Stability and instability in adiabatic shearing motions of incompressible fluids

We investigate the effect of temperature dependence of the viscosity on the stability of the adiabatic shearing flows of an incompressible Newtonian viscous fluid between two parallel plates. When the viscosity strongly decreases with temperature, the shearing flow caused by a steady motion of the upper plate (steady shearing) becomes unstable, while the shearing flow caused by a time-dependent body force is found to be stable.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.     

Parallel shear flows of fluids with a temperature dependent viscosity

We investigate whether parallel shear flows of an incompressible Newtonian fluid with a viscosity which depends linearly on temperature is possible in situations where the temperature changes along the flow direction. It is shown that parallel flow is possible only in planar or axisymmetric geometries. These two situations are investigated further. For either a plane channel or a circular pipe, we show that the temperature variation in the flow direction must be exponential.Received: December 16, 2003; revised: October 11, 2004     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.     

一维高精度离散GDQ方法

GDQ method is a kind of high order accurate numerical methods developed several years ago, which have been successfully used to simulate the solution of smooth engineering problems such as structure mechanics and incompressible fluid dynamics. In this paper, extending the traditional GDQ method, we develop a new kind of discontinuous GDQ methods to solve compressible flow problems of which solutions may be discontinuous. In order to capture the local features of fluid flows, firstly, the computational domain is divided into many small pieces of subdomains. Then, in each small subdomain, the GDQ method is implementedand some kinds of numerical flux limitation conditions will be required to keep the correct flow direction. At the boundary interface between subdomains, we also use some kind of flux conditions according to the flow direction. The numerical method obtained by the above steps has the advantages of high order accuracy and easy to treat boundary conditions. It can simulate perfectly nonlinear waves such as shock, rarefaction wave and contact discontinuity. Finally, the numerical experiments on one dimensional Burgers equation and Euler equations are given.The numerical results verify the validation of the method.     

Modeling incompressible flows using a finite particle method

This paper describes the applications of a finite particle method (FPM) to modeling incompressible flow problems. FPM is a meshfree particle method in which the approximation of a field variable and its derivatives can be simultaneously obtained through solving a pointwise matrix equation. A set of basis functions is employed to obtain the coefficient matrix through a sequence of transformations. The finite particle method can be used to discretize the Navier–Stokes equation that governs fluid flows. The incompressible flows are modeled as slightly compressible via specially selected equations of state. Four numerical examples including the classic Poiseuille flow, Couette flow, shear driven cavity and a dam collapsing problem are presented with comparisons to other sources. The numerical examples demonstrate that FPM is a very attractive alternative for simulating incompressible flows, especially those with free surfaces, moving interfaces or deformable boundaries.     

Analytical solutions to Stokes-type flows of inhomogeneous fluids

In this paper, we study the unsteady motion of an inhomogeneous incompressible viscous fluid, where the viscosity varies spatially according to various models. We study the Stokes-type flow for these types of fluids where in the first case the flow between two parallel plates is examined with one of the plates oscillating and in the second case when the flow is caused by a pulsatile pressure gradient. A general argument establishes the existence of oscillatory solutions to our problem. Exact solutions are obtained in terms of some special functions and comparisons are made with the cases of constant viscosity and the slow flow regimes.     

Heat transfer and Couette flow of a chemically reacting non‐linear fluid

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.     

A rheological model for fluid flows with arbitrary Reynolds numbers

An approach to constructing a phenomenological model of motion for a viscous fluid alternative to the Prandtl mixing-length hypothesis is suggested. The approach makes it possible to describe the motion of a fluid independently of the regime realized in a given region of the flow. On the basis of this approach, a differential one-parameter model for the flow of a viscous fluid applicable to any regimes of motion, called a uniform laminar-turbulent model, is constructed. For this purpose, the field of a scalar turbulence measure is introduced, which equals the ratio of the Reynolds stress to the total stress in the case of a simple shear flow. This makes it possible to write new expressions for turbulent viscosity. The influence of the turbulence measure field on the flow is taken into account by using an additional transport differential equation. The model is applicable to both compressible and incompressible fluids and makes it possible to obtain solutions in quadratures for steady simple shear flows. Various forms of the system of equations of motion and boundary conditions are given. Original Russian Text ￠ V.A. Pavlovskii, D.V. Nikushchenko, 2009, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2009, No. 1, pp. 104–112.     

On stability of steady-state plane–parallel shearing flows in a homogeneous in density ideal incompressible fluid

A problem on linear stability of stationary plane–parallel shearing flows in a homogeneous in density inviscid incompressible fluid between two immovable impermeable solid parallel infinite plates is studied. With the use of the direct Lyapunov method it is shown that all sufficient conditions (by Rayleigh, Fjørtoft, Arnol’d) known to this moment for stability of these flows with respect to small plane perturbations are the necessary ones as well. An a priori lower estimate is constructed; the estimate displays exponential in time growth of the considered perturbations if these conditions are not affected. An analytical example of steady-state plane–parallel shearing flows and superimposed small plane perturbations growing in time in accordance with the constructed estimate is given.     

On the Stability of Compressible Swirling Flow

The linear stability to axisymmetric perturbations of compressible non-dissipative swirling flow is shown to be insured if a suitably defined “Richardson number” depending on the basic velocity, temperature and density fields everywhere exceeds 1/4. This result combines and generalizes in a natural way some known results for incompressible swirling flow and stratified parallel flow.     

Efficient Solution of the Compressible Euler Equations for Low Mach Numbers

This paper investigates the benefits of local preconditioning for the compressible Euler equations to predict nearly incompressible fluid flow. The AUSMDV(P) upwind method by Edwards and Liou is employed to maintain the spatial accuracy of the method for low Mach numbers. The results indicate excellent solution quality and fast convergence to steady state for compressible as well as nearly incompressible fluid flow. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence of strong solution for the Cucker–Smale–Navier–Stokes system

We present a global existence theory for strong solution to the Cucker–Smale–Navier–Stokes system in a periodic domain, when initial data is sufficiently small. To model interactions between flocking particles and an incompressible viscous fluid, we couple the kinetic Cucker–Smale model and the incompressible Navier–Stokes system using a drag force mechanism that is responsible for the global flocking between particles and fluids. We also revisit the emergence of time-asymptotic flocking via new functionals measuring local variances of particles and fluid around their local averages and the distance between local averages velocities. We show that the particle and fluid velocities are asymptotically aligned to the common velocity, when the viscosity of the incompressible fluid is sufficiently large compared to the sup-norm of the particles' local mass density.     

Run up flow of an incompressible micropolar fluid between parallel plates – A state space approach

In this paper, we study the run up flow of an incompressible micropolar fluid between two horizontal infinitely long parallel plates. Initially a flow of the fluid is induced by a constant pressure gradient until steady state is reached. After the steady state is reached, the pressure gradient is suddenly withdrawn while the two plates are impulsively started with different velocities in their own plane. Using the Laplace transform technique and adopting the state space approach, we obtain the velocity and microrotation components in Laplace transform domain. A standard numerical inversion procedure is used to find the velocity and microrotation in space-time domain for various values of time, distance, material parameters and pressure gradient. The variation of velocity and microrotation components is studied and the results are illustrated through graphs. It is observed that the micropolarity parameter has a decreasing effect on velocity component. It is also found that as the gyration parameter increases there is a decrease in microrotation component and an increase in velocity component.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.     

Stability of oscillatory two-phase Couette flow

The authors investigate the stability of two-phase Couette flowof different liquids bounded between plane parallel plates.One of the plates has a time-dependent velocity in its own plane,which is composed of a constant steady part and a time-harmoniccomponent. In the absence of time-harmonic modulations, theflow can be unstable to an interfacial instability if the viscositiesare different, and the more viscous fluid occupies the thinnerof the two layers. Using Floquet theory, it is shown analyticallyin the limit of long waves that time-periodic modulations inthe basic flow can have a significant influence on flow stability.In particular, flows which are otherwise unstable for extensiveranges of viscosity ratios can be stabilized completely by theinclusion of background modulations, a finding that can haveuseful consequences in many practical applications.