Analytical solution of thermal effect on unsteady visco-elastic dusty fluid between two parallel plates in the presence of different pressure gradients

Background Thermal diffusion of dusty fluids has valuable interference in various fields, including waste-water treatment, oil transportation, and power plant pipes. Dusty fluids are used in lots of industrial fields as a result of their improved heat transfer and heat management capabilities. These industries range from renewable energy systems to aerobic plastic sheet extrusion, manufacturing, and rolling and reaching metal sheet cooling.Results The work embodied in this paper presents the analytical solution performed to predict the effects of thermal diffusion on dusty, viscous, incompressible fluid flows between two porous, parallel vertical plates with a heat source or a heat sink. The mathematical equations are solved by the separation of variables and Laplace transform techniques. The influence of temperature is investigated for various values of Prandtl number and heat source or heat sink parameters. Also, the influences of various coefficients like the thermal diffusion coefficient, Schmidt number, Prandtl number, and heat source or heat sink coefficient on the concentration are observed. The fluid velocity distribution is graphically obtained. The solutions are discussed and exhibited graphically. The influences of the thermal diffusion parameter and chemical reaction parameter on fluid and dust particles' velocities are examined. A parametric study on the effect of time on temperature and concentration is made.Conclusions The exact expressions for temperature, concentration, and velocity variation for fluid and dusty particles are obtained analytically. The temperature is inversely proportional to both the Prandtl number Pr and the heat source or heat sink parameter H-s. The concentration of the fluid is inversely proportional to the thermal diffusion parameter Td and the heat source or heat sink parameter H-s.