Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic (MHD) flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.     

Exact solutions for the incompressible viscous fluid of a porous rotating disk flow

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions with suction and injection through the surface included. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the exact velocity equations obtained. Making use of this solution, analytical formulas corresponding to the permeable wall shear stresses are extracted.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. As a result, exact formulas are obtained for the temperature field which take different forms depending on whether suction or injection is imposed on the wall. The impacts of several quantities are investigated on the resulting temperature field. In accordance with the Fourier‘s heat law, a constant heat transfer from the porous disk to the fluid takes place. Although the influence of dissipation varies, suction enhances the heat transfer rate as opposed to the injection.     

Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present investigation is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. For an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop taking into account of the rotational non-axisymmetric stationary conducting flow.Making use of the analytic solution, exact formulas for the angular velocity components as well as for the wall shear stresses are extracted. It is proved analytically that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation and the Joule heating. According to Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though decreases for small magnetic fields because of the dominance of Joule heating, it eventually increases for growing magnetic field parameters.     

Exact analytical solution of the MHD Jeffery-Hamel flow problem

This article shows that the well known nonlinear boundary value problem namely MHD Jeffery-Hamel flow problem, investigated in recent years by many numerical and semi-analytical approximative methods, is exactly solvable and furthermore, gives analytical exact solution in the implicit form for further physical interpretation.     

Series solutions of unsteady MHD flows above a rotating disk

The series solutions of unsteady flows of a viscous incompressible electrically conducting fluid caused by an impulsively rotating infinite disk are given by means of an analytic technique, namely the homotopy analysis method. Using a set of new similarity transformations, we transfer the Navier–Stokes equations into a pair of nonlinear partial differential equations. The convergent series solutions are obtained, which are uniformly valid for all dimensionless time 0 ≤ τ < ∞ in the whole spatial region 0 ≤ η < ∞. To the best of our knowledge, such kind of series solutions have never been reported. The effect of magnetic number on the velocity is investigated.     

RETRACTED ARTICLE: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present paper is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. In place of the traditional von Karman’s axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here. As a consequence, for an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop, which is influenced by a fixed point on the disk and also is bounded everywhere in the normal direction to the wall.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

MHD convective flow in a rotating fluid with Hall and ion-slip currents

Taking Hall and Ion-slip currents into account the steady magnetohydrodynamic convective flow of a partially ionized gas past an infinite vertical porous plate in a rotating frame of reference is investigated theoretically. A strong magnetic field of uniform strength is applied perpendicular to the plate and the fluid is subjected to a normal suction velocity while the heat flux at the plate is constant. An analytic solution of the problem is obtained and the effect of Hall and ion-slip currents, as well as the other parameter entering into the problem, are discussed and shown graphically.

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MHD Konvektionsströmung in einem rotierenden Fluid mit Hall- und Ionenschlupfströmen

Zusammenfassung Das Ziel dieser Arbeit ist die theoretische Untersuchung der stetigen magnet-hydrodynamischen Konvektionsströmung eines teilweise ionisierten Gases nach einer unendlichen vertikalen porösen Platte, die sich einem rotierenden Referenz-Rahmen befindet, unter Einfluß von Hall- und Ionenschlupfströmen. Ein gleichmäßig starkes Magnetfeld wurde senkrecht zur Platte errichtet, das Fluid wird senkrecht abgesaugt, während der Wärmestrom an der Platte konstant ist. Eine analytische Lösung des Problems wurde erhalten und der Einfluß von Hall- und Ionenschlupfströmen, wie auch die anderen Parameter, die dieses Problem betreffen, wurden diskutiert und graphisch dargestellt.

     

Exact solutions of a dipolar fluid flow

Exact solutions for three canonical flow problems of a dipolar fluid are obtained: (i) The flow of a dipolar fluid due to a suddenly accelerated plate, (ii) The flow generated by periodic oscillation of a plate, (iii) The flow due to plate oscillation in the presence of a transverse magnetic field. The solutions of some interesting flows caused by an arbitrary velocity of the plate and of certain special oscillations are also obtained.     

MHD rotating flow of a viscous fluid over a shrinking surface

This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.     

Approximate solutions to MHD Falkner-Skan flow over permeable wall

The magnetohydrodynamic (MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined. The approximate solutions and skin friction coefficients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method (DTM) with the Padé approximation called DTM-Padé. The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure. The approximate solutions are tabulated, plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique. It is found that the approximate solution agrees very well with the numerical solution, showing the reliability and validity of the present work. Moreover, the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.     

Hall effects on MHD flow in a rotating system with heat transfer characteristics

Closed-form solutions are derived for the steady magnetohydrodynamic (MHD) viscous flow in a parallel plate channel system with perfectly conducting walls in a rotating frame of reference, in the presence of Hall currents, heat transfer and a transverse uniform magnetic field. A mathematical analysis is described to evaluate the velocity, induced magnetic field and mass flow rate distributions, for a wide range of the governing parameters. Asymptotic behavior of the solution is analyzed for large M ² (Hartmann number squared) and K ² (rotation parameter). The heat transfer aspect is considered also with Joule and viscous heating effects present. Boundary layers arise close to the channel walls for large K ², i.e. strong rotation of the channel. For slowly rotating systems (small K ²), Hall current parameter (m) reduces primary mass flow rate (Q _x /R ρ v). Heat transfer rate at the upper plate (d θ/d η) _η=1 decreases, while at the lower plate (d θ/d η) _η=−1 increases, with increase in either K ² or m. For constant values of the rotation parameter, K ², heat transfer rate at both plates exhibits an oscillatory pattern with an increase in Hall current parameter, m. The response of the primary and secondary velocity components and also the primary and secondary induced magnetic field components to the control parameters is also studied graphically. Applications of the study arise in rotating MHD induction machine energy generators, planetary and solar plasma fluid dynamics systems, magnetic field control of materials processing systems, hybrid magnetic propulsion systems for space travel etc.     

Time-dependent MHD Couette flow of rotating fluid with Hall and ion-slip currents

The unsteady magnehydrodynamics (MHD) Couette flow of an electrically conducting fluid in a rotating system is investigated by taking the Hall and ion-slip currents into consideration.The derived fundamental equations on the assumption of a small magnetic Reynolds number are solved analytically with the well-known Laplace transform technique.The unified closed-form expressions are obtained for the velocity and the skin friction in the two different cases of the magnetic field being fixed to either the fluid or the moving plate.The effects of various parameters on the velocity and the skin friction are discussed by graphs.The results reveal that the primary and secondary velocities increase with the Hall current.An increase in the ion-slip parameter also leads to an increase in the primary velocity but a decrease in the secondary velocity.It is also shown that the combined effect of the rotation,Hall,and ion-slip parameters determines the contribution of the secondary motion in the fluid flow.     

Hall effects on the oscillatory MHD flow through porous medium in a rotating fluid

Taking Hall currents into account the unsteady magnetohydrodynamical flow through a porous medium bounded by an infinite vertical limiting surface in a rotating frame of reference is theoretically investigated when a strong magnetic field is imposed in a plane which makes an angle a with the normal to the plate. The influence of Hall currents on the velocity and temperature distribution are shown graphically for various values of .

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Hall-Effekte in oszillierender magnetohydrodynamischer Strömung durch ein poröses Medium in einer rotierenden Flüssigkeit

Zusammenfassung Bei der Betrachtung von Hallströmen wird die unstetige magnetohydrodynamische Strömung durch ein poröses Medium, begrenzt durch eine unendlich lange senkrechte Fläche in einem rotierenden Rahmen, für den Fall theoretisch untersucht, daß ein starkes magnetisches Feld in einer Ebene, die den Winkel a mit der normalen auf die Fläche einschließt, angelegt wird. Der Einfluß der Hallströme auf die Geschwindigkeit und die Temperaturverteilung wird für verschiedene Werte von graphisch dargestellt. Wärme- und Stofflibertragung 23 (1988)

     

Exact solutions of boundary-value problems for nonlinear flow in porous media

Exact solutions for flow problems in porous media with a limiting gradient in the case when the flow region in the hodograph plane is a half-strip with a longitudinal cut [1] are known only for two models of the resistance law [2–6]. The present study gives a one-parameter family of flow laws, and argues the possibility of effective determination of exact and approximate analytical solutions on the basis of successive reduction to boundary-value problems for the Laplace equation or for the equation studied in detail in [1]. It should be noted that the characteristics of the flow are determined without additional quadratures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 107–112, May–June, 1985.     

Hodograph transformation in steady plane rotating MHD flows

A hodograph transformation is employed to obtain a partial differential equation of second order which is exploited to obtain solutions for plane rotating viscous incompressible flows with orthogonal magnetic and velocity fields. Lastly radial and vortex flows are discussed.     

Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet

The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting： method. It is found that the dual solutions of the flow exist for cer- tain values of tile velocity ratio parameter. The special case of the first branch solutions （the classical Newtonian fluid model） is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.