A viscous oil is between two concentric cylinders. The inner cylinder has a radius of r, and is moving axially with a velocity, U. The outer cylinder has a much larger radius of R, and is stationary. There is a significant pressure supplied at the left end but the right end is open to the the atmosphere. Assume a purely axial fluid motion that is steady and fully developed. To find an equation for the velocity profile of the oil, which equation should you use?

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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A viscous oil is between two concentric cylinders. The inner cylinder has a radius of r, and is moving
axially with a velocity, U. The outer cylinder has a much larger radius of R, and is stationary. There is
a significant pressure supplied at the left end but the right end is open to the the atmosphere.
Assume a purely axial fluid motion that is steady and fully developed.
To find an equation for the velocity profile of the oil, which equation should you use?
U
а.
ôu
ôu
др
o'u au d'u
ди
ди
+ pgx
-v+
ôt ôx
b.
dv dv
u+
+pgy
%3D
ôt ôx
ôx? ' ôy?
C.
dw
dw
+ u
dw
+ Pgz -
+ v
+ w-
dz
dz
dz²
he
d.
1 Pv, , dv,
+ P8z
dvz
+ Ur ar
at
+
+ Vz
r a0
ar
az?
az
dz
r ar
е.
a (1
(rv,) ) +
Pv,
av, 2 dve
az?
dv,
+ Vz¯az
de
= -
1
P8r
dv,
+ v,
at
Vg dv,
dr
ar
f.
1 dve , dv,
2 ду,
合…。。需。“尝。学)--瑞 m) 制-
1 dp
r a0
v,Vo
+ pgo
+ V, ar
at
Vg dVg
+ v,
az
az? ? d0
ar
r a0
Transcribed Image Text:A viscous oil is between two concentric cylinders. The inner cylinder has a radius of r, and is moving axially with a velocity, U. The outer cylinder has a much larger radius of R, and is stationary. There is a significant pressure supplied at the left end but the right end is open to the the atmosphere. Assume a purely axial fluid motion that is steady and fully developed. To find an equation for the velocity profile of the oil, which equation should you use? U а. ôu ôu др o'u au d'u ди ди + pgx -v+ ôt ôx b. dv dv u+ +pgy %3D ôt ôx ôx? ' ôy? C. dw dw + u dw + Pgz - + v + w- dz dz dz² he d. 1 Pv, , dv, + P8z dvz + Ur ar at + + Vz r a0 ar az? az dz r ar е. a (1 (rv,) ) + Pv, av, 2 dve az? dv, + Vz¯az de = - 1 P8r dv, + v, at Vg dv, dr ar f. 1 dve , dv, 2 ду, 合…。。需。“尝。学)--瑞 m) 制- 1 dp r a0 v,Vo + pgo + V, ar at Vg dVg + v, az az? ? d0 ar r a0
Expert Solution
Step 1

From the given problem, the following assumptions can be made,

  • Flow is steady and laminar.
  • Effect of gravity can be neglected in the z-direction.
  • The rate of change of pressure is zero.
  • The density of oil is constant.
  • The motion of the oil is in the axial direction only.

For cylindrical coordinates, the generalized expression for the momentum equation in the z-direction can be written as below,

ρvzt+vrvzr+vθrvzθ+vzvzz=-pz+μ1rrrvzr+1r22vzθ2+2vzz2+ρgz ......(1)

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