I have to complete this problem to revalidate a class I took 6 years ago. I've gone through my books, notes, old tests - and it's not coming back to me as fast as I would like!
Can someone please help me with the following problem:
1. Consider a Newtonian fluid of constant density flowing in the gap between two long vertical concentric cylinders when the outer one (of radius R2) is stationary (at r = R2, vθ = 0) and the inner one (of radius R1) is turning at a constant angular velocity ω (at r = R1, vθ = ωR1). Assume that the flow is steady, that the fluid moves only in the θ direction (vr = 0 and vz = 0), that derivates of velocity and pressure with respect to q are zero (due to symmetry), and that end effects are negligible so that vθ does not vary with z.
(a) In the q component of the equation of motion, shown below, cross out the terms that are zero, explaining under each crossed out term why it is zero. Then rewrite the equation with only the remaining terms. Note that since vq is a function of r only, partial derivatives of vq with respect to r can be changed to ordinary derivatives.
I know that anything with v
r and v
z is going to be zero. I'm stuck after that part.
(b) Show that the solution to the equation remaining in part (a) for vθ, the velocity in the θ direction in cylindrical coordinates, as a function of r (the radial position) is given by
where C1 and C2 are constants.
(c) Use the boundary conditions (at r = R1, vθ = ωR1; at r = R2, vθ = 0) to find the constants C1 and C2 and the final expression for vq as a function of r, R1, R2, and ω.
Any help would be greatly, greatly appreciated! This is the last thing holding me up from being cleared for graduation!!
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