Stokes layers in oscillatory flows of viscoelastic fluids

Abstract

Oscillatory flows of viscoelastic fluids are studied from the perspective of Stokes viscoelastic layers. We identify the governing dimensionless variables, and study the flows in a general way for fluids with linear rheology. Nonlinearities can be treated perturbatively to account for reported flow instabilities.

This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

1. Introduction

Oscillatory fluid flows play an important role in industrial problems such as pumping, drilling and oil recovery, and in physiological processes such as mucous flow in respiration and blood pumping. A key concept in oscillatory flows is the Stokes layer, i.e. the region of fluid adjacent to a wall that is in motion as a result of the wall oscillatory motion along its own plane. Obtaining the flow field in this region for Newtonian fluids is a classical problem of fluid mechanics known as Stokes’ second problem [1,2]. Many fluids of industrial and physiological interest, however, exhibit complex rheological behaviours, including rate-dependent viscosity, elasticity, plasticity and thixotropy, which modify Stokes layers in very interesting ways [3–5]. The purpose of this paper is to discuss how the notion of Stokes layers sheds light on the properties of oscillatory flows of viscoelastic fluids.

In §2, we briefly summarize the historical development of the subject. In §3, we introduce Stokes’ second problem for the simplest viscoelastic fluid—modelled by a single-mode Maxwell constitutive equation—and Ferry’s work on viscoelastic shear waves. From this perspective, in §4, we look at wall-bounded zero-mean oscillatory flows of viscoelastic fluids. We pay particular attention to the presence of resonances in the one-dimensional base flow, and to the secondary vortical flows that show up as the oscillatory forcing increases. Conclusions are drawn in §5.

2. Stokes’ second problem: historical perspective

The equations of motion of a viscous fluid—the Navier–Stokes equations—were developed in the first half of the nineteenth century by Navier [6], Poisson [7], Saint-Venant [8] and Stokes [1]. Until then, ‘hydrodynamics’ was the mathematical theory of the motion of inviscid fluids, while ‘hydraulics’ dealt with hydrodynamic questions of practical interest in which viscosity played a relevant role [9]. In 1845, Stokes proposed a series of problems in which, exceptionally, it was possible to solve the dynamic equations exactly because nonlinear terms were either negligible or identically zero for geometrical reasons [1]. On the one hand, he was moved by the purpose of showing the importance of frictional forces in the movement of viscous fluids. In his own words, ‘the subsidence of the motion in a cup of tea which has been stirred may be mentioned as a familiar instance of friction, or, which is the same, of a deviation from the law of normal pressure; and the absolute regularity of the surface when it comes to rest, whatever may have been the nature of the previous disturbance, may be considered as a proof that all tangential force vanishes when the motion ceases’ [10]. On the other hand, he wanted to stress the need of a no-slip boundary condition between a viscous fluid and a solid wall (the absence of relative tangential velocity), so that the solid–fluid friction was of the same nature as the friction between adjacent layers of fluid.

Stokes’ second problem, as known today, considers an infinite flat plate beneath a semi-infinite layer of initially quiescent incompressible viscous fluid; the plate oscillates harmonically in its own plane, y–z say, along the z-axis, with amplitude z₀ and angular frequency ω₀, and the purpose is to obtain the steady periodic one-dimensional flow field, u(x, t). The solution is an exponentially decaying oscillatory field:

$$\mathit{u}(x,t)=(0,0,U_0{\text{e}}^{-x/x_0}\cos ({\omega }_{0}t-\frac{2\pi }{{\lambda }_0}x)),$$ 2.1

where U₀ = z₀ω₀ is the velocity amplitude of the driving plate, as shown in figure 1. This solution satisfies the condition of no-slip between the driving plate at x = 0 and the fluid in contact, and the far-field boundary condition lim _x→∞u = (0, 0, 0). Stokes showed that the attenuation and wave lengths of the fluid motion—for a viscous fluid of dynamic viscosity η—are given by

$$x_0=\frac{{\lambda }_0}{2\pi }=\sqrt{\frac{2\eta }{\rho {\omega }_0}},$$ 2.2

which defines the penetration depth of the Stokes layer. Note that this is the only characteristic length that can be built from the variables of the problem for a Newtonian fluid, so that x₀ and λ₀/(2π) coincide in this case. Transverse oscillations are thus overdamped, and cannot propagate into the fluid. Equation (2.1) can also be written in the more compact form $u_z=U_0\mathrm{\Re }\{\exp (\mathrm{i}\hspace{0.17em}{\omega }_{0}t-\kappa x)\}$, where the reciprocal length scale κ = (1/x₀) + i (2π/λ₀).

Figure 1.

(a) Exponentially decaying oscillatory wave (2.1) driven by the harmonic oscillation of the lateral wall. (b) Attenuation and wave lengths (upper (blue) and lower (red) curves, respectively), in units of $\sqrt{\eta \lambda /\rho }$, versus dimensionless driving frequency De. The Newtonian behaviour (for which λ = 0) can formally be included also in this representation, as it corresponds to the prefactors of equations (3.5) for arbitrary λ. The result is the universal curve shown in black (middle curve). (Online version in colour.)

3. Viscoelastic layers

In contrast with purely viscous fluids, viscoelastic fluids behave as elastic solids on short time scales. The simplest viscoelastic behaviour is reproduced by the single-mode upper-convected Maxwell constitutive equation (UCM)

$$\mathit{\tau }+\lambda {\mathit{\tau }}_{(1)}=2\eta \mathit{e},$$ 3.1

where τ and $\mathit{e}=[\mathrm{\nabla }\mathit{u}+{(\mathrm{\nabla }\mathit{u})}^{†}]/2$ are the stress and rate-of-strain tensors, respectively. This model features a constant dynamic viscosity η and a single exponential relaxation of elastic stresses in a time scale λ (fluid relaxation time). The subindex (1) denotes an upper-convected time derivative,

$${\mathit{\tau }}_{(1)}=\frac{\mathrm{\partial }\mathit{\tau }}{\mathrm{\partial }t}+\mathit{u}\cdot \mathrm{\nabla }\mathit{\tau }-\{{(\mathrm{\nabla }\mathit{u})}^{†}\cdot \mathit{\tau }+\mathit{\tau }\cdot (\mathrm{\nabla }\mathit{u})\},$$ 3.2

that makes the model frame invariant [11]. The single-mode UCM constitutive equation represents very accurately in particular the linear shear rheology of wormlike micellar (WLM) solutions, which spans however a relatively narrow range of low shear strain rates. Above this range, the flow curve becomes nonlinear and the fluid strongly shear-thins [12]. The governing equation of Stokes’ second problem for an incompressible UCM fluid reads

$$\rho (1+\lambda \frac{\mathrm{\partial }}{\mathrm{\partial }t})\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }t}-\eta \frac{{\mathrm{\partial }}^2{u}_z}{\mathrm{\partial }{x}^2}=-(1+\lambda \frac{\mathrm{\partial }}{\mathrm{\partial }t})\frac{\mathrm{\partial }p}{\mathrm{\partial }z}.$$ 3.3

Since fluid motion is driven solely by the oscillation of the wall at x = 0, ∂p/∂z = 0. Ferry showed in 1942 [3] that combination of equations (3.1) and (3.3) leads also to a solution of the form (2.1), but now

$$x_0=\frac{{\lambda }_0}{2\pi }[{\omega }_0\lambda +\sqrt{1+{({\omega }_0\lambda )}^2}].$$ 3.4

Note that the ratio λ₀/(2πx₀) is 0 for an elastic solid and 1 for a viscous liquid, while values in between correspond to viscoelastic materials [13]. The ratio between x₀ and λ₀ depends only on the dimensionless Deborah number, De ≡ ω₀λ, and since $\text{De}>0$ the penetration depth becomes longer than the wavelength for all De, as shown in figure 1. Now transverse oscillations (Ferry waves) can propagate effectively before they are attenuated [3,14]. Moreover,

$$x_0=\sqrt{\frac{2\eta \lambda }{\rho \text{De}}}\sqrt{\frac{1}{-\text{De}+\sqrt{1+{\text{De}}^2}}}\text{and}\frac{{\lambda }_0}{2\pi }=\sqrt{\frac{2\eta \lambda }{\rho \text{De}}}\sqrt{\frac{1}{\text{De}+\sqrt{1+{\text{De}}^2}}},$$ 3.5

where the square-root prefactor is the penetration depth of the purely viscous Stokes layer, given by equation (2.2). We see that the viscoelastic penetration depth is longer than the purely viscous one for all De > 0. Additionally when elasticity is dominant, i.e. for De > 1,

$$x_0\to 2\sqrt{\frac{\eta \lambda }{\rho }}\text{and}\frac{{\lambda }_0}{2\pi }\simeq \frac{x_0}{2\text{De}}.$$ 3.6

As the forcing frequency increases, the penetration depth quickly tends to a limiting value, while the wavelength decreases as 1/De allowing for an increasingly large number of oscillations before the wave decays [15,16]. Figure 1 shows that above De = 5 the expressions above can be safely taken as equalities. We will assume this to be the case in all what follows, as this is the range of driving frequencies where elastic effects are noticeable.

4. Wall-bounded zero-mean oscillatory flows of viscoelastic fluids

In this section, we study zero-mean oscillatory flows of viscoelastic liquids in wall-bounded domains. Flows are driven either by the motion of the vertical sidewalls or by an oscillatory pressure gradient. Both situations are identical for incompressible fluids of constant density in the reference frame of the sidewalls [17]. We focus on the periodic stationary state of unidirectional laminar flow, after transients have passed. This state can be explicitly derived for the single-mode UCM model in some particular geometries. We will present the results for two parallel plates and for a vertical cylinder. The oscillatory flow will be regarded as the interference in time and space of the Ferry shear waves generated by the oscillatory driving. This perspective explains the dramatic deviations of the velocity profiles of viscoelastic liquids from their Newtonian counterparts. In this linear regime, corresponding to small forcing amplitudes and low forcing frequencies, all the results apply also to the Oldroyd-B model (which in addition to the viscoelastic polymeric contribution includes the presence of a Newtonian solvent) by plugging in the corresponding expression of the complex characteristic reciprocal length κ [15]. By contrast, the nonlinear response at larger forcings requires nonlinear constitutive equations and cannot be obtained analytically. The Giesekus constitutive equation, a minimal extension of the UCM model that will be introduced at the end of this section and that is able to capture the shear-thinning behaviour of semi-dilute micellar solutions at large shear rates, has been used to investigate the nonlinear regime numerically, both in the parallel plate geometry [18–20] and in the vertical cylinder [21].

(a). Fluid motion between parallel plates

Let us consider now the fluid confined between the original plate at x = 0 and a second vertical plate, parallel to the original one, at a distance H. The dimensions of the plates are assumed to be much larger than the gap thickness H. This new length scale H is useful to write the equation of motion (3.3) in dimensionless form. To this purpose, we define the non-dimensional variables $\tilde{x}=x/H$ and $\tilde{t}=t/\lambda$. Then ${\tilde{u}}_z=u_z\lambda /H$. In the absence of a pressure gradient equation (3.3) becomes

$$\frac{1}{\text{El}}(1+\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{t}})\frac{\mathrm{\partial }{\tilde{u}}_z}{\mathrm{\partial }\tilde{t}}-\frac{{\mathrm{\partial }}^2{\tilde{u}}_z}{\mathrm{\partial }{\tilde{x}}^2}=0,$$ 4.1

where the elasticity number El = ηλ/(ρH²) is the ratio of elastic to inertial forces, El = Wi/Re. The Reynolds number is given by Re = U₀ Hρ/η and the Weissenberg number by Wi = U₀λ/H, i.e. the dimensionless amplitude of the oscillating plate velocity. Equation (4.1) must be complemented with the appropriate boundary conditions. In dimensionless form, the no-slip condition at an oscillatory wall reads

$${\tilde{u}}_z=\text{Wi}\cos (\text{De}\hspace{0.17em}\tilde{t}).$$ 4.2

The two equations together show that viscoelastic shear waves in UCM fluids are governed by the dimensionless numbers El, Wi and De.

For $\text{De}>5,$ equation (3.6) leads to $\text{El}=(1/4){(x_0/H)}^2$, where H/x₀ is twice the viscoelastic equivalent of the Stokes parameter, i.e. the ratio of half the domain gap thickness to the penetration depth of the shear waves launched by the moving plate. This ratio defines three different regimes [13,18].

• (i)A ‘gap-loading’ regime, for H ≪ x₀, where stress and strain are approximately uniform in the gap, the instantaneous velocity profiles across the gap are linear, and the shear rate is independent of position. This is the regime in which modern commercial rheometers operate.
• (ii)A ‘surface-loading’ regime, H ≫ x₀, in which shear waves decay before they reach the opposite wall. This situation is effectively the same than the semi-infinite domain. It was used by Ferry and collaborators to devise a shear wave rheometer, based on equations (3.5) and measurements of shear wave propagation in strain-birefringent polymeric liquids and gels [3,14,22–24].
• (iii)An ‘intermediate gap’ regime in between, where shear waves have room to develop, reflect off at the boundaries, and mutually interfere, setting up a complex heterogeneous flow field across the domain. Vasquez et al. have shown that this ‘intermediate gap’ regime will occur whenever λ₀/(2π) ≤ H ≤ x₀ in the case of linear viscoelastic materials [18].

Mitran and co-workers have studied the case where the oscillatory motion of the left plate sets the fluid into motion, while the right plate remains at rest. This situation emulates the mobilization of mucus by coordinated cilia oscillations in lung airways [19,25]. We will consider instead the case in which the two parallel plates oscillate synchronously in time with the same amplitude and frequency [15]. For convenience, we place the plates at x = −a and x = a, so that H = 2a. The fluid occupies the domain (x, y, z) ∈ [ − a, a] × ( − ∞, ∞) × ( − ∞, ∞). The velocity field results from the superposition of the oscillatory waves launched by the two opposite plates

$$u_z(x,t)=U_0\mathrm{\Re }\left[\{\frac{\sinh (\kappa (a-x))}{\sinh (\kappa 2a)}+\frac{\sinh (\kappa (a+x))}{\sinh (\kappa 2a)}\}{\text{e}}^{\mathrm{i}{w}_{0}t}\right]=U_0\mathrm{\Re }\left[\frac{\cosh (\kappa x)}{\cosh (\kappa a)}{\text{e}}^{\mathrm{i}{w}_{0}t}\right],$$ 4.3

where κ = (1/x₀) + i (2π/λ₀), as usual, and x₀, λ₀ are given by (3.5). Examples of these velocity profiles are shown in figure 2. For small elasticity number (surface-loading regime), the shear waves launched by the oscillatory plates cannot propagate through the entire domain, and the velocity magnitude at the centre is very small. By contrast, for elasticity numbers in the intermediate gap regime a highly reversing profile is maintained across the whole domain. The flow field organizes in parallel layers of alternating upward/downward motion that oscillate with the periodicity of the walls.

Figure 2.

Vertical velocity profiles of an UCM fluid oscillated by two synchronous parallel plates, for $\text{De}=2\pi \sqrt{\text{El}}$ (non-resonant condition, left) and $\text{De}=5\pi \sqrt{\text{El}}$ (resonant condition, right), and elasticity numbers El = 0.0025 (surface-loading regime, top) and 50 (intermediate gap regime, bottom), at time phases ω₀ t = 0 (blue curve) and π/2 (red curve), for which the velocity at the sidewalls is U₀ and 0, respectively. Note the different vertical scale for resonant conditions in the intermediate gap regime (last panel). (Online version in colour.)

(b). Fluid motion in a vertical cylinder

We consider now the zero-mean oscillatory motion of a UCM fluid in a vertical cylinder. This geometry is easier to implement experimentally than the parallel-plate geometry. Moreover, oscillatory flows of physiological and industrial interest take place often in circular pipes. In an idealized setting, the fluid is contained in an infinitely long circular pipe of radius a. The velocity field u is now given in cylindrical coordinates (r, θ, z) ∈ [0, a] × [0, 2π] × ( − ∞, ∞). The flow is induced by a pressure gradient in the axial direction that varies harmonically in time. In practice, this pressure gradient is produced either by the oscillatory motion of the sidewall or by the synchronous oscillation of the top and bottom lids in the axial direction. In the two cases, the no-slip boundary conditions are discontinuous at the junctions where the lids meet the cylindrical sidewall. The idealized boundary conditions can be satisfied approximately in cylinders of large aspect ratio. For water, the recirculation induced by the presence of the top and bottom endwalls seems not to have influence on the flow in the middle of the cylinder for aspect ratios above 20 [26]. In experiments with viscoelastic WLM solutions in a cylinder of 560 mm length and 50 mm diameter the recirculating flow at the top and bottom lids did not extend beyond 20 mm [27].

Suppose that the top and bottom lids oscillate in time according to z(t) = z₀sin(ω₀ t). The no-slip boundary conditions read u(a, θ, z, t) = (0, 0, 0) at the sidewall, and u(r, θ, ± ∞, t) = (0, 0, U₀cos(ω₀ t)) at the endwalls, where U₀ = z₀ω₀ is the velocity amplitude of the top and bottom lids. At low driving amplitudes and frequencies the laminar base flow is one dimensional and reduces to u = (0, 0, u_z(r, t)) for symmetry reasons. The governing equation in cylindrical coordinates now reads

$$\rho (1+\lambda \frac{\mathrm{\partial }}{\mathrm{\partial }t})\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }t}-\eta (\frac{1}{r}\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }r}+\frac{{\mathrm{\partial }}^2{u}_z}{\mathrm{\partial }{r}^2})=-(1+\lambda \frac{\mathrm{\partial }}{\mathrm{\partial }t})\frac{\mathrm{\partial }p}{\mathrm{\partial }z}.$$ 4.4

The pressure gradient results from the acceleration of the endwalls, $-z_0{{\omega }_0}^2\sin ({\omega }_{0}t)$, so that

$$\frac{\mathrm{\partial }p}{\mathrm{\partial }z}=-\rho {{\omega }_0}^2\mathrm{\Im }\{\delta {\text{e}}^{\mathrm{i}{\omega }_{0}t}\},$$ 4.5

where ρ is the density of the fluid and δ is a coefficient proportional to z₀ that must be determined by continuity of the velocity at the endwalls. The steady-state response u_z to this harmonically varying pressure gradient reads

$$u_z(r,t)={\omega }_0\mathrm{\Re }\left\{\mathrm{\Delta }[1-\frac{J_0(\kappa r)}{J_0(\kappa a)}]{\text{e}}^{\mathrm{i}{\omega }_{0}t}\right\},$$ 4.6

where κ = (1/x₀) + i (2π/λ₀) again, and x₀, λ₀ are given by (3.5). The factor Δ is obtained by imposing that the volume scanned by the piston in one quarter of the oscillation coincides with the volume of liquid displaced in this same time interval, computed by integration of (4.6) in space and time [28]. This leads to Δ = −z₀ J₀(κa)/J₂(κa), where J₀ and J₂ are zero- and second-order Bessel functions of the first kind, respectively. The final expression for the velocity reads [15]

$$u_z(r,t)=-U_0\mathrm{\Re }\left\{\frac{J_0(\kappa a)}{J_2(\kappa a)}[1-\frac{J_0(\kappa r)}{J_0(\kappa a)}]{\text{e}}^{\mathrm{i}{\omega }_{0}t}\right\}.$$ 4.7

With the appropriate choice of the reciprocal length scale κ, this expression is valid also for Newtonian fluids and for viscoelastic fluids of constant shear viscosity that obey the Oldroyd-B constitutive equation.

Locally u_z oscillates harmonically in time with the periodicity of the forcing, and its phase lag (with e.g. the position of the top and bottom lids) varies across the tube. The absence of additional harmonics in the response to the periodic driving is a result of the linearity of the equations. Similarly to the parallel-plate case, the velocity profiles are highly reversing in the intermediate gap regime, as a result of the mutual interference of shear waves across the fluid domain. This regime corresponds to intermediate values of the elasticity number, since the condition λ₀/(2π) ≤ H ≤ x₀ is equivalent to 1/4 ≤ El ≤ De² for De > 5. In this regime, the fluid flow organizes in concentric cylindrical layers of alternating upward/downward motion. Increasing the dimensionless driving frequency De has two effects: the number of concentric layers increases, because the wavelength of the shear waves λ₀ ∼ 1/De for De > 5, and the amplitude of the velocity field increases as well because it is proportional to ω₀. Thus, with larger and steeper velocity profiles, shear strain rates locally attain increasingly larger values at increasing De, which easily fall beyond the range of applicability of linear constitutive equations such as UCM and Oldroyd-B.

(c). Flow resonances

As we have seen, the mutual interference of the shear waves launched by oscillatory walls or driven by oscillatory pressure gradients creates an interference pattern in wall-bounded fluid domains for elasticity numbers corresponding to the intermediate gap regime [29]. At particular driving frequencies, this interference can be constructive and lead to resonances in the flow field.

In the parallel-plate geometry, the condition of constructive interference is 2a = (1/2 + n)λ₀ with n = 0, 1, 2, … [15]. For De > 5, this is equivalent to

$$\text{De}=(\frac{1}{2}+n)2\pi \sqrt{\text{El}},n=0,1,2,\dots$$ 4.8

where we recall that El depends only on plate separation and fluid properties. At those particular values of De the peak velocity increases by more than a factor 10 in magnitude and its time phase shifts abruptly by π/2, as shown in the last panel of figure 2; in the middle of the domain the fluid moves in phase (or phase opposition) with the driving plates. Since resonance frequencies are sharply defined, resonance-based interferometric measurements in this relatively simple geometry seem a promising way of measuring very accurate values of relaxation times. Though large increases in local velocity may bring the fluid rheology out of the linear regime in an actual experiment, resonances are still expected to occur precisely at the De values predicted.

In the cylindrical geometry, the fluid velocity at the symmetry axis reads, from equation (4.7),

$$u_z(0,t)=-U_0\mathrm{\Re }\left\{\left[\frac{J_0(\kappa a)-1}{J_2(\kappa a)}\right]{\text{e}}^{\mathrm{i}{\omega }_{0}t}\right\}.$$ 4.9

In the intermediate gap regime resonances occur now at Deborah numbers for which J₂(κa) is minimum. In contrast with the parallel-plate geometry, resonant peaks of u_z(0, t) are no longer equispaced in De, as shown in figure 3, and get slightly larger as De increases. At this point, it is worth noting that equation (4.7) for u_z(r, t) applies to volume-controlled (lid-driven) oscillatory flow. It differs from expressions derived also for a single-mode Maxwell fluid, but for pressure-gradient-controlled conditions [30–33]. The condition of continuity leads to a different flow field. In particular, a first resonance peak at $\text{De}\simeq 5\sqrt{\text{El}}$ (corresponding to ω₀ ≃ 19 rad s⁻¹ in the set-up of figure 3) is absent in lid-driven oscillatory flow. The phase lag between the fluid vertical velocity at the symmetry axis and the driving pressure gradient is also shown in figure 3. The phase lag changes abruptly from ϕ = π/2, 3π/2 to ϕ = 0, π at resonances.

Figure 3.

Experimental measurements of the velocity of an aqueous CPyCl/NaSal [100:60] mM WLM solution at T = 25^°C oscillated in a vertical cylinder. Fluid motion is driven by the oscillations of the bottom lid. The top panel shows the magnitude of the vertical velocity at the cylinder axis, in units of the piston velocity amplitude, against the angular frequency of the piston. The bottom panel shows the phase lag between the velocity of the fluid at the cylinder axis and the position of the driving piston. In both panels, solid dots represent experimental values measured by particle-image velocimetry, and solid lines the corresponding analytical results for an UCM fluid. Data extracted from [27].

Particle-image-velocimetry measurements of the velocity field of a semi-dilute WLM solution, oscillated vertically in a cylinder of large aspect ratio by a driving piston at the bottom end, show indeed the highly reversing nature of the base flow and the presence of resonant frequencies [27,34,35]. In particular, figure 3 shows the experimental results obtained at the symmetry axis of the cylinder. Resonance frequencies and phase lag match very accurately the predicted behaviour, with no fitting parameters. The magnitude of the velocity however is significantly smaller than the UCM prediction, particularly at resonance frequencies. This is a signature of the departure of the fluid rheology from a linear constitutive relation at the high shear rates locally attained in the flow [27].

(d). Flow instabilities

The highly reversing nature of the oscillatory base flow for intermediate elasticity numbers (intermediate gap regime) gives rise to high shear strain rates even at moderate driving amplitudes, specially at resonances. From equation (4.7), shear strain rates are given by

$$\dot{\gamma }(r,t)=\frac{\mathrm{\partial }{u}_z(r,t)}{\mathrm{\partial }r}=-U_0\mathrm{\Re }\left\{\kappa \frac{J_1(\kappa r)}{J_2(\kappa a)}{\text{e}}^{\mathrm{i}{\omega }_{0}t}\right\}={\omega }_{\theta }(r,t).$$ 4.10

The last equality indicates that shear strain rates in the flow coincide with the azimuthal components of the vorticity field $\mathit{\omega }=\mathrm{\nabla }\times \mathit{u}$, because u = (0, 0, u_z(r, t)) in this problem. For locally high strain rates, not only the rheology of most viscoelastic fluids becomes nonlinear. In the vertical cylinder, in addition, the highest shear rates localize at the boundaries of coaxial domains of alternating velocity, which implies also the localization of a large azimuthal vorticity. These two circumstances together lead to the destabilization of the one-dimensional oscillatory base flow for large enough driving amplitudes.

Experiments of oscillatory pipe flow of WLM solutions show indeed that the rectilinear flow observed at small Wi gives rise to a secondary flow at larger Wi with axisymmetric oscillatory vortex rings stacked along the vertical axis of the pipe, or eventually transits directly to a turbulent flow [36,37]. A sketch of the phase diagram reported in [37] is shown in figure 4. The instability threshold depends non-monotonically on forcing frequency, with threshold minima matching precisely the resonance frequencies of the base flow, in qualitative agreement with the mechanism outlined above.

Figure 4.

Measured stability diagram of the oscillatory pipe flow of an aqueous CPyCl/NaSal [100:60] mM WLM solution at T = 25^°C, in forcing amplitude versus forcing frequency, as reported in [37]. The different regions, blue, yellow and green (in ascending order), correspond to rectilinear, vortical, and turbulent flow, respectively, at increasing forcing amplitudes. The red line with empty dots is the threshold of instability of the base flow obtained from the divergence of normal radial stresses. (Online version in colour.)

In order to account for the nonlinear shear rheology of WLM solutions, which typically sets in at shear strain rates around 1 s⁻¹, the single-mode UCM constitutive equation can be supplemented with a quadratic term in shear stress. The resulting single-mode Giesekus constitutive equation reads

$$\mathit{\tau }+\lambda {\mathit{\tau }}_{(1)}+\frac{\alpha \lambda }{{\eta }_0}\mathit{\tau }\cdot \mathit{\tau }=2{\eta }_{\text{0}}\mathit{e},$$ 4.11

where α is a dimensionless parameter (the mobility factor) that takes values between 0 and 1, and η₀ is the zero-shear dynamic viscosity of the solution [38,39]. The quadratic term represents a stress-induced acceleration of the relaxation process which results in shear-thinning and strain softening [40]. The impact of nonlinear fluid rheology on oscillatory viscoelastic layers has been thoroughly studied numerically with the Giesekus constitutive equation by Lindley, Mitran and collaborators [18–20,25]. Here, we focus on the destabilization of the oscillatory base flow, in which the possibility of controlling Wi independently through amplitude and frequency of the forcing provides a more stringent test on theoretical and numerical approaches than other set-ups [16,37]. The velocity profile of the base flow, u_z(r, t), cannot be determined analytically in this case. However, the non-zero components of the stress tensor obey

$$\begin{array}{rl}& {\tau }_{rr}+\lambda \frac{\mathrm{\partial }{\tau }_{rr}}{\mathrm{\partial }t}+\alpha \frac{\lambda }{{\eta }_{\text{0}}}({\tau }_{rr}^2+{\tau }_{rz}^2)=0\\ & {\tau }_{zz}+\lambda \frac{\mathrm{\partial }{\tau }_{zz}}{\mathrm{\partial }t}+\alpha \frac{\lambda }{{\eta }_{\text{0}}}({\tau }_{zz}^2+{\tau }_{rz}^2)-2\lambda \frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }r}{\tau }_{rz}=0\\ \text{and}& {\tau }_{rz}+\lambda \frac{\mathrm{\partial }{\tau }_{rz}}{\mathrm{\partial }t}+\alpha \frac{\lambda }{{\eta }_{\text{0}}}{\tau }_{rz}({\tau }_{rr}+{\tau }_{rz})-\lambda \frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }r}{\tau }_{rr}={\eta }_0\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }r}\end{array}\}.$$ 4.12

For α = 0 (UCM), the first equation shows that radial normal stresses decouple from the other stress components and decay to 0 in the time scale λ. Note that this time scale spans more than one oscillation period for De > 2π. By contrast, shear stresses oscillate in time with the periodicity of the forcing, according to

$${\tau }_{rz}(r,t)=\rho U_0{\omega }_0\mathrm{\Re }\left[\frac{\text{i}}{\kappa }\frac{J_1(\kappa r)}{J_2(\kappa a)}{e}^{\text{i}{\omega }_{0}t}\right].$$ 4.13

As the flow becomes increasingly reversing at higher De, the WLM solution experiences larger shear rates as discussed before. The Giesekus constitutive equation becomes more appropriate than the UCM equation in this regime. For α ≠ 0, the presence of the mobility factor in equations (4.12) couples normal and shear stresses. This coupling makes radial normal stresses susceptible to undergoing a saddle-node bifurcation, controlled by the local shear rate ∂_r u_z. This possibility can be pursued perturbatively, by considering that the shear stresses τ_rz in the UCM base flow are a first-order approximation of the actual shear stresses in the base flow. This approximation relies on the fact that the UCM velocity profiles are in good agreement with the velocity profiles experimentally observed at small forcing amplitudes, except at resonances [27]. Following this reasoning, we have found that above a critical value of the forcing amplitude normal radial stresses diverge in a finite time, and so do all other stress components. Normal radial stresses are perpendicular to the flow direction and can bend the streamlines, giving rise to the secondary flow observed experimentally [21]. Repeating this scheme systematically, in the whole range of forcing frequencies experimentally accessible, we have found remarkably good agreement with the threshold determined experimentally, as shown in figure 4, with no adjustable parameters other than material properties and system dimensions.

Since the present scheme relies on maximum shear stresses τ_rz in the UCM base flow, it is not affected by the fact that the Giesekus model with no solvent viscosity contribution is incomplete for α > 0.5: shear-thinning is so intense for those values of α that the curve of shear stress versus shear rate exhibits a maximum (at shear rates near the inverse of the fluid relaxation time) and decreases for higher shear rates; for stress values above the peak the model makes no predictions about the shear rate [11,41]. Our approach predicts instead that for sufficiently large De there is instability for all values of α, according to

$$\sqrt{{⟨{\tau }_{rz}^2⟩}_{crit}}\simeq \frac{1}{2\alpha }\left(\frac{{\eta }_{\text{0}}}{\lambda }\right),$$ 4.14

where <· > represents a time average over one oscillation period. This being the case, the instability threshold goes to infinity as α → 0, in agreement with the fact that normal radial stresses in the rectilinear flow of the single-mode UCM constitutive equation remain finite at all driving amplitudes. Shear-thinning seems therefore an essential ingredient for the destabilization of the rectilinear flow in this framework.

This analysis however does not rule out that the zero-mean oscillatory flow of viscoelastic fluids with linear rheological constitutive equations may be unstable to infinitesimal perturbations in the radial or the azimuthal direction, without the need of invoking nonlinear rheological laws. The corresponding linear stability analysis is however cumbersome in the cylindrical geometry, because the base flow is given in terms of Bessel functions of complex argument. The facts are that shear thinning of the WLM solutions is non-negligible at the flow conditions at which instability is observed, so that the Giesekus equation is a better model of their shear rheology than the UCM equation, and that the predictions of our perturbative analysis based on the Giesekus equation for the instability threshold are in quite good agreement with the experimental results.

5. Conclusion

Elasticity enables the propagation of shear waves in viscoelastic media. Stokes layers of linear viscoelastic fluids characterized by the UCM and Oldroyd-B constitutive equations are governed by three dimensionless numbers, El, Wi and De, accounting, respectively, for the ratio of elastic to inertial forces, the dimensionless forcing amplitude and the dimensionless forcing frequency. In wall-bounded flows, three qualitatively different regimes can be distinguished in terms of El, corresponding to narrow, intermediate and wide systems. In the intermediate gap regime, the viscoelastic flow is very different from the purely viscous flow. Interference in space and time of shear viscoelastic waves in the whole fluid domain results in a highly reversing heterogeneous flow that resonates at specific, precisely defined forcing frequencies.

Spatial localization of shear stress and vorticity in the base flow at increasing forcing amplitudes and frequencies leads to destabilization, with either formation of a secondary flow consisting on oscillatory vortex rings or direct transition to turbulent flow. Treating nonlinear rheological behaviour at large shear rates perturbatively from the UCM model, it is found that the instability is triggered by the divergence of normal stresses in the transverse direction.

In conclusion, we have presented an overview of zero-mean oscillatory flows of viscoelastic fluids from the perspective of Stokes viscoelastic layers. It is a modest tribute to Sir George Gabriel Stokes foundational work on hydrodynamics, on the 200th anniversary of his birth. Borrowing a sentence that he wrote in 1845, in reference to a mathematical proof, hopefully ‘as it is, this [overview] may possibly be not altogether useless’ [1].

Data accessibility

Data used in this manuscript were previously published in refs. [27] Casanellas L, Ortín J. 2012. Experiments on the laminar oscillatory flow of wormlike micellar solutions. Rheologica Acta 51, 545–557. [37] Casanellas L, Ortín J. 2014. Vortex ring formation in oscillatory pipe flow of wormlike micellar solutions. Journal of Rheology 58, 149–181.

Competing interests

The author declares that he has no competing interests.

Funding

Financial support from MINECO (Spain) and AGAUR(Generalitat de Catalunya) through projects no. FIS2016-78507-C2-2-P and 2017-SGR-1061, respectively, is gratefully acknowledged.