Influence of magnetic field-dependent viscosity on Casson-based nanofluid boundary layers: A comprehensive analysis using Lie group and spectral quasi-linearization method

Abstract

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based $F{e}_3{O}_4$ nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of $F{e}_3{O}_4$, and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region.

Highlights

• •MFD viscosity impact on the natural convective nanofluid boundary layer is examined.
• •Lie group transformations yield non-dimensional ODEs for analysis.
• •Spectral- Quasi Linearization method is applied to solve the flow and heat transfer equations.
• •Findings reveal MFD viscosity's profound influence on flow and heat transfer.

Nomenclature

$(u^{⁎},v^{⁎})$

Velocity components ms⁻¹

$(x^{⁎},y^{⁎})$

Coordinates of the surface m

uw⁎

Stretching velocity ms⁻¹

emn

Deformation rate component

${\mathit{\pi }}_{ct}$

Critical value of the fluid

ry

Yield stress

${\mathit{\mu }}_D$

Dynamic viscosity kg (ms)⁻¹

$\overline{\mathit{\delta }}$

MFD Viscosity variation

B0

Magnetic field strength

β

Casson parameter

v

Kinetic viscosity m²s⁻¹

σ

Electric conductivity S m⁻¹

ρ

Fluid density kg m⁻³

ϕ

Volume fraction of nanoparticles

k

Fluid thermal conductivity W m⁻¹ K⁻¹

$\mathit{\rho }{C}_p$

Specific heat capacitance J m⁻³ K⁻¹

Pr

Prandtl number

Cf

Coefficient of skin friction

Nux⁎

Nusselt number

${\mathit{\tau }}_w$

Shear stress at the wall

Rex⁎

Local Reynolds number

Tw⁎

Temperature at the stretching field K

T∞⁎

Temperature far from the stretching field K

Subscript

s

Nanoparticles

f

Base fluid

nf

Nanofluid

1. Introduction

In the fields of aerodynamics, chemical engineering, manufacturing, and other engineering disciplines, it is crucial to investigate the heat transfer properties of boundary layer flows involving both Newtonian and non-Newtonian fluids. Boundary layer flows occur when a fluid moves in close proximity to a solid surface, creating velocity gradients and shear stresses within this thin layer. These flows have been extensively studied assuming Newtonian fluid behavior, where viscosity remains constant regardless of the shear rate [1], [2], [3], [4]. However, real-world fluids often deviate from this ideal behavior, displaying characteristics such as shear thinning, thickening, viscoelasticity, and yield stress, making the understanding and prediction of flow and heat transfer behaviors challenging. This is especially relevant since numerous engineering applications require the study of non-Newtonian fluids in boundary layer flow with heat transfer, as they significantly influence flow characteristics and heat transfer performance. One intriguing scenario in this context is the flow of non-Newtonian fluids over a stretching sheet. This topic finds applications in various fields such as polymer processing, thin film coating and deposition, biomedical engineering, microfluidics, heat and energy transfer systems, surface coatings and functionalization, aerospace, and aeronautics [5], [6], [7], [8], [9], [10].

Casson fluids, known for their ability to capture shear-thinning behavior, exhibit unique characteristics when subjected to stretching. These fluids flow below their yield stress, and their viscosity decreases with increasing shear rate. The stretching sheet velocity profiles, shear stresses, and heat transfer characteristics of a stretching sheet in contact with Casson fluids depend on their rheological features. Nanofluids, which are suspensions of nanoparticles in base fluids, seamlessly integrate nanotechnology into their composition. They are highly desirable for heat transfer applications because of their exceptional thermal conductivity and efficiency [11]. Investigating Casson nanofluids over a stretching sheet provides valuable insights into their flow and heat transmission properties, generating significant interest in the field of fluid dynamics and heat transfer [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].

Magnetohydrodynamics (MHD), which combines fluid dynamics and electromagnetism, is used to understand the behavior of electrically conducting fluids in magnetic fields. This interdisciplinary field uncovers a wide range of phenomena that involve the interplay between magnetic fields and fluid flows, extending beyond traditional fluid dynamics. The study of electrically conducting non-Newtonian fluids under the influence of a magnetic field holds significance in numerous engineering and technological domains. These applications span disciplines, including turbomachinery, nuclear reactor cooling, heat exchanger design, nuclear acceleration device assembly, and blood-flow measurement techniques [23], [24], [25], [26], [27], [28], [29]. For instance, Sulochana et al. [30] conducted a comparative analysis of the three-dimensional (3D) flow of Newtonian and non-Newtonian MHD Casson nanofluids over a stretching sheet under convective boundary conditions. Their study revealed that the non-Newtonian Casson nanofluid exhibited superior rates of heat and mass transfer compared to Newtonian nanofluids. Similarly, Sabir et al. [31] investigated the flow behavior of a two-phase nanofluid over a stretching sheet in the presence of a magnetic field, chemical reactions, and gyrotactic micro-organisms, with a particular focus on the boundary layer. Their findings highlighted the increasing impact of magnetic parameters on the local Nusselt number. Nadeem et al. [32] scrutinized the MHD 3D boundary layer flow of a Casson nanofluid across a linearly stretched surface. Furthermore, Mustafa et al. [33] analyzed the flow characteristics of a two-phase Casson nanofluid over a nonlinearly stretched sheet, considering the effects of varying magnetic fields and observed an increasing trend in temperature with the Casson parameter. Khan et al. [34] investigated the time-dependent homogeneous flow of sodium alginate-Casson nanofluids on a vertically-heated plate, emphasizing the influence of magnetic fields and thermal radiation. Ghadikolaei et al. [35] explored mixed convection flow of an MHD two-phase Casson nanofluid, considering the effects of nonlinear thermal radiation. Ramudu et al. [36] studied the effect of Brownian motion on heat and mass transfer in an MHD two-phase Casson nanofluid flow over a stretched sheet with nonlinear thermal radiation. Alwawi et al. [37] examined the MHD free-convection flow of sodium alginate-based nanofluids containing various nanoparticles around a sphere and found that the silver nanofluid exhibited higher temperatures than others. Panigrahi et al. [38] conducted a comprehensive review of the heat and mass transfer properties of MHD two-phase Casson nanofluids in a porous medium with chemical reactions, convective boundary conditions, and non-uniform heat sink/source, considering the influence of thermal radiation. Their analysis revealed a decreasing temperature profile with an increase in the Casson parameters. Jamshed et al. [39] investigated the flow characteristics of a single-phase Casson nanofluid based on ethylene glycol with copper and molybdenum disulfide nanoparticles over a stretching sheet, considering an inclined magnetic field, viscous and ohmic dissipations, and entropy generation. Kumar et al. [40] examined the two-phase MHD flow of a Casson nanofluid over a surface with exponential stretching velocity, incorporating activation energy, thermal radiation, and Hall current effects. Reddy and Maddileti [41] studied the MHD two-phase flow of a Casson nanofluid with temperature-dependent viscosity and thermal conductivity over a stretching sheet, reporting enhanced temperature and concentration profiles using the Casson parameter. Sekar et al. [42] focused on the impact of the Soret and Dufour effects on the boundary layer flow of an MHD two-phase Casson nanofluid over a nonlinearly stretched, inclined surface. Rani et al. [43] explored the effects of various nanoparticle shapes, including spherical, brick, blade, and platelet-shaped copper nanoparticles, in a sodium alginate-based Casson nanofluid, considering the implications of the magnetic field. They concluded that an increase in the Casson parameter led to elevated temperatures for all the nanoparticle shapes in the presence of a magnetic field.

Magnetic-field-dependent (MFD) viscosity shows how magnetic fields influence fluid flow. In classical fluid dynamics, viscosity is typically assumed to be constant and unaffected by external variables. The strength and orientation of the magnetic field can affect the viscosity of the conducting fluid. MFD arises from the interaction of the magnetic field with charged particles in the fluid, altering internal friction and flow behavior. Understanding and quantifying MFD viscosity is of great importance with broad scientific and engineering implications. In fields such as astrophysics and plasma physics, accurate models that account for MHD and viscosity fluctuations are necessary to explain magnetized plasmas. In engineering and industry, magnetic fields can be harnessed to control and manipulate conducting fluid flows, thereby enhancing efficiency and performance. To illustrate this, Sheikholeslami et al. [44] explored the effects of MFD viscosity on a two-dimensional (2D), incompressible water-based alumina nanofluid flow within a square container. Their findings indicated a decrease in temperature within the enclosure with increasing magnetic viscosity. Similarly, Sheikholeslami and Sadoughi et al. [45] examined the effects of the MFD viscosity and shape factor of $F{e}_3{O}_4$ nanoparticles on the behavior of water-based $F{e}_3{O}_4$ nanofluids within a porous square enclosure. Their investigation revealed that platelet-shaped nanoparticles exhibit the highest Nusselt number in the presence of high MFD viscosity. Molana et al. [46] delved into the effects of the MFD viscosity on the natural convective flow of a water-based $F{e}_3{O}_4$ nanofluid in a novel porous cavity.

However, after an extensive review of the existing literature, it becomes evident that no prior studies have explored the impact of viscosity dependence on the magnetic field in the context of natural convection flow of a sodium alginate Casson-based magnetite nanofluid over a stretching sheet. This is surprising given the considerable potential of this area of research to enhance heat transfer, improve energy efficiency, optimize industrial processes, facilitate material processing, and advance renewable energy. In light of this gap, the current study conducted fundamental experiments to investigate the influence of MFD viscosity on the natural convection flow of a steady non-Newtonian fluid composed of sodium alginate and $F{e}_3{O}_4$ nanoparticles over a stretched surface. To address this problem, the governing equations were non-dimensionalized using Lie group transformations. The resulting set of nonlinear coupled non-dimensional equations was solved using a highly efficient spectral quasi-linearization technique (SQLM).

2. Problem description

The problem involves examining the laminar, steady, 2D, incompressible flow of a Casson-based $F{e}_3{O}_4$ nanofluid over an impermeable stretching surface that experiences a stretching velocity $u_w^{⁎}=a{x}^{⁎}$ (see Fig. 1). The Casson fluid model is described by the following rheological equation [15]:

$${\tau }_{mn}=\{\begin{array}{c}\hfill 2({\mu }_D+r_y{\left(2{\pi }_{ct}\right)}^{-0.5})e_{mn},\pi <{\pi }_{ct},\hfill \\ \hfill 2({\mu }_D+r_y{\left(2\pi \right)}^{-0.5})e_{mn},\pi >{\pi }_{ct},\hfill \end{array}$$ (1)

where ${\mu }_D$, $e_{mn}$, and ${\pi }_{ct}$ indicate the dynamic viscosity, deformation rate component, critical value (dependent on the fluid model), and yield stress $(r_y)$, respectively. Additionally, $\pi =e_{mn}{e}_{mn}$. The working fluid under consideration consists of electrically-conducting sodium alginate and $F{e}_3{O}_4$ nanoparticle suspension. The dynamic plastic viscosity of a Casson fluid was described as $\mu =(r_y{\left(2\pi \right)}^{-0.5}+{\mu }_D),({\pi }_{ct}<\pi )$. By employing a rheological model (see Equation (1)), the kinematic viscosity of the Casson fluid is expressed as $\nu =\frac{{\mu }_D}{\rho }(1+\frac{1}{\beta })$, where $\beta =\left(\frac{{\mu }_D{\left(2\pi \right)}^{0.5}}{r_y}\right)$ denotes Casson parameter. The kinematic viscosity of the sodium alginate Casson-based $F{e}_3{O}_4$ nanofluid is derived as ${\nu }_{nf}=\frac{{\mu }_{nf}}{{\rho }_{nf}}(1+\frac{1}{\beta })$. The nanofluids viscosity is expected to vary due to the magnetic field and is described as follows: [44], [45]

$${\eta }_{nf}=(1+\overline{\delta }.\overline{B}){\mu }_{nf},$$ (2)

where $\overline{\delta }$ represents the viscosity variation caused by the applied magnetic field, and it is assumed to be isotropic. Furthermore, $(\overline{\delta }=\delta )$, and $\overline{B}$ denotes the magnetic field intensity. Using Equation (2), the MFD kinematic viscosity of the Casson-based nanofluid is expressed as follows:

$${\nu }_{nf}=\frac{{\eta }_{nf}}{{\rho }_{nf}}(1+\frac{1}{\beta })=\frac{(1+\overline{\delta }.\overline{B}){\mu }_{nf}}{{\rho }_{nf}}(1+\frac{1}{\beta }).$$ (3)

Figure 1.

Physical configuration of the present model.

The axes labeled $x^{⁎}$ and $y^{⁎}$ remain fixed relative to the stretching surface and are oriented perpendicular to it. Temperature at the stretching and far fields is denoted by $T_w^{⁎}$ and $T_{\infty }^{⁎}$, respectively. A uniform magnetic field of intensity $B_0$ is applied transversely, with negligible electric field intensity and magnetic Reynolds number. Furthermore, it is assumed that there was no slip between the base fluid and $F{e}_3{O}_4$ nanoparticles, and both are in a state of thermal equilibrium. Table 1 lists the thermophysical characteristics of the working fluid and nanoparticles.

Table 1.

Thermo-physical properties of fluid and nanoparticles.

	ρ (kg/m3)	Cp (J/kg K)	k (W/m K)	σ (Ω m)−1
Fe${}_{3}O_4$	5180	670	9.7	25000
Sodium alginate	989	4175	0.6376	0.05

By considering the aforementioned assumptions and applying the conventional boundary layer approximation alongside Equation (3), the governing equations that describe the mass, momentum, and energy of a non-Newtonian (Casson)-based homogeneous $F{e}_3{O}_4$ nanofluid are derived, considering the influential effects of the MFD viscosity as follows [17]:

$$\frac{\partial u^{⁎}}{\partial x^{⁎}}+\frac{\partial v^{⁎}}{\partial y^{⁎}}=0,$$ (4)

$$u^{⁎}\frac{\partial u^{⁎}}{\partial x^{⁎}}+v^{⁎}\frac{\partial u^{⁎}}{\partial y^{⁎}}-(1+\frac{1}{\beta })\frac{{\eta }_{nf}}{{\rho }_{nf}}\frac{{\partial }^2{u}^{⁎}}{\partial y^{⁎2}}+\frac{{\sigma }_{nf}{B}_0^2{u}^{⁎}}{{\rho }_{nf}}=0,$$ (5)

$$u^{⁎}\frac{\partial T^{⁎}}{\partial x^{⁎}}+v^{⁎}\frac{\partial T^{⁎}}{\partial y^{⁎}}-\frac{k_{nf}}{{(\rho C_p)}_{nf}}\frac{{\partial }^2{T}^{⁎}}{\partial y^{⁎2}}=0,$$ (6)

where the velocities $u^{⁎}$ and $v^{⁎}$ are along the $x^{⁎}$ and $y^{⁎}$ directions, respectively, and $T^{⁎}$ represents the local temperature of the fluid. The corresponding dimensional boundary conditions for Equations (4)–(6) are as follows:

$$u^{⁎}=u_w^{⁎}(x^{⁎})=a{x}^{⁎},v^{⁎}=0,T^{⁎}=T_w^{⁎},\text{at}{y}^{⁎}=0,u^{⁎}\to 0,T^{⁎}\to T_{\infty }^{⁎},\text{as}{y}^{⁎}\to \infty .$$ (7)

3. Thermo-physical properties of nanofluid

The effective density $({\rho }_{nf})$, effective dynamic viscosity $({\mu }_{nf})$, specific heat capacitance ${(\rho C_{\rho })}_{nf}$, effective thermal conductivity $(k_{nf})$ and effective electric conductivity $(\frac{{\sigma }_{nf}}{{\sigma }_f})$ of the nanofluid are all expressed as functions of the nanoparticle volume fraction ϕ, as documented in [44]:

$$\frac{{\rho }_{nf}}{{\rho }_f}=(1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f},\frac{{\mu }_{nf}}{{\mu }_f}={(1-\varphi )}^{-2.5},$$ (8)

$$\frac{{(\rho C_{\rho })}_{nf}}{{(\rho C_{\rho })}_f}=(1-\varphi )+\varphi \frac{{(\rho C_{\rho })}_s}{{(\rho C_{\rho })}_f},$$ (9)

$$\frac{k_{nf}}{k_f}=\left[\frac{k_s+2k_f-2\varphi (k_f-k_s)}{k_s+2k_f+\varphi (k_f-k_s)}\right],$$ (10)

and

$$\frac{{\sigma }_{nf}}{{\sigma }_f}=1+\frac{3(\frac{{\sigma }_s}{{\sigma }_f}-1)\varphi }{(\frac{{\sigma }_s}{{\sigma }_f})+2-(\frac{{\sigma }_s}{{\sigma }_f}-1)\varphi }.$$ (11)

4. Non-dimentionalization and Lie group transformations

Now, incorporating the aforementioned nondimensional variables,

$$x={(v_f/a)}^{-\frac{1}{2}}{x}^{⁎};y={(v_f/a)}^{-\frac{1}{2}}{y}^{⁎};u={(v_{f}a)}^{-\frac{1}{2}}{u}^{⁎};v={(v_{f}a)}^{-\frac{1}{2}}{v}^{⁎};\theta =(T^{⁎}-T_{\infty }^{⁎}){(T_w^{⁎}-T_{\infty }^{⁎})}^{-1}.$$ (12)

Using Equations (8) - (12), the Equations (4), (5) and (6) transforms to

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$ (13)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-\frac{(1+\frac{1}{\beta })(1+{\delta }^{⁎})}{{(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})}\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\sigma }_{nf}}{{\sigma }_f}\frac{Mu}{((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})}=0,$$ (14)

$$u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}-\frac{k_{nf}}{k_f{P}_r}\frac{1}{((1-\varphi )+\varphi \frac{{(\rho C_{\rho })}_s}{{(\rho C_{\rho })}_f})}\frac{{\partial }^2\theta }{\partial y^2}=0,$$ (15)

and the boundary conditions in Equation (7) transformed to

$$u=x,v=0,\theta =1,\text{at}y=0,u\to 0,\theta \to 0,\text{as}y\to \infty ,$$ (16)

where ${\delta }^{⁎}=\delta B_0$ represents the viscosity parameter, $M={\sigma }_f{B}_0^2{({\rho }_{f}a)}^{-1}$ is the magnetic parameter, and $P_r=\frac{v_f}{{\alpha }_f}$ denotes the Prandtl number of the base fluid. The stream function, ψ, which satisfies Equation (13) and is specified as $u=\frac{\partial \psi }{\partial y}$ and $v=-\frac{\partial \psi }{\partial x}$. The Equations (14)- (16) can be expressed in terms of stream function as follows:

$$\frac{\partial \psi }{\partial y}\frac{{\partial }^2\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}-\frac{(1+\frac{1}{\beta })(1+{\delta }^{⁎})}{{(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})}\frac{{\partial }^3\psi }{\partial y^3}+\frac{{\sigma }_{nf}}{{\sigma }_f}\frac{Mu}{((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})}\frac{\partial \psi }{\partial y}=0,$$ (17)

$$\frac{\partial \psi }{\partial y}\frac{\partial \theta }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial \theta }{\partial y}-\frac{k_{nf}}{k_f{P}_r}\frac{1}{((1-\varphi )+\varphi \frac{{(\rho C_{\rho })}_s}{{(\rho C_{\rho })}_f})}\frac{{\partial }^2\theta }{\partial y^2}=0,$$ (18)

with the boundary conditions

$$\frac{\partial \psi }{\partial y}=x,\frac{\partial \psi }{\partial x}=0,\theta =1,\text{at}y=0,\frac{\partial \psi }{\partial y}\to 0,\theta \to 0,\text{as}y\to \infty .$$ (19)

By utilizing the following Lie-group transformations [47]

$$y=\eta ,\psi =xf(\eta ),\theta =\theta (\eta ),$$

in Equations (17) - (19), the following non-dimensional ODE's and the boundary conditions are obtained:

$$(1+\frac{1}{\beta })(1+{\delta }^{⁎})f^{‴}+{(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})[f{f}^{″}-f^{\prime 2}]={(1-\varphi )}^{2.5}\frac{{\sigma }_{nf}}{{\sigma }_f}M{f}^{\prime },$$ (20)

$$\frac{k_{nf}}{k_{f}Pr}\frac{1}{((1-\varphi )+\varphi \frac{{(\rho C_{\rho })}_s}{{(\rho C_{\rho })}_f})}{\theta }^{″}+f{\theta }^{\prime }=0,$$ (21)

$$f(\eta )=0,f^{\prime }(\eta )=1,\theta (0)=1,\text{at}\eta =0,{\text{f}}^{\prime }(\eta )\to 0,\theta (\eta )\to 0,\text{as}\eta \to \infty .$$ (22)

5. Skin friction and heat transfer coefficients

For the analysis of surface drag and wall heat transfer, the key physical parameters of interest are the coefficient of skin friction $C_f$ and the reduced Nusselt number $N{u}_{x^{⁎}}$. The wall shear stress is expressed as follows:

$${\tau }_w=-(1+\frac{1}{\beta }){\eta }_{nf}{\left(\frac{\partial u^{⁎}}{\partial y^{⁎}}\right)}_{y^{⁎}=0}.$$ (23)

The coefficient of skin friction is expressed as follows:

$$C_f=\frac{{\tau }_w}{{\rho }_f{U}^2},$$ (24)

from the Equations (12), (19), (23) and (24), we obtain

$$C_f\frac{{(1-\varphi )}^{2.5}R{e}_{x^{⁎}}^{1/2}}{(1+\frac{1}{\beta })(1+{\delta }^{⁎})}=-f^{″}(0).$$ (25)

The rate of heat transfer at the surface is represented by

$$q_w=-k_{nf}{\left(\frac{\partial T^{⁎}}{\partial y^{⁎}}\right)}_{y^{⁎}=0}.$$ (26)

The Nusselt number is defined as

$$N{u}_{x^{⁎}}=\frac{x^{⁎}{q}_w}{k_f(T_w-T_{\infty })}.$$ (27)

Thus, the dimensionless rate of wall heat transfer using Equations (26) and (27) is derived as

$$\frac{N{u}_{x^{⁎}}}{R{e}_{x^{⁎}}^{1/2}}\left(\frac{k_f}{k_{nf}}\right)=-{\theta }^{\prime }(0),$$ (28)

where $R{e}_{x^{⁎}}=x^{⁎}{u}_w^{⁎}(x^{⁎})/v_f$ in Equations (25) and (28) denotes the local Reynolds number, $R{e}_{x^{⁎}}^{1/2}{C}_f$ and $R{e}_{x^{⁎}}^{-1/2}N{u}_{x^{⁎}}$ represent the local skin friction coefficient and the reduced Nusselt number, respectively.

6. Numerical solution using SQLM

The coupled system of Equations (20) and (21), along with the boundary conditions specified in Equation (22), is solved using the SQLM [48], [49], [50], [51], [52]. These equations are expressed using nonlinear ordinary differential systems. The SQLM combines quasi-linearization and spectral (Chebyshev) collocation methods, which are reliable techniques. The quasi-linearization method (QLM), proposed by Bellman and Kalaba [53] as an extension of the Newton-Raphson approach, is employed for numerically solving nonlinear boundary-value problems. The core principle of the QLM is to linearize nonlinear variables related to the basic equations. Successive iterations at $\mathit{\text{r}}+1$ and r are assumed to be relatively small and are performed using the Taylor series. The functions F and Θ are denoted as follows:

$$F=(1+\frac{1}{\beta })(1+{\delta }^{⁎})f^{‴}+{(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})[f{f}^{″}-f^{\prime 2}]-{(1-\varphi )}^{2.5}\frac{{\sigma }_{nf}}{{\sigma }_f}M{f}^{\prime },$$ (29)

$$\mathrm{\Theta }=\frac{k_{nf}}{k_{f}Pr}\frac{1}{((1-\varphi )+\varphi \frac{{(\rho C_{\rho })}_s}{{(\rho C_{\rho })}_f})}{\theta }^{″}+f{\theta }^{\prime }.$$ (30)

Using Equations (29) and (30), the error function for this iterative procedure is designed as follows:

$$R_f=a_{3,r}{f}_{r+1}^{‴}+a_{2,r}{f}_{r+1}^{″}+a_{1,r}{f}_{r+1}^{\prime }+a_{0,r}{f}_{r+1}-F_r,R_{\theta }=b_{2,r}{\theta }_{r+1}^{″}+b_{1,r}{\theta }_{r+1}^{\prime }+b_{0,r}{\theta }_{r+1}+b_{0,r}^{⁎}{f}_{r+1}-{\mathrm{\Theta }}_r,$$ (31)

with relevant boundary conditions

$$f_{r+1}(\eta )=0,f_{r+1}^{\prime }(\eta )=1,{\theta }_{r+1}(0)=1,\text{at}\eta =0,f_{r+1}^{\prime }(\eta )\to 0,{\theta }_{r+1}(\eta )\to 0,\text{as}\eta \to \infty .$$ (32)

The corresponding coefficients of Equation (31) are as follows

$$a_{0,r}={(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})f_r^{″},a_{1,r}=-2{(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})f_r^{\prime }+{(1-\varphi )}^{2.5}[1+\frac{3(\frac{{\sigma }_s}{{\sigma }_f}-1)\varphi }{(\frac{{\sigma }_s}{{\sigma }_f}+2)-(\frac{{\sigma }_s}{{\sigma }_f}-1)\varphi }]M,a_{2,r}={(1-\varphi )}^{2.5}((1-\varphi )+\varphi \frac{{\rho }_s}{{\rho }_f})f_r,a_{3,r}=(1+\frac{1}{\beta })(1+{\delta }^{⁎}),b_{0,r}=0,b_{1,r}=f_r,b_{2,r}=\frac{k_{nf}}{k_{f}Pr}{((1-\varphi )+\varphi \frac{{(\rho C_{\rho })}_s}{{(\rho C_{\rho })}_f})}^{-1},\text{and}{b}_{0,r}^{⁎}={\theta }_r^{\prime }.$$

The initial approximation for this iterative procedure is applied as

$f_0(\eta )=1-e^{-\eta }$ and ${\theta }_0=e^{-\eta }$.

The QLM is applied to Equations (20) and (21), and the SQLM, a competent iterative numerical method, is executed using the following procedure for a linearly coupled differential system of equations with a variable coefficient. The transformation of η into [0,J], where $J\in Z^{⁎}$ is the boundary region limit, is mapped as $\eta =\text{J}(\bar{\chi }+1)/2$ in the computational domain [-1,1]. Chebyshev interpolating polynomials are used to approximate the unknown functions $f_{r+1}$ and ${\theta }_{r+1}$, followed by Gauss-Lobatto collocation points to find the derivatives of the approximated functions as follows:

$${\bar{\chi }}_i=cos\left(\frac{\pi i}{N}\right),\bar{\chi }=[-1,1],i=0,1,2,3,....,N,$$ (33)

where N is the number of collocation points.

The Chebyshev differentiation matrix D is given as

$$\frac{d^n{f}_{r+1}({\bar{\chi }}_i)}{d\bar{\chi }}=\sum _{k=0}^N{D}_{ik}^n{f}_{r+1}({\bar{\chi }}_i)=D^n\mathbf{\text{F}},\frac{d^n{\theta }_{r+1}({\bar{\chi }}_i)}{d\bar{\chi }}=\sum _{k=0}^N{D}_{ik}^n{\theta }_{r+1}({\bar{\chi }}_i)=D^n\theta ,$$ (34)

where D = $\frac{2D}{\text{J}}$, $\mathbf{\text{F}}={[f_{r+1}({\bar{\chi }}_0),f_{r+1}({\bar{\chi }}_1),f_{r+1}({\bar{\chi }}_2),....,f_{r+1}({\bar{\chi }}_N)]}^T$, and

$\theta ={[{\theta }_{r+1}({\bar{\chi }}_0),{\theta }_{r+1}({\bar{\chi }}_1),{\theta }_{r+1}({\bar{\chi }}_2),....,{\theta }_{r+1}({\bar{\chi }}_N)]}^T$.

Using Equations (20), (21) and (31)-(34), the following simultaneous system is obtained at the collocation points:

$$C_{11}{\mathbf{\text{F}}}_{r+1}+C_{12}{\theta }_{r+1}={\mathbf{\text{R}}}_f,C_{21}{\mathbf{\text{F}}}_{r+1}+C_{22}{\theta }_{r+1}={\mathbf{\text{R}}}_{\theta },$$

and the matrix set up of the SQLM technique having derivatives of anonymous functions are as follows

$$\left[\begin{array}{cc}\hfill C_{11}\hfill & \hfill C_{12}\hfill \\ \hfill C_{21}\hfill & \hfill C_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\mathbf{\text{F}}}_{r+1}\hfill \\ \hfill {\theta }_{r+1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\mathbf{\text{R}}}_f\hfill \\ \hfill {\mathbf{\text{R}}}_{\theta }\hfill \end{array}\right],$$ (35)

where

$C_{11}=\text{diag}[a_{3,r}]{\mathbf{\text{D}}}^3+\text{diag}[a_{2,r}]{\mathbf{\text{D}}}^2+\text{diag}[a_{1,r}]\mathbf{\text{D}}+\text{diag}[a_{0,r}]I$,

$C_{12}=0_{N+1⁎N+1}$,

$C_{21}=\text{diag}[b_{0,r}^{⁎}]I$, and

$C_{22}=\text{diag}[b_{2,r}]{\mathbf{\text{D}}}^2+\text{diag}[b_{1,r}]\mathbf{\text{D}}$.

Here the diagonal matrix labeled as diag[], I is the identity matrix and 0 is the zero matrix with order (N+1) x (N+1). The solution is obtained by solving the matrix system (35) in accordance with the boundary conditions (32):

When addressing practical challenges in fluid dynamics, it was observed that these issues are correlated with various physical processes. Consequently, these challenges compelled the researchers to tackle intricate, multi-ordered nonlinear differential equations. The process of finding solutions became notably more time-consuming and intricate over time. Occasionally, this approach exhibited slow convergence or even failed to yield a solution. However, it was reported that the utilization of SQLM demonstrates a significantly higher convergence rate compared to existing numerical techniques, and its iterative steps are relatively straightforward to implement [50], [51], [52]. The distinctive feature of SQLM, contributing to its widespread acceptance as a numerical technique, was reported to be its ability to break down a complex set of nonlinear equations into smaller subsystems that are inherently more manageable. The residual errors are constructed as follows:

$$Res(f)=||{△}_f[F_r,{\theta }_r]|{|}_{\infty },Res(\theta )=||{△}_{\theta }[F_r,{\theta }_r]|{|}_{\infty },$$ (36)

where ${△}_f$ and ${△}_{\theta }$ represents the nonlinear Equations (21) and (22), and $F_r$ and ${\theta }_r$ represents the SQLM solutions at the nodes. Also the residue errors in Equation (36) are clearly shown in Fig. 2. It is noticed that after the $5^{th}$ or $6^{th}$ iteration the related error is consistent and the rapid convergence rate falls after the $5^{th}$ or $6^{th}$ iteration.

Figure 2.

The residues of velocity and temperature profiles.

7. SQLM validation

The governing system of equations describing the mass, momentum, and energy of a non-Newtonian (Casson)-based homogeneous $F{e}_3{O}_4$ nanofluid is solved using a spectral QLM. To validate the accuracy of this study, it is compared with the results of Wang [54] and Gorla et al. [55] (see Table 2). The results presented in Table 2 demonstrate a good agreement with the benchmark results. In this study, a tolerance error of $\epsilon ={10}^{-10}$ was employed, and the residues were used to test accuracy had an error value of ${10}^{-10}$ for momentum and ${10}^{-12}$ for the energy equations (see Fig. 2).

Table 2.

Comparison of variations of – θ′(0).

Pr	Wang [54]	Gorla et al. [55]	Present Study
0.07	0.0656	0.0656	0.065806
0.2	0.1691	0.1691	0.169088
0.7	0.4539	0.4539	0.453916
2	0.9114	0.9114	0.911354
7	1.8971	1.8904	1.897186

8. Results and discussion

The computational results of the dimensionless governing ordinary differential equations obtained using the SQLM are discussed in detail. Graphical plots are used to investigate the impacts of the isotropic MFD viscosity (δ), nanoparticle volume fraction (ϕ), magnetic parameter (M), and Casson parameter (β) on the natural convection of the boundary layer flow of a sodium alginate-based $F{e}_3{O}_4$-nanofluid over an impermeable stretching surface. The following values were held constant throughout the investigation while studying the impacts of individual parameters: $Pr=6.45$, $M=1$, $\varphi =0.1$, $\beta =0.1$ and ${\delta }^{⁎}=0.15$. Additionally, $N=100$ collocation points were used to emphasize individuality.

Figs. 3(a) and 3(b) illustrate the effects of the isotropic MFD viscosity parameter on the velocity and thermal profiles of the Casson-based $F{e}_3{O}_4$ nanofluid, respectively. The MFD viscosity parameter primarily influences the velocity and momentum boundary layers than the temperature and thermal boundary layer due to the negligible effects of viscous and ohmic dissipations in the model. Fig. 3(a) presents the results for two cases: fluid flow with viscosity dependent on the magnetic field, and fluid flow with viscosity independent of the magnetic field. Increasing the MFD viscosity accelerated the nanofluid velocity, resulting in an expanded momentum boundary layer. This is attributed to the increased MFD viscosity when exposed to a transverse magnetic field. This led to a reduction in the shear stress of the $F{e}_3{O}_4$ nanofluid and a stretching of the surface interface parallel to the magnetic field, potentially increasing velocity and momentum boundary layers. Fig. 3(b) shows that an increase in the MFD viscosity parameter leads to a slight decrease in both the temperature profile and thickness of the thermal boundary layer.

Figure 3.

Effect of MFD viscosity parameter on (a) velocity profile & (b) temperature profile.

The influence of the volume fraction of $F{e}_3{O}_4$ nanoparticles on velocity and temperature profiles of Casson-based nanofluid is depicted in the existence of magnetic field dependent viscosity (see Figs. 4(a) and 4(b)). The volume fraction of nanoparticles affected the thermal profile and thermal boundary layers more than the fluid flow and momentum boundary layer. A marginal increase in the velocity profile was noted, whereas an enhancement was detected due to changes in the nanoparticle volume fraction. This was attributed to the increase in suspension volume fraction, which resulted in a marginal reduction in the wall shear stress and a simultaneous increase in thermal conductivity within the boundary layer region.

Figure 4.

Effect of volume fraction of Fe₃O₄ nanoparticles on (a) velocity profile & (b) temperature profile.

Figs. 5(a) and 5(b) demonstrate the impact of the magnetic parameters on the velocity and temperature profiles of the Casson-based $F{e}_3{O}_4$ nanofluid. Increased magnetic parameters resulted in an improved temperature profile and reduced velocity. This was because the Lorentz force, acting perpendicular to the magnetic field and opposite to the fluid viscosity, affects the nanofluid as the magnetic parameter increased, while the magnetic field strength remained constant.

Figure 5.

Effect of magnetic parameter on (a) velocity profile & (b) temperature profile.

Figure 6, Figure 7 illustrate the impact of the magnetic, MFD viscosity, and Casson parameters on significant physical quantities: the local skin friction coefficient and reduced Nusselt number. The numerical outcomes of the local skin friction coefficient and reduced Nusselt number are displayed in Table. 3. The findings indicated an inverse relationship between the local skin friction coefficient and the MFD viscosity, whereas a direct relationship existed with the magnetic field and Casson parameters (see Figs. 6(a) and 6(b)). The increased viscosity parameter, dependent on the magnetic field, led to an increase in the reduced Nusselt number (see Figs. 7(a) and 7(b)). Furthermore, the Nusselt number decreased as the magnetic and Casson parameters increased. It was observed that an increase in the MFD viscosity parameter has the potential to enhance the heat transfer rate at the stretching boundary, while reducing skin friction can increase the flow velocity at the stretching sheet. The MFD viscosity was found to contribute to skin friction reduction and enhancement of the rate of heat transfer within the boundary layer of the Casson-based nanofluid containing $F{e}_3{O}_4$, generated by a stretching sheet.

Figure 6.

The local skin friction coefficient variations (a) Effect of viscosity parameter & Casson parameter (b) Effect of viscosity parameter & magnetic parameter.

Figure 7.

The reduced Nusselt number variations (a) Effect of MFD viscosity parameter & Casson parameter & (b) Effect of MFD viscosity parameter & magnetic parameter.

Table 3.

Values of local Skin friction and reduced Nusselt number.

M	β	δ⁎	−f″(0)	−θ′(0)
1	0.05	0.05	0.3099734	1.7175150
		0.10	0.3028466	1.7190461
		0.15	0.2961898	1.7204759
	0.10	0.05	0.4282898	1.6920365
		0.10	0.4184428	1.6941611
		0.15	0.4092451	1.6961450
	0.15	0.05	0.5130159	1.6737259
		0.10	0.5012208	1.6762781
		0.15	0.4902037	1.6786611
2	0.05	0.05	0.3775573	1.7029748
		0.10	0.3688767	1.7048444
		0.15	0.3607685	1.7065901
	0.10	0.05	0.5216705	1.6718526
		0.10	0.5096764	1.6744486
		0.15	0.4984734	1.6787254
	0.15	0.05	0.6248695	1.6494757
		0.10	0.6105027	1.6525951
		0.15	0.5970834	1.6555076

9. Conclusion

This study investigated the impact of MFD viscosity on the boundary layer flow of a nanofluid across a stretched surface. The SQLM and Lie group approaches were adopted to explore the impact of the MFD on the boundary layer flow of a non-Newtonian-based $F{e}_3{O}_4$ nanofluid over an impermeable stretching surface. The key findings of the current numerical investigations were:

• •The presence of MFD viscosity led to significant variations in the velocity profile of the Casson-based $F{e}_3{O}_4$ nanofluid. Increasing the MFD viscosity parameter accelerated the nanofluid velocity, resulting in an expansion of the momentum boundary layer thickness.
• •The thermal boundary layer thickness was suppressed as the magnetic field parameter increased, owing to the effects of MFD viscosity.
• •Increasing the MFD viscosity parameter resulted in an increase in the dimensionless quantities, specifically the local skin friction and the reduced Nusselt number.
• •The numerical results for skin friction and reduced Nusselt number presented in this study can serve as benchmark results for future comparative studies investigating the effects of MFD viscosity on boundary layer flows.

CRediT authorship contribution statement

N. Vishnu Ganesh: Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. B. Rajesh: Writing – review & editing, Writing – original draft, Validation, Methodology, Formal analysis, Conceptualization. Qasem M. Al-Mdallal: Writing – review & editing, Software, Methodology, Funding acquisition, Conceptualization. Hillary Muzara: Writing – review & editing, Supervision, Methodology, Formal analysis, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.