SUMMARYThe magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow, has been investigated. One of the duct's boundaries which is perpendicular to the magnetic field is taken partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 100. All the infinite series obtained are transformed to infinite integrals first and then to finite integrals which contain modified Bessel functions of the second kind. In this way, the difficulties associated with the computation of infinite integrals with oscillating integrands and slowly converging infinite series, the convergence of which is further affected for large values of M , have been avoided. It is found that, as M increases, boundary layers are formed near the non-conducting boundaries and in the interface region, and a stagnant region is developed in front of the conducting boundary for velocity field. The maximm value of magnetic field takes place on the conducting part. These behaviours are shown on some graphs.

SEZGIN, M. ; AGGARWALA, B. D.; ARIEL, P. D. : Electrically Driven Flows in MHD 267 -~~ ZAMM . Z. angew. Math. Mech. 68 (1988) ' I , 267 -280 SEZGIN, M.; AGGARWALA, B. D.; ARIEL, P. D.Flow of viscous, incompressible, electrically conducting fluid, driven by imposed electric currents has been investigated in the presence of a transverse magnetic field. The boundary perpendicular to the magnetic field is perfectly conducting partly along its length. Three cases have been considered: a ) flow in the upper half plane when the boundary to the right of origin i s insulating and that to the left is perfectly conducting, b) flow in the upper half plane when a finite length of the boundary i s perfectly conducting, and c) flow in a flat channel when a finite length of boundary in each plane, symmetrically situated, is perfectly conducting. I n case a), an exact analytical solution is derived, from which the existence of a boundary layer, parabolic in shape and emanating from the point of discontinuity in electrical boundary conditions, is established. I n cases b) and c), the problem is reduced analytically to a Fredholm's integral equation of second kind, which is solved numerically. A number of transformations and valid approximations are used to simplify the task of numerical computation. Velocity contours and current lines have been plotted for each problem. For large values of Hartmann number the parabolic boundary layer mentioned above is demarcated. Die Stromung einer viskosen, inkompressibkn, elektrisch leitenden Fliissigkeit, bewegt durch angelegte elektrische Strome, wird in Gegenwart eines transversalen Magnetfeldes untersucht. Die Grenzfliiche senkrecht zum Magnetfeld ist entlang ihrer Lange teilweise vollkommen leitend. Drei Falle werden betrachtet : a ) Stromung in der oberen Halbebene, wenn die Grenzflache rechts vom Ursprung isolierend ist und links vollkommen leitend, b) Stromung in der oberen Halbebene, wenn ein endliches Stuck der Urenzfliiche vollkommen leitend ist und c) Stromung in einem flachen Kanal, wenn ein endliches Stuck der Grenzfliiche in jeder symrnetrisch zum Ursprung liegenden Ebene vollkommen leitend ist. I m Fall a ) wird eine exakte analytische Liisung abgeleitet, von der die Existenz einer Grenzschicht abgeleitet wird, die von parabolischer Gestalt ist und vom Unstetigkeitspunkt in den elektrischen Randbedingungen ausstrht. In den Fallen b) und c) wird das Problem analytisch reduziert auf eine Fredholmache Integralgleichung zweiter Art, die numerisch gelost wird. Eine Anzahl con Transformationen und giiltigen Approximationen wird benutzt, die Aufgabe der numerischen Berechnung zu vereinfachen. Geschwindigkeitsund lgtromlinien werden fur jedes Problem gezeichnet. Fur groJ3e Werte der Hartmannzahl wird die oben erwahnte parabolische Grenzschicht &gegrenzt. lIccnenyeTcH TeqeHHe B I I~K O~, Hecmmae~of , npononsiwet XHAKOCTH, Asumyrrletcfl B p e a y n b~a~e npu-JIOHteHHbIX ~J I~I E T~H Y~C K I I X TOKOB npu Hamwm nonepemoro MarHuTHoro ~O J I H . rpaamHax noBepxaocTb, nepneHauHynnpHan K MarHu...

SUMMARYWe investigate the magnetohydrodynamic flow (MHD) on the upper half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour. KEY WORDS MHD Flow Half-plane FORMULATION OF THE PROBLEMWe consider the steady flow of an incompressible fluid with uniform prroperties driven by the interaction of imposed electric currents and a uniform transverse magnetic field. Imposed currents enter the fluid at l = f a , through external circuits and move up on the plane. We assume that all the physical variables, including pressure, and the boundary conditions are functions of < and q only. The pressure gradient is zero. There is only one component of velocity and of magnetic field (in the z-direction). The equations describing such flows are the same as those of MHD duct flow problems when pressure gradient is taken as zero. Figure 1 shows the geometry of the problem.A uniform magnetic field of strength H , is directed along the axis of q. The wall is electrically insulated except for a length 2a, in the middle, where a perfectly conducting electrode is placed so that this part is conducting. Thus the partial differential equations describing the flow (in nondimensionalized form) are', VzV+M(aB/ay)=O,V 2 B + M ( a v/aq) = 0,

SUMMARYThe magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.

SUMMARYIn Sezgin'. the problems considered are the magnetohydrodynamic (MHD) flows in an electrodynamically conducting infinite channel and in a rectangular duct respectively, in the presence of an applied magnetic field. In the present paper we extend the solution procedure of these papers to two rectangular channels connected by a barrier which is partially conductor and partially insulator. The problem has been reduced to the solution of a pair of dual series equations and then to the solution of a Fredholm's integral equation of the second kind. The infinite series obtained were transformed to finite integrals containing Bessel functions of the second kind to avoid the computations of slowly converging infinite series and infinite integrals with oscillating integrands. The results obtained compared well with those of Butsenieks and Shcherbinin3 which were obtained for the perfectly conducting barrier separating the flows.

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The magnetohydrodynamic (MHD) flow of an electrically conducting fluid is studied in an array of identical parallel ducts stacked in the direction of external magnetic field and are separated by conducting walls of arbitrary thickness. Such arrangement of electromagnetically coupled ducts arises in fusion blanket applications in which a liquid metal is used both as coolant and tritium generation. The finite element method (FEM) with SUPG stabilisation is used for solving the set of coupled MHD equations. Numerical results show that, there is a significant effect of coupling the ducts with conducting walls of varying thickness, on the flow and induced current behaviours especially near the walls and for increasing values of Hartmann number. The results are presented for one, two and three coupled ducts in both coand counter-flow configurations which induce reversal and counter-current flows.