(50 points) To reduce pumping power requirements in oil transport, water is introduced into the
pipe line yielding a concentric flow of oil and water. The less-viscous water forms a uniform annulus
around the oil core as shown in the figure. Oil and water are assumed to be immiscible, and
gravitational effects can be neglected. The viscosity of water is mu _(w) and that of oil is mu _(o). The radius of
the pipe is R and that of the oil core is R_(o) and the flow is steady state. The pressure drop over length L
of the pipe is Delta p, so the axial pressure gradient is del(p)/(d)elz=-Delta (p)/(L). Our goal is to find the velocity
profile for oil flow.
(a) (10 points) The Navier-Stokes equation for z-direction is given by
ho ((delu_(z))/(delt)+u_(r)(delu_(z))/(delr)+(u_( heta ))/(r)(delu_(z))/(del heta )+u_(z)(delu_(z))/(delz))=-(delp)/(delz)+
ho g_(z)+mu [(1)/(r)(del)/(delr)(r(delu_(z))/(delr))+(1)/(r^(2))(del^(2)u_(z))/(del heta ^(2))+(del^(2)u_(z))/(delz^(2))].
By deleting the zero terms, reduce the equation as much as you can. Briefly specify the reasons
why the terms are zero.
(b) (10 points) What are the boundary conditions defining the flow of both the water and oil?
(c) (30 points) Obtain the velocity profile for oil flow.
1. (50 points) To reduce pumping power requirements in oil transport, water is introduced into the
pipe line yielding a concentric flow of oil and water. The less-viscous water forms a uniform annulus
around the oil core as shown in the figure. Oil and water are assumed to be immiscible, and
gravitational effects can be neglected. The viscosity of water is w and that of oil is o. The radius of
the pipe is R and that of the oil core is R, and the flow is steady state. The pressure drop over length L
of the pipe is p, so the axial pressure gradient is Op/Oz = -p/L). Our goal is to find the velocity
profile for oil flow.
Water, Oil,
R
Water
Oi
Side View
End View
(a) (10 points) The Navier-Stokes equation for z-direction is given by
au
u Ou
Ou -u Oz
Ou.
1 ou. Ou
20
-
Ot
or
Oz
ar
r2 o02
az
By deleting the zero terms, reduce the equation as much as you can. Briefly specify the reasons
why the terms are zero.
(b) (10 points) What are the boundary conditions defining the flow of both the water and oil?
(c) (30 points) Obtain the velocity profile for oil flow.