting to the experimental data of Patarin et al. [7].

	Healthy	CF
$G^{\ast }(\text{Pa})$	0.1096	2.8314
${\tau }_y^{\ast }(\text{Pa})$	0.4999	14.857
$n$	0.7479	0.7603
$K^{\ast }(\text{Pa}\cdot {\mathrm{s}}^n)$	0.0389	0.2605
$q$	0.00044	0.05877
$W{i}_{\text{La}=100}$	144.18	41.288
$W{i}_{\text{La}=200}$	242.13	69.935
$W{i}_{\text{La}=300}$	327.90	95.187
$B{i}_{\text{La}=100}$	1.250 × 10−2	3.714 × 10−1
$B{i}_{\text{La}=200}$	6.249 × 10−3	1.857 × 10−1
$B{i}_{\text{La}=300}$	4.166 × 10−3	1.238 × 10−1

The fitting results are presented in Fig. 2 where the Saramito-HB-IKH model (dashed lines) is compared with the Saramito-HB model (solid lines) to highlight the effects of adding isotropic kinematic hardening to the EVP model for the mucus. Compared to the experimental data, it is clearly seen that the Saramito-HB model is insufficient in capturing $G^{″}$ accurately in low strain amplitude regions as also reported by Erken et al. [27] and Fraggedakis et al. [15]. However, when the isotropic kinematic hardening is included, the model is able to capture $G^{″}$ much better in all regions. The main reason is that the yield stress evolves with the flow allowing effective energy dissipation before yielding in the kinematic hardening case whereas the fluid behaves as an elastic solid where no energy dissipation is possible in the Saramito-HB model. Overall, the fitting results for the Saramito-HB-IKH model are found to be in good agreement with the experimental data, and the model is thus expected to better capture the behavior of mucus in the context of the airway closure.

Fig. 2.

Variation of the storage ($G^{\prime }$) and loss ($G^{″}$) moduli against the strain amplitude ${\gamma }_o\%$ for the healthy and CF mucus. The symbols denote the experimental data Patarin et al. [7] while the solid and dashed lines represent the fitted results for the Saramito-HB and Saramito-HB-IKH models, respectively.

4.2. Evolution of Effective Yield Stress

The effective yield stress is a distinctive feature of the Saramito-HB-IKH model. Thus, we first examine evolution of effective yield stress ${\tau }_{y,e}^{\ast }$ in the entire airway closure process. Simulations are performed for the initial conditions of ${{\tau }_{y,e}^{\ast }\mid }_{t=0}=0$ and ${{\tau }_{y,e}^{\ast }\mid }_{t=0}={\tau }_y^{\ast }$ corresponding to the initially no-pre-energy-defect and high-pre-energy-defect, respectively. Figure 3 shows the evolution of ${\tau }_{y,e}^{\ast }$ for the CF case in comparison with the healthy case at $\epsilon =0.35$ and $La=300$ for two distinctly different initial conditions. In the figure, ${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$ is provided for the pre-coalescence, near-coalescence and post-coalescence phases. The figure indicates very different evolution of effective yield stress for the no-pre-energy-defect (top plots) and high-pre-energy-defect (bottom plots) initial conditions. In the case of the no-pre-energy-defect condition (${{\tau }_{y,e}^{\ast }\mid }_{t=0}=0$), it is observed that, rather than a constant yield stress as would be the case in the Saramito-HB model, ${\tau }_{y,e}^{\ast }$ is distributed non-uniformly in the mucus region. It is observed that, especially before the coalescence, ${\tau }_{y,e}^{\ast }$ has the highest values near the airway wall. This is expected because evolution of ${\tau }_{y,e}^{\ast }$ from its initial zero condition is strongly dependent on stresses as seen in Eq. (6). Since stresses are higher near the walls, ${\tau }_{y,e}^{\ast }$ shows a similar pattern. After the plug formation, there is also higher accumulation of ${\tau }_{y,e}^{\ast }$ concentrated around the centerline due to high shear stresses around the tip of the plug during this phase. For the high-pre-energy-defect initial condition (${{\tau }_{y,e}^{\ast }\mid }_{t=0}={\tau }_y^{\ast }$), the effective yield stress evolves differently for the healthy and the CF cases. For the healthy, the effective yield stress is constant throughout the domain with a small decrease in the narrow region near the centerline. However, for the CF mucus, the ratio of ${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$ decreases immediately and evolves to follow a distribution pattern similar to what is seen in the initially no-pre-energy-defect condition. Additionally, the change in the initial conditions for the effective yield stress has strong impact on the airway closure time for the CF case. This situation is not observed for the healthy mucus. This phenomena is discussed in more detail in upcoming sections.

Fig. 3.

Evolution of constant contours of the non-dimensionalized effective yield stress (${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$) for the CF mucus (right portion) and the healthy mucus (left portion) during the airway closure process for the no-pre-energy-defect, ${{\tau }_{y,e}^{\ast }\mid }_{t=0}=0$, (top row) and high-pre-energy-defect, ${{\tau }_{y,e}^{\ast }\mid }_{t=0}={\tau }_y^{\ast }$, (bottom row) initial conditions. The snapshots are taken at times as indicated in each plot. The air-liquid interface is indicated by the solid magenta line. [$\epsilon =0.35$ and $La=300$].

The results clearly show the importance of the initial conditions that are not easy to determine exactly in physiological conditions due to unknown amount of residual stresses in the airway’s coating layer that depends on breathing history.

4.3. Evolution of Yielding Patterns

Simulations are performed to compare the Saramito-HB and Saramito-HB-IKH models in terms of yielded regions of the liquid layer throughout the airway closure process. The simulation conditions are the same as those of subsection 4.2.

For the Saramito-HB model, the following criteria is used to determine whether the liquid region is yielded or not [45, 27, 46]

$$\{\begin{array}{cc}\max {\left(0,\frac{\mid {\tau }^{\ast d}\mid -{\tau }_y^{\ast }}{K^{\ast }{\mid {\tau }^{\ast d}\mid }^n}\right)}^{\frac{1}{n}}\ge {10}^{-3}\hfill & \text{yielded region}\hfill \\ \max {\left(0,\frac{\mid {\tau }^{\ast d}\mid -{\tau }_y^{\ast }}{K^{\ast }{\mid {\tau }^{\ast d}\mid }^n}\right)}^{\frac{1}{n}}<{10}^{-3}\hfill & \text{unyielded region}.\hfill \end{array}\phantom{\}}$$ (11)

For the Saramito-HB-IKH model, ${\tau }_y^{\ast }$ is replaced with ${\tau }_{y,e}^{\ast }$ in Eq. (11) for the yielding criteria. The limit of 10⁻³ is selected to make sure that the results are not influenced by the numerical uncertainties.

Figures 4 and 5 highlight differences between the yielded and unyielded regions for the Saramito-HB-IKH (left of centerline) and the Saramito-HB (right of centerline) models and presents evolution of these regions throughout the closure process for the healthy and the CF cases. For the Saramito-HB-IKH model, the no-pre-energy-defect and the high-pre-energy-defect initial conditions are shown in the top and bottom rows, respectively. The first thing to notice is, for all the visuals, before and after the closure, the liquid layer is always yielded when the Saramito-HB-IKH model (left side) is used with the no-pre-energy-defect initial condition (top row). This is due to the fact that the effective yield stress ${\tau }_{y,e}^{\ast }$, is initially set to zero and it evolves according to the flow as explained in Section 4.2. Therefore, the yield criteria given in Eq. (11) is almost always satisfied. On the other hand, the results for the Saramito-HB model show that yielding of the liquid layer does not occur immediately but rather occurs in certain areas as closure is neared as also reported by Erken et al. [27]. Approaching the breakup time, yielded regions emerge near the walls and around the centerline, i.e., in the regions where shear stresses are relatively high. After the breakup, the yielded region enlarges as stresses accumulate around the walls and the edge of the plug. When the Saramito-HB-IKH model is initialized to have high-pre-energy-defect, i.e., ${{\tau }_{y,e}^{\ast }\mid }_{t=0}={\tau }_y^{\ast }$, yielding occurs gradually as shown in the bottom row of Figures 4 and 5, which is in contrast with the case of ${{\tau }_{y,e}^{\ast }\mid }_{t^{\ast }=0}=0$ initial condition. Similar to the Saramito-HB model, yielding occurs as the closure is approached for the Saramito-HB-IKH model under this condition. For the healthy case, Fig. 4 shows almost identical yielding patterns for both models, as well as identical closure times. However, for the CF case, it is observed from Fig. 5 that yielding occurs earlier and grows faster for the Saramito-HB-IKH model than that of the Saramito-HB model. The evolution of ${\tau }_{y,e}^{\ast }$ plays an important role in this phenomena, as highlighted in the previous section. Moreover, the convective component of Eq. (6) is responsible for the qualitative difference in the yielding front between the Saramito-HB and Saramito-HB-IKH models. For the CF case, very different evolution patterns of the yielded zone for different initial conditions demonstrate the importance of the residual stresses from the previous breathing cycles when pathological conditions are considered. For full understanding of the effects of the residual stresses requires simulations of several breathing cycles involving repeating airway closure and reopening processes, which is not in the scope of the present study.

Fig. 4.

Comparison of yielded regions of the liquid layer between Saramito-HB-IKH model (left portion) and Saramito-HB model (right portion) for the healthy case for the no-pre-energy-defect ${{\tau }_{y,e}^{\ast }\mid }_{t=0}=0$ (top row) and high-pre-energy-defect ${{\tau }_{y,e}^{\ast }\mid }_{t=0}={\tau }_y^{\ast }$ (bottom row) initial conditions. Air is indicated in dark red and the air-liquid layer interface is shown by the solid magenta line. Yielded and unyielded regions are indicated in yellow and blue, respectively. [$\epsilon =0.35$ and $La=300$].

Fig. 5.

Comparison of yielded regions of the liquid layer between Saramito-HB-IKH model (left portion) and Saramito-HB model (right portion) for the CF case for the no-pre-energy-defect ${{\tau }_{y,e}^{\ast }\mid }_{t=0}=0$ (top row) and high-pre-energy-defect ${{\tau }_{y,e}^{\ast }\mid }_{t=0}={\tau }_y^{\ast }$ (bottom row) initial conditions. Air is indicated in dark red and the air-liquid layer interface is shown by the solid magenta line. Yielded and unyielded regions are indicated in yellow and blue, respectively. [$\epsilon =0.35$ and $La=300$].

Another important result observed for the CF case is that there is a significant difference between the closure time for the case of the initially no-pre-energy-defect condition, i.e., the airway closure occurs much earlier in the case of the Saramito-HB-IKH model than that of the Saramito-HB model. In the case of the high-pre-energy-defect initial conditions, the difference between the closure times diminishes significantly although the closure still occurs earlier in the Saramito-HB-IKH model. It is also important to notice that, while there is a significant difference in the closure time, shape of the air-liquid interface is very similar regardless of the model used, indicating that the isotropic kinematic hardening does not have a significant effect on the shape of the interface.

4.4. Effects of Kinematic Hardening on Wall Stresses and Closure Time

In this section, effects of isotropic kinematic hardening are examined in terms of airway wall stresses and airway closure time for the healthy and the CF mucus conditions using the no-pre-energy-defect initial conditions. Simulations are performed using both the Saramito-HB and the Saramito-HB-IKH models for the range of the parameters in Table 1. Wall shear stress and pressure excursions as well as their local gradients are plotted to evaluate the differences between the two EVP models. Simulations are performed for $La\in [100,200,300]$ and $\epsilon \in [0.25,0.30,0.35]$ to mimic various conditions of surfactant deficiency and mucus hypersecretion. Note that increasing $La$ represents more surfactant deficiency while increasing $\epsilon$ mimics more severe hypersecretion of mucus. The study of Romano et al. [25] has investigated the single-liquid-layer closure in details, including surface tension gradients due to a complex interfacial surfactant dynamics. Their study showed that the mean effect of the surfactant in lowering the average surface tension is much more important than the Marangoni stress characterizing the soluto-capillary flow. For this reason, we mimic the surfactant deficiency only in an average sense, solely by changing the Laplace number. We stress, however, that non-linear interactions between viscoplastic films subject to soluto-capillary effects may lead to a complex dynamics that is out of scope for this study [47, 48].

4.4.1. Normal Surfactant Condition ($La=100$)

Simulations are first performed at $La=100$ that represents approximately the normal surfactant level in the airway. The corresponding Bingham and Weissenberg numbers are $(Bi,Wi)=(1.250\times {10}^{-2},144.18)$ and (3.714 × 10⁻¹,41.288) for the healthy and the CF cases, respectively.

Figure 6 shows the differences between the Saramito-HB and the Saramito-HB-IKH models in terms of the wall shear stress excursion, defined as $\Delta {\tau }_w=\max ({\tau }_w)-\text{min}({\tau }_w)$ for $\epsilon \in [0.25,0.30,0.35]$. The Newtonian and extra-stress components of the shear stress excursion $\Delta {\tau }_w$ are also plotted in Fig. 6. Note that the mechanical stresses and non-dimensional time $t$ are re-scaled by $La$ to correct for the effects of surface tension on the results and facilitate an easier interpretation.

Fig. 6.

Evolution of the wall shear stress excursion, $\Delta {\tau }_w$, for $La=100$ and $\epsilon =0.25$ (top), $\epsilon =0.30$ (middle), and $\epsilon =0.35$ (bottom) using the no-pre-energy-defect initial conditions. $\Delta {\tau }_w$ is partitioned into (a) the Newtonian portion, (b) the extra-stress portion, and its total value is plotted in (c). Solid and dashed lines represent the results for the Saramito-HB and the Saramito-HB-IKH models, respectively. $\Delta {\tau }_w$ and time are re-scaled by $La$ to eradicate the effects of the surface tension. The closure times are plotted in (d) for all conditions. Gray symbols refer to conditions for which closure did not occur.

Airway closure is indicated by the sudden stress peaks shown in Fig. 6. It can be seen that the closure occurs only in the healthy (black lines) case for $La=100$ for all $\epsilon =0.25$, $\epsilon =0.30$, and $\epsilon =0.35$ values. As reported by Erken et al. [27], the highly EVP characteristics of the mucus in the CF (blue lines) cases inhibit the closure process at these $\epsilon$ values.

The particular effects of using an isotropic kinematic hardening model on wall shear stress excursion can be observed specifically for the healthy case for $\epsilon =0.25$. Here, it can be seen that the initial stress peaks occur significantly sooner in the Saramito-HB-IKH model compared to the Saramito-HB model. As $\epsilon$ increases, the closure times become almost identical, independent of the EVP model used. It is thus understood that the effects of the pathological conditions on closure time become less significant when the initial film thickness is higher than a certain value, which is consistent with the findings of Erken et al. [27].

In addition to the shear stress excursion, Fig. 7 also shows evolution of pressure excursion on the wall, defined as $\Delta p_w=\max (p_w)-\text{min}(p_w)$, and the local gradients of the stresses for the healthy case. Once again, the solid and dashed lines correspond to the Saramito-HB and the Saramito-HB-IKH models, respectively. The different line thicknesses indicate different values of $\epsilon$, i.e., the thinnest line is used for $\epsilon =0.25$ and thickest line for $\epsilon =0.35$. Similar to what is observed in Fig. 6, the isotropic kinematic hardening has a remarkable impact on the space and time location of the peaks in stresses and their gradients. Figure 7 also shows an increasing trend for the maximum wall pressure gradient ${\mid {\partial }_z{p}_w\mid }_{\max }$ in both models and it starts growing gradually following a sudden drop after the closure. This trend is caused by the bifrontral plug growth and the resulting compression of the liquid layer on the airway wall [23]. Another interesting feature observed in the figure is the emergence of the oscillations in maximum shear stress gradient for the healthy case after the closure. These oscillations are attributed to the elasto-inertial instability driven by both the inertial and elastic effects. This instability occurs for cases of high $Wi\sqrt{1-({\mu }_{L,S}^{\ast }∕{\mu }_L^{\ast })}∕La$ and may cause a secondary stress peak as reported by Romanò et al. [24].

Fig. 7.

Evolution of the wall shear stress excursion $\Delta {\tau }_w$ (top left), the maximum absolute value of the wall shear stress gradient ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ (top right), the wall pressure excursion $\Delta p_w$ (bottom left), and the maximum absolute value of the wall pressure gradient ${\mid {\partial }_z{p}_w\mid }_{\max }$ (bottom right) for the healthy mucus case at $La=100$ using the no-pre-energy-defect initial conditions. Thin and thick lines show $\epsilon =0.25$ and $\epsilon =0.30$ cases, respectively. Solid and dashed lines represent the results for the Saramito-HB and the Saramito-HB-IKH models, respectively. All the results are re-scaled by $La$ to eradicate the effects of surface tension.

4.4.2. Moderate Surfactant Deficiency Condition ($La=200$)

Simulations are then performed for $La=200$ to mimic the case of a moderate surfactant deficiency while keeping the other parameters fixed at the baseline case. The corresponding Bingham and Weissenberg numbers are $(Bi,Wi)=(6.249\times {10}^{-3},242.13)$ and (1.857 × 10⁻¹,69.935) for the healthy and the CF cases, respectively.

Figure 8 shows evolution of $\Delta {\tau }_w$ and differences between the Saramito-HB and the Saramito-HB-IKH models. Compared to the results for $La=100$, it is seen that for the healthy case, the closure times are shorter, and the stresses are higher, which is consistent with the findings of Erken et al. [27]. For $\epsilon =0.25$, it is seen that the closure occurs earlier using the Saramito-HB-IKH model for the healthy condition, similar to what is observed for the $La=100$ case. However, the differences between the closure times of two models are less evident. For $\epsilon =0.30$ and $\epsilon =0.35$, evolution of $\Delta {\tau }_w$ becomes almost identical for both models for the healthy case. This result suggests that, for thicker films, the rheological features become less relevant. For $La=200$, the most crucial effect of the kinematic hardening can be spotted in the $\epsilon =0.35$ case. While the initial stress peak is not observed for the CF case in the Saramito-HB model, there is a stress peak when the Saramito-HB-IKH model is utilized. This demonstrates that, under moderate deficiency of the surfactan, airway closure can occur even for a mucus with highly EVP characteristics, which is not the case in the traditional EVP models. It is also observed that, while the magnitude of the total stress peaks (Fig. 8c) for the healthy case is due to the Newtonian portion of the stresses (Fig. 8a), for the CF mucus under the Saramito-HB-IKH model, the stress peaks are also highly dependent on the extra-stress portion (Fig. 8b). This shows that the closure mechanisms for varying mucus conditions may not be necessarily the same [27].

Fig. 8.

Evolution of the wall shear stress excursion, $\Delta {\tau }_w$, for $La=200$ and $\epsilon =0.25$ (top), $\epsilon =0.30$ (middle), and $\epsilon =0.35$ (bottom) using the no-pre-energy-defect initial conditions. $\Delta {\tau }_w$ is partitioned into (a) the Newtonian and (b) the extra-stress portions, and its total value is plotted in (c). Solid and dashed lines represent the results for the Saramito-HB and the Saramito-HB-IKH models, respectively. $\Delta {\tau }_w$ and time are re-scaled by $La$ to eradicate the effects of the surface tension. The closure times are plotted in (d) for all conditions. Gray symbols refer to conditions for which closure did not occur.

4.4.3. Severe Surfactant Deficiency Condition ($La=300$)

Simulations are finally performed for $La=300$ to mimic a severe surfactant deficient case. The corresponding Bingham and Weissenberg numbers are $(Bi,Wi)=(4.166\times {10}^{-3},327.90)$ and (1.238 × 10⁻¹,95.187) for the healthy, and the CF cases, respectively.

Evolution of $\Delta {\tau }_w$ corresponding to this condition is plotted in Fig. 9. As seen, closure times have become even shorter and magnitude of stress peaks get larger as $La$ is increased. For $\epsilon =0.25$, considering the healthy case, it is observed that the evolution of $\Delta {\tau }_w$ is approximately identical for both the Saramito-HB and the Saramito-HB-IKH models. Considering the results for $La=100$ and $La=200$ cases at $\epsilon =0.25$, it is seen that, as $La$ increases, the differences between the closure times of both models decrease for the healthy case. Therefore, it can be interpreted that the effects of kinematic hardening become less significant as the surface tension between the air-liquid interface increases, i.e., in the case of surfactant deficiency. For $\epsilon =0.30$, while the healthy case shows almost the identical results for the both models, it can be seen that the closure occurs for the CF case only when the Saramito-HB-IKH model is used, similar to what is observed for $\epsilon =0.35$ where $La=200$ (Fig. 8). For $\epsilon =0.35$, the airway closure occurs for all the mucus conditions. The results for the healthy case are almost identical for both models. However, for the CF case, the airway closure occurs significantly earlier in the Saramito-HB-IKH model.

Fig. 9.

Evolution of the wall shear stress excursion, $\Delta {\tau }_w$, for $La=300$ and $\epsilon =0.25$ (top), $\epsilon =0.30$ (middle), and $\epsilon =0.35$ (bottom) using the no-pre-energy-defect initial conditions. $\Delta {\tau }_w$ is partitioned into (a) the Newtonian and (b) the extra-stress portions, and its total value is plotted in (c). Solid and dashed lines represent the results for the Saramito-HB and the Saramito-HB-IKH models, respectively. $\Delta {\tau }_w$ and time are re-scaled by $La$ to eradicate the effects of the surface tension. The closure times are plotted in (d) for all conditions. Gray symbols refer to conditions for which closure did not occur.

4.5. Sensitivity to Residual Stress

In order to assess the sensitivity of the results to the initial condition of the effective yield stress ${\tau }_{y,e}^{\ast }$, and to account for the pre-yielding of the mucus leftover from the previous breathing cycle, a set of simulations using the Saramito-HB-IKH model are performed with the value of ${\tau }_{y,e}^{\ast }$ ranging between $[0,{\tau }_y^{\ast }]$. The simulations are performed using the mucus properties of the CF case at $La=300$ and $\epsilon =0.35$. Figure 10 shows the results in terms of the wall shear stress excursion $\Delta {\tau }_w$, the wall pressure excursion $\Delta p_w$ and the maximum absolute value of their gradients, ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ and ${\mid {\partial }_z{p}_w\mid }_{\max }$ for varying initial values of ${\tau }_{y,e}^{\ast }$. It is observed that, as the initial value of ${\tau }_{y,e}^{\ast }$ is increased, there is a delay of the airway closure, as can be seen from the timing of the sudden stress peaks. For the Saramito-HB-IKH model, the airway closure occurs earliest when ${\tau }_{y,e}^{\ast }$ is set to ${\tau }_{y,e}^{\ast }=0$ and latest when it is set to ${\tau }_{y,e}^{\ast }={\tau }_y^{\ast }$ initially. This figure clearly demonstrates that the initial value of ${\tau }_{y,e}^{\ast }$ has a very dominant influence on the closure time, and hence on the entire closure process. Therefore, in the case of disturbed or non-ideal mucus conditions caused by the repetitive breathing cycle and resulting perturbations of the liquid layer, there could be a serious difference between the predicted and actual closure times. It is therefore interpreted that, any changes in the mucus conditions caused by the continuous breathing cycle may alter the airway closure mechanism in a significant manner.

Fig. 10.

Evolution of the wall shear stress excursion $\Delta {\tau }_w$ (top left), the maximum absolute value of the wall shear stress gradient ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ (top right), the wall pressure excursion $\Delta p_w$ (bottom left), and the maximum absolute value of the wall pressure gradient ${\mid {\partial }_z{p}_w\mid }_{\max }$ (bottom right) plotted for the CF case at $La=300$ and $\epsilon =0.35$ with varying initial conditions for ${\tau }_{y,e}^{\ast }$. Solid lines represent results for Saramito-HB-IKH model whereas the dashed black line represents the results for Saramito-HB model. All results are re-scaled with $La$ to eradicate the effects of surface tension. Insets show the variation of the peak values as a function of the initial effective yield stress.

It is emphasized here that the Saramito-HB-IKH and the Saramito-HB models result in different closure times even when the initial yield stress is set to ${\tau }_{y,e}^{\ast }={\tau }_y^{\ast }$ in the Saramito-HB-IKH model, i.e., the Saramito-HB-IKH case still yields an earlier closure as seen in Fig. 10. Although not shown here due to space considerations, when ${\tau }_{y,e}^{\ast }$ is initialized as ${\tau }_{y,e}^{\ast }={\tau }_y^{\ast }$ in the Saramito-HB-IKH model, it slightly decreases in the early time and then relaxes back to the initial value as the closure is approached. It seems that this earlier decrease in ${\tau }_{y,e}^{\ast }$ eventually leads to an earlier closure in the Saramito-HB-IKH model when compared to the Saramito-HB model.

For a more detailed analysis, $\Delta {\tau }_w$ and $\Delta p_w$ at the moment of closure are plotted against ${\tau }_{y,e}^{\ast }$ for varying initial conditions in the insets of Fig. 10. It is seen that, while the difference in the magnitudes of $\Delta {\tau }_w$ and $\Delta p_w$ are not significant for the different initial conditions of ${\tau }_{y,e}^{\ast }$, there is an overall decreasing trend for the magnitude of $\Delta {\tau }_w$ whereas the magnitude of $\Delta p_w$ shows an increasing trend as ${\tau }_{y,e}^{\ast }$ increases.

4.6. Sensitivity to Model Parameters

The model parameters are determined by fitting the models to the available experimental data. Thus, the model parameters involve significant uncertainties stemming from the experimental data that exhibit large variability depending on many factors, as well as the fitting procedure. In this section, we evaluate the sensitivity of the results to the variations in the model parameters of the mucus rheology. Wall stresses and their local gradients are used for this purpose for the range of parameters in Table 1. The properties of the CF mucus are used as the baseline case. Simulations are performed for $La=300$ and $\epsilon =0.35$ to ensure that the closure occurs for a wide range of the parameters.

To determine the sole effects of the model parameters, only one parameter is varied between −75% and +75% from its baseline value while the other parameters are kept constant at their values in Table 1. The parameters considered include the elastic modulus ($G^{\ast }$), the yield stress (${\tau }_y^{\ast }$), the polymeric viscosity (${\mu }_{L,P}^{\ast }$), and the yield strain indicator ($q$). The polymeric viscosity is evaluated as ${\mu }_{L,P}^{\ast }=K^{\ast }{\left(\frac{a^{\ast }{\mu }_{L,S}^{\ast }}{{\sigma }^{\ast }}\right)}^{1-n}$, therefore the consistency parameter, $K^{\ast }$ is varied in order to induce changes in ${\mu }_{L,P}^{\ast }$. The Saramito-HB model is used for the simulations regarding ${\tau }_y^{\ast }$ while the Saramito-HB-IKH model is used for the remaining parameters. The simulations are performed using the no-pre-energy-defect initial conditions. Figures 11, 12, and 13 show the results for variations in $G^{\ast }$, ${\mu }_{L,P}^{\ast }$ and ${\tau }_y^{\ast }$, respectively. Only the results for these three parameters are presented because changes in other parameters are found not to lead to a significant change in the evolution of stresses.

Fig. 11.

The effects of elastic modulus ($G^{\ast }$) on the evolution of the wall shear stress excursion $\Delta {\tau }_w$ (top left), the maximum absolute value of the wall shear stress gradient ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ (top right), the wall pressure excursion $\Delta p_w$ (bottom left), and the maximum absolute value of the wall pressure gradient ${\mid {\partial }_z{p}_w\mid }_{\max }$ (bottom right) plotted for $La=300$ and $\epsilon =0.35$. For the base case, $G^{\ast }$ corresponding to the CF mucus is used. $G^{\ast }$ is varied between −75% and +75% of the base value and variations are indicated with different colors. The Saramito-HB-IKH model is used here. Insets show the variation of the peak values as a function of $G^{\ast }$.

Fig. 12.

The effects of polymeric viscosity (${\mu }_{L,P}^{\ast }$) on the evolution of the wall shear stress excursion $\Delta {\tau }_w$ (top left), the maximum absolute value of the wall shear stress gradient ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ (top right), the wall pressure excursion $\Delta p_w$ (bottom left), and the maximum absolute value of the wall pressure gradient ${\mid {\partial }_z{p}_w\mid }_{\max }$ (bottom right) plotted for $La=300$ and $\epsilon =0.35$. For the baseline case, ${\mu }_{L,P}^{\ast }$ corresponding to the CF mucus is used. ${\mu }_{L,P}^{\ast }$ is varied between −75% and +75% of the base value and variations are indicated with different colors. Saramito-HB-IKH model is utilized for the simulations. Values of $\Delta {\tau }_w$ and $\Delta p_w$ are plotted at $t∕La=5$ against ${\mu }_{L,P}^{\ast }$ in the insets.

Fig. 13.

The effects of yield stress (${\tau }_y^{\ast }$) on the evolution of the wall shear stress excursion $\Delta {\tau }_w$ (top left), the maximum absolute value of the wall shear stress gradient ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ (top right), the wall pressure excursion $\Delta p_w$ (bottom left), and the maximum absolute value of the wall pressure gradient ${\mid {\partial }_z{p}_w\mid }_{\max }$ (bottom right) plotted for $La=300$ and $\epsilon =0.35$. For the base case, ${\tau }_y^{\ast }$ corresponding to CF mucus is used. ${\tau }_y^{\ast }$ is varied between −75% and +75% of the base value and variations are indicated with different colors. Solid lines represent results for Saramito-HB model whereas the dashed black line represents the results for Saramito-HB-IKH model with conditions identical to the base case, yet with no-pre-energy defects. Insets show the variation of peak values as a function of percentage change in ${\tau }_y^{\ast }$ from its baseline value.

From Fig. 11, it is seen that $G^{\ast }$ has a significant effect on the airway closure time. The plots demonstrate a delay of the closure with an increase in $G^{\ast }$, i.e., the closure occurs the earliest for the −75% case and the latest for the +75% case. This finding suggests that one of the main reasons for the differences between the closure times of the healthy and the CF cases is the significant variations in their $G^{\ast }$ values. Another effect can be seen in the magnitudes of the stress and gradient peaks. For $\Delta {\tau }_w$, changing $G^{\ast }$ has a negligible effect on the magnitude of the initial peak. However, for $\Delta p_w$, the magnitude of the initial peak increases as $G^{\ast }$ is increased. For ${\mid {\partial }_z{\tau }_w\mid }_{\max }$ and ${\mid {\partial }_z{p}_w\mid }_{\max }$, the opposite occurs, i.e., the magnitude of the peaks decrease with increasing $G^{\ast }$.

To analyze the effects of varying $G^{\ast }$ in more detail, $\Delta {\tau }_w$ and $\Delta p_w$ are plotted for different $G^{\ast }$ values at the moment of the closure. The results shown in the insets in Fig. 11 indicate that, there is no clear increasing or decreasing trend in the magnitude of $\Delta {\tau }_w$ with increasing $G^{\ast }$. The overall trend is an oscillation around approximately the same values. For the magnitude of $\Delta p_w$, there is a clear increasing trend with increasing value of $G^{\ast }$.

Figure 12 shows that ${\mu }_{L,P}^{\ast }$ has a notable but less significant effect on the closure time. Once again, there is a delay of the closure with an increase in ${\mu }_{L,P}^{\ast }$. evolution of stresses after the closure but the closure times are not much affected. There is a decreasing trend for $\Delta {\tau }_w$ and $\Delta p_w$ right after the closure as the value of ${\mu }_{L,P}^{\ast }$ is decreased. Similar to what is seen in Fig. 11 the magnitude of $\Delta {\tau }_w$ is not affected. However, the magnitude of $\Delta p_w$ increases with increasing ${\mu }_{L,P}^{\ast }$. For the stress gradients, the magnitude of the peaks decrease as ${\mu }_{L,P}^{\ast }$ is increased. There is also a visible effect on how the stresses evolve after the closure point. The stresses are observed to decrease more rapidly for lower values of ${\mu }_{L,P}^{\ast }$. Therefore, it is understood that ${\mu }_{L,P}^{\ast }$ plays an important role in the dissipation of the wall stresses after the closure.

To further quantify the effects of ${\mu }_{L,P}^{\ast }$ on mechanical stresses, values of $\Delta {\tau }_w$ and $\Delta p_w$ are plotted at the closure time against ${\mu }_{L,P}^{\ast }$ in the insets in Fig. 12. For the magnitude of $\Delta {\tau }_w$, it is seen that there is first an increasing and then a decreasing trend, indicating that a clear correlation is not very recognizable. For $\Delta p_w$, the magnitude monotonically increases as ${\mu }_{L,P}^{\ast }$ increases.

Figure 13 shows the effects of ${\tau }_y^{\ast }$ on the closure process in terms of stresses and stress gradients. As seen, ${\tau }_y^{\ast }$ has a significant effect on the closure time similar to $G^{\ast }$, i.e., closure times reduce as ${\tau }_y^{\ast }$ decreases. The Saramito-HB-IKH model results (dashed black line) are also plotted in Fig. 13 for the same baseline parameters as a reference. It can be seen that the closure occurs earliest in the case of the Saramito-HB-IKH model and the Saramito-HB results move toward the Saramito-HB-IKH model results as ${\tau }_y^{\ast }$ is reduced. In other words, the Saramito-HB-IKH model acts similarly to the Saramito-HB model at low yield stresses. The peak values of $\Delta {\tau }_w$ and $\Delta p_w$ are also plotted against ${\tau }_y^{\ast }$ in the insets in Fig. 13. It is seen that $\Delta p_w$ monotonically increases as ${\tau }_y^{\ast }$ increases but there is not a clear trend for $\Delta {\tau }_w$.

5. Conclusions

In this study, the airway closure phenomenon is numerically studied by considering the healthy and the cystic fibrosis (CF) conditions. The airway is modeled as a rigid tube with a single EVP liquid layer covering inside the airway wall. Extensive simulations are performed to evaluate the effects of the kinematic hardening properties of mucus polymers on the airway closure process. For this purpose, an isotropic kinematic hardening model is coupled with the Saramito-Herschel-Bulkley (Saramito-HB) elastoviscoplastic (EVP) fluid as a model for the pulmonary mucus. Then, the airway closure is simulated using both the new model (Saramito-HB-IKH) and the conventional Saramito-HB model, and the results are compared in terms of the airway closure time and the mechanical stresses exerted on the airway wall.

Prior to the airway closure simulations, the model parameters are determined by fitting the models to the experimental data of Patarin et al. [7] for the healthy and the CF mucus cases. It is found that the Saramito-HB-IKH model fits significantly better to the experimental data especially for the loss modulus at low strain amplitude regions than the Saramito-HB model, i.e., the kinematic hardening appears to remedy the well-known deficiency of the Saramito-HB model.

The evolution of the normalized effective yield stress (${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$) is examined for the healthy and the CF cases during the entire airway closure process. Simulations are repeated for the no-pre-energy-defect and high-pre-energy-defect initial conditions. It is found that ${\tau }_{y,e}^{\ast }$ is concentrated near the walls in the pre-coalescence phase and also near the centerline after the coalescence when the mucus is initialized to have no-pre-energy-defect. The stress distribution also exhibits similar patterns for this initial condition. When the mucus is initialized to have high-pre-energy-defect, the evolution of ${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$ exhibits a different behavior. The ${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$ value remains approximately constant throughout the mucus with a slight decrease near the centerline for the healthy case. In the CF case, ${\tau }_{y,e}^{\ast }∕{\tau }_y^{\ast }$ decreases immediately and later evolves in a pattern similar to the no-pre-energy-defect case. The different initial conditions have a significant impact on the closure time for the CF case, which marks the importance of the residual stresses from the previous breathing cycles.

The evolution of yielded/unyielded regions is examined for the healthy and the CF cases for both the Saramito-HB-IKH and Saramito-HB models throughout the airway closure process for the no-pre-energy-defect and high-pre-energy-defect initial conditions. For both cases under the no-pre-energy-defect initial conditions, it is found that, while the yielding occurs after sufficient stress buildup first near the shoulders of the liquid bulge and then in the necking region at the centerline in the Saramito-HB model, it occurs immediately throughout the liquid region in the Saramito-HB-IKH model. In the case of initially high-pre-energy-defect case, the yielding starts near the neck region prior to the coalescence and grows further into the liquid in the post-coalescence phase in a qualitatively similar way to the Saramito-HB model. For the healthy case, the speed of this growth is essentially similar in both models whereas for the CF case, the Saramito-HB-IKH model demonstrates faster growth than the case of the Saramito-HB model.

It is found that the kinematic hardening has a larger effect on the airway closure time rather than the mechanical stresses on the airway wall. In general, the airway closure occurs earlier in the Saramito-HB-IKH model than that in the Saramito-HB model for both the cases but this effect is significantly pronounced in the CF case. Airway closure is not observed in the Saramito-HB model for the CF mucus where the elastic modulus is high except for very high $La$ and $\epsilon$ values in contrast with the Saramito-HB-IKH model which results in an airway closure in the same conditions. The effect of kinematic hardening on the closure time becomes more pronounced as the initial film thickness ($\epsilon$) decreases but it is not very sensitive to the variations in the Laplace number ($La$).

Simulations are also performed by varying the initial value of the effective yield stress in the range of ${\tau }_{y,e}^{\ast }\in [0,{\tau }_y^{\ast }]$ to mimic the residual stresses remaining from the previous breathing cycles. It is observed that airway closure is delayed for the CF case as the initial value of ${\tau }_{y,e}^{\ast }$ is increased, indicating that the disturbances on the liquid layer caused by the breathing cycle could lead to a significant effect on the airway closure mechanism and resulting mechanical stresses.

A parametric study is also conducted to examine the sensitivity of airway closure to the uncertainties in the model parameters. It is found that variations in $G^{\ast }$, ${\mu }_{L,P}^{\ast }$ and ${\tau }_y^{\ast }$ have a significant effect on the airway closure and the resulting mechanical stresses. In particular, $G^{\ast }$ significantly affects the closure time, the wall stresses and their gradients. The polymeric viscosity ${\mu }_{L,P}^{\ast }$ has similar effects as $G^{\ast }$, with the effects being less prominent. However, ${\mu }_{L,P}^{\ast }$ also influences the dissipation of the stresses at the wall after the closure. The yield stress is found to have a considerable effect on the closure time. Airway closure occurs earlier as ${\tau }_y^{\ast }$ decreases. It is also worth noting that the effects of the kinematic hardening diminish as the yield stress decreases.

This study provides important results regarding the complex behavior of airway pulmonary mucus and the importance of taking the energy dissipation into account before yielding via the concept of the kinematic hardening. The utilization of an isotropic kinematic hardening remedies an important deficiency of the EVP model, and the coupled model thus provides a more realistic evaluation of the closure process. Using the Saramito-HB-IKH model allows to understand how sensitive a numerical prediction is to the modeled initial conditions, hence how predictive it can be. In a future work, additional physics of the closure phenomenon can be uncovered through quantification of the dissipated energy. Additionally, considering a two-layer approach with the kinematic hardening model, and including the effects of surfactant could provide a deeper understanding of this phenomenon. While this study focuses mainly on the numerical and fluid dynamics aspects of the airway closure phenomenon, the effects of mucus rheology on airway physiology will be investigated in details in a future study for a wide range of diseased conditions. Finally, it is also important to realize that results presented in this paper are highly dependent on the accuracy of the rheological measurements for the pulmonary mucus properties. Accurately capturing the complex non-Newtonian behavior of the pulmonary mucus is a challenging but essential task for more accurate model predictions which would be highly beneficial for understanding of the closure process in the human lungs.

Highlights.

• Integrating kinematic hardening into an elastoviscoplastic (EVP) model captures the complex non-Newtonian behavior of pulmonary mucus more accurately.

• Kinematic hardening of mucus polymers induces earlier closure of the airway with effects being more apparent at low Laplace numbers and initial film thicknesses.

• The mucus yield stress is a prominent factor in the timing of airway closure.

• The relaxation of stresses after closure is particularly affected by the polymeric viscosity.

Appendix A. Rheological Fitting Methodology

In the fitting procedure, a one-dimensional version of the constitutive law for the Saramito-HB-IKH model is considered. Thus, the stress components evolve by

$$\begin{gathered} \frac{d{\tau }_{xx}^{\ast }}{d{t}^{\ast }}-\frac{2{\tau }_{xy}^{\ast }{\stackrel{.}{\gamma }}^{\ast }}{G^{\ast }}+\max {\left(0,\frac{\mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }}{K^{\ast }{\mid {\tau }^{\ast d}\mid }^n}\right)}^{\frac{1}{n}}{\tau }_{xx}^{\ast }=0, \\ \frac{d{\tau }_{xy}^{\ast }}{d{t}^{\ast }}+\max {\left(0,\frac{\mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }}{K^{\ast }{\mid {\tau }^{\ast d}\mid }^n}\right)}^{\frac{1}{n}}{\tau }_{xy}^{\ast }={\stackrel{.}{\gamma }}^{\ast }, \end{gathered}$$ (A.1)

where ${\tau }_{xx}^{\ast }$ is the normal stress component, ${\tau }_{xy}^{\ast }$ is the shear stress component, $G^{\ast }$ is the elastic modulus, $K^{\ast }$ is the consistency parameter and $n$ is the power-law index. The magnitude of deviatoric part of the extra stresses $\mid {\tau }^{\ast }d\mid$ is calculated as

$$\mid {\tau }^{\ast d}\mid =\sqrt{{(0.5{\tau }_{xx}^{\ast })}^2+{({\tau }_{xy}^{\ast })}^2}.$$ (A.2)

At each time step, effective yield stress ${\tau }_{y,e}^{\ast }$ is evaluated according to Eq. (5) where the parameter $A$ is obtained from

$$\frac{dA}{d{t}^{\ast }}=\frac{\mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }}{\mid \mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }\mid }{\left(\frac{\mid \mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }\mid }{K^{\ast }}\right)}^{\frac{1}{n}}-qA{\left(\frac{\mid \mid {\tau }^{\ast d}\mid -{\tau }_{y,e}^{\ast }\mid }{K^{\ast }}\right)}^{\frac{1}{n}},$$ (A.3)

which is solved numerically using a 4th order Runge-Kutta method. It should be noted that ${\tau }_{y,e}^{\ast }$ and $A$ are initially set to zero in the fitting procedure. In the LAOS tests [14], the deformation in strain amplitude sweep testing procedure takes the following form

$$\gamma ={\gamma }_{o}sin({\omega }^{\ast }{t}^{\ast }),$$ (A.4)

where ${\gamma }_o$ is the constant strain amplitude and ${\omega }^{\ast }$ is the oscillation angular frequency. For the Saramito-HB-IKH model, strain is not directly taken as an input. Instead, strain rate is considered, which is defined as $\stackrel{.}{\gamma }=\partial \gamma ∕\partial t$. The input velocity field is defined as a simple shear flow, i.e.,

$${\mathit{u}}^{\ast }=({\stackrel{.}{\gamma }}^{\ast }(t^{\ast })y^{\ast },0,0).$$ (A.5)

For the fitting procedure, following Erken et al. [27], the cost function is defined as

$$C(G^{\ast },{\tau }_y^{\ast },n,K^{\ast },q)=\sum _{i=1}^M\left({\left[\frac{G_{fit}^{\prime }}{G_{exp}^{\prime }}-1\right]}^2+{\left[\frac{G_{fit}^{″}}{G_{exp}^{″}}-1\right]}^2\right),$$ (A.6)

where $M$ is the number of strain amplitude values provided in the experimental data of Patarin et al. [7] for each condition.

The fitting procedure involves the following steps:

1. Initial values for $G^{\ast }$, ${\tau }_y^{\ast }$, $n$, $K^{\ast }$ and $q$ are assumed.

2. Using the Fourier transform, the resulting stress response is analyzed for each strain amplitude.

3. For a specified strain amplitude and the assumed material properties, the updated values of $G_{fit}^{\prime }$ and $G_{fit}^{″}$ are acquired and the cost function is re-evaluated from Eq. (A.6) using these new values.

4. If the calculated cost function reaches a local minimum, the iteration is terminated and the final values for $G^{\ast }$, ${\tau }_y^{\ast }$, $n$, $K^{\ast }$ and $q$ are obtained. Otherwise, the values in the first step are updated and the iteration is repeated until a local minimum is reached. MATLAB’s fmincon function [49] is used to solve the minimization problem.

It should be noted that, before determining the final values of the mucus properties, ${\tau }_y^{\ast }$ is limited for each pathological condition in accordance with the ranges provided in Patarin et al. [7] in order to comply with the experimental data and model the mucus behavior accurately. So, even if better fits are obtained for values outside the specified ranges, they are disregarded. Similarly, $n$ is kept within a range of $0<n\le 1$ since mucus is expected to exhibit a shear-thinning behavior.

We finally note that the value of ${\mu }_{L,S}^{\ast }$ is set to zero in the fitting process. In addition, when ${\mu }_{L,S}^{\ast }$ is included among the fitting parameters, its value appears to be very close to zero. However, following Tai et al. [21], ${\mu }_{L,S}=0.013\text{Pa}\cdot \mathrm{s}$ is used in all the simulations presented in the paper to account for the water content especially in the serous layer which is very watery and included in the liquid layer via homogenization as suggested by Romanò et al. [23, 24].

Appendix B. Validation of the isotropic kinematic hardening model

The isotropic kinematic hardening model has been incorporated into the Saramito model [9] by Fraggedakis et al. [15] and the combined model has been shown to remedy the deficiency of the Saramito model in modelling the loss modulus through fitting the model to the experimental data for a 0.07% w/w Carbopol solution under a constant frequency sweep test [50]. In the present paper, the isotropic kinematic hardening model is incorporated into the Saramito-Herschel-Bulkley (Saramito-HB) model [10] and its performance is examined using the same experimental data and the procedure developed by Fraggedakis et al. [15]. For this purpose, both the Saramito-HB-IKH and Saramito-HB models are fitted to the experimental data and the results are compared in Fig. B.1. It is clearly seen that the Saramito-HB-IKH model remedies the known deficiency of the the Saramito-HB model in modeling the loss modulus at low shear rates and fits to the experimental data very well for both $G^{\prime }$ and $G^{″}$ values in the entire range of shear rates.

Fig. B.1.

Fit to experimental data of storage, $G^{\prime }$, and the loss, $G^{″}$, moduli for 0.07% w/w Carbopol gel [50] using the Saramito-HB model (left side) and the Saramito-HB-IKH model (right side).

Appendix C. Grid Independence

A grid convergence study is performed to determine the grid resolution required to obtain grid-independent results. For this purpose, simulations are performed for the CF case at $\epsilon =0.35$ and $La=300$ using the Saramito-HB-IKH model with grid resolutions of 64 × 386, 80 × 480, 96 × 576 and 112 × 672 in the radial and axial directions, respectively. Results are plotted in Fig. C.1 for $\Delta {\tau }_w$. The inset in Fig. C.1 shows the variation of $\Delta {\tau }_w$ against square of non-dimensional grid size at the non-dimensional times indicated by vertical dotted lines in Fig. C.1, i.e., at $t∕La=0.5$, $t∕La=1.0$, $t∕La=1.125$ and $t∕La=1.5$. A nearly linear relationship between computed values and ${\left(\frac{\Delta x}{a}\right)}^2$ indicates that the asymptotic range is reached and the numerical method is second order accurate. This figure also shows that the grid resolution of 96 × 576 is sufficient to reduce the maximum error below 1.5%. Note that the maximum error is computed with respect to the spatial error-free values predicted using the Richardson’s extrapolation in the limit as $\Delta x\to 0$ and assuming a second order accuracy. The most critical point of consideration would be when the closure occurs, corresponding to $t∕La\approx 1.125$, where the error is found to be even lower. Therefore, this grid resolution is used in all the simulations presented in this paper.

Fig. C.1.

Grid independence performed for the case of CF considering the evolution of $\Delta {\tau }_w$ using the Saramito-HB-IKH model [$\epsilon =0.35$ and $La=300$]. The vertical dashed lines indicate the non-dimensional times of $t∕La=0.5$, $t∕La=1.0$, $t∕La=1.125$ and $t∕La=1.5$ where the spatial error is quantified and plotted in the inset. The solid lines in the inset are the least-squares fits to the numerical data.

Data availability

Data will be made available on request.