Victor
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 Posted: Wed Feb 27, 2008 11:21 am Post subject: ýA Couette flow over a cylinderý |
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Hi folks
I>ve been trying to solve a flow problem of a cylinder immersed in a
Couette flow (two dimensional problem). I assume a creeping viscous
flow. I use the stream function psi in polar coordinates, i.e. Vr = -
(1/r)*diff(psi,theta) and Vtheta = diff(psi,r). The boundary
conditions in infinity are u=gamma_dot*y (gamma_dot is the shear rate)
and v = 0 or in polar coordinates Vr =
r*gamma_dot*sin(theta)*cos(theta) and Vtheta = -
r*gamma_dot*sin^2(theta). From this I reckon that the stream function
is of a form psi = f(r)*sin^2. And here I have a problem - when I do
the nabla^4 operation I obtain a differential equation for both r and
theta and not just for r as in uniform flow. Any suggestions?
I also tried to go in other way. In Probstein>s book, "Physiochemical
hydrodynamics: An Introduction" is written that "a small spherical
particle must rotate around the z axis with an angular velocity
gamma_dot/2" and that velocity of a point on the sphere is u =
0.5*gamma_dot*y and v=0.5*gamma_dot*c. I assume that this is also the
case in two-dimensions. So I guessed a solution - u = gamma_dot*y +
A*x^2*y/r^4 and v = A*x*y^2/r^4, where A is an unknown constant. These
guesses satisfy the continuity equation and the vorticity equation
(i.e. vorticity = 0). However, if I try to find the A constant from
the boundary conditions on the cylinder I get different values for
different places.
I always could solve it numerically, but it>s not the idea.
Victor Chernov |
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