Some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids

The problems dealing with some simple unsteady unidirectional flows of a mixture of two incompressible Newtonian fluids are investigated. By using the constitutive equations appeared in the literature for binary mixtures of chemically inert incompressible Newtonian fluids, the equations governing the motion of the binary mixture are reduced to a system of coupled partial differential equations. By means of integral transforms, the exact solutions of these equations are obtained for the following three problems: (i) unsteady Couette flow, (ii) unsteady plane Poiseuille flow, (iii) unsteady axisymmetric Poiseuille flow.     

Flow of a binary mixture of incompressible Newtonian fluids in a rectangular channel

The problems concerning some simple steady and unsteady flows of a mixture composed of two incompressible Newtonian fluids in an infinitely long channel of rectangular cross-section are examined. By means of finite Fourier sine transforms, the exact solutions of the field equations are obtained for the following four problems: (i) steady Couette flow in a rectangular channel, (ii) unsteady Couette flow in a rectangular channel, (iii) steady Poiseuille flow in a rectangular channel, (iv) unsteady Poiseuille flow in a rectangular channel.     

Numerical study of axisymmetric vortex breakdown in an annulus

Summary The steady axisymmetric Navier-Stokes equations describing the flow in a cylindrical container and in an annulus with one or two rotating endwalls were solved by the continuation method. The unsteady flow regime was obtained from steady solution by following the temporal development of a small axisymmetric disturbance introduced into the initial conditions for either the full or the linearized system of equations. By variation of the boundary condition for vorticity at the axis the transition of the flow in a conventional container to that in an annulus with infinitesimal inner radius was continuously followed; the flow in the latter case has been found to differ only insignificantly from that in the container. By further continution in the radii ratio (the ratio between the inner and the outer radii), flow patterns in an annulus were obtained. An annulus with the stress-free (slip) boundary condition at the inner wall was also considered in order to examine the influence of viscous effects on the formation of separation bubble. While for the solid inner cylinder the recirculation region turned out to persist even at very large values of the radii ratio, the stress-free condition imposed at this boundary leads to vanishing of the separation bubble already at relatively small values of this parameter. The transition to unsteady motion (which, in application to a cylindrical container, was shown by Tsitverblit [14] to be due to the supercritical Hopf bifurcation) was studied in comparison with the results for a cylindrical container.     

Numerical Solutions for Unsteady Flows of a Magnetohydrodynamic Jeffrey Fluid Between Parallel Plates Through a Porous Medium

In the present article, the numerical solutions for three fundamental unsteady flows (namely Couette, Poiseuille, and generalized Couette flows) of an incompressible magnetohydrodynamic Jeffrey fluid between two parallel plates through a porous medium are presented using differential quadrature method. The equations governing the flow of Jeffrey fluid are modeled in Cartesian coordinate system. The resulting non-dimensional differential equations are approximated by using a new scheme that is trigonometric B-spline differential quadrature method. The scheme is based on the differential quadrature method in which the weighting coefficients are obtained by using trigonometric B-splines as a set of basis functions. This scheme reduces the equation into the system of first-order ordinary differential equation which is solved by adopting strong stability-preserving time-stepping Runge–Kutta scheme. The effects of the sundry parameters of interest on the velocity profiles are studied and the results are presented through graphs. It is observed that, the velocity increases from the horizontal channel to vertical channel. The velocity is a decreasing function of magnetic parameter. With an increase in time, the velocity increases.     

Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative

Summary In this paper the unsteady Couette flow of a generalized Maxwell fluid with fractional derivative (GMF) is studied. The exact solution is obtained with the help of integral transforms (Laplace transform and Weber transform) and generalized Mittag-Leffler function. It was shown that the distribution and establishment of the velocity is governed by two non-dimensional parameters η, b and fractional derivative α of the model. The result of classical (Newtonian fluid and standard Maxwell fluid) Couette flow can be obtained as a special case of the result given by this paper, and the decaying of the unsteady part of GMF displays power law behavior, which has scale invariance.     

Flows induced by the surface motions of a cylindrical sheet

Fluid flows induced by the surface stretching or shearing motion of cylindrical sheets are investigated. Steady and unsteady exact solutions of the Navier–Stokes equations are found for periodic axial shearing of an impermeable sheet and for periodic azimuthal stretching of a permeable sheet. Steady Stokes-flow solutions induced by the periodic axial stretching and the periodic azimuthal stretching of impermeable cylindrical sheets are also reported. In each case flows interior and exterior to a cylinder are considered, as well as the flow in the annulus between concentric cylinders.     

Development of a scalable finite element solution to the Navier–Stokes equations

A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its subdomain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance.The authors wish to acknowledge Mr. W. K. Kwan of The University of Hong Kong for his help in using the IBM SP2 supercomputer.     

Rapid couette flow of cohesionless granular materials

Summary Presented is an analysis on the Couette flow of cohesionless granular materials between two co-axial rotating cylinders. The constitutive equations employed have been postulated on the basis of available experimental and theoretical results which take into account the particle collisions as well as dynamic pressures induced by the trace of the unsemble phase average of the square of flow fluctuations. These constitutive equations loosely resemble the Reiner-Rivlin fluid behavior, and predict normal stress effects.New non-Newtonian effects in striking manners have been predicted in the cases of outer cylinder rotating-inner cylinder fixed as well as outer cylinder fixed-inner cylinder rotating. The theoretical predictions for the free surface profile for these two cases agree with our experimental observations and point to the validity of the proposed constitutive equations. All our results are based on no-slip conditions on the boundary surfaces. Furthermore, the results obtained are different from the classical results obtained for the Couette flow of simple non-Newtonian fluids.With 4 Figures     

流体力学问题基于核重构思想的最小二乘配点法

将基于核重构思想的最小二乘配点法应用于流体力学问题,给出了离散二维不可压缩粘性流体非线性偏微分方程的最小二乘配点格式。为了检验该方法的有效性,以二维Stokes问题——Couette流动为典型算例,分别研究了正压与负压两种工况作用下Couette流动的速度分布。数值模拟结果表明,无论离散点是均匀分布还是随机分布,均给出了较准确的数值结果。     

Stability of axial Poiseuille-Couette flow between concentric cylinders

Summary The linear stability of axial parallel Poiseuille-Couette flow in an annulus between concentric circular cylinders is considered. Using a long-wave version of the axisymmetric Orr-Sommerfeld equation the stability chart of this flow in the velocity ratio-radius ratio plane is derived. It is shown that pure sliding Couette flow can become unstable if the radius ratio is below a specific threshold value. Finally, applying the results to other flow geometries, it is shown that the boundary layer along a slender cylinder can become unstable in a confined region downstream the leading edge only.     

Deceleration of a porous rotating disk in a viscous fluid

The flow due to a rotating disk decelerating with an angular velocity inversely proportional to time with either surface suction (or injection) which again varies with time is investigated. The unsteady Navier-Stokes equations are transformed to non-linear ordinary differential equations using similarity transformations. The resulting equations are solved numerically using a globally convergent homotopy method. The flow depends on two non-dimensional parameters, namely an unsteadiness parameter $\text{S}$ and a suction (or injection) parameter $\text{A}$. Some interesting numerical results are presented graphically and discussed.     

Prediction of incidence effects on oscillating airfoil aerodynamics by a locally analytical method

A complete mathematical model is formulated to analyse the effects of mean flow incidence angle on the unsteady aerodynamics of an oscillating airfoil in an incompressible flow field. A velocity potential formulation is utilized. The steady flow is independent of the unsteady flow field. However, the unsteady flow is coupled to the steady flow field through the boundary conditions on the oscillating airfoil. The numerical solution technique for both the steady and unsteady flow fields is based on a locally analytical method. In this method, analytical solutions are incorporated into the numerical technique, with the discrete algebraic equations which represent the differential flow field equations obtained from analytic solutions in individual local computational grid elements. This flow model and locally analytic numerical solution method are then verified through the excellent correlation obtained with the Theodorsen oscillating flat plate and Sears transverse gust classical solutions. The effects of mean flow incidence on the steady and oscillating airfoil aerodynamics are then investigated.     

Implications of Holm's Finite Temperature Corrections to the HVBK Equations

The macroscopic equations used to model helium II are the Hall–Vinen–Bekharevich–Khalatnikov (HVBK) equations. Recently Holm suggested a finite-temperature adjustment to these equations, which effects the mutual friction parameters. In this paper we investigate the effect of this adjustment on the linear stability of Couette flow. Using the original HVBK equations, good agreement between the predicted and observed Reynolds number at which Couette flow becomes unstable has been found, particularly at temperatures close to the lambda temperature. Performing the same test on Holm's finite temperature corrections, we find no such agreement. The stability curves all predict that long-wavelength axial disturbances are unstable. We conclude that with the current values of the mutual friction parameters, Holm's finite-temperature corrections do not constitute a good model of helium II. We also discuss the possibility that, since the method of experimentally determining the mutual friction parameters depends on the form of the mutual friction, our interpretation of these parameters needs to be amended.     

Semianalytical solution of unsteady quasi-one-dimensional cavitating nozzle flows

Unsteady quasi-one-dimensional bubbly cavitating nozzle flows are considered by employing a homogeneous bubbly liquid flow model, where the nonlinear dynamics of cavitating bubbles is described by a modified Rayleigh–Plesset equation. The model equations are uncoupled by scale separation leading to two evolution equations, one for the flow speed and the other for the bubble radius. The initial-boundary value problem of the evolution equations is then formulated and a semianalytical solution is constructed. The solution for the mixture pressure, the mixture density, and the void fraction are then explicitly related to the solution of the evolution equations. In particular, a relation independent of flow dimensionality is established between the mixture pressure, the void fraction, and the flow dilation for unsteady bubbly cavitating flows in the model considered. The steady-state compressible and incompressible limits of the solution are also discussed. The solution algorithm is first validated against the numerical solution of Preston et al. [Phys Fluids 14:300–311, 2002] for an essentially quasi-one-dimensional nozzle. Results obtained for a two-dimensional nozzle seem to be in good agreement with the mean pressure measurements at the nozzle wall for attached cavitation sheets despite the observed two-dimensional cavitation structures.     

Travelling Taylor vortices in closed systems

Summary Travelling Taylor vortices between conical cylinders and a cone-cylinder combination are described. The flow results from a rotating inner cone and an outer shell at rest. Due to different centrifugal forces on the cone's surface one obtains a three-dimensional laminar basic flow and regions of subcritical and supercritical flows in the annulus. By some specific bifurcation sequences a rich variety of flow patterns occurs. The generated Taylor vortices can be steady or unsteady, toroidal or helical. Depending on different initial and boundary conditions, toroidal or helical vortices travel through a closed system. Furthermore, combinations of steady toroidal and unsteady helical vortices are possible. The influence of the governing parameters is discussed.     

Isothermal slip of a binary gas mixture along a solid surface

This paper investigates Couette flow of a binary mixture of a rarefied gas between two flat plates. An analytical expression for the rate of isothermal slip of a binary mixture of the rarefied gas along the surface is obtained.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 4, pp. 683–686, April, 1977.     

Numerical and experimental study on the flow history effects of axial flow on the Couette–Taylor flow

In this research, experimental and numerical techniques are used to study the flow history effects of axial flow on the Couette–Taylor flow. For the experimental investigation, the flow is visualized using the PIV technique with reflective particles with a density of 1.62 g/cm³. Dispersed in a solution, the particles have a strong refraction index equal to 1.85. In this study, two protocols are adopted to study the effect of an axial flow superimposed on a Couette–Taylor flow, and of the history of the flow. The first one, the direct protocol, consists of imposing an azimuthal flow to the inner cylinder. In this case, when the regime is established, the axial flow is superimposed. The second protocol, the inverse protocol, consists of imposing first the axial flow in the gap of the system, after which an azimuthal flow is conveyed. The Couette–Taylor flow with axial flow is strongly dependent on the flow history (the protocol). Thus, the flow structures and development for different protocols are studied and analyzed here experimentally and numerically. In addition, from the numerical results, mathematical models for the two protocols are presented. For the direct protocol, a new relation between the axial Reynolds number, which stabilizes the Couette–Taylor flow, and the Taylor number is presented; for the inverse protocol, a new mathematical model for the critical Taylor number is developed as a function of the axial Reynolds number and also the first critical Taylor number without axial flow.     

Numerical study of plane couette flow in a rotating framework

Summary A numerical study of laminar plane Couette flow subjected to a steady spanwise rotation is conducted. The full nonlinear Navier-Stokes equations in a steadily rotating framework are solved by a finite difference method for a long, large-aspect-ratio rectangular channel where the outer wall moves at a constant velocity. In this manner, nonlinear and wall end effects which are present in any real laboratory Couette flow experiment are taken into account. The computations demonstrate the existence of a roll instability, at intermediate rotation rates, when the Reynolds number exceeds a critical value of 42. The associated secondary flow in the form of longitudinal rolls is shown to have a severe distortional effect on the primary axial velocity in the interior of the channel which is not linear like its counterpart in an inertial framework. Comparisons are made with previously conducted linear stability analyses as well as with other analogous numerical and experimental studies.With 15 Figures     

Computer study of the effect of the turbulence of an oncoming flow on the V-shaped combustion of a homogeneous methane-air mixture

Numerical simulation of the V-shaped turbulent combustion of a homogeneous methane-air mixture has been performed using the program Fluent from the ANSYS-CFD software package. The combustion process was described by the two-dimensional unsteady Navier-Stokes equations and the one-step kinetic mechanism. The unsteady profiles of the velocity components at the input domain of the calculation area were modeled by means of the artificial turbulence algorithms. These algorithms made it possible to set the pulsation intensity and the integral turbulence scale in the oncoming flow. Comparison of the calculated and experimental data has shown that experimentally observed broadening of the combustion front can be a result of its instability to the perturbations of the velocity in the oncoming flow.     

Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder

The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinite circular cylinder is investigated when both the free stream velocity and the velocity of the cylinder vary arbitrarily with time. The cylinder moves either in the same direction as that of the free stream or in the opposite direction. The flow is initially (t=0) steady and then at t>0 it becomes unsteady. The semi-similar solution of the unsteady Navier–Stokes equations has been obtained numerically using an implicit finite-difference scheme. Also the self-similar solution of the Navier–Stokes equations is obtained when the velocity of the cylinder and the free stream velocity vary inversely as a linear function of time. For small Reynolds number, a closed form solution is obtained. When the Reynolds number tends to infinity, the Navier–Stokes equations reduce to those of the two-dimensional stagnation-point flow. The shear stresses corresponding to stationary and the moving cylinder increase with the Reynolds number. The shear stresses increase with time for the accelerating flow but decrease with increasing time for the decelerating flow. For the decelerating case flow reversal occurs in the velocity profiles after a certain instant of time.