On MHD flow of an incompressible viscous fluid

In this paper, we apply Homotopy Perturbation Method (HPM) to find the analytical solutions of nonlinear MHD flow of an incompressible viscous fluid through convergent or divergent channels in presence of a high magnetic field. The flow of an incompressible electrically conducting viscous fluid in convergent or divergent channels under the influence of an externally applied homogeneous magnetic field is studied both analytically and numerically. The graphs are presented to reveal the physical characteristics of flow by changing angles of the channel, Hartmann and Reynolds numbers.     

Mixed boundary value problems for steady-state magnetohydrodynamic equations of viscous incompressible fluid

The inhomogeneous boundary value problem for the steady-state magnetohydrodynamic equations of viscous incompressible fluid under the Dirichlet conditions for the velocity and mixed boundary conditions for the electromagnetic field is considered. Sufficient conditions for the data that ensure the global solvability of this problem and the local uniqueness of its solution are found.     

Steady plane-parallel flow of an anomalously viscous fluid between parallel plates

The authors consider the plane-parallel flow of an anomalously viscous fluid between two parallel plates, one of which is fixed while the other moves at constant velocity. An empirical power law is used as the rheological equation of state. It is shown that in the presence of a counterpressure exceeding a certain value, which is a function of the rheological properties of the polymer, the geometry and the flow regime, a counterflow develops near the bottom of the channel. The existence of a counterflow leads to the appearance of a minimum on the V_X(y) curve. This additional boundary condition makes it possible to obtain an approximate solution of the differential equations of motion and to describe the velocity distribution in the flow by means of two exponential equations. Integration of the velocity distribution thus obtained over the height of the channel gives an equation for the volume flow rate in plane-parallel flow in the presence of a counterpressure. It is shown that if the counterpressure gradient exceeds a critical value, the volume flow rate decreases much more sharply than for a Newtonian fluid.Mekhanika Polimerov, Vol. 1, No. 6, pp. 138–145, 1965     

Steady two-dimensional magnetohydrodynamic flow

Résumé Pour illustrer les effets d'un champ magnétique sur un écoulement plant stationnaire d'un fluide non visqueux idéal, parfaitement conducteur de l'électricité, on envisage ici les cas particuliers où le champ magnétique a une composante différente de zéro. Si on considère les régimes d'onde simple où le champ magnétique est en chaque point normal au plan de l'écoulement, qualitativement tout se passe en gross comme dans le cas de la dynamique des gas classique. Enfin, on utilise l'analyse linéaire pour étudier les faibles perturbations non-isentropiques d'un écoulement initialement uniforme.

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This research was supported by a National Aeronautics and Space Administration general research grant to the University of Wisconsin.     

Steady axi-symmetric magnetohydrodynamic flow

Résumé Pour illustrer les effets d'un champ magnétique sur un écoulement de révolution stationnaire d'un fluide non visqueux idéal, parfaitement coducteur de l'électricité, on envisage ici les cas particuliers où le champ magnétique a une composante différente de zéro. On utilise l'analyse linéaire pour étudier les faibles perturbations non-isentropiques d'un écoulement initialement uniforme.     

Steady flow of incompressible fluid between two co-rotating disks

This article provides an analytical solution of the Navier–Stokes equations for the case of the steady flow of an incompressible fluid between two uniformly co-rotating disks. The solution is derived from the asymptotical evolution of unknown components of velocity and pressure in a radial direction – in contrast to the Briter–Pohlhausen analytical solution, which is supported by simplified Navier–Stokes equations. The obtained infinite system of ordinary differential equations forms recurrent relations from which unknown functions can be calculated successively. The first and second approximations of solution are solved analytically and the third and fourth approximations of solutions are solved numerically. The numerical example demonstrates agreements with results obtained by other authors using different methods.     

On the optimal control of viscous incompressible fluid flow

The framework of the Navier-Stokes (N-S) equations is used to study flow past an arbitrary body on whose surface the tangential or normal velocity is under control. The necessary conditions are obtained for the minimum rate of energy dissipation. Exact analytical solutions of the corresponding problems are found for the case of flow past an ellipsoid in the Stokes approximation.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

Steady two-dimensional magnetohydrodynamic flow II

Résumé Pour illustrer les effets d'un champ magnétique sur un écoulement plan stationnaire d'un fluide non visqueux idéal, parfaitement conducteur de l'électricité, on envisage ici les cas particuliers où le champ magnétique a une composante différente de zéro. Si l'on considère les régimes d'onde simple où le champ magnétique est en chaque point normal au plan de l'écoulement, qualitativement tout se passe en gros comme dans le cas de la dynamique des gaz classique. On utilise l'analyse linéaire pour étudier les faibles perturbations non-isentropiques d'un écoulement initialement uniforme.

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This research was supported by a National Aeronautics and Space Administration general research grant to the University of Wisconsin.     

On the control of layered flow of a viscous incompressible fluid within MHD

The control of a viscous incompressible flow within MHD is considered. The phenomenon of internal heat release is explored as applied to a layered plane flow in which the kinetic energy dissipates strictly into heat. The influence exerted by the electromagnetic field of the flowing fluid on the surrounding medium is examined, and methods for flow control in the context of digital deposit field technology are considered.     

On the flow of a viscous incompressible fluid between two coaxial rotating porous cylinders

Exact solution of the Navier-Stokes equations for the laminar flow of a viscous incompressible fluid between two coaxial rotating porous cylinders, kept at constant temperatures, has been studied. The rate of injection at one cylinder is taken to be the same as the rate of suction at the other. Expressions for the velocity and temperature distributions and for the torque required to turn the outer cylinder are obtained. The effects of λ (injection parameter), σ (the ratio of the radii of the cylinders) and Pé (Péclet number = λPr) on them are shown graphically.     

An improved nonlinear Reynolds equation for a thin flow of a viscous incompressible fluid

A scalar nonlinear second-order equation describing a fluid flow between two closely spaced rigid walls is constructed on the basis of an asymptotic analysis of the Navier-Stokes equations. This equation takes into account the convective term and the surface curvature; it improves the Reynolds equation and provides a two-term asymptotics for the solution to the initial problem. Integral and pointwise error estimates are obtained. It is demonstrated that the one-dimensional model cannot be improved further.     

High-order approximations for an incompressible viscous flow on a rough boundary

In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flow domain is denoted by ?, being a small number. We impose periodic boundary conditions on the flow. We generalize the boundary layers needed for the construction of flow approximations of higher order with respect to ?. The existence of the layers and their features are discussed. Finally we give the error estimates for the approximations and establish an explicit wall law.     

Numerical modelling of aeroelastic behaviour of an airfoil in viscous incompressible flow

In this paper, the problem of the numerical approximation of a two-dimensional incompressible viscous fluid flow interacting with a flexible structure is considered. Due to high Reynolds numbers in the range 10⁴ − 10⁶ the turbulent character of the flow is considered and modelled with the aid of Reynolds equations coupled with the k − ω turbulence model. The structure motion is described by a system of ordinary differential equations for three degrees of freedom: vertical displacement, rotation and rotation of the aileron. The problem is discretized in space by the Galerkin Least-Squares stabilized finite element method and the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian method.     

Optimal control of a magnetohydrodynamic viscous heat-conducting gas flow

The control problem for a one-dimensional flow of a polytropic viscous heat-conducting perfect gas through an interval is considered. The density of external currents is taken as the control. The existence of an optimal control function is proved. Necessary optimality conditions are derived. The compactness of the set of solutions is established.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

Efficient computation of non-isothermal highly viscous incompressible flow

We consider the numerical solution of the non–isothermal incompressible Navier–Stokes equations using a discrete projection method. The computation of velocity and temperature subproblems is carried out on different meshes chosen with respect to the physical behavior of these quantities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)