Analytical solution of the problem on MHD flow in duct with perfectly conducting Hartmann walls and slip condition on conducting side walls

Abstract

Abstract. An analytical solution has been obtained for a problem on magnetohydrodynamic flow in a rectangular duct with perfectly conducting Hartmann walls and side walls of finite conductivity. The slip boundary conditions for velocity were applied to the duct side walls. The exact analytical solution of the problem has been obtained in the form of convergent series. On the base of the obtained analytical solution the velocity field of the flow has been studied numerically.Keywords: MHD duct flow, slip boundary condition. IntroductionA flow of electrically conducting fluid in the presence of external magnetic field produces a variety of new effects, which are not realizable in usual hydrodynamics. These effects are studied by magnetohydrodynamics (MHD), the discipline combining the classical fluid mechanics and electrodynamics.Electromagnetic methods of action on electrically conducting medium are widely exploited in industrial processes in metallurgy, material processing, chemical industry, industrial power engineering and nuclear engineering in order to control and manipulate various conducting materials. Besides, the MHD effects are exploited in technical devices such as MHD pumps, flow meters, MHD generators and accelerators, where channels with electrically conducting fluids are their common parts. Thus, investigation of MHD phenomena in channels with conducting fluids is quite important.In magnetohydrodynamics the number of exact analytical solutions is limited due to the nonlinearity of the Navier-Stokes equations. Therefore, numerical methods are widely used for solving these problems, although analytical solutions still do not lose their importance (even if these solutions are often obtained for simplified flows). The present analytical study was motivated by the recent publication [2], in which three classical MHD problems were considered, but the applied boundary condition at the interface between the electrically conducting fluid and the insulating solid walls was the hydrodynamic slip condition. Despite the no-slip boundary condition is usually applied for the majority of flows of viscous fluids, there exist situations when no-slip boundary conditions lead to unrealistic behaviour, for example, if viscous effects at the wall are negligible, then the slip boundary condition must be applied (see [1; 2]). This paper presents an exact analytical solution obtained for a new problem on a MHD flow in the rectangular duct with perfectly conducting Hartmann walls and the side walls of finite conductivity. The boundary condition applied to the Hartmann walls of the duct is the Dirichlet (no-slip) boundary condition, which is the simplest boundary condition for the velocity of viscous fluid, while the boundary condition applied to the side walls is the 3rd kind (slip) boundary condition. This solution seems absent in literature. Some numerical results for the velocity of the flow calculated on the base of the obtained analytical solution are also presented. It is to be noted that the study of the present prob...

The DRBEM solution of Cauchy MHD duct flow with a slipping and variably conducting wall using the well-posed iterations

“…The collocation of the inhomogeneities at N + L points results in the systems F α = b 1 , F β = b 2 . Then, the right hand sides of the equations (3)- (4) are rewritten as…”

“…Analytical solutions in terms of asymptotic expansions are given by Ligere et. al [3,4] for MHD duct flow with perfectly conducting Hartmann walls and slipping side walls. The MHD equations are commonly solved using numerical methods in no-slip and most general form of the wall conductivity conditions [5][6][7][8].…”

“…There are quite a number of BEM or DRBEM solutions of MHD duct flow problems with no-slip velocity condition and different combinations of wall conductivities [5,6,8,11,12]. In the studies carried by S. Smolentsev and E. Ligere [4,13], MHD flow problems with slip velocity conditions and with a known slip length in a channel are solved analytically on various parts of the duct walls. The DRBEM solution of Cauchy MHD duct flow equations with a perturbed slipping upper boundary is given in [14] by solving the MHD equations as a whole.…”