Stability Of Viscous Flow Research Articles

Stability of Viscous Flow in a Curved Porous Channel with Radial Flow

In this paper, we present a linear hydrodynamic stability analysis of the fluid, flowing in a porous curved channel. The motion is due to Pressure gradient acting round the curved channel and an imposed radial flow. The analytical solution of the eigen value problem is obtained by using the Galerkin’s method, for the wide gap case. Results for critical wave number and Dean Number are obtained and are compared with earlier result. The agreement is very good. Also, the stability curve, amplitude of the radial velocity and the cell-pattern are shown on graphs. The results show that the flow is strongly stabilized by an outward radial flow and weakly stabilized by a strong inward radial flow, while it is destabilized by a weak inward radial flow. In presence of outward flow, wide gap systems show stronger stability than the small gap system.

Radial Velocity Effects on the Convective Instability Induced by Centrifugal Buoyancy

The effect of radial velocity on the stability of viscous flow between two arbitrarily spaced concentric porous cylinders in the presence of a radial temperature gradient has been examined by numerically solving the resulting differential equation with variable coefficients. The combined influence of suction (or injection) and the temperature gradient has been presented graphically.

Linear stability of viscous flow induced by surface stretching

The stability of the two-dimensional flow in a semi-infinite fluid induced by the stretching of a planar surface is investigated by a normal-mode linear stability analysis. The base flow is described by Crane’s exact analytical solution of the Navier–Stokes equation. A finite-difference method and a shooting method are implemented to solve the relevant eigenvalue problem for three-dimensional perturbations that vary sinusoidally with arbitrary wave number in the spanwise direction normal to the plane of the flow. The Crane flow is found to be linearly stable, with the least-damped mode arising in the limit of two-dimensional perturbations. Eigenfunctions for the disturbance velocity are presented and discussed for selected values of the transverse wave number. The numerical results for the dispersion relation corresponding to the least-damped mode for each transverse wave number are described by a simple analytical expression. The results of the stability analysis are contextualized by superposing a planar stagnation-point flow toward the stretching surface. As the ratio of the rate of elongation of the stagnation-point flow to the rate of stretching of the surface tends to zero, the dispersion curves smoothly asymptote to those found for the Crane flow. The results for planar stagnation-point flow toward a stationary surface are recovered in the opposite limit. The analysis is extended to account for suction or injection through a porous surface undergoing in-plane stretching.

Stability of Viscous Flow between Two Concentric Rotating Porous Cylinders

The stability of a viscous flow between two concentric rotating porous cylinders has been examined when the difference in radii of the cylinders is small in comparison with their mean radius. Critical Taylor numbers have been calculated for various wave numbers and velocity ratios of the cylinders. Theroetical results show that injection at the outer cylinder improves the stability of the flow whereas suction has an opposite effect.

Stability of viscous flow driven by an azimuthal pressure gradient between two porous concentric cylinders with radial flow and a radial temperature gradient

The effect of a radial temperature gradient on the stability of viscous flow between two porous concentric circular cylinders driven by a constant azimuthal pressure gradient is studied when a radial flow through the permeable walls of the cylinders is present. The radial Reynolds number β based on the radial velocity at the inner cylinder and the inner radius R1 is varied from −130 to 30 and both positive and negative values of the parameter N are taken, where N depends on the temperature difference T2 − T1 between the outer and inner cylinders. The linearized stability equations form an eigenvalue problem, which are solved by using a classical Runge-Kutta scheme combined with a shooting method, termed unit disturbance method. It is found that for a given value of N the radially outward flow (β > 0) has a stabilizing effect and the stabilization is more as the gap between the cylinders increases. But the inward throughflow (β 0, the flow becomes more and more unstable with an increase in N.

Differential geometry on diffeomorphism groups and Lagrangian stability of viscous flows

A differential geometrical formulation of the motion of an incompressible viscous fluid is presented. The geodesic equation on a manifold is extended with an additional term consisting of an endomorphism of the tangent space, and a class of curves defined by the resultant equation is introduced. Based on this extension, the motion of an incompressible viscous fluid is formulated as curves of this class in a diffeomorphism group, and the expression for the variational equation of the curves is derived. The expression is then shown to coincide with the governing equations for the Lagrangian displacement, which interprets the physical meaning of variation vector fields of the curves. The variational equation is reduced to a more simplified form which can be used to study evolution of the distances between fluid particles advected by a given basic flow.

Hydrodynamic stability of viscous flow between curved porous channel with radial flow

A linear stability analysis has been presented for the flow between long concentric stationary porous cylinders driven by constant azimuthal pressure gradient, when a radial flow through the permeable walls of the cylinders is present. The radial Reynolds number, based on the radial velocity at the inner cylinder and the inner radius is varied from −100 to 30. The linearized stability equations form an eigenvalue problem which are solved using a numerical technique based on classical Runge–Kutta scheme combined with a shooting method, termed as unit disturbance method. It is observed that radially outward flow and strong inward flow have a stabilizing effect, while weak inward flow has a destabilizing effect on the stability. Profiles of the relative amplitude of the perturbed radial velocities show that radially outward flow shifts the vortices toward the outer cylinder, while radially inward flow shifts the vortices toward the inner cylinder.

ONSET OF INSTABILITY IN HYDROMAGNETIC COUETTE FLOW

The stability of viscous flow between rotating cylinders in the presence of a constant axial magnetic field is considered. The boundary conditions for general conductivities are examined. It is proved that the Principle of Exchange of Stabilities holds at zero magnetic Prandtl number, for all Chandrasekhar numbers, when the cylinders rotate in the same direction, the circulation decreases outwards, and the cylinders have insulating walls. The result holds for both the finite gap and the narrow gap approximation.

The stability of the narrow-gap Taylor-Couette system with an axial flow

The stability of viscous flow between concentric rotating cylinders with an axial flow due to an axial pressure gradient is considered. The governing equations with respect to three-dimensional disturbances are derived and solved by a direct numerical procedure. Results are given for the case of small-gap approximation. Three typical cases μ=−1,0 and 0.5 are studied, where μ represents the ratio of angular velocity of the outer cylinder to that of the inner cylinder. The value of the axial Reynolds numberR is up to 100. It is found that the critical disturbance is a non-axisymmetric mode when the value ofR is sufficiently large, and the transition of the onset mode withR is demonstrated in detail. Results for the critical Taylor number, wave number, vortex incline angle, and relative wave velocity are also determined. The present stability analysis is found to be in agreement with previous experimental studies and particularly reveals the stability characteristics with the variation of μ.

Effects of Radial Temperature Gradient on the Stability of a Viscous Flow between Two Rotating Porous Cylinders with a Narrow Gap

The linear stability of viscous fluid flow between two rotating concentric porous cylinders at temperature θ 1 (of the inner cylinder) and θ 2 (of the outer cylinder) is studied in a narrow gap case. The numerical values of the critical wave number a c and the critical Taylor number T c are listed in a table for different values of μ (-0 only the inner cylinder rotating), μ (≷0, both cylinders co-rotating or counter-rotating) ratio of angular velocities Ω 2 /Ω 1 , and for different values of ±N, ±S where N is a radial temperature gradient parameter based on the temperature difference θ 2 - θ 1 >/ 0) or injection (<0) velocity. The effects of these parameters on the stability of flows are discussed.

Solution Structure and Stability of Viscous Flow in Curved Square Ducts

The bifurcation structure of viscous flow in curved square ducts is studied numerically and the stability of solutions on various solution branches is examined extensively. The solution structure of the flow is determined using the Euler-Newton continuation, the arc-length continuation, and the local parameterization continuation scheme. Test function and branch switch technique are used to monitor the bifurcation points in each continuation step and to switch branches. Up to 6 solution branches are found for the case of a flow in the curved square channel within the parameter range under consideration. Among them, three are new. The flow patterns on various bifurcation branches are also examined. A direct transient calculation is made to determine the stability of various solution branches. The results indicate that, within the scope of the present work, at given set of parameter values, the arbitrary initial disturbances lead all solutions to the same state. In addition to stable steady two-vortex solutions and temporally periodic solutions, intermittent and chaotic oscillations are discovered within a certain region of the parameter space. Temporal intermittency that is periodic for certain time intervals manifests itself by bursts of aperiodic oscillations of finite duration. After the burst, a new periodic phase starts, and so on. The intermittency serves as one of the routes for the onset of chaos. The results show that the chaotic flow in the curved channel develops through the intermittency. The chaotic oscillations appear when the number of bursts becomes large. The calculations also show that transient solutions on various bifurcation branches oscillate chaotically about the common equilibrium states at a high value of the dynamic parameter.

On the stability of viscous flow over a stretching sheet

The linear stability of two-dimensional boundary layer flow of an incompresible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with an outward velocity proportional to the distance from a point on it. Using Galerkin’s method the stability equations are solved for three-dimensional disturbances periodic in a direction normal to the plane of the basic flow and it is shown that the flow is stable.

Stability of viscous flow in a rotating porous medium in the form of an annulus: The small‐gap problem

AbstractThe paper deals with the linear stability analysis of laminar flow of a viscous fluid in a rotating porous medium in the form of an annulus bounded by two concentric circular impermeable cylinders. The usual no‐slip condition is imposed at both the boundaries. The resulting sixth order boundary value, eigenvalue problem has been solved numerically for the small‐gap case by the Runge‐Kutta‐Gill method, assuming that the marginal state is stationary. The results of computation reveal that the critical Taylor number increases with decreasing permeability of the medium. The problem is found to reduce to the case of ordinary viscous flow in the annulus obtained by Chandrasekhar,1 when the permeability parameter tends to zero.

Local estimates and stability of viscous flows in an exterior domain

Local estimates and stability of viscous flows in an exterior domain

Experiments on the stability of viscous flow between two concentric rotating spheres

Experimental results concerning the onset of instability in a spherical gap are presented, for the case in which the gap width is not too large with respect to the radius. The two spheres may be rotated in the same direction or in opposite directions. The resulting flow field and some special observations are described. The first onset of instability is recorded and the results are summarized in stability diagrams. It is shown that Taylor's calculations for rotating cylinders are also valid for rotating spheres. Finally, further experimental results by several authors concerning the onset of instabilitty between rotating cylinders are compared with those for rotating spheres.

The stability of viscous flow between rotating concentric cylinders with an axial flow

The stability of viscous flow between concentric cylinders with the inner cylinder rotating and with an axial flow due to an axial pressure gradient is considered. The analysis is motivated, in part, by recent papers by Hasoon &amp; Martin (1977) and Chung &amp; Astill (1977). Results are given for radius ratiosɳ=R1/R2= 1 (the small-gap limit), 0.95, and 0.9, and for values of the axial Reynolds numberRup to 100. HereR1andR2are the radii of the inner and outer cylinders, respectively. For the small-gap problem, the results are compared with those obtained by approximating the angular velocity and/or the axial velocity by average values. Two of the conclusions of the paper are that the small-gap problem is a valid approximation to theɳ≠ 1 problem forɳnear 1, and that the approximation of the axial velocity by its average value leads to qualitatively and quantitatively incorrect results. In addition, we find that for axial Reynolds numbers of about ninety a second mode of instability arises and there is a discontinuity in the axial wavenumber of the critical disturbance as a function of the Reynolds number.

Linear stability of slowly varying unsteady flows in a curved channel

The linear stability of viscous flows in a curved channel due to time dependent slowly varying pressure gradients is considered by an approach of the W.K.B. type. The asymptotic behaviour of small perturbation waves is determined, allowing their characteristics (amplitude, transverse structure, amplification rate) to be slowly varying with time. The evolution of such disturbances is followed and the instantaneous marginal states are determined according to a ‘momentary’ criterion for stability which is based on the definition of a ‘growth rate’ of the disturbance. An asymptotic representation for the growth rate is found which, at lowest order, reduces to the quasi-steady amplification rate of the disturbance velocity field associated with the lowest order component of the instantaneous basic flow. A quasi-steady correction term appears at the next order together with terms which arise from the slow variation with time of the actual disturbance structure. Slowly accelerated basic flows are also investigated. The effect of the quasi-steady correction is found to be stabilizing whereas part of the ‘slowly varying’ terms exhibit the opposite behaviour. However the influence of the correction terms appears to be small, as expected since the characteristic scale of the instability is much bigger than the ‘slow’ scale of the basic state. Under such circumstances the quasi-steady approximation may be considered a fairly accurate approach. Low frequency modulated basic flows are also investigated by using the periodicity criterion to define the marginal state. The modulation is always found to destabilize the mean flow and the ‘critical’ wavenumber is found to decrease from its unmodulated value when the amplitude of the oscillation increases. Some discussion follows.

Finite-difference solutions of Taylor instabilities in viscous plane flow

Abstract An integration algorithm is presented for the numerical solution of a linear, sixth-order eigenvalue problem that is associated with hydrodynamic stability of viscous flow between rotating planes. The solution method approximates the highest derivative of the eigensolutions D 6 phi(y) by passing a third-degree polynomial through three backward points and one forward point. Formulas for the lower derivatives are obtained by integration of the polynomial approximation. Linearity of the differential operator results in an explicit form for the highest derivative at the forward mesh point; a similar, two-point algorithm is used to start the computations. Neutrally stable eigensolutions associated with vortex instability of Taylor's type have been calculated for plane Couette and Poiseuille flow in a rotating system; these solutions agreed well with accepted solutions found by series-expansion methods.

On the stability of viscous flow between two rotating nonconcentric cylinders

The stability of viscous flow between two rotating nonconcentric cylinders is investigated with respect to infinitesimal disturbances. After the finite gap disturbance equations are formulated, the small gap disturbance equations are obtained. An approximate solution to the resulting eigenvalue problem is obtained by employing a perturbation method. A formula for calculating the critical Taylor number is deduced which is valid for small gap, small eccentricity and for the case when the outer cylinder is fixed. The mathematical model formulated gives rise to a local stability analysis. For θ = 0 o the eccentricity destabilizes the flow for small values of eccentricity ratio—the flow is less stable than when the cylinders are concentric. The results of a number of experimental and theoretical investigations are discussed.

Effect of constant heat flux at outer cylinder on stability of viscous flow in a narrow-gap annulus with radial temperature gradient

In this paper, the stability of the Couette flow of a viscous incompressible fluid between two concentric rotating cylinders is studied in the presence of a radial temperature gradient, when the outer cylinder is maintained at a constant heat flux. The analytical solution of the eigen-value problem is obtained by using the trigonometric series method, when the gap between the cylinders is narrow. The numerical values of the critical wave number and critical Taylor number are computed from the obtained analytical expressions for the first, second and third approximations. It is found that the difference between the numerical values of the critical Taylor number corresponding to the second and third approximations are very small as compared to the difference between first and second approximations. The critical Taylor numbers obtained by the third approximation agree very well with the earlier results computed numerically by using the shooting method. This clearly indicates that for the better result one should obtain the numerical values by taking more approximations. Also, the amplitude of the radial velocity and the cell-pattern are shown on the graphs for different values of the ratio of the angular velocities.Keywords: Radial temperature gradient, trigonometric series method, constant heat flux, Taylor number.