Math 275 Homework 2
Due: 11/18/05

Compute the terminal velocity of an infinite train of aluminum cylinders falling through a column of honey flowing between two rigid vertical plates in response to gravity and the motion of the cylinders. Assume the flow of honey is governed by the Stokes equations

$$-\mu\Delta\mathbf{u}+\nabla p = \rho\mathbf{g}, \qquad \nabla\cdot\mathbf{u}=0.$$

The viscosity of honey is $\mu=3000$ centipoise, which is $30\, g/$cm$\,s$. The density of honey is $\rho=1.5\,g/$cm$^3$ and the density of aluminum is $2.7\,g/$cm$^3$. The acceleration due to gravity is $\mathbf{g}=(0,-980)\,$cm$/s^2$. The plates are $5\,$cm apart; the cylinders are $2\,$cm in diameter and are vertically separated from one another by $10\,$cm as shown below. Assume they fall steadily through the center of the column of honey (with no rotation) and remain equally spaced. Use quadratic elements for the velocity components and linear elements for the pressure components. Don't bother using isoparametric elements on the curved boundaries (unless you're a skilled programmer and are feeling ambitious).

Hint: go into the reference frame moving with the cylinders. Use periodic boundary conditions in the vertical direction to reduce the computation to a single period. Since the Stokes equations are linear, the answer may be obtained by solving two auxiliary problems. In the first, hold the walls and cylinder fixed and compute the downward drag force $F_1$ exerted by the honey on the cylinder as it flows due to gravity. In the second, set $\mathbf{g}=0$ and compute the upward drag force $F_2$ exerted by the honey on the cylinder if the walls are moving upward with unit speed. The final solution will be a superposition of these ( $\mathbf{u}=\mathbf{u}_1 + V\mathbf{u}_2$, $p=p_1+Vp_2$), where $V$ is the correct steady state velocity of the walls (which is the terminal velocity of the cylinders in the lab frame). To determine $V$, solve the force balance equation $VF_2 = mg+F_1$, where $m$ is the mass of the cylinder. Note: this mass and the two forces $F_i$ are ``per unit length'' in the out of plane direction.

• meshes.zip