The Combined Effects of Variable Viscosity and Thermal Conductivity of An Unsteady Casson Fluid Flow Along a Vertical Porous Channel With Convective Cooling Walls, Using the Bivariate Spectral Local Linearization Method

ABSTRACT This study examines the combined impact of different thermal conductivity and viscosity on unsteady non‐Newtonian Casson fluid flow of incompressible, electrical conductivity in a porous vertical channel with convective cooling walls, uniform magnetic field, and constant pressure gradient. All the properties of the fluid are taken to be constant, except for its viscosity and thermal conductivity. The primary goal of this research is to analyze the unsteady flow of a non‐Newtonian Casson fluid with variable viscosity and thermal conductivity in a porous vertical channel with convective cooling on both channel walls. Unlike previous studies that primarily focus on steady Casson fluid flows this study introduces an unsteady framework to examine the effects of temperature‐dependent viscosity and thermal conductivity, alongside convective cooling walls. The governing partial differential equations for momentum, energy, and mass concentration are transformed into dimensionless forms. The bivariate spectral local linearization method is then implemented to solve numerically. The behavior of the flow, temperature, and concentration distributions is discussed for various key variables, including the Casson parameter (), the magnetic parameter (), the variable viscosity (), and the variable thermal conductivity (), respectively. It becomes apparent that variables like the Casson fluid, the variable viscosity, the Prandtl number (), and the Grashoff and solutal Grashoff numbers , respectively, were observed to amplify velocity and temperature distributions as their values were increased; conversely, variations in thermal conductivity and the magnetic parameter resulted in a decline in both velocity and temperature profiles as they were increased. These findings provide valuable insights for optimizing engineering systems involving unsteady, electrically conductive Casson‐like fluids in porous media.