Multi-layer flows of immiscible fractional second grade fluids in a rectangular channel

Abstract

Multi-layer laminar unsteady flows of immiscible fractional second grade fluids in a rectangular channel made by two parallel plates are studied. The fluid motion is produced by the motion of parallel walls in their plane and by the timedependent pressure gradient in the presence of the linear fluid-fluid interface conditions. The mathematical model is based on the generalized constitutive equations for the shear stress described by the time-fractional Caputo derivative. Integral transforms (finite Fourier sine transform and Laplace transform) have been used to obtain analytical and semi-analytical solutions for velocity, shear stress and the temperature fields. In the case of semi-analytical solutions, the Talbot's algorithms are used for the inverse Laplace transform. The numerical calculations are carried out with the help of Mathcad software, and the results are illustrated graphically. It has been found that the memory effects have a significant influence on the motion of the fluids. Keywords n-layered immiscible fluids • Fractional second grade fluids • Analytical and semi analytical solutions • Integral transforms Mathematics Subject Classification 76-XX • 76T30 • 76D50 Nomenclature i Density i Dynamic viscosity i Kynamic viscosity u 0 Characteristic velocity G i Elastic modulus i Shear stress u i (y, t) Velocity P Pressure h Distance between two plates X (y,) Laplace transform of the function X(y, t) E i , i (⋅) Mittag-Leffler function G 1 , 2 , 3 (t,) G-Lorenzo Hartely function

Heat transfer analysis of generalized second-grade fluid with exponential heating and thermal heat flux

“…The exact solution is acquired via Laplace transform (LT). References 6 , 7 investigated of fractional immiscible multi-layer flow of second-grade fluid in rectangular channel. The remedy is acquired via finite Fourier sine and LT. Siddique et al 8 studied the heat transfer analysis of fractional second-grade fluid with Newtonian heating.…”

Simultaneous Flow of n-Immiscible Fractional Maxwell Fluids with Generalized Thermal Flux and Robin Boundary Conditions

“…It has been found that the heat transfer in ordinary fluids is higher than those of thermal memory fluids; however, the memory parameters affect the fluids' velocities as accelerating factors. Some other important multilayer flow problems are studied in [14][15][16]. The parallel flow of the fractional Maxwell fluid inside a cylindrical domain has been studied by Rauf et al [17] in the presence of pressure gradient in the flow direction.…”