Analytical and numerical solutions to an electrohydrodynamic stability problem
Introduction
Nowadays the industry and ecology need more and more results from hydrodynamic stability theory for sophisticated fluid motions occurring in complicated circumstances. In certain of these situations, a direct application of numerical methods can lead one to false results due to the bifurcation problems of the stationary solutions set of the Navier–Stokes equations (or of some more general models) and to the dependence of the eigenvalue on physical parameters. That is why, an analytic along with a numerical study of the eigenvalue problems from hydrodynamic stability theory is highly requested. This is in fact one of the most difficult topic in hydrodynamic stability. The occurrence of false secular points in the linear stability of continua was first pointed out by Collatz in 1981 in his paper [4].
A considerable number of theoretical and numerical studies have been devoted to the interaction of electromagnetic fields with fluids. Rosenswieg [24] pointed out that there are three main categories on this subject, i.e. electrohydrodynamics (EHD), magnetohydrodynamics (MHD) and ferrohydrodynamics (FHD). Here we are interested in an eigenvalue problem in EHD which implies the presence of electric forces. The problem at hand consists into an eighth order differential equation, containing only even order derivatives, supplied with homogeneous boundary conditions for the even order derivatives up to sixth order, i.e. the so called hinged boundary conditions.
The main aim of this paper is to solve analytically as well as numerically this problem in order to get confidence in the later. In fact we are mainly interested in the low part of the spectrum of this problem.
With respect to the numerical methods we have to observe two important facts.
First, a straightforward application of the tau method based on Chebyshev polynomials leads to extremely ill conditioned matrices which are also fully populated (the asymptotic order of the entries varies between zero and O(N2×8) where N is the spectral parameter). The eighth order Chebyshev differentiation matrices are also fairly bad conditioned. For N = 29 the condition number of these matrices attains something of order O(1030) (see our paper [15]). Consequently, as we have shown, a direct application of the Chebyshev collocation method to this eighth order problem leads to huge instabilities.
Second, due to the fact that the boundary conditions imply a fairly challenging lacunar interpolation problem, the Galerkin and the collocation (pseudospectral) methods become directly inapplicable. It is worth noting in connection with this second point that Huang and Sloan, in their papers [20], [21], consider a fairly general non lacunar interpolation problem with multiple nodes and then solve some fourth and sixth order eigenvalue problems. We also succeeded (see [13]) in solving a non-standard eigenvalue problem, i.e. an Orr-Sommerfeld equation supplied with boundary conditions containing the spectral parameter, concerning flows driven by surface tension gradients. As it is not the case with our problem, we follow our “D2” strategy from [15], i.e. we rewrite the problem as a second order differential system supplied only with Dirichlet boundary conditions (also called clamped boundary conditions). This way, all the three spectral methods, namely tau, Galerkin and collocation can be efficiently applicable. They provide fairly accurate approximations for the low part of the spectrum at a modest computational cost.
The rest of the paper is organized as follows. In Section 2 we introduce the problem and show that whenever the physical parameters satisfy an “ellipticity” condition the smallest eigenvalue is real and positive. In Section 3, a direct analytical method based on the characteristic equation is used in order to solve the eigenvalue problem. In Section 4, we carry out the above transformation of the problem into a second order system supplied with Dirichlet boundary conditions and solve this problem by analytical methods which use Fourier expansions. In Section 5 we solve this eigenvalue problem, first by standard Chebyshev tau method and try to explain its lack of accuracy. We observe that the matrices involved in this method are highly non-normal. In the last subsection we solve the eigenvalue problem using our “D2” strategy along with Galerkin methods. For various values of the physical parameters, the computed values of the critical Rayleigh number, are displayed in a table. For the considered numerical values of them, the numerical results confirm the analytical ones.
Section snippets
The statement of the problem
The linear stability of the stationary solution in an electrohydrodynamic convection model in a layer situated between the walls z = ±0.5, against normal mode perturbations, is governed by the following eigenvalue problem from [10]Here F, which represents the amplitude of the temperature field perturbation, stands for the eigenfunction in (1). The physical parameter a represents the wavenumber, L is a parameter effectively
The direct method
Most of the problems of hydrodynamic, electrohydrodynamic or hydromagnetic stability as well as bifurcation of solutions can be reduced to eigenvalue problems defined by systems of ordinary differential equations including a large set physical parameters. In the presence of more than one physical parameter and with a high order of differentiation in the system, it is, however, difficult to analyze how the most relevant eigenvalue of the system depends on parameters, since, in general, these
Methods based on Fourier series expansions
In order to apply these methods we rewrite the two-point boundary value problem (1) in a different form. First, we introduce the new vector functionwithThis substitution turns the eighth order equation from (1) into the following system of second order ordinary differential equationsConsequently, the boundary conditions simplify considerably and become
The Chebyshev tau method
The classical (standard) Chebyshev tau method has been applied successfully to many eigenvalue problem from hydrodynamic stability also (see for instance [1], [5], [8] to quote but a few). The pioneering paper was that of Orszag [22]. In the monograph [2] Section 6.4, it is thoroughly analyzed for the Orr–Sommerfeld problem. However, it is fairly clear that this method is universally applicable, i.e., for any type of boundary conditions.
At first, we shift the problem in [−1, 1] by the
Conclusions
The linear stability problem of electrohydrodynamic convection between two parallel walls was accurately solved by two classes of analytical methods as well as by spectral methods. The eighth order eigenvalue problem was transformed into a second order system of differential equations supplied with (only!) Dirichlet boundary conditions. This new problem was first solved analytically by two methods based on Fourier series. The Fourier approximation was split into an even and an odd part. The
Acknowledgement
The first author would like to thank Prof. Adelina Georgescu for her encouragement, advice and support during the years of the author’s Ph.D. study. The work of the second author was partly supported by the Grant 2-CEx06-11-96.
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