Stokes' second problem and oscillatory Couette flow for a two-Layer fluid: Analytical solutions

TL;DR

This paper provides analytical solutions for the unsteady motion of a two-layer fluid under oscillatory boundary conditions, covering both Stokes' second problem and oscillatory Couette flow, with explicit velocity and shear stress expressions.

Contribution

It extends classical single-layer flow solutions to two-layer fluids, deriving explicit analytical solutions for both steady and transient states using Laplace transforms.

Findings

Explicit velocity fields for two-layer flows are derived.

Transient and steady shear stresses are calculated.

Results generalize known single-layer flow solutions.

Abstract

The unsteady motion of a two-layer fluid induced by oscillatory motion of a flat plate along its length is mathematically analyzed. Two cases are considered: (i) the two-layer fluid is bounded only by the oscillating plate (Stokes' second problem), (ii) the two-layer fluid is confined between two parallel plates, one of which oscillates while the other is held stationary (oscillatory Couette flow). In each of the Stokes' and Couette cases, both cosine and sine oscillations of the plate are considered. It is assumed that the fluids are immiscible, and that the flat interface between the fluids remains flat for all times. Solutions to the initial-boundary value problems are obtained using the Laplace transform method. Steady periodic and transient velocity fields are explicitly presented. Transient and steady-state shear stresses at the boundaries of the flows are calculated. The results derived in this paper retrieve previously known results for corresponding single-layer flows. Further, illustrative example of each of the Stokes' problem and the Couette flow is presented and discussed. Again, the results obtained could also be applicable to a problem of heat conduction in a composite solid with sinusoidal temperature variation on the surface.