Stability Of Viscous Flow Research Articles

The stability of spiral flow between rotating cylinders

The effect of an axial pressure gradient on the stability of viscous flow between rotating cylinders is discussed on the basis of the narrow gap approximation, the assumption of axisymmetric disturbances, and the assumption that the cylinders rotate in the same direction. The onset of instability then depends on both the Taylor number ( T ) and the axial Reynolds number (R). For large values of R, the dominant mechanism of instability is of the Tollmien-Schlichting type and the present theory is based therefore on a generalization of the asymptotic methods of analysis that have been developed for the Orr-Sommerfeld equation. The present results, when combined with previous results for small values of R, give the complete stability boundary in the -plane. Only limited agreement is found with existing experimental data and it is suggested therefore that it may be necessary to consider either non-axisymmetric disturbances or nonlinear effects.

On the stability of viscous flow between eccentric rotating cylinders

The stability of viscous flow between eccentric cylinders is analysed for the case in which the inner cylinder rotates while the outer cylinder remains stationary, and where the difference in radii of the cylinders is small in comparison with their mean radius. The linearised equations governing the marginal stability of axially periodic disturbances are derived in general for the case where the cylinders are infinitely long, and are solved approximately to give estimates of the critical Taylor number at which vortex flow occurs for a range of relative eccentricity of the cylinders.The results give an upper bound to the stability boundary, and certain results of DiPrima are used to establish a lower bound, and consequently the stability boundary is well established for eccentricity ratios less than about 0·6. One important conclusion is that for a considerable range of eccentricity ratio the flow is less stable than when the cylinders are concentric.

Stability of viscous flow over concave cylindrical surface

The perturbation equations for the stability of viscous flow over a concave surface have been solved only to the first approximation. The criteria for the stability has been obtained. The flow becomes unstable when the dimensionless critical parameter\(R_\theta = \sqrt {\theta /r} \) exceeds the value 0.57 at αθ = 0.11.

On the analogy between thermal and rotational hydrodynamic stability

The correspondence between the eigenvalues for the problem of the onset of convection in a fluid confined between two horizontal plates and for the stability of viscous flow between two cylinders rotating at almost the same angular velocity has been known for some time. The recent work of Chandrasekhar (1961) has prompted the extension of the analogy to a larger group of rotating cylinder problems and their associated convection cases in which the primary temperature distribution is parabolic. This paper shows the analogy between these two problems and presents data which give the corresponding temperature distribution for a given ratio of angular velocities between the two cylinders. The equivalent Rayleigh numbers are listed for the Taylor numbers given by Chandrasekhar (1954). The eigenfunctions for several of the parabolic temperature profiles are determined. These results show that the single vortex convection pattern becomes a double vortex for certain initial temperature distributions. The critical Rayleigh numbers for the stability of a layer of water which is near 4 °C is also found by analogy.

Experiments on the stability of viscous flow between rotating cylinders. - V. The theory of the ion technique

The theory underlying the ion technique described in part IV of this series is developed: the equations are written down and solved under certain simplifying assumptions which appear physically reasonable. The results explain observations made in part IV by showing, in particular, that the current changes are linearly proportional to the radial component of the perturbations on the flow.

Experiments on the stability of viscous flow between rotating cylinders - IV. The ion technique

The development of an instrument which responds to the radial component of a Taylor vortex is described. The method is to measure the current passing through the vortex in a fluid of low conductivity, such as carbon tetrachloride. Experiments performed to check the usefulness of the technique are reported.

Experiments on the stability of viscous flow between rotating cylinders - VI. Finite-amplitude experiments

The ion technique is applied to a variety of experiments in Couette flow: determination of the critical Taylor number; determination of wavenumber and waveform of the vortices; establishment of the laws that the square of the amplitude and the amplification factor both vary as the difference between the Taylor number and the critical Taylor number; and finally certain studies of the growth and decay of vortices as the Taylor number is changed. The experimental data are compared with the results of integration of the exact equations for the linear stability problem by means of a high-speed computer. These theo­retical values were obtained by P. H. Roberts and are presented in an appendix to this paper.

On the stability of viscous flow between rotating cylinders Part 2. Numerical analysis

A simple numerical method is presented for solving the eigenvalue problem which governs the stability of Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. When the cylinders rotate in opposite directions, these eigenfunctions exhibit an exponentially damped oscillatory behaviour for sufficiently large values of − μ, where μ = Ω2/Ω1. In terms of the stream function which describes the motion in planes through the axis of the cylinders, this means that weak, viscously driven cells appear in the outer layes of the fluid which, according to Rayleigh's criterion, are dynamically stable. For μ = − 3, for example, four cells are present, the amplitudes of which are in the ratios 1·0:0·0172:0·013:0·00125.

On the stability of viscous flow between rotating cylinders Part 1. Asymptotic analysis

The stability of Couette flow is discussed in the case in which the cylindes rotate in opposite directions by an asymptotic method in which the Taylor number is treated as a large parameter. On assuming the principle of exchange of stabilities to hold, the problem is then governed by a sixth-order differential equation with a simple turning point. It is shown how the solutions of this equation can be represented asymptotically in terms of the solutions of the comparison equation yvi = xy. The solutions of this comparison equation have recently been tabulated and we thus have an explicit representation of the solution of the stability problem in terms of tabulate functions. Detailed results for the critical Taylor number and wave-number at the onset of instability and the associated eigenfunctions are given for the limiting case μ → − ∞, where μ = Ω2/Ω1, and Ω1 and Ω2 are the angular velocities of the inner and outer cylinders respectively. In this limiting case it is found that there exists and infinite number of cells between the cylinders, but that the amplitude of the secondary motion in all but the innermost cell is small.

Experiments on the stability of viscous flow between rotating cylinders III. Enhancement of stability by modulation

The onset of instability in Couette flow can be inhibited by modulating the rate of rotation of the inner cylinder. The origin of the inhibition has been shown experimentally to be due to the viscous wave propagated across the annulus by the modulated cylinder.

The stability of a viscous fluid between rotating cylinders with an axial flow

The stability of viscous flow between rotating cylinders with an axial flow has been investigated theoretically by Goldstein (1937), Chandrasekhar (1960, 1962), and Di Prima (1960); and experimentally by Cornish (1933), Fage (1938), Kaye & Elgar (1957), Donnelly & Fultz (1960) and Snyder (1962a). As was pointed out by Di Prima (1960) there were a number of discrepancies in the early work of the 1930's which were clarified in part by the papers of the 1960's. In turn, there appear to be certain small detailed differences in the more recent papers. In part it is these differences with which the present paper is concerned. In addition, the results of the previous theoretical investigations which are limited to the case in which the cylinders rotate in the same direction, are extended to the case of counter rotation.

Stability of Viscous Flow between Rotating Cylinders with Finite Gap

Stability of Viscous Flow between Rotating Cylinders with Finite Gap

Stability of viscous flow between rotating cylinders

Ein Beweis des Rayleighschen Kriteriums der Stabilitat einer zahen Stromung zwischen zwei unendlichen, coaxialen, rotierenden Zylindern, der auf einer einfachen Abanderung des Energieintegrals fur die Perturbationsgeschwindigkeit begrundet ist, wird dargestellt.

The stability of viscous flow between two co-axial rotating cylinders

In this paper a new method is described for solving the characteristic value problem underlying the theory of the stability of viscous flow between two co-axial rotating cylinders. The method depends on solving a simpler adjoint system of equations. It is revealed that the present method which yields a new and more precise secular equation than the one obtained by Steinman, is much simpler and at the same time saves considerable labour in numerical work.

On the principle of exchange of stabilities for the viscous flow between two co-axial rotating cylinders

In the first part of this paper we have derived an adjoint system of equations for the set of equations characterising the solution of the stability of viscous flow between two rotating cylinders, when the marginal stability is assumed not to be stationary. Then the adjoint system of differential equations has been solved to arrive at a simpler secular equation than the one obtained by Chandrasekhar. By a different approach than that of Chandrasekhar's, an attempt is made to show that for μ, which is defined as the ratio of the velocities Ω1 and Ω2 with which the inner and outer cylinders are rotated, greater than zero, there is no possibility of the instability setting in as overstability.

The stability of spiral flow between rotating cylinders

The stability of viscous flow under an axial pressure gradient between two co-axial cylinders rotating in the same direction is considered. The critical Taylor number (T c) for the onset of instability is then a function of the Reynolds number (R) of the mean axial flow. A perturbation theory valid in the limit P-> 0 is developed and the formula T c (B) = T c (0) + 26.5P 2 (R-> 0) is established.

The stability of viscous flow in a curved channel in the presence of a magnetic field

The stability of viscous flow between two coaxial cylinders maintained by a constant transverse pressure gradient is considered when the fluid is an electrical conductor and a uniform magnetic field is impressed in the axial direction. The problem is solved and the dependence of the critical number for the onset of instability on the strength of the magnetic field and the coefficient of electrical conductivity of the fluid is determined.

The stability of viscous flow between rotating cylinders in the presence of a magnetic field. II

The theory developed in an earlier paper (Chandrasekhar 1953) is extended to allow for counter-rotation of the two cylinders. Explicit results are given for the case when the two cylinders rotate in opposite directions with equal angular velocities.

The stability of viscous flow between rotating concentric cylinders with a pressure gradient acting round the cylinders

The stability of a viscous fluid between concentric cylinders is analysed, for the case in which the basic velocity distribution is the sum of a velocity distribution due to the rotation of the cylinders (Taylor 1923), and a ‘pumping’ velocity distribution due to a pressure gradient acting round the cylinders (Dean 1928). The critical Taylor number is computed for a wide range of values of the ratio of average velocity of pumping to average velocity of rotation for the case in which the outer cylinder is stationary. It is assumed that the spacing between the cylinders is small.

The stability of viscous flow between horizontal concentric cylinders

The critical conditions for the formation of Taylor vortices between horizontal concentric cylinders are considered in detail. (1) Where the flow is unidirectional round the annular space and is caused entirely by the rotation of the inner cylinder. (2) Where the flow is caused entirely by pumping round the annular space. (3) Where a liquid is caused to reverse its flow at a free surface, the flow being entirely caused by the rotation of the inner cylinder. The last case is analyzed by the method of small disturbances and the conditions under which Taylor vortices will form are found. Results for the first two cases are already available in the literature. From examination of the three criteria a dimensionless number is proposed to correlate the critical values of the various parameters at the onset of these Taylor vortices. The proposed number has the advantage that it is insensitive to the velocity distribution in the annulus. It is subsequently used to predict the critical conditions for the onset of Taylor vortices under conditions that have not been analyzed, i.e. where flow is due to pumping and rotation of the inner cylinder. The criterion is found to predict successfully the results of various experiments carried out. In addition experimental verification of the theoretical work of Dean (1928) and of the additional analysis carried out in this paper is also given.