Comparison of unsteady MHD flow of second grade fluid by two fractional derivatives

A study is conducted on the unsteady motion of a free convective flow of second grade fluid, energy transfer, and Darcy's law over an oscillatory smooth vertical plate. The study compares two different approaches in developing a fractional model: the Caputo-Fabrizio operator with nonsingular kernel and the constant proportional Caputo fractional operator with Fourier's and Fick's laws. By applying the Laplace method and transforming the provided set of equations into nondimensional form, we obtained semi-analytical results and presented these results through graphical analysis. The study examines how different flow parameters, including the fractional parameters, affect the velocity, mass, and heat profiles of the physical system. The results suggest that the physical model using constant proportional Caputo derivative leads to higher temperature, stronger concentration, and increased velocity compared to the Caputo-Fabrizio model. This highlights the importance of selecting an appropriate fractional model when studying complex physical systems. It is also depicted from the whole analysis that field variables with novel hybrid fractional derivative constant proportional Caputo exhibits a more effective and declining tendency than Caputo-Fabrizio.