Instability of cylindrical compressible gas jets in viscous liquid streams

Summary A linear stability study of cylindrical compressible gas jets in a moving incompressible viscous liquid medium subject to varicose disturbances is described. It was found that the gas jet is always unstable for a range of wavenumbers at any flow condition. When the gas and liquid velocity are not equal, temporal instability is enhanced by surface tension effects for small Weber numbers, and by aerodynamic interaction between the gas and liquid phase for high Weber numbers (where surface tension has a stabilizing influence). Increasing liquid viscosity always reduces the growth rate and the dominant wavenumber, whereas increasing gas density always increases gas jet instability. It was also found that the relative, rather than the absolute, velocity of the gas and liquid controls temporal instability. Increasing gas compressibility always increases the maximum growth rate and dominant wavenumber. On the other hand, for equal gas and liquid velocities, increasing surface tension always destabilizes, while increasing gas density always stabilizes, the gas jet. For absolute and spatial (or convective) instability, it was shown that the critical Weber number, separating the region of absolute from that of spatial instability, decreases monotonically as the liquid velocity is increased. For a stationary liquid medium, the gas jet is always absolutely unstable, and spatial instability does not exist, in contrast to liquid jets in a stationary gas medium. For sufficiently large liquid velocities, the gas jet is spatially unstable, whereas absolute instability disappears. Further, the absolute velocity of gas and liquid flow controls not only the growth of unstable disturbances, but also the characteristics of the instability. Increasing viscous effects tends to suppress absolute instability, while increasing both gas density and compressibility promotes absolute instability for small liquid velocities (however, their effect diminishes as liquid velocity is increased).     

On a method for vibration analysis of viscous compressible fluid–structure systems

A fluid–structure interaction formulation for viscous compressible fluid is under consideration. The formulation involves finite element approximation of linearized Navier–Stokes equations and response determination made by means of modal superposition analysis. Standard and simplified schemes of the viscous compressible fluid–structure interaction problem solution are developed. The schemes are based on the frequency condensation method of a complex eigenvalue problem solving. Free and forced oscillations of several fluid–structure systems are studied by the standard and simplified schemes. The analysis of the results obtained shows that the simplified scheme provides a saving of 90% of the computational time required to define oscillation of the structure with viscous compressible fluid in the lowest frequency range. A certain influence of the fluid viscosity on the transient response of the fluid–structure system is also demonstrated. Copyright © 2004 John Wiley & Sons, Ltd.     

Propagation of thickness-twist waves in a piezoelectric ceramic plate in contact with viscous fluids

We study theoretically the propagation of thickness-twist waves in an unbounded piezoelectric ceramic plate with unattached electrodes and viscous fluids between the plate surfaces and the electrodes. Based on the theories of piezoelectricity and viscous fluids, an equation that determines the dispersion relations of the waves is obtained, showing the dependence of the phase velocity on material and geometric parameters. Due to the viscosity of the fluid, the dispersion relations are complex in general, representing damped waves with attenuation. The dispersion relations obtained can reduce to the results of a few special cases with known results. The results of the present paper are useful for developing and designing fluid sensors for measuring fluid viscosity or density.     

Specially orthotropic panel flutter control using PID controller

Controlling the flutter speed of a specially orthotropic plate of rectangular cross-section through which an inviscid compressible fluid flows is considered. A state-space model of the coupled aeroelastic system helps us to determine flutter boundaries and provides a method for applying modern control theory to the problem. So in the present study, flow is modelled by piston theory and the coupled state-space model of panel motion and flow is solved. After that, the coupled system is controlled with Proportional-Integral-Derivative controller. This makes it possible to compute the flutter dynamic pressure and so the flutter velocity at which unstable plate oscillations occur as a function of the gain coefficient. It is found that by changing the gain coefficient the flutter of the plate can indeed be pushed to higher velocities.     

Hydrodynamic braking

A viscous fluid lies between two parallel plates. The bottom plate, moving laterally in its own plane, is affected by increasing viscous resistance when the top plate squeezes downward. The problem is solved by a power series expansion in a squeeze number S. Braking characteristics are obtained for several different states: the top plate moves with constant velocity, constant force, or constant power; the bottom moves with constant velocity or when it is free.     

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.     

MHD flow between two parallel plates with heat transfer

Summary In the present paper, the steady flow of an electrically conducting, viscous, incompressible fluid bounded by two parallel infinite insulated horizontal plates and the heat transfe through it are studied. The upper plate is given a constant velocity while the lower plate is kept stationary. The viscosity of the fluid is assumed to vary with temperature. The effect of an external uniform magnetic field as well as the action of an inflow perpendicular to the plates together with the influence of the pressure gradient on the flow and temperature distributions are reported. A numerical solution for the governing non-linear ordinary differential equations is developed.     

连续粘变条件下的速度场分析

流体的流动状态将直接影响轴承的润滑特性，轴承的润滑特性主要由润滑剂的粘度决定。工作中润滑剂的粘度变化影响其运动速度的变化，从而影响润滑剂的温升及内剪切稀化的程度等。尤其在薄膜润滑中，润滑膜的厚度极小，润滑剂流体粒子间的相对运动速度增大，其润滑剂特性的变化更为突出。以连续变化的粘度修正模型来研究在微小间隙内润滑剂的速度特性，得出间隙内速度的分布规律，为研究薄膜流体温度场和剪切稀化问题提供计算数据。     

Initiation and propagation of cell collapse in dynamic compression of cellular materials

A one-dimensional model of a cellular material is used to simulate deformation in compression at constant velocity. Each cell has a characteristic load-strain curve with a saddle transition to simulate cell collapse. Different cells may have different collapse forces. A damping term is included in the dynamic equations of motion, which are numerically integrated. Particular attention is given to the initiation of collapse and to its propagation, either to adjacent cells (band propagation) or in a random fashion. The actual behaviour depends on the imposed velocity and on the parameters of the characteristic equations. Low velocities tend to favour random propagation while high velocities originate a band that initiates at the moving end. At intermediate velocities two bands form. Equations are derived for the force for propagation of collapse under quasi-static and dynamic conditions. When there is a weaker element, collapse tends to initiate in it and may propagate to adjacent elements, under an approximately constant force which is affected by damping. When there are large differences among the collapse forces of the various elements, collapse occurs under an increasing load.     

Pressure and flow in fluid films constrained between translating and vibrating flexible surfaces

In this paper, the dynamic pressure and flow developed in a two-dimensional, viscous fluid film constrained between flexible surfaces are analyzed. The problem formulation assumes that the response of the flexible surface is governed by linear equations of motion, and the fluid motion is governed by linearized momentum equations including the unsteady inertia. Three states of the model are developed to describe the coupled fluid-structural response problem. The fluid dynamic pressure is derived in the frequency domain as a function of the fluid impedances and the surface transverse vibrations. The perturbed, coupled problem is described by an integral equation (in state vector form) that governs the coupled responses of the flexible surfaces. The integral equation is solved by a discretization method. The analysis is applied to a rigid slider bearing with a flexible, translating plate surface under the excitation of a harmonic point load. The accuracy of the discretization method is evaluated, and numerical results for the dynamic pressure and the plate response are presented.     

Helmholtz decomposition revisited: Vorticity generation and trailing edge condition

The use of the Helmholtz decomposition for exterior incompressible viscous flows is examined, with special emphasis on the issue of the boundary conditions for the vorticity. The problem is addressed by using the decomposition for the infinite space; that is, by using a representation for the velocity that is valid for both the fluid region and the region inside the boundary surface. The motion of the boundary is described as the limiting case of a sequence of impulsive accelerations. It is shown that at each instant of velocity discontinuity, vorticity is generated by the boundary condition on the normal component of the velocity, for both inviscid and viscous flows. In viscous flows, the vorticity is then diffused into the surroundings: this yields that the no-slip conditions are thus automatically satisfied (since the presence of a vortex layer on the surface is required to obtain a velocity slip at the boundary). This result is then used to show that in order for the solution to the Euler equations to be the limit of the solution to the Navier-Stokes equations, a trailing-edge condition (that the vortices be shed as soon as they are formed) must be satisfied. The use of the results for a computational scheme is also discussed. Finally, Lighthill's transpiration velocity is interpreted in terms of Helmholtz decomposition, and extended to unsteady compressible flows.     

Crack propagation velocities in bi-phase plates under static and dynamic loading

An investigation of the dependence of the crack-propagation velocity in complex bimaterial plates at different loading rates was undertaken. The specimens were fractured under the influence of both static and dynamic loadings and the crack-propagation velocities were detected by high speed photography with the optical method of caustics.The investigation was concentrated in detecting the influence that the different loading rates have on the fracture velocities in both phases of the plates and how this influence interferes—counteracting or superimposing—with the other factors that determine the crack propagation process in bi-material specimens. These factors are the effect of interface, the influence of the mechanical characteristics of each phase on the crack-propagation velocity etc.The results show that for constant and given material characteristics of the bi-phase plate the crack propagation velocity in the first (notched) phase tends to increase with increasing strain rates.The same is valid for the crack propagation velocity in the second phase, but only for the case when fracture occurs under the influence of a dynamic load. A significant discrepancy of the latter statement occurs, however, in the case of fracture under a continuously-increasing static load. In this case the crack-propagation velocity in the second phase reaches some remarkably high values, which are of the order of high strain-rate dynamic crack propagation velocities.In this way, the crack-arrest effect on the crack propagation velocity appears to be more significant in the case of a static loading than it is for the case of dynamic loading.     

Dynamics of the Moving Ring Load Acting in the System “Hollow Cylinder + Surrounding Medium” with Inhomogeneous Initial Stresses

This paper studies the influence of the inhomogeneous initial stress state in the system consisting of a hollow cylinder and surrounding elastic medium on the dynamics of the moving ring load acting in the interior of the cylinder. It is assumed that in the initial state the system is compressed by uniformly distributed normal forces acting at infinity in the radial inward direction and as a result of this compression the inhomogeneous initial stresses appear in the system. After appearance of the initial stresses, the interior of the hollow cylinder is loaded by the moving ring load and so it is required to study the influence of the indicated inhomogeneous initial stresses on the dynamics of this moving load. This influence is studied with utilizing the so-called three-dimensional linearized theory of elastic waves in elastic bodies with initial stresses. For solution of the corresponding mathematical problems, the discrete-analytical solution method is employed and the approximate analytical solution of these equations is achieved. Numerical results obtained within this method and related to the influence of the inhomogeneous initial stresses on the critical velocity of the moving load and on the response of the interface stresses to this load are presented and discussed. In particular, it is established that the initial inhomogeneous initial stresses appearing as a result of the action of the aforementioned compressional forces cause to increase the values of the critical velocity of the moving load.     

The measurement of bulk viscosity and the elastic-viscous analogy

Summary The motion of a liquid containing a substantial volume fraction of gas bubbles can often be calculated by considering a suitably averaged single phase continuum. This averaged material will be compressible as well as viscous and the problem arises to the determination (by experiment) of its shear and bulk viscosity coefficients. The direct measurement of bulk viscosity would be difficult, and the usual approach has been to measure an apparent elongational viscosity and then appeal to an analogy between elasticity and viscosity, claiming a connection similar to that which holds between the various elastic moduli. It is shown that this analogy does not hold in compression and that the experiments must be reinterpreted more carefully.     

Entropy generation in the flow system generated in between two parallel plates due to bivertical motion of the top plate

Thermodynamic irreversibility in the flow system provides information on the energy and power losses in the system. Minimization of entropy generation in the flow system enables for the parametric optimization of the system operation. In the present study, parallel plates, in between, filled with the fluid are considered. The fluid motion resulted from the bi-vertical compression of the top plate of the parallel plates is examined. The entropy generation rate in the flow system is formulated after considering the constant movement of the top plate, constant applied load, and the combination of the constant velocity and applied load to the top plate. The optimum operating conditions related to the fluid motion in between the parallel plates is determined through the entropy analysis. It is found that the combination of the constant velocity and the constant applied load resulted in the low entropy generation rate.     

轴向运动黏弹性夹层板的多模态耦合横向振动

基于薄板小挠度理论和Kelvin-Voigt黏弹性本构方程, 建立了轴向运动黏弹性夹层板横向振动控制方程, 研究了其横向振动特性。采用一阶和二阶Galerkin截断得到夹层板横向振动的特征方程, 讨论了两种夹心层所占总厚度比率下轴向运动速度对其横向振动特性的影响。研究表明: 在未超过临界速度前, 无论一阶还是二阶截断, 在定性描述系统特征上二者相同, 但一阶截断不适合描述轴向运动速度超过临界速度的情形; 对四边简支黏弹性夹层板, 临界速度和发生耦合模态颤振的速度随着夹心层比率的减少逐渐增大。     

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.     

On the flow of an electrically conducting nonlocal viscous fluid between parallel plates in the presence of a transverse magnetic field in magnetohydrodynamics

A continuum theory is constructed for the flow of an electrically conducting nonlocal viscous fluid between two nonconducting parallel plates. The flow is subject to the influence of a transverse magnetic field. The effects of long range or nonlocal interactions at a material point in the fluid arising from all material points in the rest of the fluid are taken into account by means of a nonlocal influence function. Equations of motion governing the nonlocal viscous flow are derived from localized forms of global balance laws and constitutive equations appropriate for electromagnetically active media. These field equations are analytically solved for the nonlocal velocity and the nonlocal stress fields. The effects of varying the magnetic field strength on the shear stress are investigated. The effects of such variations on the shear stress exerted on the walls of microscopic channels are also determined. Numerical computations are provided for these results.     

Radiation and thermal diffusion effects on an unsteady MHD free convection mass-transfer flow past an infinite vertical porous plate with the hall current and a heat source

An analysis is performed to study the effects of radiation and thermal diffusion on an unsteady MHD free convection heat- and mass-transfer flow of an incompressible, electrically conducting, viscous fluid past an infinite vertical porous plate with the Hall current and a heat source. The flow is considered under the influence of a constant suction velocity and a uniform magnetic field applied normally to the flow. The dimensionless governing equations are solved numerically by the Galerkin finite element method. The effects of the flow parameters on the primary and secondary velocities, temperature, species concentration, shearing stresses, Nusselt number, and Sherwood number are calculated and presented in figures and tables. The results obtained show that a decrease in the temperature boundary layer thickness occurs when the Prandtl number and radiation parameter are increased and an increase in the Schmidt number leads to a decrease in the concentration boundary layer thickness.     

Motion of a rigid sphere in a shear field in a micropolar fluid

The motion of a rigid sphere, suspended in a micropolar fluid which is undergoing a shearing motion, is discussed. The expressions for the pressure, velocity and spin in the fluid and those for the force and torque on the sphere are obtained. A compromise boundary condition, relating the spin of the particle with the vorticity vector at the boundary, is employed. The results are compared with the classical values and apart from other interesting observations, it is noted that the torque on the sphere depends upon the various parameters in a complicated manner. By extending the definitions of the effective viscosity for the viscous fluids, an expression for the viscosity of the suspension in the micropolar fluid is derived.