On the analytical solution of the two-phase Couette flow with wall transpiration

Abstract

The two-phase Couette flow with transpiration through both walls is considered, where there is a constant blowing v0 at the lower wall and a corresponding suction at the upper wall. The interface between both fluids is initially flat and, hence, stays flat as it moves upward at the constant speed of the transpiration velocity v0. The corresponding initial value problem is subject to three dimensionless numbers consisting of the Reynolds number Re and the viscosity and density ratios, ϵ and γ. The solution is obtained by the unified transform method (Fokas method) in the form of an integral representation depending on initial and all boundary values including the Dirichlet and Neumann values at the interface. The unknown values at the moving interface are determined by a system of linear Volterra integral equations (VIEs). The VIEs are of the second kind with continuous and bounded kernels. Hence, the entire two-phase spatiotemporal 1 + 1 system has dimensionally reduced. The system of VIEs is solved via a standard marching method. For the numerical computation of the complex integral contours, a parameterized hyperbola is used. The influence of the dimensionless numbers Re, γ, and ϵ is studied exemplarily. The most notable effect results from ϵ that gives rise to a kink in the velocity at the moving interface. Both ratios, ϵ and γ, allow for very different flow regimes in each fluid phase such as nearly pure Couette flows and transpiration dominated flows with strongly curved velocity profiles. Those regimes are mainly determined by the effective Reynolds number in the respective phases.

Solution to the 1D Stefan problem using the unified transform method

“…In the following, lemma 1 in [28] for the spatial derivatives of the temperature distributions at the interface x = g(t) is derived for our present case in an analogous way as done by Friedman in order to prove the statement (23) and thus the correctness of the heat fluxes (25) and (27).…”

“…ds, (23) with i = 1, 2 representing the kernel for each phase. The sign of the additional term 1 2 √ α i ρ(t) depends on the direction in which the interface is being approached.…”

“…The complete derivation of ( 23) can be found in the appendix. Thus by applying (23) the derivative of the temperature distribution at the interface of the first phase is given as…”

“…First, we present analytical expressions for the temperature distributions and the interface position for the one-dimensional two-phase case. Second, we lay out a numerical scheme to solve the resulting integro-differential equations in a similar way as done in [23], where the authors used the UTM, to solve the two-phase Couette flow with transpiration through both walls. The solution is thus obtained in the form of integrals depending on initial and boundary values.…”