Stokes modeled the flow field created by an oscillating clock$$s pendulum, which we

now outline. Consider an infinite plate lying along the x$-$axis with a semi$-$infinite body of an

incompressible fluid of constant viscosity above it$.$ The fluid occupies the upper half of the

xy$-$plane. The plate undergoes a simple harmonic oscillation in the x$-$direction with a frequency

and velocity Uplate$($t$)=$ U cos$($t$)$ as indicated in the figure below $($the plate is oscillating in its

own xz$-$plane, with $+$z$-$direction out of the page$).$ Stokes assumed that the pendulum had been

swinging for a long time so that all startup transient motion has died out and the system is in

steady$-$state dynamic equilibrium $$ no initial conditions are needed. Stokes$$s also assumed the

continuum hypothesis, constant density, and constant viscosity.

In order to arrive at the simplified governing equation for determining the velocity field, Stokes

made the following additional assumptions:

$($i$)$ The flow field is planar$/$columnar;

$($ii$)$ The flow field is fully$-$developed in the x$-$direction, and there is no physical mechanism to

support a horizontal pressure gradient in the x$-$direction, so $$P$/$x $=0$;

$($iii$)$ Gravity points in the negative z$-$direction $$g $=$g$$k and can be neglected since it does not

affect the phenomenon under investigation;

Starting from the three$-$dimensional, three$-$directional, Navier$-$Stokes equations and continuity

equation given on the worksheet, answer the following questions.

$[2$ pts$]($a$)$ Determine the appropriate no$-$slip, no$-$penetration boundary conditions along the plate,

and the far field boundary conditions as y $+$$.$

$[2$ pts$]($b$)$ Starting from the Navier$-$Stokes and continuity equations, use assumptions $($i$)-($iii$)$ to

simplify the equations as far as possible $($including gravity$).$ Be sure to label which assumption

you use to eliminate which term by drawing an arrow through it with the assumption number next

to it like we did in class. There may be some terms that can be eliminated by more than one

assumption. To avoid this ambiguity, do parts $($i$)-($iii$)$ in the order in which they are listed. Do not

solve the resulting simplified equations yet!

$[2$ pts$]($c$)$ Apply the appropriate boundary condition$($s$)$ to solve the reduced continuity equation

and use your result to further simplify the momentum equations down to a single equation. Explic$-$

itly write out the remaining x or y momentum equation. List the appropriate boundary conditions

that are to accompany the solution of this equation.

$[2$ pts$]($d$)$ Seek a solution of the form: u$($y$,$ t$)=$ U f $($y$)$ cos$($ky $$ t$)$ for some dimensional param$-$

eter k and non$-$dimensional function f $($y$)$ to be determined. Make sure it satisfies the boundary

conditions.

$[2$ pts$]($e$)$ Use the velocity profile u$($y$,$ t$)$ to compute the vorticity field z and stress field yx

throughout the fluid. What is the connection between the vorticity field and viscous diffusion

throughout the flow field?

Hint: When you substitute this ansatz$1$ into the PDE, you should get terms with sines and cosines.

For the right balance of terms, this will lead to two equations for f $($y$),$ both of which must be sat$-$

isfied by f $,$ so the resulting equations will seem to be over$-$constrained, but for the correct choice

of f $,$ they are not. One equation will determine f $,$ and the second will determine k in terms of

and other parameters. This leads to a dispersion relation.