Hydromagnetic flow near an oscillating porous flat plate

A solution in the closed form has been obtained for the flow of a viscous incompressible fluid of small electrical conductivity near an infinite insulated porous flat plate oscillating harmonically in its own plane in the presence of a transverse magnetic field of uniform strength fixed relative to the fluid. Small uniform suction has been imposed along the surface of the plate. This is a generalisation of the result of Ong and Nicholls for the hydromagnetic flow near an oscillating solid flat plate.     

Unsteady hydromagnetic rotating flow of a conducting second grade fluid

The purpose of this work is to investigate the hydromagnetic oscillatory flow of a fluid bounded by a porous plate, when the entire system rotates about an axis normal to the plate. The fluid is assumed to be non-Newtonian (second grade), incompressible and electrically conducting. The magnetic field is applied transversely to the direction of the flow. Such a flow model has great significance not only of its theoretical interest, but also for applications to geophysics and engineering. The resulting initial value problem has been solved analytically for steady and unsteady cases. The analysis of the obtained results showed that the flow field is appreciably influenced by the material parameter of the second grade fluid, the applied magnetic field, the imposed frequency, rotation and suction and blowing parameters. It is observed in a second grade fluid that a steady asymptotic hydromagnetic solution exists for blowing and resonance which is different from the hydrodynamic situation.     

On the energetic balance for the flow of an Oldroyd-B fluid induced by a constantly accelerating plate

Exact and approximate expressions are established for dissipation, the power due to the shear stress at the wall and the boundary layer thickness corresponding to the motion of an Oldroyd-B fluid induced by a constantly accelerating plate. The similar expressions for Maxwell, Newtonian and second grade fluids, performing the same motion, are obtained as limiting cases of our general results. The specific features of the four models are emphasized by means of the asymptotic approximations.     

Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate

The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

On the non-orthogonal Stagnation flow of the Oldroyd-B fluid

This paper is concerned with a non-orthogonal stagnation flow of an Oldroyd-B fluid between two parallel plates. We reduce the problem to a set of ordinary differential equations (ODE's), which is then solved with finite differences using a parameter continuation method. Perturbation analyses are also carried out for small Reynolds numbers and small Weissenberg numbers respectively. The solution of the set of ODE's is discussed. It is known that for a Newtonian fluid, the stagnation point shifts from the potential flow case in the opposite direction of the tangential velocity. The effect of the fluid elasticity is to reduce this shift. It is also shown that the Oldroyd-B model has a limiting Weissenbeg number, depending on the angle of the injected flow.     

Starting solutions for oscillating motions of an Oldroyd-B fluid over a plane wall

In this paper, we establish the starting solutions for oscillating motions of an Oldroyd-B fluid between two side walls perpendicular to a plane wall. The expressions for the velocity field and the associated tangential stress at the bottom wall are obtained, presented under integral and series form. These satisfy all imposed initial and boundary conditions. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. In the absence of side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Moreover, the obtained solutions can be specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motions.     

A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid

The velocity field and the adequate tangential stress that is induced by the flow due to a constantly accelerating plate in an Oldroyd-B fluid, are determined by means of Fourier sine transforms. The solutions corresponding to a Maxwell, Second grade and Navier–Stokes fluid appear as limiting cases of the solutions obtained here. However, in marked contrast to the solution for a Navier–Stokes fluid, in the case of an Oldroyd-B fluid oscillations are set up which decay exponentially with time.     

Unsteady hydromagnetic motion of a rotating conducting fluid

The combined effects of stratification and magnetic field on the unsteady motion of a viscous, electrically conducting fluid between two rotating disks are analysed. Solutions are obtained for the linearized equations under Boussinesq approximation and steady state solutions are deduced from them. The results are compared with those obtained by Loper and Benton and Balanet al. Graphs are presented for the steady state velocity, magnetic field and temperature distributions.     

Non-isothermal spherical Couette flow of Oldroyd-B fluid

The present paper is concerned with non-isothermal spherical Couette flow of Oldroyd-B fluid in the annular region between two concentric spheres. The inner sphere rotates with a constant angular velocity while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum and energy equations. An approximate analytical solution is obtained through the expansion of the dynamical variable fields in power series of Nahme number. Non-homogeneous, harmonic for axial- velocity and temperature equations and biharmonic for stream function equations, have been solved up to second order approximation. In comparison of the present work with isothermal case; [1,2], two additional terms; a first order velocity and a second order stream function are stem as a result of the interaction between the fluid viscoelasticity and temperature profile. These contributions prove to be the most important results for rheology in this work.     

Non-isothermal spherical Couette flow of Oldroyd-B fluid

The present paper is concerned with non-isothermal spherical Couette flow of Oldroyd-B fluid in the annular region between two concentric spheres. The inner sphere rotates with a constant angular velocity while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum and energy equations. An approximate analytical solution is obtained through the expansion of the dynamical variable fields in power series of Nahme number. Non-homogeneous, harmonic for axial- velocity and temperature equations and biharmonic for stream function equations, have been solved up to second order approximation. In comparison of the present work with isothermal case; [1,2], two additional terms; a first order velocity and a second order stream function are stem as a result of the interaction between the fluid viscoelasticity and temperature profile. These contributions prove to be the most important results for rheology in this work.     

The slow flow of a rotating fluid past a split plate

Summary The flow of a rotating fluid past a split plate is examined in the limit of vanishing Rossby number to determine the nature of the StewartsonE ^1/4-layers in the region of the gap between the plates. When the gap width is order unity it is established that the end regions have a radius ofO(E ^1/4) and that they do not interact. However, when the gap width isO(E ^1/4) there is an interaction between the two end regions, and the solution is calculated numerically for this case.

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Zusammenfassung Es wird eine Drehströmung mit der Rossby-Zahl Null entlang zwei parallelen Platten untersucht, wobei die Spaltbreite die Größenordnung Eins hat. Es erhebt sich die Frage, ob in der Region zwischen den Platten eine StewartsonscheE ^1/4-Schicht existieren kann. Es wird festgestellt, daß so eine Schicht nicht vorhanden sein kann; wenn jedoch die SpaltbreiteO(E ^1/4) ist, dann wird der Spalt durch eine Stewartson-Schicht ausgefüllt. Die Lösung wird für diesen Fall berechnet, und der Effekt der Zähigkeit auf die Strömung zwischen den Platten wird gezeigt.

     

On accelerated flows of an Oldroyd-B fluid in a porous medium

In this work, the problems dealing with unsteady unidirectional flows of an Oldroyd-B fluid in a porous medium are investigated. By using modified Darcy's law of an Oldroyd-B fluid, the equations governing the flow are modelled. Employing Fourier sine transform, the analytic solutions of the modelled equations are developed for the following two problems: (i) constant accelerated flow, (ii) variable accelerated flow. Explicit expressions for the velocity field and adequate tangential stress are obtained in each case. The solutions for Newtonian, second grade and Maxwell fluids in a porous medium appear as the limiting cases of the present analysis.     

Squeeze-film flow of an Oldroyd-B fluid: Similarity solution and limiting Weissenberg number

Summary In this paper we consider the motion of an Oldroyd-B fluid being squeezed between two parallel disks of infinite extent. We show that a similarity solution exists provided that the fluid inertia is neglected and that the squeezing velocity varies exponentially with time. We prove that there is a critical Weissenberg number above which at least one component of the stresses grows unboundedly with time. This critical Weissenberg number is of the order of 0.67. This latter conclusion is also valid in the continuous squeeze-film mode.

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Zusammenfassung In diesem Artikel betrachten wir die Bewegung einer Oldroyd-B Flüssigkeit welche zwischen zwei parallele Scheiben von unendlicher Dimension gequetscht wird. Es wird gezeigt, daß eine Similaritätslösung existiert, vorausgesetzt, daß die Flüssigkeitsträgheit vernachlässigt wird und die Quetschgeschwindigkeit sich exponentiell mit der Zeit verändert. Der Beweis wird erbracht, daß eine kritische Weissenbergzahl existiert, oberhalb welcher wenigstens eine Komponente der Spannungen mit der Zeit anwächst. Diese kritische Weissenbergzahl ist in der Größenordnung von 0.67. Dasselbe Resultat gilt für den kontinuierlichen Quetsch-Film.

     

Numerical investigation of laminar fluid flow in an oscillating sphere

The paper deals with the numerical simulation of three dimensional oscillating fluid flow of filled and partial filled hollow spheres. We use a program based on finite volume method developed by the DLR. Our main interest is on the secondary flow in the middle plane. Results are presented for Newtonian fluids in dependence of different boundary conditions which are compared with analytical and experimental investigations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the flow of an electrically conducting,incompressible fluid near an accelerated plate in the presence of a parallel plate,under transverse magnetic field

This paper deals with hydromagnetic flow of an electrically conducting, incompressible viscous fluid near an accelerated flat, non-conducting plate, in the presence of another parallel plate, when there is a transversely applied magnetic field. Induced magnetic field is neglected in comparison with the applied magnetic field. Laplace transform techniques are used. The equations are integrated by applying residue principle, and expressions for velocity profiles and skin-friction at both plates are derived for different values of Hartmann number M. It is observed that, with the increase of the value of the Hartmann number M, the velocity profiles are flattened, the shear stress at the stationary plate decreases, as the value of the time T and Hartmann number M increases, but the shear stress at the accelerated plate increases directly in proportion with the increase in time and Hartmann number.     

Hydromagnetic flow near an accelerated plate with suction

The two-dimensional unsteady flow of a conducting viscous incompressible fluid past, an infinite flat plate with uniform suction, is considered in the presence of a uniform magnetic field. For a constant time, it is shown that for a given Hartmann numbera, as the cross Reynolds number β (corresponding to the suction velocity of the plate) increases, the velocity at any point of the fluid decreases and the skin friction at the plate increases. The results also hold good for a given β, asa increases if the magnetic lines of force are fixed relative to the fluid and are just opposite for the magnetic lines of force fixed relative to the plate.