Time fractional analysis of channel flow of couple stress Casson fluid using Fick\u2019s and Fourier\u2019s Laws

Shafiq Ahmad,Kottakkaran Sooppy Nisar,Sami Ul Haq,Farhad Ali,Ilyas Khan

doi:10.1038/s41598-022-06897-y

Abstract

This study aim to examine the channel flow of a couple stress Casson fluid. The flow is generated due to the motion of the plate at y=0, while the plate at y=d is at rest. This physical phenomenon is derived in terms of partial differential equations. The subjected governing PDE’s are non-dimensionalized with the help of dimensionless variables. The dimensionless classical model is generalized by transforming it to the time fractional model using Fick’s and Fourier’s Laws. The general fractional model is solved by applying the Laplace and Fourier integral transformation. Furthermore, the parametric influence of various physical parameters like Casson parameter, couple stress parameter, Grashof number, Schmidt number and Prandtl number on velocity, temperature, and concentration distributions is shown graphically and discussed. The heat transfer rate, skin friction, and Sherwood number are calculated and presented in tabular form. It is worth noting that the increasing values of the couple stress parameter lambda deaccelerate the velocity of Couple stress Casson fluid.

Highlights

This study aim to examine the channel flow of a couple stress Casson fluid
Fractional calculus is the expansion of classical calculus, and it has around three centuries-old histories
B aleanu[19] studied discrete constrained systems and presented the fractional dynamics for the said system using Caputo derivatives. In20, Atangana and Baleanu developed the Mittag Leffler function in 2016, which is very useful in finding the solution of fractional order integral equation or fractional order differential equations

Summary

Mathematical formulation

Consider the motion of couple stress Casson fluid between in a infinite parallel plates.The flow is considered in x direction. Applying the Finite Sine Fourier Transform to Eq (24) and using the conditions in Eq (25), we obtain:. Applying the Finite Sine Fourier Transform to Eq (28) and using the conditions in Eq (25), we can write:. Applying Finite Fourier Sine Transform to Eqs. Taking Inverse Laplace Transform, Eq (32) can be written as:. By taking Inverse Fourier Finite Sine transformation of Eq (33) , the final exact solution of the Eq (31) is:. The mathematical form of Nusselt number for Couple stress Casson fluid is defined as: Nu. Sherwood number. The mathematical form of Sherwood Number for Couple stress Casson fluid is defined as: Sh

Results and discussion

Conclusion

Additional information