Effects of Hall current and slip condition on unsteady flow of a viscous fluid due to non-coaxial rotation of a porous disk and a fluid at infinity

The unsteady hydromagnetic flow due to non-coaxial rotations of a porous disk with slip condition and a fluid at infinity has been studied on taking Hall currents into account. An exact solution of the governing equation has been obtained by the Laplace transform technique. Asymptotic solution is obtained for large time. It is found that for large time there exists a thin boundary layer near the disk. The thickness of this layer decreases with increase in either suction or magnetic parameter.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

Hall effects on the unsteady hydromagnetic oscillatory flow of a second-grade fluid

An exact solution of an oscillatory flow is constructed in a rotating fluid under the influence of an uniform transverse magnetic field. The fluid is considered as second-grade (non-Newtonian). The influence of Hall currents and material parameters of the second-grade fluid is investigated. The hydromagnetic flow is generated in the uniformly rotating fluid bounded between two rigid non-conducting parallel plates by small amplitude oscillations of the upper plate. The exact solutions of the steady and unsteady velocity fields are constructed. It is found that the steady solution depends on the Hall parameter but is independent of the material parameter of the fluid. The unsteady part of the solution depends upon both (Hall and material) parameters. Attention is focused upon the physical nature of the solution, and the structure of the various kinds of boundary layers is examined. Several results of physical interest have been deduced in limiting cases.     

Three-dimensional motion of a viscous incompressible fluid in a narrow tube

An approximate solution of the problem of unsteady motion of a viscous incompressible fluid in a long narrow deformable tube at low Reynolds numbers is obtained. Pressure oscillations and tube deformation are shown to be related by an integrodifferential equation. The solution obtained extends the Poiseuille solution in elliptic tubes to the case of comparatively arbitrary small deformations in terms of the tube length and angle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 28–32, July–August, 2009.     

Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number.     

Oscillatory flow of second grade fluid in cylindrical tube

The unsteady oscillatory flow of an incompressible second grade fluid in a cylindrical tube with large wall suction is studied analytically. Flow in the tube is due to uniform suction at the permeable walls, and the oscillations in the velocity field are due to small amplitude time harmonic pressure waves. The physical quantities of interest are the velocity field, the amplitude of oscillation, and the penetration depth of the oscillatory wave. The analytical solution of the governing boundary value problem is obtained, and the effects of second grade fluid parameters are analyzed and discussed.     

Flow on oscillating rectangular duct for Maxwell fluid

This paper presents an analysis for the unsteady flow of an incompressible Maxwell fluid in an oscillating rectangular cross section.By using the Fourier and Laplace transforms as mathematical tools,the solutions are presented as a sum of the steady-state and transient solutions.For large time,when the transients disappear,the solution is represented by the steady-state solution.The solutions for the Newtonian fluids appear as limiting cases of the solutions obtained here.In the absence of the frequency of oscillations,we obtain the problem for the flow of the Maxwell fluid in a duct of a rectangular cross-section moving parallel to its length.Finally,the required time to reach the steady-state for sine oscillations of the rectangular duct is obtained by graphical illustrations for different parameters.Moreover,the graphs are sketched for the velocity.     

Unsteady boundary layer flow due to a stretching surface in a rotating fluid

The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution.     

Pulsatile electroosmotic flow of a Maxwell fluid in a parallel flat plate microchannel with asymmetric zeta potentials

The pulsatile electroosmotic flow (PEOF) of a Maxwell fluid in a parallel flat plate microchannel with asymmetric wall zeta potentials is theoretically analyzed. By combining the linear Maxwell viscoelastic model, the Cauchy equation, and the electric field solution obtained from the linearized Poisson-Boltzmann equation, a hyperbolic partial differential equation is obtained to derive the flow field. The PEOF is controlled by the angular Reynolds number, the ratio of the zeta potentials of the microchannel walls, the electrokinetic parameter, and the elasticity number. The main results obtained from this analysis show strong oscillations in the velocity profiles when the values of the elasticity number and the angular Reynolds number increase due to the competition among the elastic, viscous, inertial, and electric forces in the flow.     

Determination of the hydrodynamic characteristics of a partially filled cavity containing a pendulum

Equations are given for the small oscillations of a viscous liquid-filled cavity containing a pendulum and the dissipative and inertial coefficients are determined on the basis of a systematic application of the finite element method (FEM). In order to improve the accuracy of the values obtained for these coefficients the use of a nonlinear Shanks transformation is proposed. This makes it possible to achieve the required accuracy using much less machine time and memory. The properties of the inertial hydrodynamic characteristics associated with an ideal fluid are studied in relation to cavities lacking a pendulum but fitted with annular ribs. It is shown that as a result of the presence of these rings the estimate given in [6] for the generalized mass of the fundamental mode of the fluid oscillations is inaccurate and must be modified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 91–100, November–December, 1987.     

Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

Tangential oscillations of a differentially rotating spherical fluid layer

The natural harmonic oscillations of a differentially rotating fluid layer under the action of a potential force are considered. The rise in the layer level is assumed to be negligible. The oscillations satisfy an ordinary second-order differential equation with singular coefficients that depend on the spatial coordinate. This equation is solved by the method of local separation of singularities based on the use of the properties of the Fuchs series for a bounded solution. Various laws for the latitude dependence of the angular rate of ocean rotation and the effect of these laws on the problem spectrum are considered. An equation is obtained for the streamlines of the oscillations investigated. Two cases in which the latitude dependence of the base flow velocity coincides with the real dependence for a celestial body are considered and the corresponding modes are found.     

Impulsive motion of a non-Newtonian fluid between two oscillating parallel plates

An exact solution for the flow of an incompressible viscoelastic fluid between two infinitely extended parallel plates, due to the harmonic oscillations of the upper plate and the impulsively started harmonic oscillations of the lower plate from rest, in the respective planes of the plates, has been obtained. The momentum transfer towards the central region and the skin friction of the lower plate are found to be greater for the viscoelastic fluid than that for viscous fluid. The effect of out-of-phase oscillations of the plates with different amplitudes on the flow characteristics has also been investigated.     

The influence of Hall current on the rotating oscillating flows of an Oldroyd-B fluid in a porous medium

In this article, analysis is presented to study the effect of Hall current on the rotating flow of a non-Newtonian fluid in a porous medium taking into consideration the modified Darcy's law. The Oldroyd-B fluid model is used to characterize the non-Newtonian fluid behavior. The governing equations for unsteady rotating flow have been modeled in a porous medium. The analysis includes the flows induced by general periodic oscillations and elliptic harmonic oscillations of a plate. The effect of the various emerging parameters is discussed on the velocity distribution. The analytical results are confirmed mathematically by giving comparison with previous studies in the literature. It is observed that the velocity distribution increases with an increase of Hall parameter. The behavior of permeability is similar to that of the Hall parameter.     

Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution

This paper studies the propagation of a plane-strain fluid-driven fracture with a fluid lag in an elastic solid. The fracture is driven by a constant rate of injection of an incompressible viscous fluid at the fracture inlet. The leak-off of the fracturing fluid into the host solid is considered negligible. The viscous fluid flow is lagging behind an advancing fracture tip, and the resulting tip cavity is assumed to be filled at some specified low pressure with either fluid vapor (impermeable host solid) or pore-fluids infiltrating from the permeable host solid. The scaling analysis allows to reduce problem parametric space to two lumped dimensionless parameters with the meaning of the solid toughness and of the tip underpressure (difference between the specified pressure in the tip cavity and the far field confining stress). A constant lumped toughness parameter uniquely defines solution trajectory in the parametric space, while time-varying lumped tip underpressure parameter describes evolution along the trajectory. Further analysis identifies the early and large time asymptotic states of the fracture evolution as corresponding to the small and large tip underpressure solutions, respectively. The former solution is obtained numerically herein and is characterized by a maximum fluid lag (as a fraction of the crack length), while the latter corresponds to the zero-lag solution of Spence and Sharp [Spence, D.A., Sharp, P.W., 1985. Self-similar solution for elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. A (400), 289–313]. The self-similarity at small/large tip underpressure implies that the solution for crack length, crack opening and net fluid pressure in the fluid-filled part of the crack is a given power-law of time, while the fluid lag is a constant fraction of the increasing fracture length. Evolution of a fluid-driven fracture between the two limit states corresponds to gradual expansion of the fluid-filled region and disappearance of the fluid lag. For small solid toughness and small tip underpressure, the fracture is practically devoid of fluid, which is localized into a narrow region near the fracture inlet. Corresponding asymptotic solution on the fracture lengthscale corresponds to that of a crack loaded by a pair of point forces which magnitude is determined from the coupled hydromechanical solution in the fluid-filled region near the crack inlet. For large solid toughness, the fluid lag is vanishingly small at any underpressure and the solution is adequately approximated by the zero-lag self-similar large toughness solution at any stage of fracture evolution. The small underpressure asymptotic solutions obtained in this work are sought to provide initial condition for the propagation of fractures which are initially under plane-strain conditions.     

On thin film flow of a third-grade fluid down an inclined plane

An exact solution for the thin film flow of a third-grade fluid on an inclined plane is presented. This is a corrected version of the solution obtained by Hayat et al. (Chaos Solitons Fractals 38:1336–1341, 2008). An alternative parametric form for the solution is also derived. The variation of the dimensionless velocity and average velocity is given for a wide range of parameter values. An asymptotic solution for large parameter values is obtained giving rise to a boundary-layer structure at the free surface.     

Motion of a slender body near the free surface of a compressible fluid

An analytic solution to the problem of motion of a slender rigid body in a semi-infinite domain of a compressible fluid is obtained for the case when the body moves in parallel to the free surface at a constant velocity. This problem is similar to the problem of motion of a hydrofoil ship whose wing-like device allows it to lift its hull above the water surface and to decrease the friction and drag forces limiting the speed of usual ships. During its motion in water, a hydrofoil produces a lift force. The obtained analytic solution allows one to derive explicit expressions for the drag force and for the lift force in the limiting cases of relatively small and large depths. When depth is small, the drag force is greater than that in an infinite medium, since the wave drag is additionally evolved. When the velocity increases and approaches the sound velocity, the forces exerted on the body increase without limit, which is typical for a linear formulation of the problem.     

Permeable plate at angle of attack in steady flow of a viscous fluid

An approximate solution is presented for the problem of the resistance of a permeable plate of widthl at an angle of attack in a steady plane flow of an incompressible viscous fluid for the case of both small and very large Reynolds numbers with different permeability laws. The results obtained in the case of large Reynolds numbers are compared with the corresponding results for flow past plane rod grids.     

Interaction of a One-Dimensional Continuum with an Inertial Object Moving over the Continuum

Free transverse oscillations in a system consisting of an infinite moment continuum, such as the Euler-Bernoulli beam lying on the Winkler foundation, and a rigid body moving along the beam with a constant velocity and having a point contact with the guide are studied. The range of the considered velocities of the concentrated inertial object along the continuum is limited by the requirement of a finite energy of elastic deformation of the infinite continuum, corresponding to cojoint free osillations of an unbounded system. An analytical solution of the corresponding spectral problem in a system with a mixed spectrum is constructed. Limiting situations are analyzed, where the inertial rigid object moving along the beam is devoid of one “oscillatory” degree of freedom for some reasons. In particular, an inertial object devoid of mass but having a nonzero tensor of inertia is considered. Dependences of all characteristics of the discrete spectrum of oscillations and their shapes on the magnitude of object velocity along the moment elastoinertial guide are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 88–97, July– August, 2005.     

The flow of an incompressible viscous fluid under a periodic pressure gradient through the annular space between two porous concentric circular cylinders subjected to suction and injection

Summary The flow of an incompressible viscous fluid due to a periodic pressure gradient through the annular space between two porous concentric circular cylinders with uniform injection into the outer cylinder and uniform suction into the inner cylinder has been considered. The expressions for the pressure and velocity are found. In view of the presence of the Bessel function in the axial component of velocity, we have discussed the two special cases of very small and very large oscillations. An approximate expression for the temperature, including viscous dissipation, when the oscillations are small is also found.