Nonhomogeneous Incompressible Herschel–Bulkley Fluid Flows Between Two Eccentric Cylinders

The equations for the nonhomogeneous incompressible Herschel–Bulkley fluid are considered and existence of a weak solution is proved for a boundary-value problem which describes three-dimensional flows between two eccentric cylinders when in each two-dimensional cross-section annulus the flow characteristics are the same. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous stress-strain law. A fluid volume stiffens if its local stresses do not exceed τ*, and a fluid behaves like a nonlinear fluid otherwise. The flow equations are formulated in the stress–velocity–density–pressure setting. Our approach is different from that of Duvaut–Lions developed for the classical Bingham viscoplastic fluids. We do not apply the variational inequality but make use of an approximation of the generalized Bingham fluid by a non-Newtonian fluid with a continuous constitutive law.     

Oil low to a lateral well in the presence of bottom water

Exact solutions are obtained for a number of two-dimensional problems of steady-state fluid flow to a lateral hole in a reservoir with a quiescent bottom fluid of higher density or with a fluid of lower density at the reservoir top __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 114–126, September–October, 2008.     

Filtration of a three-component mixture in a nonhomogeneous porous medium

We shall consider the problem of injecting a mixture of two incompressible fluids having different viscosities into an infinite nonhomogeneous porous stratum which is initially filled with a third fluid. The filtration rate of each of the phases depends basically on its concentration and viscosity, and therefore in the displacement process in the general case their rates of movement will be different, and as a result of this, zones of three-, two-, and single-phase flow are formed. These zones will be separated from one another by moving interfaces (fronts) at which there are jumps of the corresponding concentration levels.We shall assume for simplicity that in the entire region where there is combined flow of several fluids they are incompressible and insoluble, and outside this region, in the external zone, a homogeneous elastic fluid moves.     

Parameters analysis of fluid diffusion

The influences of fluid density, diffusivity, viscosity, width of the flow channel, travel distance, and flow velocity on fluid diffusion are analyzed theoretically and numerically. Concentration boundary layer is taken as the quantitative index of fluid diffusion in this work. The results show that diffusion is a function of travel distance, diffusivity, fluid density, and flow velocity. Diffusion is independent of the width of the channel. Viscous effect determines the velocity gradient and does not affect diffusion directly. The usually used Péclet number uL/D cannot govern the full condition of fluid diffusion. For two-fluids co-flowing in a two-dimensional straight channel with relative low viscous effect, diffusion is proportional to the square root of travel distance and diffusivity, and is inversely proportional to the square root of flow velocity.     

Orientation tensors in simple flows of dilute suspensions of non-Brownian rigid ellipsoids,comparison of analytical and approximate solutions

General analytical solutions are obtained for the planar orientation structure of rigid ellipsoid of revolutions subjected to an arbitrary homogeneous flow in a Newtonian fluid. Both finite and infinite aspect ratio particles are considered. The orientation structure is described in terms of two-dimensional, time-dependent tensors that are commonly employed in constitutive equations for anisotropic fluids such as fiber suspensions. The effect of particle aspect ratio on the evolution of orientation structure is studied in simple shear and planar elongational flows. With the availability of analytical solutions, accuracies of quadratic closure approximations used for nonhomogeneous flows are analyzed, avoiding numerical integration of orientation distribution function. In general, fourth-order orientation evolution equations with sixth-order quadratic closure approximations yield more accurate representations compared to the commonly used second-order evolution equations with fourth-order quadratic closure approximations. However, quadratic closure approximations of any order are found to give correct maximum orientation angle (i.e., preferred direction) results for all particle aspect ratios and flow cases.     

The classification of orthogonal electromagneto-fluid-dynamic flows

A theoretical analysis based on the equations of electromagneto-fluid-dynamics is undertaken in order to completely classify the flow geometries admitted by these equations. The steady two-dimensional flow of a viscous incompressible fluid of finite electrical conductivity and non-zero electric charge density is considered. The flow equations are formulated in terms of the streamfunction and magnetic flux function as independent variables. The exact analytical solution of the resulting equations is obtained when the magnetic field and the velocity field are everywhere orthogonal to each other. It is shown that the only possible flow is a uniform parallel flow.     

Laminar axisymmetric flow of a weakly compressible viscoelastic fluid

The combined effects of weak compressibility and viscoelasticity in steady, isothermal, laminar axisymmetric Poiseuille flow are investigated. Viscoelasticity is taken into account by employing the Oldroyd-B constitutive model. The fluid is assumed to be weakly compressible with a density that varies linearly with pressure. The flow problem is solved using a regular perturbation scheme in terms of the dimensionless isothermal compressibility parameter. The sequence of partial differential equations resulting from the perturbation procedure is solved analytically up to second order. The two-dimensional solution reveals the effects of compressibility and the other dimensionless numbers and parameters in the flow. Expressions for the average pressure drop, the volumetric flow rate, the total axial stress, as well as for the skin friction factor are also derived and discussed. The validity of other techniques used to obtain approximate solutions of weakly compressible flows is also discussed in conjunction with the present results.     

Effect of density extremum on the solidification of water on a vertical wall of a rectangular cavity

Experiments were performed in a two-dimensional rectangular cavity to study the transient flow in an initially isothermal and motionless fluid due to a step decrease in temperature on one of the two vertical end walls. In the experiments water was used as the phase-change medium, with the cold-wall temperature maintained below the freezing temperature. The opposite vertical wall was kept at the initial temperature, greater than the temperature where the density extremum occurs. The growth of ice and the transient flow in the cavity were visualized with the aid of a tracer technique to examine the effect of density inversion. The temperature field was continuously recorded by an array of thermocouples. It was found that the density inversion of water strongly influences both the growth of ice and the convective flow in the liquid region of the test cavity.     

Stagnation-point flow of a second-grade fluid with slip

The steady two-dimensional stagnation-point flow of a second-grade fluid with slip is examined. The fluid impinges on the wall either orthogonally or obliquely. Numerical solutions are obtained using a quasi-linearization technique.     

On the Arnold Stability of a Solid in a Plane Steady Flow of an Ideal Incompressible Fluid

We study the stability of a rigid body in a steady rotational flow of an inviscid incompressible fluid. We consider the two-dimensional problem: a body is an infinite cylinder with arbitrary cross section moving perpendicularly to its axis, a flow is two-dimensional, i.e., it does not depend on the coordinate along the axis of a cylinder; both body and fluid are in a two-dimensional bounded domain with an arbitrary smooth boundary. Arnold's method is exploited to obtain sufficient conditions for linear stability of an equilibrium of a body in a steady rotational flow. We first establish a new energy-type variational principle which is a natural generalization of the well-known Arnold's result (1965a, 1966) to the system “body + fluid.” Then, by Arnold's technique, a general sufficient condition for linear stability is obtained. Received 21 February 1997 and accepted 23 June 1997     

Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

Impact of a figure of revolution in a stream of incompressible fluid with separation of a jet

The impact interaction of bodies with a fluid in a flow with jet separation has been considered in [1–3], for example. This investigation was in the two-dimensional formulation. The present paper considers the three-dimensional problem of impact of a figure of revolution in a stream of an ideal incompressible fluid with separation of a jet in accordance with Kirchhoff's scheme. A boundary-value problem is formulated for the impact flow potential and solved by the Green's function method. A method for constructing the Green's function is described. Expressions are given for the coefficients of the apparent masses. The results are given of computer calculations of these coefficients in the case of a cone using the flow geometry of the corresponding two-dimensional problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 176–180, November–December, 1980.     

Optimization of Viscous Mixing in a Two-Dimensional Cavity Transfer Mixer

Following up ideas put forward by J.M. Ottino and colleagues, the possibility of designing a computational tool to optimize the mixing of viscous fluids in industrial devices is studied. It is shown that an efficient method to characterize and quantify a mixing process is to apply the statistical measures introduced by Danckwerts (e.g., intensity of segregation and scale of segregation) on the coarse-grained density distribution of points in Poincaré sections and advection patterns, that can be obtained by tracking the positions of marked fluid elements numerically. This method is not computationally excessively costly and, as is demonstrated here, can be applied easily to experimental dye advection studies. The model system used is the Stokes flow in a two-dimensional cavity transfer mixer: two rectangular cavities which are periodically driven by a solid wall and by the passage of the cavities over each other. This system shares with many industrial devices the complexity that the geometry of the flow is time-dependent. These changes in the geometry of the flow impose difficulties on the techniques of calculating the fluid velocity field (a boundary element method) and the advection of marked fluid elements. Ways of overcoming these difficulties are described.     

Numerical simulation of time‐dependent hydrodynamic removal of a contaminated fluid from a cavity

The time‐dependent hydrodynamic removal of a contaminated fluid from a rectangular cavity on the floor of a duct is analysed numerically. Laminar duct flows are considered for Reynolds numbers of 50 and 1600 where the characteristic length is the duct height. Two cases are considered where: (1) the fluid density in the cavity is the same as that for the duct fluid and (2) the cavity fluid has a higher density than the duct fluid but the two fluids are miscible. The flow is solved by a numerical solution of the time‐dependent Navier–Stokes equations. Attention is focused on the convective transport of contaminated fluid out from the cavity and the effect of duct flow velocity profile on the cleaning process. Passive markers are used in the numerical simulation for the purpose of identifying the contaminated cavity fluid. The results show that the flow patterns in the cavity are influenced by the type of duct flow. From a cleaning perspective, the results suggest that it is easier for the duct flow to penetrate a cavity and to remove contaminated cavity fluid when the duct flow is of the Poiseuille type and the aspect ratio is large. Copyright © 2003 John Wiley & Sons, Ltd.     

The distribution of solid particles suspended in a turbulent flow: A stochastic approach

Basic fluid mechanics and stochastic theories are applied to show that the concentration distribution of suspended solid particles in a direction normal to the mean streamlines of a two-dimensional turbulent flow is greatly influenced by the lift force exerted on them in the vicinity of the wall. Analytic solution shows that, when the direction of the mean flow is horizontal, the probability density functionp (y, t) for random displacements of the particles will have a maximum value at a point from the wall where the perpendicular component of the lift force precisely balances particle gravity. Interpretation of experimental observations is presented using this theory.     

Dynamics of particles floating on liquid flowing in a semicircular open channel

The dynamic characteristics of surface-floating particles in liquids flowing in a two-dimensional, semicircular open channel is studied experimentally. For high visibility in the experiments, relatively large particles are employed whose particle-liquid density ratio is either equal to or less than unity. Particles of different size and geometry are tested in a water-glycerin mixture. A video camera traces the pathline of each particle from which the velocity and direction of particle motion are evaluated. Liquid velocity distribution is determined by hot-film anemometry. A modified dynamics (Basset-Boussinesq-Oseen) equation is derived and numerically solved by means of a finite-difference technique to determine fluid velocity. A new dimensionless parameter is disclosed which is pertinent to both particle geometry and fluid flow conditions. It correlates particle trajectory and velocity, trajectory dispersion and fluid-particle velocity ratio.Visiting Scholar on leave from Department of Mechanical Engineering, Fukuyama University, Fukujama, Japan     

Numerical and experimental study of the fine structure of a stratified fluid flow over a circular cylinder

The fine structure of the flow field of a continuously stratified fluid around a circular cylinder for small values of the Froude number was investigated in laboratory and numerical experiments. The parameters of the leading perturbation, the internal-wave field, and the cylinder wake were calculated using a two-dimensional model. The existence of the previously experimentally observed high-gradient density layers in the wake that are parallel to the flow axis was for the first time confirmed by numerical calculations. Results of the numerical and experimental studies are in good agreement with each other and with analytical models for small values of the Froude number. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 43–54, January–February, 2007.     

The unsteady hydromagnetic thermal boundary layer with suction

The unsteady two-dimensional laminar flow of a viscous incompressible and electrically conducting fluid near an oscillating porous plate in the presence of uniform suction, is investigated. The solutions for the velocity, magnetic field, electric current density, temperature and Nusselt number are given in a closed form for the case of the magnetic Prandtl number being equal to unity. The other significant constants are the Eckert number, the fluid Prandtl number and the frequency of oscillation. The influence of these parametres on the solutions is given in both tabulated and graphical forms.     

Micropolar fluid flow over a shrinking sheet

An analysis is carried out for the steady two-dimensional flow of a micropolar fluid over a shrinking sheet in its own plane. The shrinking velocity is assumed to vary linearly with the distance from a fixed point on the sheet. The features of the flow and heat transfer characteristics are analyzed and discussed. It is found that the solution exists only if adequate suction through the permeable sheet is introduced. Moreover, stronger suction is necessary for the solution to exist for a micropolar fluid compared to a classical Newtonian fluid. Dual solutions are obtained for certain suction and material parameters.     

Magnetohydrodynamics convective flow in a vertical slot through a porous medium

An analysis is presented with magnetohydrodynamics natural convective flow of a viscous Newtonian fluid saturated porous medium in a vertical slot. The flow in the porous media has been modeled using the Brinkman model. The fully-developed two-dimensional flow from capped to open ends is considered for which a continuum of solutions is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. The asymptotic behaviour and the volume flux are analyzed and incorporated graphically for the three-parameter family of solution.