Application of caputo fraction derivatives to the thermai radiative convective casson flow in a microchannel

Abstract

The application of fractional derivative is currently convenient and anticipated in the industrial and technological fields due to its unique properties. Therefore, the goal of this research is to learn more about the characteristics of the Caputo fractional derivative, which is one of the most often used fractional derivative operators. Additionally, microchannels exist in many industries and engineering process equipment, and their geometrical structure is one of the most important factors influencing fluid flow. Therefore, in this thesis, the Casson fluid behavior flowing in three different forms of microchannel which are static, accelerated, and oscillating is investigated. The effect of thermal radiation on the Casson fluid is also considered. The formulation of the governing equation for the problems is thoroughly discussed. First, the partial differential equations and boundary conditions are transformed into dimensionless equations by using appropriate dimensionless variables. Second, the resultant dimensionless governing equations are transformed into fractional form by using Caputo fractional derivatives. The equations are then reduced to linear ordinary differential equations by using the Laplace transform technique and solved by using appropriate methods. Finally, the numerical solution is obtained by using the inverse Laplace transform technique with the help of Zakian’s explicit formula approach. The result of velocity and temperature profiles are plotted by using Mathcad software. The obtained solutions are reduced to the published results for such problem for verification and accuracy, and have achieved excellent agreement. The influence of key physical parameters on the velocity and temperature profiles is analyzed and discussed in depth. The results reveal that as the fractional and radiation parameters are increased, the velocity and temperature profiles for all three geometries of the microchannel increase. On the contrary, high Prandtl numbers have increased the viscous force, resulting in a reduction in both profiles. Since the Grashof number has a positive influence on the buoyancy force, it has caused the velocity profile to increase. Meanwhile, the velocity profile reveals a contrasting pattern, with the Casson fluid parameter increasing due to increased viscous forces compared to thermal forces.