MHD Flow of an Oldroyd-B fluid between eccentric rotating disks

An exact solution is obtained for the magnetohydrodynamics (MHD) flow of a conducting, incompressible Oldroyd-B fluid between two infinite, parallel, insulated disks rotating about non-coincident axes normal to the disks in the presence of a uniform transverse magnetic field. The effects of the Hartmann number M, the Deborah number D, the Reynolds number R and the elastic number on the velocity field and the force are discussed. It is found that the value of the torque exerted by the fluid on one of the disks is zero.     

Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid

The influence of Hall currents and rotation on the oscillatory flows of an infinite plate is investigated. Exact solutions for the two problems are obtained.The fluid considered is a homogeneous Oldroyd-B. During the mathematical analysis it is found that governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of viscous fluid. Further, it is observed that in absence of the strength of transverse magnetic field (B₀) the solution in resonance case does not satisfy the boundary condition at infinity. Physical significance of mathematical results is also discussed.     

Steady circulatory flow of a second grade fluid about a rotating porous cylinder

An exact solution for the circulatory flow of an incompressible second grade fluid about a rotating porous cylinder is given. The solution is expressed in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number, the elastic number and the ratio of the circulation at infinity to that on the surface of the cylinder. The velocity, the vorticity and the torque exerted by the fluid on the cylinder are calculated. It is shown that there are some discrepancies between the results obtained by the exact solution and those obtained by the perturbation solution which is valid for small values of the elastic number.     

Exact flow of a third grade fluid past a porous plate using homotopy analysis method

The flow of a third grade fluid past a porous plate is considered. An exact analytical solution of the governing non-linear differential equation is constructed using homotopy analysis method. It is observed that the relevant perturbation solution corresponds to a special case of the presented solution.     

Unsteady hydromagnetic couette flow in a rotating system

Unsteady hydromagnetic flow of an electrically conducting viscous incompressible fluid in a rotating system under the influence of a uniform transverse magnetic field is investigated when one of the plates is set into motion with the time dependent velocity $\text{U(t)}$ in its own plane. Two cases of interest, namely, impulsive start as well as accelerated start of the moving plate are discussed. The asymptotic behaviour of the solution is also analysed for both small and large time to highlight the transient approach to the final steady state and effects of rotation parameter as well as Hartmann number. The shear stresses at the moving plate due to the primary and secondary flows are derived in both cases. It is found that the shear stress components due to the primary flow decrease, whereas that due to the secondary flow increase with the increase in rotation parameter.     

Hydromagnetic flow past a rotating porous plate in a conducting fluid rotating about a noncoincident parallel axis

Summary The hydromagnetic flow of an incompressible viscous electrically conducting fluid past a porous plate is investigated when the plate rotates with a uniform angular velocity about an axis normal to the plate and the fluid at infinity rotates with the same angular velocity about a non-coincident parallel axis. It is shown that in the presence of a uniform magnetic field parallel to the axis of rotation the boundary layer thickness decreases with an increase in either the suction at the plate or the magnetic parameter M. In the presence of suction at the plate, the velocity component u in the direction normal to the plane containing the axis of rotation of the plate and that of the fluid increases with an increase in M, while the velocity component v in the transverse direction parallel to the plane of the plate decreases with an increase in M. For a fixed value of M, at a given location u increases with an increase in the suction parameter S while v decreases with increasing S. For a fixed value of M, at a given location both u and v decrease with an increase in the blowing parameter S₁. Further, for a fixed value of S₁, at a given position u increases with an increase in M but v decreases with increasing M. It is shown that no torque is exerted by the fluid on the plate, and non-coaxial rotations of the plate and the fluid at infinity have no influence on the torque.The solution of the heat transfer equation reveals that for given values of the suction parameter S, Prandtl number P and Eckert number E, the temperature at a given point in the flow increases with increasing M. On the other hand, for fixed values of M, P and E, the temperature at a given point decreases with increasing S. No steady distribution of temperature exists when there is blowing at the plate.     

Stability of ferrofluid flow between concentric rotating cylinders with an axial magnetic field

A linear stability analysis for the ferrofluid flow between two concentric rotating cylinders in the presence of an axial magnetic field is implemented in this study. Both of the wide-gap and small-gap cases are considered and the governing equations with respect to three-dimensional disturbances including axisymmetric and non-axisymmetric modes are derived and solved by a direct numerical procedure. A parametric study covering wide ranges of ?, the volume fraction of colloidal particles; ξ, the strength of axial magnetic field; μ, the ratio of angular velocity of the outer cylinder to that of the inner cylinder; and ε, the ratio of radius of the inner cylinder to that of the outer cylinder, is conducted. Results show that the stability characteristics depend heavily on these factors. It is found that the increases of ? and ξ, and decrease of ε tend to stabilize the basic flow for an assigned value of μ. The variations of the onset mode with these parameters are discussed in detail. An example for the practical application of present results is given to help the understanding of stability behaviour of this flow.     

Numerical study of plane couette flow in a rotating framework

Summary A numerical study of laminar plane Couette flow subjected to a steady spanwise rotation is conducted. The full nonlinear Navier-Stokes equations in a steadily rotating framework are solved by a finite difference method for a long, large-aspect-ratio rectangular channel where the outer wall moves at a constant velocity. In this manner, nonlinear and wall end effects which are present in any real laboratory Couette flow experiment are taken into account. The computations demonstrate the existence of a roll instability, at intermediate rotation rates, when the Reynolds number exceeds a critical value of 42. The associated secondary flow in the form of longitudinal rolls is shown to have a severe distortional effect on the primary axial velocity in the interior of the channel which is not linear like its counterpart in an inertial framework. Comparisons are made with previously conducted linear stability analyses as well as with other analogous numerical and experimental studies.With 15 Figures     

Taylor stability of a thermoviscoelastic fluid in couette flow

Taylor stability in Couette flow of a thermoviscoelastic fluid between two rotating coaxial circular cylinders is investigated employing the constitutive theory of Eringen and Koh. The stability problem is reduced to a characteristic value problem which is solved to obtain stability criteria in terms of critical Values of Taylor numbers. The results are compared with those of the classical viscous fluids and viscoinelastic fluids and the stabilizing effects of the thermoviscoelastic nature of the fluid are discussed.     

Hydromagnetic stability of plane Poiseuille flow of an Oldroyd fluid

Summary The linear stability of plane Poiseuille flow of a conducting Oldroyd liquid in the presence of a transverse magnetic field is studied. The fourth-order Orr-Sommerfeld equation governing the stability analysis in this case is solved numerically by a spectral method.     

Decay of a potential vortex in an Oldroyd-B fluid

Analytical expressions for the velocity field and the associated tangential tension corresponding to a potential vortex in an Oldroyd-B fluid are determined by means of the Hankel transform. The well-known solutions for a Navier–Stokes fluid as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, some comparative diagrams are presented for the circular motion of the glycerine. In each case, the velocity fields as well as the adequate tangential tensions are going to zero for t or r → ∞. Consequently, the potential vortex is damping in time and space.     

Heat transfer in generalized couette flow of a nonlinear viscoplastic fluid

The steady-state heat-transfer problem is solved for dissipative pressure flow of a nonlinear viscoplastic fluid between two parallel isothermal plates, one of which is moving at a constant velocity while the other is stationary.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 6, pp. 1070–1080, June, 1978.     

Some properties of models for generalized Oldroyd-B fluids

A class of generalized Oldroyd-B fluids is studied with reference to properties that are relevant in obtaining energy estimates. The generalization takes the form of a viscosity that depends on shear rate, with shear-thinning behaviour being of particular interest. Conformation stress tensors for the generalized problem are defined, and the conditions for positive-definiteness of the stress, and for its determinant to exceed unity, are obtained. It is shown that these properties, established for standard Oldroyd-B fluids, carry over provided that an expression depending on the Weissenberg number, a relaxation time, and the rate of change of viscosity, is positive. Various alternative energy estimates are obtained. The consequences of the positivity constraint are investigated for a specific viscosity function due to Yeleswarapu et al. [K.K. Yeleswarapu, M.V. Kameneva, K.R. Rajagopal, J.F. Antaki, The flow of blood in tubes: theory and experiment, Mechanics Research Communications 25(3) (1998) 257-262], first in general and then for a set of special flows.     

Stability of the circular couette flow of a power-law fluid

The stability of the circular Couette flow of a non-Newtonian power-law fluid is analyzed in the small-gap approximation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 4, pp. 719–724, April, 1973.     

On the comparison of the methods used for the solutions of the governing equation for unsteady unidirectional flows of second grade fluids

In order to solve the governing equation for unsteady unidirectional flows of second grade fluids, the use of the Laplace and the Fourier transform methods are discussed. Three characteristic examples which are unsteady flow between two parallel plates, unsteady pipe flow and unsteady flow over a plane wall are considered. It is found that the solution for unsteady flow in bounded regions obtained by the Laplace and the Fourier transform methods are exactly the same as the case of the unsteady flow of a Newtonian fluid. It is shown that the Laplace transform method for small values of time is useful for flows of Newtonian fluids but it is not convenient for flows in unbounded regions of second grade fluids. Furthermore, it is explained that for some unsteady flows of second grade fluids, the solution obtained by using the Laplace transform does not satisfy the initial condition and therefore the Fourier transform method is used.     

Nonisothermal generalized couette flow of a fluid with a power-law rheological characteristic

The nonisothermal steady flow of a power-law fluid between two parallel plates is analyzed for different kinds of temperature boundary conditions and, moreover, with energy dissipation taken into account. It is assumed that the fluidity of the substance is a linear function of the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.22, No. 5, pp. 872–880, May, 1972.     

Radial stagnation flow on a rotating circular cylinder with uniform transpiration

Radial stagnation flow of strain rate k impinging on a cylinder with uniform transpiration U₀ and rotating at constant angular velocity is investigated. An exact reduction of the Navier-Stokes equations to a primary nonlinear equation for the meridional flow similar to that found by Wang [1] and a secondary linear equation for the azimuthal flow is obtained. The governing parameters are the stagnation-flow Reynolds number R = ka²/2v, the dimensionless transpiration S = U⁰/ka, and the dimensionless rotation rate = /k, where a is the cylinder radius and v is the kinematic viscosity of the fluid. The boundary-value problem is solved by numerical integration and by asymptotic analysis in certain limits. The results are succinctly summarized in plots of the axial and azimuthal shear-stress parameters as functions of R and S. Sample velocity profiles, meridional streamfunction plots, and projections of particle paths for both suction and blowing are given. An interesting double-layer structure in the azimuthal velocity profile, consisting of a removed free shear layer connected to a wall boundary layer, is observed at large values of blowing. This feature is consistent with results obtained from the asymptotic analysis.     

Hydromagnetic stagnation point flow by a perturbation technique

An analysis is performed to study the momentum, heat and mass transfer characteristics of MHD natural convection flow and heat generating/absorbing fluid at the stagnation point of an isothermal two-dimensional porous body immersed in a fluid saturated porous medium. The results are obtained by solving the coupled non-linear partial differential equations describing the conservation of mass, momentum and energy by a perturbation technique [A. Aziz, T.Y. Na, Perturbation Methods in Heat Transfer, Springer-Verlag, Berlin, 1984 (pp. 1–184), R. Kenneth Cramer, Shih-I Pai, Magnetofluid Dynamics for Engineers and Applied Physicists, McGraw-Hill Book Company, New York, 1973 (pp. 164–171).]. These results are presented to illustrate the influence of the Hartmann number, Prandtl number, and dimensionless heat generation/absorption coefficient and suction injection parameter. Numerical results for the dimensionless velocity profiles, the temperature profiles, the local friction coefficient and the local Nusselt number are presented for various parameters. These effects of the different parameters on the velocity and temperature as well as the skin friction and wall heat transfer are presented graphically.     

Unsteady flow from a source in a rotating fluid

Summary A source, which is situated on the axis of a rotating fluid, commences to expel fluid with constant rate at the time t=0. We describe how the geostrophic forces lead to the formation of a narrow column along the axis, before the eventual development of the viscous Stewartson column along the axis, and how the final steady state is achieved. An understanding of the role of the non-linear inertial forces in the neighbourhood of the source is given. The results are also extended by considering the effect of placing the source between two infinite discs situated perpendicular to the axis of rotation.