Question

A fluid is contained between two parallel plates of infinite extent in the xz plane. One is stationary and the other moves at a velocity U as shown in the figure below. The velocity profile across the gap of width b is described by u/U = y/b +P(y/b)(1− by p/b) where P=−(b²/2μU)(∂p/∂x). Find the value of (y/b) corresponding to the velocity maximum when P=2.75. (y/b)

Answer 1

The dimensionless value of (y/b) corresponding to the maximum velocity is about 0.1818.

Step-by-step explanation:

The question involves the concept of laminar flow in a fluid and relates to the measurement of viscosity. When a top plate moves at a constant speed U over a fluid resting on a stationary bottom plate, the fluid particles closest to each moving surface will move with the same velocity as that surface due to the no-slip condition.

To find the value of (y/b) corresponding to the maximum velocity when P=2.75, we can use the given velocity profile:

u/U = y/b + P(y/b)(1 - y/b).

Since we are looking for the maximum of the velocity, we need to take the derivative of u/U with respect to (y/b) and set it to zero:

d(u/U)/d(y/b) = 1 - 2P(y/b).

Setting this derivative equal to zero and solving for (y/b) when P=2.75 gives:

0 = 1 - 2*2.75*(y/b),

(y/b) = 1/(2*2.75).