Nonlinear-Mixed Convection Flow with Variable Thermal Conductivity Impacted by Asymmetric/Symmetric Heating/Cooling Conditions

Abstract

The current article investigates the effects of nonlinear mixed convection flow in an upright channel with asymmetric or symmetric heating and cooling conditions, considering the influence of temperature-dependent thermal conductivity. The momentum equation is approximated using the nonlinear Boussinesq approximation in the buoyancy force term. Computational solutions for the dimensionless partial differential equations are obtained through the use of an unconditionally stable and convergent implicit finite difference technique. A regular perturbation series approach is employed to ascertain steady-state solutions, facilitating the assessment of the correctness of the numerical approach. During the numerical computing process, it is observed that the nonlinearity in density variation with temperature, along with the mixed convection parameter and the symmetric or asymmetric heating and cooling of the plates, significantly influences flow generation. It is also noted that in the scenario of asymmetric heating, fluid motion is stronger at the bottom plate, whereas in the case of symmetric heating/cooling settings, the highest velocity is observed in the center of the channel.

Appendix

$$\begin{aligned} & B_{3} = - \frac{Gr}{{{\text{Re}} M^{4} }}Q_{3} ,C_{5} = - \frac{Gr}{{{\text{Re}} M^{2} }}2k_{2},\\ & C_{6} = \frac{Gr}{{{\text{Re}} M^{2} }}\left[ {k_{1} + k_{2} + \frac{{12k_{2} }}{{M^{2} }}} \right],C_{7} = \frac{Gr}{{{\text{Re}} M^{2} }}Q_{2} \\ & k_{1} + \frac{{6k_{1} }}{{M^{2} }} + \frac{{6k_{2} }}{{M^{2} }} + \frac{{72k_{2} }}{{M^{4} }} = Q_{2} ,C_{8} = \frac{Gr}{{{\text{Re}} M^{4} }}Q_{3},\\ & C_{4} = \frac{Gr}{{{\text{Re}} M^{2} }}K_{2} ,k_{1} = \frac{{\left( {R - 1} \right)^{2} }}{2},k_{2} = N\frac{{\left( {R - 1} \right)^{4} }}{4} \end{aligned}$$

$$\begin{aligned} & B_{4} = \frac{Gr}{{{\text{Re}} M^{2} }}\left[ {\frac{{\left[ {\frac{{Q_{3} }}{{M^{4} }}} \right]\left[ {\cosh \left( m \right) - 1} \right] + k_{2} - Q_{1} - Q_{2} }}{\sinh \left( m \right)}} \right],\\ & B_{2} = \frac{Gr}{{{\text{Re}} m^{2} }}f_{4} + \frac{\xi }{{m^{2} }}f_{5} ,\frac{dP}{{dx}} = \xi \\ & f_{3} = 2N(R - 1)^{2} + m^{2} N + m^{2},\\ & \boxed{C_{3} = \frac{Gr}{{{\text{Re}} m^{4} }}\left[ {2N(R - 1)^{2} + m^{2} N + m^{2} } \right] - \frac{\xi }{{m^{2} }}} \end{aligned}$$

$$\begin{aligned} & let,f_{4} = \frac{{\frac{{f_{3} \cosh (m)}}{{m^{2} }} - f_{1} - f_{2} - f_{3} }}{\sinh (m)},f_{5} = \frac{1 - \cosh (m)}{{\sinh (m)}},\\ & B_{1} = - \frac{Gr}{{{\text{Re}} m^{4} }}f_{3} + \frac{\xi }{{m^{2} }},C_{3} = \frac{Gr}{{{\text{Re}} m^{4} }}f_{3} - \frac{\xi }{{m^{2} }}\end{aligned}$$

$$C_{1} = \frac{{GrN(R - 1)^{2} }}{{{\text{Re}} m^{2} }},C_{2} = \frac{Gr(R - 1)}{{\text{Re}}}\left[ {1 + 2N} \right]$$