Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel

1. Introduction

A Bingham fluid is a category of non-Newtonian fluid that exhibits solid-like behavior at low-stress levels and flows as a viscous liquid under elevated stress levels. They demonstrate yield stress, indicating that the fluid remains static until the applied stress is above a specific critical threshold. Below this yield stress, the material exhibits rigid body behavior. Upon surpassing the yield stress, the flow behavior becomes linear, characterized by a constant viscosity. The linear correlation between stress and strain rate manifests similarly to that observed in Newtonian fluids. Typical instances encompass toothpaste, mayonnaise, and concrete. These materials necessitate a certain force to initiate flow. Bingham fluids are significant in numerous industrial and scientific applications, especially where materials necessitate exact regulation of flow characteristics. Bird et al.¹ examined the behaviour of a Bingham fluid within a hard circular tube. Kapur² introduced several mathematical models, such as Cais-son, the Bingham, and Herschel-Bulkley models. Rathy³ examined the fluid dynamics of a Bingham fluid within a channel and an annulus featuring impermeable barriers. Vajravelu et al.⁴ conducted an experimented on the flow of Bingham fluid in an annular tube with a porous wall. Ramakrishna et al⁵ investigated the flow behaviour of a Bingham fluid on a permeable bed in an enclosed channel. Goverdhan⁶ examined the flow of Bingham fluid in a porous channel. Narahari et al.⁷ Investigate the movement of a Bingham fluid between two porous substrates, focusing on its unsteady behaviour. Recently, Murthy et al.⁸ focused on analysing the flow of a Bingham fluid, which is not stable, in contact with a Newtonian fluid within two parallel plates. The objective is to determine the velocity field, mass flow rates, and interface velocity. Tsangaris et al.⁹ conducted on the movement of a Bingham fluid between two porous walls that are parallel to each other. One wall moves steadily in the same direction as the other wall, which remains stationary. At the same time, there is a pressure difference throughout the length of the wall, and there is a flow of fluid across the walls due to their porous nature. Adnan and Abdulhadi¹⁰ performed an investigation on the influence of an inclined magnetic field on the flow of incompressible Bingham plastic fluid within an inclined symmetric channel. The study also considered the effects of mass transfer and heat transfer. Slip circumstances are utilised for heat transmission and focus. Lakshminarayana et al.¹¹ examined the impact of wall slip circumstances, elasticity wall characteristics, and heat transfer on the movement of conducting Bingham fluid in an irregular channel using the presumptions of a lengthy wavelength and low Reynolds number. The current research by Mahabaleshwar et al.¹² investigates how radiation and chemical reactions affect the two-dimensional boundary layer flow of a bi-viscous Bingham fluid on a thermosolutal Marangoni boundary that is accompanied by a magnetic field and thermal source or sink. A mathematical model is developed using the Navier-Stokes equations to represent the physical flow problem. In order to convert these nonlinear PDEs into a system of nonlinear ODEs, they employed a similarity transformation. The study conducted by Basavarajappa et al.¹³ examines the multilayer flow of a bi-viscous Bingham fluid within a vertical slab with a hybrid nanofluid, using the nonlinear Boussinesq approximation.

Heat transport in Bingham fluids entails intricate interplay between the fluid’s rheological features and thermal aspects. The existence of a yield stress, which regulates flow commencement, profoundly influences heat transfer processes. Vradis et al.¹⁴ numerically resolved the concurrent evolution of thermal fields and hydrodynamic at the entry region of a cylindrical pipe for a non-Newtonian Bingham-type fluid by employing the completely elliptic mathematical models of continuity, momentum, and energy. Mustafa et al.¹⁵ conducted a heat transfer simulation for the swirling flow of a Bingham fluid constrained by a permeable rotation disk. The influence of concentration and temperature variations on magnetohydrodynamic (MHD) oscillatory flow in a porous media has numerous practical implications in the fields of engineering, industry, medical research, and issues related to the extraction and transportation of petroleum. Hamza et al.¹⁶ conducted a study on the impact of slip condition, radiative heat transfer, and transverse magnetic field on the unsteady flow of a conducting optical thin fluid via a channel equipped with a porous media. Khudair and Al-Khafajy¹⁷ proposed a heat transfer model for MHD oscillating flow of Williamson fluid over a porous plate, considering two different forms of flow. Al-Aridhee and Al-Khafajy¹⁸ examined the impact of mass and heat transfer on the peristaltic movement of MHD flow of a non-Newtonian Jeffrey fluid via a cylindrical porous media channel. The investigation focuses on the flow within a wave frame of reference that is travelling at the velocity of the wave. Al-Khafajy and Labban¹⁹ conducted a study on the combined impact of concentration and thermo-diffusion on the fluctuating flow of an incompressible Carreau fluid via an angled permeable channel. Liu et al.²⁰ investigate the steady flow and thermal transfer of Bingham fluid over a spinning disk of limited radius with radially varying thickness in the boundary layer. Eldabe et al.²¹ examined the flow of non-Newtonian Bingham blood fluid down an irregular conduit. The fluid exhibits electrical conductivity, and an external uniform magnetic field is added to this motion. Heat and mass transmission are considered, leading to an examination of the Dufour and Soret effects. Al-Khafajy and Mohammed²² examined a mathematical model elucidating the effects of thermal transfer on the oscillating flow of Bingham fluid with changing viscosity in a porous channel within the context of magnetohydrodynamics. Salahuddin et al.²³ performed a study analyzing numerical behavior utilizing the Adams-Bashforth predictor-corrector method of numerical analysis for the Williamson fluid model, considering variable viscosity, natural convection, and an angled magnetic field. Additionally, thermal radiation, Joule heating, and heat source/sink effects are incorporated into the thermal considerations. Akram et al.²⁴ studied the peristaltic flow of Bingham fluids under an inclined magnetic field, while Humnekar and Darbhasayanam²⁵ investigated variable-viscosity nanofluid flow in inclined porous media. These recent studies highlight the ongoing need for improved formulations that incorporate thermal, magnetic, and viscous variations simultaneously.

In an inclined channel, the influence of gravity is crucial in propelling the flow. The gravitational force acting along the slope of the channel affects the overcoming of yield stress, allowing the fluid to flow. In the case of Bingham fluids, flow takes place when the shear stress surpasses the yield stress. The angle of inclination in a channel influences the critical shear stress required to initiate flow. Lakshminarayana et al.²⁶ investigated the concurrent impacts of heated Joules and slip on the peristaltic flow of a Bingham fluid within an inclination tapered permeable channel featuring elastic walls. Mohammed and Al-Khafajy²⁷ examined the impact of temperature and concentration on magnetohydrodynamic oscillatory flow of Bingham fluid with varying viscosity in a sloped channel. umbinarasaiah²⁸ conducted a numerical investigation of entropy generation in an incompressible Casson fluid moving through an inclined permeable channel subjected to magnetic influence. Jha and Aina²⁹ offered a mathematical model for the complete magnetohydrodynamic mixed convection flow of an electrically conducting, viscous, incompressible fluid in an inclined permeable channel subject to time-periodic boundary constraints.

Prior studies stimulate interest in examining the MHD flow of a Bingham fluid under no-slip conditions within an inclined porous channel, influenced by variations in viscosity, temperature, and concentration at the channel wall. This study comprises five sections. The initial section is the introduction, which encompasses a historical review of the factors being examined. The second involves constructing the mathematical model. The third portion presents the resolution to the issue. The fourth portion encompasses an analysis of the outcomes via the function diagrams we acquired. The investigation was ended with a discussion and conclusions.

2. Mathematical formulation

Let us consider the flow of a non-Newtonian (Bingham) fluid with variable viscosity under the effects of radioactive heat transfer and electrically-applied magnetic field as depicted through an inclined porous channel with a width of $h$ (Figure 1). Fluids are supposed to have minimal electromagnetic power produced with a low electrical conductivity. We think of the system of Cartesian coordinates so that is the velocity vector.

Figure 1. Physical model of the inclined porous channel.

The basic equations governing the problem are provided as:

The continuity equation is given by:

The momentum equations:

The concentration equation:

The temperature equation:

The basic equation for the Bingham fluid,²⁶ given by:

Where $\overline{p}$ “pressure”, $\mathit{I}$ “unit tensor”, ${\mu }_c$ “fluid viscosity”, ${\tau }_0$ “yield stress”, and $\overline{\dot{\gamma }}$ “shear rate”.

When compensating for the velocity vector and the concentration and temperature functions, taking into account the magnetic field generated by the passage of a simple electric current over a porous wall of the inclined flow channel, and by compensating for the shear stress (Equation 6), and after simplifying, we rewrite the nonlinear partial differential system (1)-(4) as follows:

The stress components are given by:

By substituting the equation (12) into the governing equations and after simplifying, we obtain:

3. Method of solution

The equation (14) shows that the pressure does not depend on $y$ . To solve the above system of equations, we use its non-dimensional conditions as follows:

By substituting equations from Table 1 into equations (13), (15), (16) and the equations of boundary conditions equation (5), we have the following non-dimensional equations:

3.2 Bingham regularization and yield surface

The momentum equation was revised to explicitly include the yield-stress impact through the Papanastasiou regularization [Papanastasiou, 1987]. This formulation eliminates discontinuities between yielded and unyielded regions, facilitating seamless numerical analysis. The effective shear stress is written as:

$$\mathit{\tau }={\mathit{\tau }}_{\mathrm{o}}[1-exp(-\mathit{m}\dot{\mathit{\gamma }})]+\mathit{\mu }\mathrm{c}\dot{\mathit{\gamma }}\mathbf{,}$$

where τ₀ represents the yield stress. Shear rate is denoted as $\dot{\gamma }$ , and m represents a significant regularization parameter that governs the abruptness of the transition between the unyielded and yielded states. As m approaches infinity, this statement simplifies to the traditional Bingham model.

The yield surface $y=y_p$ is defined as the location where the local shear stress equals the yield stress, i.e., $\left|\tau \left(y_p\right)\right|={\tau }_0$ . For $\left|\tau \right|<{\tau }_0$ , the material behaves as a rigid plug region, while for $\left|\tau \right|>{\tau }_0$ , it flows as a viscous fluid.

This regularized form was substituted into the non-dimensional momentum equation (17), allowing explicit inclusion of the Bingham number $B_n={\tau }_{0}h/\left({\mu }_c{U}_0\right)$ . The modified governing equation therefore becomes:

$$\mathit{Re}\frac{\partial \mathit{u}}{\partial t}=-\frac{\partial p}{\partial x}+\mu \left(T\right)\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial \mu \left(T\right)}{\partial y}\frac{\partial u}{\partial y}-(M^2{sin}^2\varphi +\frac{\mu \left(T\right)}{\mathit{Da}})u+B_n\frac{\partial u}{\partial y}+sin\varphi (G_{T}T+G_{C}C)+\frac{\mathit{Re}}{\mathit{Fr}}sin\phi .$$

This expression correctly accounts for yield-stress effects and ensures the influence of both yielded and unyielded regions is captured in the flow field. The velocity profiles in Figures 8–24 were reinterpreted under this framework, showing clear flattening within the central plug region, consistent with expected Bingham behaviour.

Figure 2. Temperature graph for values ${\mathcal{K}}_{\mathcal{T}}$ and ${\mathcal{Q}}_H$ with $\omega =1,\mathit{Pe}=0.7$ .

Figure 3. Temperature graph for values $\mathit{Pe}$ and $\omega$ with ${\mathcal{K}}_{\mathcal{T}}=2,{\mathcal{Q}}_H=2$ .

Figure 4. Concentration graph for values $\mathit{Pe}$ and $\omega$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,S_{\mathcal{C}}=S_{\mathcal{T}}={\mathcal{K}}_{\mathcal{C}}=0.7$ .

Figure 5. Concentration graph for values ${\mathcal{K}}_{\mathcal{T}}$ and ${\mathcal{Q}}_H$ with $\omega =1,\mathit{Pe}=S_{\mathcal{C}}=S_{\mathcal{T}}={\mathcal{K}}_{\mathcal{C}}=0.7$ .

Figure 6. Concentration graph for values $S_{\mathcal{C}}$ and $S_{\mathcal{T}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\mathit{Pe}={\mathcal{K}}_{\mathcal{C}}=0.7$ .

Figure 7. Concentration graph for values ${\mathcal{K}}_{\mathcal{C}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\mathit{Pe}=S_{\mathcal{C}}=S_{\mathcal{T}}=0.7$ .

Figure 8. Velocity graph for values ${\mathcal{Q}}_H$ and ${\mathcal{K}}_{\mathcal{T}}$ with $\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0$ .

Figure 9. Velocity graph for values ${\mathcal{Q}}_H$ and ${\mathcal{K}}_{\mathcal{T}}$ with $\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0.2$ .

Figure 10. Velocity graph for values $\mathit{Pe}$ and $G_{\mathcal{C}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0$ .

Figure 11. Velocity graph for values $\mathit{Pe}$ and $G_{\mathcal{C}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0.2$ .

Figure 12. Velocity graph for values $R_e$ and $G_{\mathcal{T}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{C}}=0.5,F_r=0.8,\phi =\varphi =\pi /4,ϵ=0$ .

Figure 13. Velocity graph for values $R_e$ and $G_{\mathcal{T}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{C}}=0.5,F_r=0.8,\phi =\varphi =\pi /4,ϵ=0.2$ .

Figure 14. Velocity graph for values $\lambda$ and $\varphi$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,R_e=2,\phi =\pi /4,ϵ=0$ .

Figure 15. Velocity graph for values $\lambda$ and $\varphi$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,R_e=2,\phi =\pi /4,ϵ=0.2$ .

Figure 16. Velocity graph for values $\mathit{Da}$ and $\phi$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,R_e=2,\varphi =\pi /4,ϵ=0$ .

Figure 17. Velocity graph for values $\mathit{Da}$ and $\phi$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,R_e=2,\varphi =\pi /4,ϵ=0.2$ .

Figure 18. Velocity graph for values $S_{\mathcal{C}}$ and $S_{\mathcal{T}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,\mathit{Pe}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,R_e=2,\phi =\varphi =\pi /4,ϵ=0$ .

Figure 19. Velocity graph for values $S_{\mathcal{C}}$ and $S_{\mathcal{T}}$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,\mathit{Pe}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,R_e=2,\phi =\varphi =\pi /4,ϵ=0.2$ .

Figure 20. Velocity graph for values ${\mathcal{K}}_{\mathcal{C}}$ and $M$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,$ $\mathit{Pe}=S_{\mathcal{C}}=\mathit{Da}=0.7$ , $G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0$ .

Figure 21. Velocity graph for values ${\mathcal{K}}_{\mathcal{C}}$ and $M$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,$ $\mathit{Pe}=S_{\mathcal{C}}=\mathit{Da}=0.7$ , $G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0.2$ .

Figure 22. Velocity graph for values $F_r$ and $\omega$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0$ .

Figure 23. Velocity graph for values $F_r$ and $\omega$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\lambda =0.6,S_{\mathcal{T}}=0.3,\mathit{Pe}=S_{\mathcal{C}}={\mathcal{K}}_{\mathcal{C}}=\mathit{Da}=0.7$ , $M=1.1,G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,$ $R_e=2,\phi =\varphi =\pi /4,ϵ=0.2$ .

Figure 24. Velocity graph for values $t$ and $ϵ$ with ${\mathcal{K}}_{\mathcal{T}}={\mathcal{Q}}_H=2,\omega =1,\lambda =0.6,S_{\mathcal{T}}=0.3,$ $\mathit{Pe}={\mathcal{K}}_{\mathcal{C}}=S_{\mathcal{C}}=\mathit{Da}=0.7$ , $G_{\mathcal{T}}=G_{\mathcal{C}}=0.5,F_r=0.8,$ $R_e=2,\phi =\varphi =\pi /4,M=1.1$ .

4. Solution analysis

This part includes analyzing the solutions we obtained. We start with the temperature function Figures 2 and 3, then the concentration function Figures 4–7, and then the fluid velocity function Figures 8–24.

Figure 2 The data indicates that the temperature of the fluid rises as ${\mathcal{Q}}_H$ and ${\mathcal{K}}_{\mathcal{T}}$ increase. In contrast, Figure 3 shows that temperature fluid decreases by increasing $\omega$ and $\mathit{Pe}$ .

Figures 4–7, Fluid concentration rises with higher values of ω and Pe, but decreases with increased ${\mathcal{Q}}_H$ , ${\mathcal{K}}_{\mathcal{T}}$ , $S_{\mathcal{C}}$ , $S_{\mathcal{T}}$ , and ${\mathcal{K}}_{\mathcal{C}}$ .

Figures 8–24, Exhibit that fluid velocity escalates with an increase in ${\mathcal{Q}}_H$ , ${\mathcal{K}}_{\mathcal{T}}$ , $G_{\mathcal{C}}$ , $\mathit{Pe}$ , $R_e$ , $G_{\mathcal{T}}$ , $\lambda$ , $\varphi$ , $\phi$ , and $\mathit{Da}$ , while diminishing with an increase in $S_{\mathcal{C}}$ , $S_{\mathcal{T}}$ , ${\mathcal{K}}_{\mathcal{C}}$ , $M$ , $\omega$ , and $F_r$ . It is evident that the velocity fluctuates more significantly in the scenario of changing viscosity compared to that of constant viscosity.

5. Discussions and conclusion

This study analyzed the transient, incompressible magnetohydrodynamic (MHD) oscillatory flow of a non-Newtonian Bingham fluid through an inclined porous channel, considering variable viscosity, heat generation, and concentration effects. The governing nonlinear equations were solved using the separation of variables technique with computational assistance from Wolfram Mathematica 13. The obtained results were interpreted in terms of the influence of several dimensionless parameters on temperature, concentration, and velocity distributions.