An MHD Stokes eigenvalue problem and its approximation by a spectral collocation method
Introduction
The magnetohydrodynamic (MHD) flow, that is, the flow of electrically conducting fluids in the presence of a magnetic field has important industrial application areas. Among them are magnetic power generation systems, geothermal and solar energy systems, liquid metal productions, plasma physics, and nuclear fusion reactors (see, [1], [2], and the references therein). The Stokes equations that govern the incompressible creeping flow of viscous fluids are the constituents of the incompressible Navier–Stokes equations. They are also form-identical to the equations of the (isotropic) incompressible elasticity theory which govern the linear response of elastic materials [3]. These theoretical and practical aspects constitute the strong basis for accurate determination of the eigenvalues and eigenmodes of the Stokes operator.
Analytical solutions to the Stokes eigenvalue problems do not exist when the eigenfunctions are constrained to satisfy the velocity no-slip conditions which are of great importance from the physical point of view in fluid dynamics [1], [4]. Thus, one has to resort to numerical procedures to calculate the eigenvalues and eigenfunctions in the discrete spectrum of the operator.
The number of works in the literature concerning the Stokes operator is tremendous. A number of recent studies devoted to the source problem can be found in [5], [6], [7], [8], and the references therein. The associated eigenproblem, on the other hand, has been tackled by a variety of methods including spectral methods [2], [9], [10], [11], finite element methods [12], [13], [14], and mesh-free methods [15].
In this study, a novel eigenvalue problem associated with the MHD Stokes equations describing the flow of an electrically conducting fluid is presented. The flow takes place in a duct of constant cross section that is subjected to an externally applied uniform and stationary magnetic field, and the fluid is assumed to be viscous and incompressible. A two-dimensional case study under a transversely applied magnetic field is implemented in order to evaluate the influence of the applied magnetic field on the eigenstructure of the Stokes operator. The governing equations are defined in terms of the primitive variables, namely, velocity and pressure (see, e.g., [16], [17]). The solution of the eigenvalue problem that is referred as the MHD Stokes eigenproblem is approximated by the Chebyshev spectral collocation method (CSCM). The method is based on forcing the residual equation to be zero at the collocation points on the physical domain that are set to be the Chebyshev–Gauss–Lobatto points (also known as Chebyshev-extrema points). It is implemented on the physical domain, and it allows the direct evaluation of the derivatives of the unknowns. This convenience and ease of implementation explicates the extensive use of the method in approximating the solutions of various problems involving partial differential equations.
As the governing equations are formulated in the primitive variables, due to the incompressibility constraint, they include the continuity equation and the momentum equations. As a consequence, the CSCM discretization originally results in a generalized eigenvalue system with zero diagonal entries. A penalization approach is adopted as a remedy where a perturbed form of the problem is presented, and a zero mean pressure is set forth. The use of the classical penalty technique and its variant (namely, the iterative approach) to handle constraints has been proposed in [18] for approximating the eigensolutions of the (hydrodynamic) Stokes operator. In particular, the discretization by CSCM has been specifically selected to allow the model to be straightforwardly established in [18]. The present study extends the applicability of CSCM designed as a direct approach in its essential features to consider the MHD Stokes eigenproblem proposed for the first time to the best of the author’s knowledge. The numerical scheme is initially evaluated by a number of experiments to assess its numerical aspects. Then, a primary focus is given to examine the effects of introducing a magnetic field on the eigenspectrum from the results visualized in terms of the velocity and pressure plots for different intensities of the magnetic field. The physical implications of these effects are investigated numerically with an emphasis on the variation of the fundamental eigensolutions, and it is shown that the flow field is considerably affected upon the onset of a magnetic field.
CSCM is traditionally considered to be a very convenient method for approximating the solutions of (nonlinear) partial differential equations since it combines the crucial features that are already mentioned before. It is important to note, however, that its application requires a careful handling of dense matrices whose condition numbers rapidly grow with the degree of the polynomials involved, that is the number of the collocation points. The need arises to use specifically designed preconditioners when the system is ill conditioned for problems particularly of large scales (see for instance [19] and [20]). For such systems, there exists another option which has gained interest recently, involving the use of the ultraspherical polynomials that leads to banded matrices with better conditioning [21], [22]. One of the main objectives of the present work, on the other hand, is to provide a feasible approach to appropriately determine reasonably accurate approximations to the selected eigensolutions of the problem introduced. Thus, CSCM in the uncomplicated form (in terms of implementation complexity) employed, serves very well for the fundamental purpose of the present study.
Section snippets
The problem definition
The transient flow of a viscous, incompressible and electrically conducting fluid in an externally applied uniform and stationary magnetic field is considered. The fundamental equations governing the fluid motion, namely the Navier–Stokes equation with the continuity equation, are considered with the electromagnetic field which is subjected to the simplified Maxwell equations coupled with Ohm’s law. Therefore, the continuity equation, the Navier–Stokes equation incorporated with the Lorentz
Application of CSCM to the MHD Stokes EVP
The discretization of the MHD Stokes EVP (7) carried out with the use of a collocation approach is presented in this section. The method consists of requiring the residual of the discrete problem to be exactly zero at the abscissae of the points where the Chebyshev polynomials take their extreme values. The procedure is of global approximation nature, and thus, the approximations of the unknowns as well as their derivatives depend on the entire discretization of the problem domain under
Numerical results
The numerical results obtained from the CSCM procedure with the penalization idea described in Section 3 applied for the MHD Stokes EVP are presented in this section. The reduced penalized system (15) that is a generalized matrix eigenvalue problem is solved by a computer program using MATLAB making use of its generalized matrix eigenvalue problem solvers. In the specification of the solution to (15), a normalization is imposed such that . As there is no danger of confusion, in the
Conclusions
An eigenvalue problem is introduced for the MHD Stokes equations describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field for the first time in the present study. The solution of this eigenproblem is approximated by using CSCM that is based on vanishing the residual equation at the extreme points of the Chebyshev polynomials. The simplicity and efficiency of the method in the computations have been exploited for approximating
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