Download 57:020 Mechanics of Fluids and Transport

57:020 Mechanics of Fluids and Transport
November 1, 2010
NAME
Fluids-ID
Quiz 8. An incompressible, viscous fluid is placed between horizontal,
infinite, parallel plates as is shown in the figure at the right. The two
plates move in opposite directions with constant velocities, U1 and U2,
as shown. The pressure gradient in the x direction is zero and the only
body force is due to the fluid weight. Use the Navier-Stokes equations
to derive an expression for the velocity distribution between the plates.
Assume the flow is steady, laminar, and parallel to the plates.
•
Navier-Stokes equations:
Solution:
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑝
𝜕2𝑢 𝜕2𝑢 𝜕2𝑢
𝜌� +𝑢
+𝑣
+𝑤 �=−
+ 𝜇 � 2 + 2 + 2�
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑧
For this geometry there is no velocity in the 𝑦 or 𝑧 direction, i.e., 𝑣 = 𝑤 = 0, as the flow is parallel to
the plates. In this case, 𝜕𝑢⁄𝜕𝑥 = 0, from the continuity equation. Furthermore, 𝜕𝑢⁄𝜕𝑧 = 0, for infinite
plates, and 𝜕𝑢⁄𝜕𝑡 = 0 for steady flow. With these conditions and 𝜕𝑝⁄𝜕𝑥 = 0, the Navier-Stokes equation reduces to
𝜕2𝑢
=0
𝜕𝑦 2
By integrating the above equation,
𝑢 = 𝐶1 𝑦 + 𝐶2
For 𝑦 = 0, 𝑢 = −𝑈2 , so that
For 𝑦 = 𝑏, 𝑢 = 𝑈1 , so that
Thus,
(+5 points)
(+2 points)
𝐶2 = −𝑈2
𝐶1 =
𝑢=�
𝑈1 + 𝑈2
𝑏
𝑈1 + 𝑈2
� 𝑦 − 𝑈2
𝑏
(+3 points)