Time-dependent fractional second-grade fluid flow through a channel influenced by unsteady motion of a bottom plate

This investigation theoretically describes the exact solution of an unsteady fractional a second-grade fluid upon a bottom plate constrained by two walls at the sides which are parallel to each other and are normal to the bottom plate. The flow in the fluid is induced by the time dependent motion of the bottom plate. Initially the flow equation along with boundary and initial conditions are considered which are then transformed to dimensionless notations using suitable set of variables. The Laplace as well as Fourier transformations have been employed to recover the exact solution of flow equation. The time fractional differential operator of Caputo-Fabrizio has been employed to have constitutive equations of fractional order for second-grade fluid. After obtaining the general exact solutions for flow characteristics, three different cases at the surface of bottom plate are discussed; namely (i) Stokes first problem (ii) Accelerating flow (iii) Stokes second problem. It has noticed in this study that, for higher values of Reynolds number the flow characteristics have augmented in all the three cases. Moreover, higher values of time variable have supported the flow of fractional fluid for impulsive and constantly accelerated motion and have opposeed the flow for sine and cosine oscillations.