Flow due to parallel disks rotating about non-coincident axis with one of them oscillating in its plane

An exact solution of the Navier–Stokes equations is obtained for the flow between two eccentric disks rotating with the same angular velocity and one of them executing non-torsional oscillations. An analytical solution describing the flow at large and small times after the start is given. The solutions depend on the ratio of the frequency of oscillation to the angular velocity of the disks and the ratio of the amplitude of oscillation to the angular velocity of the disks and to the distance between the axes of rotation, and the Reynolds number based on the distance between the disks and the angular velocity of the disks. The solutions for three cases when the angular velocity is greater than the frequency of oscillation or it is smaller than the frequency or it is equal to the frequency are discussed.     

On unsteady hydromagnetic flows of a dusty fluid between two oscillating plates

An initial value investigation is made of the motion of an incompressible viscous conducting fluid with embedded small spherical particles bounded by two infinite rigid non-conducting plates. The flow is generated in the fluid-particle system due to rectilinear oscillations of given frequencies superimposed on the plates in presence of an external transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities and the wall shear stress. It is shown that the effect of the dust particles on the fluid velocity depends on the time periods of the oscillating plates. When the time-periods are small, i.e., when the plates oscillate with high frequency, the fluid motion is found to be retarded by the particles. However, when the plates oscillate with larger time periods (smaller frequencies), the fluid velocity is increased by the presence of the particles at the early stage of the motion, and this effect persists until the equilibrium is reached when the particles exert their influence to resist the flow.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

Unsteady flow in the Ekman layer of an elastico-viscous liquid

The transient flow in the Ekman layer of an elastico-viscous liquid near a flat plate is discussed. Initially the fluid and the plate were rotating together and the plate then suddently starts moving with a uniform velocity in its own plane relative to the rotating frame of reference. It is shown that the ultimate steady state is reached through decay of inertial oscillations whose frequency decreases with increase in the elastic parameter.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Unsteady Couette flow in a rotating system

We have studied the unsteady Couette flow of a viscous incompressible fluid confined between parallel plates, rotating with an uniform angular velocity about an axis normal to the plates. The flow is induced by the motion of the upper plate and the fluid and plates rotate in unison with the same constant angular velocity. An exact solution of the governing equations have been obtained for small and large time $\tau$ by applying Laplace transform technique. It is found that the primary velocity decreases with increase in rotation parameter for small as well as large time. It is interesting to note that a back flow occurs in the region $0.0⩽\eta ⩽0.7$ for large time with increase in K when $K=4$ and 5. The secondary velocity increases in magnitude for small time with increase in rotation parameter. It is observed that the secondary velocity increases in magnitude for small values of rotation parameter. On the other hand, for large values of rotation parameter $K^2$, it decreases near the stationary plate and increases near the moving plate. The shear stress due to primary flow decreases with increase in rotation parameter $K^2$. On the other hand, it increases due to secondary flow with increase in rotation parameter for small time. It is noticed that for large time there exists separation in the primary and secondary flows due to high rotation.     

Hall effects on unsteady flow of a viscous fluid due to non-coaxial rotation of a porous disk and a fluid at infinity

An exact solution of the unsteady hydromagnetic flow due to non-coaxial rotations of a porous disk and a fluid at infinity is obtained on taking Hall currents into account. An analytical solution of the problem is obtained for small and large times after the start by the Laplace transform method. It is found that for small values of time there is no inertial oscillations while for large time the steady state is reached through inertial oscillations. The frequency of these oscillations first increases, reaches a maximum and then decreases with increase in Hall parameter.     

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–169, January–February, 1984.     

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.     

On the stability of the simple shear flow of a Johnson–Segalman fluid

We solve the time-dependent simple shear flow of a Johnson–Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress/shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.     

MHD viscous flow between torsionally oscillating disks

The unsteady motion of an incompressible, viscous, stratified fluid between two parallel infinite disks maintained at different temperatures is studied under the influence of a uniform transverse magnetic field. The whole system is under rigid rotation in the initial state and perturbations are created by the small amplitude torsional oscillations of the disks. The time required for the transient velocity and temperature to decay is found for various ranges of the values of the forcing frequency of the disks. The steady state velocity and temperature distributions represent boundary layers on the disks and an interior flow. The interplay between the Hartmann number and the Ekman number in determining the boundary layers on the disks is discussed.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms.The motion of the fluid is due to the inner cylinder that applies a time dependent torsional shear to the fluid.The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions.They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is highlighted by graphical illustrations.     

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.     

The Rayleigh and Stokes problems with an incompressible non-Newtonian fluid

Summary The Rayleigh problem or impulsive motion of a flat plate has been solved using a perturbation scheme when the surrounding fluid is representable by the constitutive equations of Oldroyd or Coleman and Noll. The shear stress and normal stress at the wall were expressed analytically for this unsteady motion. Further, an exact solution of the equations was found for a special case of the constitutive equations.The motion of the fluid above a harmonically oscillating plate or the Stokes problem has been determined for a special non-Newtonian fluid. The penetration of the shear wave into the fluid, the energy dissipation, the velocity profiles and the shear and normal stresses at the wall were expressed and compared to an equivalent Newtonian fluid.Some of the features of these non-Newtonian fluids were examined in simple shearing flows, and techniques to calculate some of the material constants discussed.     

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. The flow near a wall suddenly set in motion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model. Exact solutions of velocity and stress are obtained by using the discrete inverse Laplace transform of the sequential fractional derivatives. It is found that the effect of the fractional orders in the constitutive relationship on the flow field is significant. The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate, for large times the viscoelastic effects become weak. The project supported by the National Natural Science Foundation of China (10002003), Foundation for University Key Teacher by the Ministry of Education, Research Fund for the Doctoral Program of Higher Education     

剪切流作用下层合梁非线性振动特性研究

针对剪切流中层合梁的大变形非线性振动问题, 采用高阶剪切变形锯齿理论和冯·卡门应变描述层合梁的变形模式和几何非线性效应, 构建了大变形层合梁非线性振动有限元数值模型; 采用基于任意拉格朗日?欧拉方法的有限体积法求解不可压缩黏性流体纳维-斯托克斯方程, 结合层合梁和流体的耦合界面条件建立了剪切流作用下层合梁流固耦合非线性动力学数值模型, 采用分区并行强耦合方法对层合梁的流致非线性振动响应进行了迭代计算. 研究了不同速度分布的剪切流作用下单层梁和多层复合材料梁的振动响应特性, 并验证了本文数值建模方法的有效性. 结果表明: 剪切流作用下单层梁的振动特性与均匀流作用下的情况不同, 梁的运动轨迹受剪切流影响向下偏斜, 随着速度分布系数增加, 尾部流场中的涡结构发生改变; 刚度比对剪切流作用下层合梁的振动特性有显著影响, 随着刚度比的增加, 层合梁振动的振幅增大, 主导频率下降, 运动轨迹由‘8’字形逐渐变得不对称; 发现了不同厚度比和铺层角度情况下, 层合梁存在定点稳定模式、周期极限环振动模式和非周期振动模式三种不同的振动模式, 改变层合梁铺层角度可实现层合梁周期极限环振动模式向非周期振动模式转变.      

A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed.The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection.The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations.Making use of this solution,analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived.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.As a result,exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.     

A note on sinusoidal motion of a viscoelastic non-Newtonian fluid

In this note, the exact solutions of velocity field and associated shear stress corresponding to the flow of second-grade fluid in a cylindrical pipe, subject to a sinusoidal shear stress, are determined by means of Laplace and finite Hankel transform. These solutions are written as sum of steady-state and transient solutions, and they satisfy governing equations and all imposed initial and boundary conditions. The corresponding solutions for the Newtonian fluid, performing the same motion, can be obtained from our general solutions. At the end of this note, the effects of different parameters are presented and discussed by showing flow profiles graphically.     

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.