A generalized dusty Brinkman type fluid of MHD free convection two phase flow between parallel plates

Fractional order derivative operators are regarded to be very complex mathematical tools for implementing more realistic solutions in a wide range of physics and engineering problems. Due to significance applications of fractional derivatives, the aim of this article is analyzed the combinational effect of the magnetic field and heat transfer on Magneto hydrodynamic, free convective, two-phase flow of generalized Brinkman type dusty fluid among two parallel plates. The buoyant force causes the flow, which is aided by natural convection to transfer heat. Furthermore, the left plate travels at a constant speed and the right plate remains fixed, and all spherically dust particles are homogeneously dispersed among the fluid. The problem is modeled mathematically via set of partial differential equations. The Caputo-Fabrizio fractional derivative is employed to generalize the derived PDE's. A combination of Laplace and finite sine Fourier transformations is utilized to get the closed form solution of the problem. The impacts of many factors on temperature, Brinkman fluid, and dust particle velocity have also been examined, including a Grashof number, magnetic parameter, Reynold number, Peclet number and Dusty fluid parameter. Mathcad-15 is used to plot the graphical findings for the dusty fluid, Brinkman fluid, and temperature profiles. For various embedded parameters, the Brinkman fluid and dusty fluid behave similarly. It is concluded that the fractional Brinkman type dusty fluid is more realistic feature as compared to the classical one. The rate of heat transfer can be enhance with the passage of time.