Exact Solutions and Analysis for a Class of Extended Stokes’ Problems

Abstract

This paper studies a class of unsteady flows as extensions of the classic Stokes' problems to consider influence of solid walls, effect of pressure gradients, and situation of two-layer fluids. The flows are solved using the method of separating variables and the eigenfunction expansion method. With simplifications, the derived solutions will degenerate to solutions to the classic Stokes' problems, the Couette flow, and the Poiseuille flow. The exact solutions of these flows clearly illustrate the complexity of the involved physics including evolution of flow velocity profiles and energy transferring at fluid boundaries. For a single-layer flow driven by a plate moving at a constant speed, energy transferred from the plate decreases with time and tends to a non-zero constant as a result of wall effect. In a single-layer flow with an oscillatory boundary, negative energy input may appear at the boundary. For an air-water flow with a finite depth, the interface velocity is proportional to the air velocity, which is a well-known observation in physical oceanography. In addition, there is no energy transferring at the interface between the two fluids in a purely pressure driven two-layer flow.

Fluid Driven by Tangential Velocity and Shear Stress: Mathematical Analysis, Numerical Experiment, and Implication to Surface Flow

“…Solution (9) is restricted with infinite depth of the fluid. A solution for a finite depth is given in [12].…”

“…It is applicable if the thickness of lower and upper layers is large, and the flow is free from the effect of walls. Analytical solutions considering finite depth of air and water have been presented in [12].…”

“…Analytical solutions for steady flows between two parallel plates such as the Couette flow can be found in literature [8]. Unsteady flows caused by a sudden movement of a plate at a constant or a time-dependent speed, namely, the Stokes' first and second problems, can be analytically solved [9][10][11][12]. Discussions on flows introduced by velocity boundary conditions have been extended to non-Newtonian fluids [13].…”