Applied Scientific Research

, Volume 26, Issue 1, pp 53–67 | Cite as

Stability of a radially bounded rotating fluid heated from below

  • G. H. Homsy
  • J. L. Hudson
Article

Abstract

The stability of a bounded rotating cylinder of fluid heated from below is treated mathematically under the assumptions of stationary onset and axisymmetry. Critical Rayleigh numbers are computed by Galerkin's method as a function of the Taylor number and cylinder aspect ratio for Taylor numbers,τ≤106. The constraining effect of the side walls is shown to decrease with either increasingτ or increasing radius/height ratios. Forτ>106, most cylinders, excluding extremely tall ones, will appear infinite in horizontal extent as far as stability characteristics are concerned. The form of the motion at onset is discussed in relation to previous work.

Keywords

Aspect Ratio Rayleigh Number Side Wall Stability Characteristic Critical Rayleigh Number 

Nomenclature

a

cylinder radius

ac

critical wave number for layers

a(i, j)

integral array, see Appendix

A

square matrix of orderM×N

b(i, j)

integral array, see Appendix

Bij

stream function expansion coefficients

C(i, j)

integral array, see Appendix

Cij

tangential velocity expansion coefficients

Cj

axial trial functions

d

cylinder height

D(i, j)

integral array, see Appendix

F(i, j)

integral array, see Appendix

g

acceleration of gravity

G(i, j)

integral array, see Appendix

J0,J1

Bessel functions of the first kind

i

unit vector in the radial direction

I

the identity matrix

k

unit vector in the axial direction

M

limit of the radial expansion

N

limit of the axial expansion

p, p

pressure, reduced pressure

r

radial coordinate

R

Rayleigh number,gαΔTd3/νκ

S

square matrix of orderM×N

T

temperature

ΔT

imposed temperature difference

u

radial component of velocity

u

two dimensional velocity vector, (u, w)

v

tangential component of velocity

w

axial component of velocity

X

vector composed ofBij

Yi

radial trial functions

z

axial coordinate

α

fluid coefficient of thermal expansion

αn

roots of eitherJ0 orJ1

γ

aspect ratio,a/d

δn

roots ofJ1

δi,j

Kronecker delta

ε1,ε2,ε3

errors in Galerkin treatment

κ

fluid thermal diffusivity

λn

eigenvalues of (15)

μn

eigenvalues of (16)

ν

fluid kinematic viscosity

ρ0

fluid reference density

τ

Taylor number, 4ω2d4/ν2

ψ

stream function

ω

rotational frequency

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Copyright information

© Martinus Nijhoff 1972

Authors and Affiliations

  • G. H. Homsy
    • 1
  • J. L. Hudson
    • 1
  1. 1.Dept. of Chemical EngineeringUniversity of IllinoisUrbanaUSA

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