Experiments on the Motion of Solid

Experiments on the Motion of Solid

Bodies in Rotating Fluids

Emily Meyer

In this paper, Taylor considers the notion put forward by Proudman that all slow steady motions of a rotating liquid must be two dimensional. If one attempts to create a slow, steady motion by moving a three-dimensional object with small, uniform velocity, either the motion of the fluid will never become stable, the motion will be steady but not small in the region immediately surrounding the body, or the motion will become steady and two-dimensional. Through experimentation, Taylor demonstrated that this last possibility is the one that occurs. What follows is much of the math leading to the Taylor-Proudman Theorem as well as a related problem.

² Summary of Equations for Rotating Flows:

The equation of motion for fluid rotating at frequency W a distance r from the axis of rotation is given by:

[1] .

Note that , so that equation [1] can be re-written as:

[2]

where .

The characteristic parameters of this flow are the Rossby number, R, and the Eckman number, E, which are given by the following ratios of characteristic magnitudes:

Nondimensionalizing in terms of the characteristic velocity, U, and the characteristic length scale, L, yields:

[3] .

If E is neglected, it results in a singular perturbation and one type of boundary layers. If E is small, neglecting R also leads to a singular perturbation and different boundary layers. In the limit as both E and R approach zero and the flow is steady, equation [3] reduces to

[4] ,

which is known as the geostrophic balance. Taking the curl of each side of equation [4] yields the relation:

[5] .

Assuming that W is constant (thus neglecting gradients of W ), this can be written as:

[6]

For a rotating tank with axis of rotation in the z-direction, change in velocity with respect to the z-axis, is zero. This implies that the flow is two-dimensional, or

(Taylor-Proudman Theorem).

For a fluid with velocity v = (u, v, w), the rotating flow is given by:

[8] Þ .

Pressure, then, plays the role of the stream function, so velocity moves along lines of constant pressure. This means that a ball in a rotating tank, moving relative to the rotating flow such that the assumptions made above remain valid, the only way for a two-dimensional flow to be maintained is for a cylinder of fluid to move with the ball. Taylor writes that if this were indeed the case, "the motion would be a remarkable one….The boundary of such a cylinder would act as a solid body, and the liquid outside would behave as though a solid cylindrical body were being moved through it. No fluid would cross the boundary, and the liquid inside it would, in general, be at rest relative to the solid body. This idea appears fantastic, but the experiments now to be described show that the true motion does, in fact, approximate to this curious type."

² Rotating Boundary Layers: The Problem of Coffee-Cup Spin-Down

When the boundaries of a flow are moving relative to the flow, the time for the zero-velocity state to diffuse in from the wall to the center (the time for W to decrease by a factor of e) for a cup of radius R and height L is proportional to (R²/n ) = 900 seconds for the case of water in a coffee cup. This is clearly not correct, and the reason for this is that the cup has a bottom that contribute to the spin down, so that the correct formula is given by

which depends on viscosity. This viscosity dependence is a result of the fact that, in regions I and III of Figure 2, the geometry of the flow is changing from two dimensional to three-dimensional. Continuing to assume steady flow and that second derivatives in the x- and y-directions vanish, the equations of motion are now written as:

Replacing the pressure gradients by their values in region II (given by Taylor-Proudman), these equations become:

Combining these two equations yields a fourth-order differential equation in z, which can be integrated to determine the flow in these regions.