University The of Michigan
Department of Mechanical Engineering
ME 320 - Sections 1 & 2
Final Exam - December 15, 2011
Problem 1 (23 points)
Two thin rectangular plates, of length L and width w into the page, are connected at one end
by a closed-end exible bellows arrangement, as shown in the gure. The parallel plates are
separated by a distance h(t) which is decreasing at a constant rate dh/dt. The ow between
the plates can be assumed to be highly viscous (inertia free) and incompressible. The
plate width w is much greater than the length L and dh/dt is small, so that it can be
assumed the ow between the plates is unidirectional along the x direction with volume
ow rate Q(x, t). The pressure at the outlet x = L is atmospheric pressure, pa , and the
uid density is .
The given quantities are , L, w, h(t), dh/dt, and pa .
(3 pnts) (a) Find the volume ow rate per unit width, Q(x, t)/w.
(9 pnts) (b) Find the pressure distribution, p(x, y, t) between the two plates. You may
neglect gravity.
(3 pnts) (c) Find the uid velocity prole, u(x, y, t), between the two plates.
(6 pnts) (d) Find the force per unit width, F/w, required to keep the plates from moving
horizontally.
(2 pnts) (c) Under what conditions can the ow be assumed to be highly viscous (inertiafree).
Problem 2 (12 points)
Rain drops are formed in clouds by condensation of water vapor around nuclei. As they
fall through the air, larger drops fall faster than smaller ones, catching up and merging
with the latter. As a result, a large drop becomes even larger by this agglomeration process
until the pressure forces set up by the airow around it overcomes the surface tension forces
which holds the drop together and causes it to break up into small drops. This scenario is
repeated until the drop reaches the earth.
(3 pnts) (a) For large rain drops, the diameter D at which break-up begins to occur
depends on the drops free-fall speed U , air density a and the surface tension . Use
dimensional analysis to derive an expression for the diameter D at which rain drops begin
to break up.
(3 pnts) (b) For very small the drops, free-fall speed V depends on the drop diameter D ,
the air viscosity a , and the gravity force per unit volume of the droplet w = w g. Use
dimensional analysis to derive an expression for the free-fall velocity, V , of small drops.
(6 pnts) (c) Using the data for drag coecient on a sphere, nd the free-fall velocity of a
solid sphere of density f and diameter D falling through air. Compare your answer to (b)
and explain the dierences.
Problem 3 (15 points)
(3 pnt) (a) For a thin at plate moving through a stationary uid at zero angle of attack,
how does the total drag force vary with velocity and viscosity in (i) laminar ow regime,
(ii) fully rough turbulent ow regime?
(2 pnt) (b) For a thin at plate positioned perpendicular to an incoming stream, how does
the total drag force vary with velocity and viscosity in (i) laminar ow regime, (ii) fully
rough turbulent ow regime?
(1.5 pnt) (c) A cable repair man is concerned about the humming noise due to vortex
shedding behind the cables he is installing. The cables are 2 in diameter and are placed
in an environment with wind speeds of up to 50 miles/hr. Is there any vortex shedding
past the cables at these wind speeds (explain)? A friend suggests that any potential vortex
shedding can be eliminated by roughening the surface of the cables. Is he correct (explain)?
(2.5 pnt) (d) Consider ow past a sphere at transitional Reynolds numbers, where the ow
can be either laminar or turbulent at a given Reynolds number depending on the surface
roughness. Of the laminar and turbulent ow regimes at the same Reynolds number, which
has the higher viscous drag (explain why)? Which has the higher pressure drag (explain
why)? Which has the higher total drag (explain why)?
(6 pnt) (e) A high pressure reservoir is connected to a low pressure reservoir using two
smooth parallel pipes of identical diameter. The length of one pipe is twice the other.
Neglecting all minor losses, compute the ratio of the volume ow rate in the longer pipe
to that in the shorter pipe in (i) laminar ow regime, and (ii) fully rough turbulent ow
regime.