Consider the steady, laminar flow of two liquids, A and B, with = μ and μB viscosities A 2μ, respectively, between infinite parallel plates located at z = 1 and z = -ɛ, as shown in the diagram below. The plates are stationary and the liquids do not mix. There is an applied pressure gradient acting on liquid A, given by Vp = -Gi (where G> 0 is constant), and the effects of gravitation can be assumed to be negligible. =
Consider the steady, laminar flow of two liquids, A and B, with = μ and μB viscosities A 2μ, respectively, between infinite parallel plates located at z = 1 and z = -ɛ, as shown in the diagram below. The plates are stationary and the liquids do not mix. There is an applied pressure gradient acting on liquid A, given by Vp = -Gi (where G> 0 is constant), and the effects of gravitation can be assumed to be negligible. =
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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![Consider the steady, laminar flow of two liquids, A and B, with
viscosities MA = μ and MB 2μ, respectively, between infinite parallel
plates located at z = 1 and z = −ɛ, as shown in the diagram below.
The plates are stationary and the liquids do not mix. There is an
applied pressure gradient acting on liquid A, given by Vp = -Gi
(where G> 0 is constant), and the effects of gravitation can be
assumed to be negligible.
Z
0
MA = μ
U₁(z) =
μB = 2µ
UB(Z) =
=
(a) Write down the mathematical consequences of the following
assumptions:
(i) The flow is two-dimensional.
(ii) There is no variation in the direction into the page.
(iii) The flow is steady.
(iv) There is no variation of velocity parallel to the plates.
Hence write down the continuity equation for either of the two
liquids and show that the fluid velocities in liquid A and in liquid
B are given by uд = µ₁ (²) i and uß = uß(z) i, respectively.
(b) State the boundary conditions at the upper and lower plates, and
terface z =
at
(c) Solve the Navier-Stokes equations for uд and up to show that
==0
(2²
a (z + E)
2+ ε
Vp=-Gi
2z+ ε
2+ ε
X
where a =
G/2μ.
(d) Show that u₁(0)/uA (0) is proportional to ɛ.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c0b2a3d-410e-4110-bb5c-52cbbf4f21f4%2Fbd95baa1-4891-46f4-bb69-188304b1c817%2Fsbo7d4q_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the steady, laminar flow of two liquids, A and B, with
viscosities MA = μ and MB 2μ, respectively, between infinite parallel
plates located at z = 1 and z = −ɛ, as shown in the diagram below.
The plates are stationary and the liquids do not mix. There is an
applied pressure gradient acting on liquid A, given by Vp = -Gi
(where G> 0 is constant), and the effects of gravitation can be
assumed to be negligible.
Z
0
MA = μ
U₁(z) =
μB = 2µ
UB(Z) =
=
(a) Write down the mathematical consequences of the following
assumptions:
(i) The flow is two-dimensional.
(ii) There is no variation in the direction into the page.
(iii) The flow is steady.
(iv) There is no variation of velocity parallel to the plates.
Hence write down the continuity equation for either of the two
liquids and show that the fluid velocities in liquid A and in liquid
B are given by uд = µ₁ (²) i and uß = uß(z) i, respectively.
(b) State the boundary conditions at the upper and lower plates, and
terface z =
at
(c) Solve the Navier-Stokes equations for uд and up to show that
==0
(2²
a (z + E)
2+ ε
Vp=-Gi
2z+ ε
2+ ε
X
where a =
G/2μ.
(d) Show that u₁(0)/uA (0) is proportional to ɛ.
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