A Newtonian fluid with constant density flows in a parallel-plate apparatus that separated by a distance d and length L as shown in Figure 1. The top plate is moving in z-direction with a velocity uw. Derive the velocity distribution of the fluid, vz as a function of y using equation of motion in Appendix 1. List the postulates and you may neglect the gravity force.
A Newtonian fluid with constant density flows in a parallel-plate apparatus that separated by a distance d and length L as shown in Figure 1. The top plate is moving in z-direction with a velocity uw. Derive the velocity distribution of the fluid, vz as a function of y using equation of motion in Appendix 1. List the postulates and you may neglect the gravity force.
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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A Newtonian fluid with constant density flows in a parallel-plate apparatus that separated by a distance d and length L as shown in Figure 1. The top plate is moving in z-direction with a velocity uw. Derive the velocity distribution of the fluid, vz as a function of y using equation of motion in Appendix 1. List the postulates and you may neglect the gravity force.
Reference:
Transcribed Image Text:d
y
Moving wall
u
Stationary wall
dP
dx
u
W
Figure 1: A sketch of Newtonian fluid in wide parallel plates
![a
Cartesian coordinates (x, y, z):
ôt
av
of
at
Et
+
av
The Equation of Motion for a Newtonian Fluid with Constant p and u
[p Dv/Dt= -(V p)+ V²v+pg]
êt
x
Ov₂
Əx
(3x + + 0₂ !
+ V₂
av,
Ox
dv.
Əx
by
Oy
ave
ar
av.
by
Cylindrical coordinates (1,8, z):
OV, V v,
+
ar r 80
2+1=
Ve Ove
r 20
APPENDIX 1(1)
Ov- v₂ Ov
+
êr r 20
+ V₂
Ov₂
dz
y
cz
*)-
==
av,
dz
==
V,Va
ave
ac 1"
Ex
2=J=-=-=
3)
7"
др
2 + 120 3².
ap
dz
) --
o²v.
2x²
op
ar
8²v.
1 ap
r 20
14
op 1 8
+
az
+ fl
2
8²v, 8²v,
Oy2 Oz
or rar
[OFF]
8²v
cy² ²
a
ar
+
2 Ove
1-2 ce
]
(16
1 / -(1,₁)) + — 3 0 0 ²
8²v
2 Ov,
1-² 00
+
+Pg₂
+ po
+ PB:
(10 (₂)) +;
1-² 80²
1 8²v, ²v,
8₂²
8²
[(*). +
છે.
rər
1 0² v
F80²
8²v. 8²v
+
02²
+ Pgr
+ Pse](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F377e5f92-a51f-4603-b799-a60233f873b4%2F4f6df8d3-1e74-4b1c-b172-b9f578276aed%2Foqu2v9l_processed.png&w=3840&q=75)
Transcribed Image Text:a
Cartesian coordinates (x, y, z):
ôt
av
of
at
Et
+
av
The Equation of Motion for a Newtonian Fluid with Constant p and u
[p Dv/Dt= -(V p)+ V²v+pg]
êt
x
Ov₂
Əx
(3x + + 0₂ !
+ V₂
av,
Ox
dv.
Əx
by
Oy
ave
ar
av.
by
Cylindrical coordinates (1,8, z):
OV, V v,
+
ar r 80
2+1=
Ve Ove
r 20
APPENDIX 1(1)
Ov- v₂ Ov
+
êr r 20
+ V₂
Ov₂
dz
y
cz
*)-
==
av,
dz
==
V,Va
ave
ac 1"
Ex
2=J=-=-=
3)
7"
др
2 + 120 3².
ap
dz
) --
o²v.
2x²
op
ar
8²v.
1 ap
r 20
14
op 1 8
+
az
+ fl
2
8²v, 8²v,
Oy2 Oz
or rar
[OFF]
8²v
cy² ²
a
ar
+
2 Ove
1-2 ce
]
(16
1 / -(1,₁)) + — 3 0 0 ²
8²v
2 Ov,
1-² 00
+
+Pg₂
+ po
+ PB:
(10 (₂)) +;
1-² 80²
1 8²v, ²v,
8₂²
8²
[(*). +
છે.
rər
1 0² v
F80²
8²v. 8²v
+
02²
+ Pgr
+ Pse
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