Effect of Magnetic Field on Couette Flow in a Fluid-Saturated Porous-Filled Duct Under the Local Thermal Non-equilibrium with Viscous Dissipation

Abstract

In our present analysis, we explore the impact of the magnetic field on the Couette flow occurring within a duct that contains a porous material. This investigation incorporates considerations of viscous dissipation and local thermal non-equilibrium (LTNE). The lower plate experiences movement and is exposed to isoflux boundary conditions, whereas the upper plate remains stationary and adiabatic. To describe the one-directional flow within the porous region, we utilize the Darcy–Brinkman (DB) model. The investigations also aim to quantify the effects of the Hartmann number (M_W), thermal conductivity ratio (κ), Brinkman number (Br_W), and Biot number (Bi_W) on enhancing heat transfer. Analytical solutions are derived for the governing equations, providing fully developed profiles of Nusselt numbers and dimensionless temperatures for both the fluid and solid phases. In the Couette flow model, the presence of a magnetic field impacts the temperature distribution in both phases. Furthermore, irrespective of the Hartmann number (M_W) in the Couette flow, the temperature in the solid phase is consistently higher than that in the fluid phase, thereby confirming the existence of local thermal non-equilibrium (LTNE).

Appendix

$$\begin{aligned} A_{1} & = \varepsilon \left( {1 + {\text{Da}}M^{2} } \right),A_{2} = {{\sqrt {A_{2} } } \mathord{\left/ {\vphantom {{\sqrt {A_{2} } } 2}} \right. \kern-0pt} 2},A_{3} = {\text{csch}} \left[ {\sqrt {A_{2} } } \right]A_{4} = 4{\text{Da}}_{W} \left( {A_{2} A_{3} } \right)^{2} , \\ A_{5} & = \left( {\varepsilon A_{3}^{2} + A_{4} } \right),A_{6} = \left( {\varepsilon A_{3}^{2} - A_{4} } \right) \\ B_{1} & = \sqrt {{{\kappa {\text{Bi}}_{W} } \mathord{\left/ {\vphantom {{\kappa {\text{Bi}}_{W} } 2}} \right. \kern-0pt} 2}} ,B_{2} = \left( {1 + e^{{2\sqrt {2\kappa {\text{Bi}}_{W} } }} } \right),B_{3} = \left( {\kappa {\text{Bi}}_{W} - 2A_{2}^{2} } \right), \\ B_{4} & = \left( {\kappa {\text{Bi}}_{W} - 4A_{2}^{2} } \right),B_{5} = \left( {\kappa {\text{Bi}}_{W} - 8A_{2}^{2} } \right) \\ B_{6} & = \left( {\kappa {\text{Bi}}_{W} - 16A_{2}^{2} } \right),B_{7} = 2\varepsilon {\text{Bi}}_{W} \sinh \left[ {2A_{2}^{2} } \right]A_{3} B_{5} ,B_{8} = \kappa {\text{Bi}}_{W} cosh\left[ {4A_{2} } \right]A_{5} \\ \end{aligned}$$

$$\begin{aligned} C_{1} & = B_{7} + {\text{Br}}_{W} A_{2}^{2} B_{3} \left[ {B_{8} - A_{6} B_{5} } \right],C_{2} = - 4A_{2} A_{6} B_{5} + A_{5} B_{6} \sinh \left[ {4A_{2} } \right] \\ C_{3} & = - 12\varepsilon \sinh \left[ {4A_{2} } \right]A_{3} B_{4} B_{5} + \kappa {\text{Br}}_{W} \sinh \left[ {3A_{2} } \right]^{2} A_{5} B_{3} B_{6} \\ C_{4} & = - 4\varepsilon {\text{Bi}}_{W} A_{3} \sinh \left[ {2A_{2} } \right] + {\text{Br}}_{W} \left( {1 + \kappa {\text{Bi}}_{W} - cosh\left[ {2A_{2} } \right]} \right)A_{6} B_{2} B_{3} B_{5} \\ C_{5} & = 8\varepsilon {\text{Bi}}_{W} A_{2}^{2} A_{3} - \kappa {\text{Br}}_{W} \left( {{\text{Br}}_{W} B_{3} \sinh \left[ {2A_{2} } \right]A_{6} + 4\kappa Bi_{W} A_{3} } \right)B_{5} + A_{2} C_{4} \\ C_{6} & = \kappa {\text{Bi}}_{W} {\text{Br}}_{W} A_{4} B_{3} \left( {B_{6} - 8B_{5} } \right) + \kappa \varepsilon {\text{Bi}}_{W} {\text{Br}}_{W} A_{3}^{2} B_{3} \left( {B_{6} + 8B_{5} } \right) \\ & \quad - 16\left( {1 + \kappa {\text{Bi}}_{W} } \right)B_{7} + 2{\text{Br}}_{W} B_{3} B_{5} B_{8} \\ C_{7} & = \sinh \left[ {2B_{1} } \right]A_{2} - \sinh \left[ {2A_{2} } \right]B_{1} ,C_{8} = \sinh \left[ {2A_{2} } \right]\sinh \left[ {2B_{1} } \right]B_{1} + A_{2} \\ C_{9} & = 32\kappa \varepsilon {\text{Bi}}_{W}^{2} cosh\left[ {A_{2} } \right]A_{3} B_{5} + \sinh \left[ {A_{2} } \right]C_{6} \\ C_{10} & = A_{2} B_{2} \left[ {\varepsilon 8A_{3} \left( {\cosh \left[ {2A_{2} } \right]B_{4} - \kappa {\text{Bi}}} \right)B_{5} - \kappa {\text{Br}}B_{3} C_{2} } \right] + \varepsilon 16B_{2} A_{2}^{2} A_{3} \\ \end{aligned}$$