Answered: Velocity Profiles for Flow Between…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Velocity Profiles for Flow Between Parallel Plates. In Example 8.3-2, a fluid is flowing between vertical parallel plates with one plate moving. Do as follows:

1. Determine the average velocity and the maximum velocity.

2. Make a sketch of the velocity profile for three cases where the surface is moving upward, moving downward, and stationary.

Transcribed Image Text:EXAMPLE 8.3-2. Laminar Flow Between Vertical Plates with One Plate Moving A Newtonian fluid is confined between two parallel and vertical plates as shown in Fig. 8.3-2 (W6). The surface on the left is stationary and the other is moving vertically at a constant velocity v.. Assuming that the flow is laminar, solve for the velocity profile. Solution: The equation to use is the Navier-Stokes equation for the y coordinate, Eq. (8.2-37): ὃν. Ət + V₂ Əv.. əx +v Əv Əy Əv y + V: az = μl (8²v₁ 8²v a²v, Əx² Əy² Əz² d²v. dx² Әр Əy + pgy At steady-state, ov,/t = o. The velocities vand v₂ = o. Also, av, /ay = o from the continuity equation, av/az = o, and pg, pg. The partial derivatives become derivatives and Eq. (8.2-37) becomes pg = 0 (8.2-37) dp dy This is similar to Eq. (8.3-2) in Example 8.3-1. The pressure gradient dp/dy is constant. Integrating Eq. (8.3-10) once yields == (8.3-10)

Integrating again gives
dv
dx
x²
2μ
x dp
μdy
dp
dy
+ pg
= C₁
·+pg|= C₁x + C₂
(8.3-11)
1 dp
2 / 4 (12 + 198 ) (H x - x ²) + 10 =/17
+pg
2μ dy
(8.3-12)
V
The boundary conditions are at x =o, V, = 0, and at x = H, v,= v.. Solving for the constants, C₁ = v/H - (H/zu)(dp/dy + pg)
and C₂ = o. Hence, Eq. (8.3-12) becomes
(8.3-13)

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