Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow

Munson, B. R.; Joseph, Daniel D.

doi:10.1017/s0022112071002064

Cited by 79 publications

(30 citation statements)

References 6 publications

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“…The essence of this is to check the results of Greenspan [2] as well as Schultz and Greenspan [13] which differ substantially from those of Pearson [11] for Re = 1000. A plot of the stream functions for Re < 1000 agrees qualitatively with those of Pearson and are consistent with the results of Munson and Joseph [7,8] and other published results. Secondary flow patterns were clearly visible in the meridional plane and a recirculation region developed near the equator.…”

Section: Resultssupporting

confidence: 91%

Numerical studies of viscous incompressible flow between two rotating concentric spheres

Ifidon

2004

Journal of Applied Mathematics

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The problem of determining the induced steady axially symmetric motion of an incompressible viscous fluid confined between two concentric spheres, with the outer sphere rotating with constant angular velocity and the inner sphere fixed, is numerically investigated for large Reynolds number. The governing Navier-Stokes equations expressed in terms of a stream function-vorticity formulation are reduced to a set of nonlinear ordinary differential equations in the radial variable, one of second order and the other of fourth order, by expanding the flow variables as an infinite series of orthogonal Gegenbauer functions. The numerical investigation is based on a finite-difference technique which does not involve iterations and which is valid for arbitrary large Reynolds number. Present calculations are performed for Reynolds numbers as large as 5000. The resulting flow patterns are displayed in the form of level curves. The results show a stable configuration consistent with experimental results with no evidence of any disjoint closed curves.

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Section: Resultssupporting

confidence: 91%

Numerical studies of viscous incompressible flow between two rotating concentric spheres

Ifidon

2004

Journal of Applied Mathematics

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“…An incompressible fluid with constant kinematic viscosity v fills the gap between the two concentric spheres with radii R 1 < R 2 • The inner sphere rotates with angular velocity w. Since we consider only axisymmetric flows, we can introduce a stream function '1', a vorticity function ,, and an angular velocity function q, as follows (Krause & Bartels 1980;Bonnet & Alziary de Roquefort 1976;Munson & Joseph 1971 ;Rosenhead 1963;Lamb 1932;Stokes 1842) :…”

Section: Governing Equations and Numerical Methodsmentioning

confidence: 99%

The first instability in spherical Taylor-Couette flow

Schrauf

1986

J. Fluid Mech.

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287In this paper continuation methods are applied to the axisymmetric Navier-Stokes equations in order to investigate how the stability of spherical Couette flow depends on the gap size 0'. We find that the flow loses its stability due to symmetry-breaking bifurcations and exhibits a transition with hysteresis into a flow with one pair of Taylor vortices if the gap size is sufficiently small, i.e. if u ~ uB.In wider gaps, i.e. for CTB < u ~ CTF, both flows, the spherical Couette flow and the flow with one pair of Taylor vortices, are stable. We predict that the latter exists in much wider gaps than previous experiments and calculations showed. Taylor vortices do not exist if u > CTF. The numbers CTB and uF are computed by calculating the instability region of the spherical Couette flow and the region of existence of the flow with one pair of Taylor vortices. IntroductionWe consider a flow of a viscous incompressible fluid contained between two concentric spheres. The outer sphere is fixed while the inner one rotates. If the inner sphere rotates slowly the flow contains two large cells ranging from each pole to the equator. This flow is uniquely determined by the Reynolds number, and is called 'spherical Couette flow' ( cf. figure 1). By increasing the Reynolds number, Khlebutin to produce the Taylor-vortex flow by rotating both spheres and stopping the outer one when the vortices appeared. They observed a transition back into spherical Couette flow if the Reynolds number was decreased. However, the Taylor-vortex flow cannot be produced by rotating only the inner sphere. In that case, the spherical Couette flow remains stable until it encounters the so-called 'wide-gap instability'

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“…Proudman [1], Stewartson [2], Carrier [3], Haberman [4], and Munson and Joseph [5] obtained an approximate analytical solution to the problem involving the flow in an annulus between two spheres rotating with prescribed constant angular velocities. Pedlosky [6] extended the problem to include temperature effects.…”

Section: Introductionmentioning

confidence: 99%

Study of the axially symmetric motion of an incompressible viscous fluid between two concentric rotating spheres

Gagliardi¹,

Nigro

Elkouh

et al. 1990

J Eng Math

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The research reported herein involves the study of the steady state and transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research is to study the use of an approximate analytical method for analyzing the transient motion of the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems include cases where: (a) one (or both) spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive torque.In this research, the coupled solid and fluid equations of motion are linearized by employing the perturbation technique. The meridional dependence in these equations is removed by expanding the dependent variables in a series of Gegenbauer functions with variable coefficients and employing the orthogonality property of these functions. The equations for the variable coefficients are solved by separation of variables and Laplace transform methods. Results for the stream function, circumferential function, angular velocity of the spheres and torque coefficient are presented as a function of time for various values of the dimensionless system parameters.

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Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow

Cited by 79 publications

References 6 publications

Numerical studies of viscous incompressible flow between two rotating concentric spheres

Numerical studies of viscous incompressible flow between two rotating concentric spheres

The first instability in spherical Taylor-Couette flow

Study of the axially symmetric motion of an incompressible viscous fluid between two concentric rotating spheres

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