Abstract
Couple Stress Fluid (CSF) incorporates a new material constant whose, in contrast to a Newtonian fluid, regulates both couple stress and lubricant viscosity. Due to the fourth-order spatial derivative term that this material constant introduces into the momentum equation, this fluid (CSF), even for classical fluid problems, has received limited research. The aim of this article is to examine the joint impact of heat transmission and magnetic field on a time-fractional model of two-phase free convection flow of dusty fluid that is electrically conducting. Among parallel plates, one of which is stationary and the other of which is oscillating with constant velocity, the CSF is supposed to flow. Heat is transferred by free convection and buoyant force, which generates the flow. Furthermore, all spherical dust particles are uniformly dispersed through the fluid. The flow is modeled mathematically in terms of PDEs. The provided derived system of PDEs is generalized by applying a recently invented fractional derivative, the Caputo-Fabrizio fractional derivative. Finite sine Fourier and Laplace transformations are jointly applied to handle the problem. The velocity and temperature profiles have closed form solutions. The CSF outcomes for stimulating fluid parameters are shown in numerous graphs for CF time fractional derivatives. Additionally, the influence of various parameters has been discussed. Mathcad-15 is used to plot the graphical outcomes for the CSF, dust particle, and temperature profiles. Furthermore, the skin friction and Nusselt number are calculated. Table 1 demonstrates how the rate of heat transmission reduces as the Peclet number's value rises. Similarly, Table 2 demonstrates that the skin friction increases as the magnetic parameter and couple stress parameter are raised. Table 3 and 4, shows the Regression analysis that the variation in the velocity for Couple stress and dusty fluid parameter are statistically significant. By increasing the couple stress parameter lambda, retarde the both velocities profile.