Question

A Newtonian fluid is placed in a slit of thickness, H, between two flat, plates that are horizontal to gravity. The upper plate moves at velocity U parallel to the lower plate, which is stationary. Find the velocity profile in the slit.

Answer 1

Proof with Step-by-step explanation:

From the Newton's law of viscosity we have

`\tau =\mu (du)/(dy).........(i)`

where
`\tau` is the wall shear stress

`\mu` is the coefficient of dynamic viscosity

`(du)/(dy)` is the velocity gradient in the flow

Now from the principle of boundary layer condition we know that the velocity of the fluid that is in contact with a surface the velocity of the fluid is same as that of the boundary itself.

Hence from the attached figure we can infer that

Velocity at
`y=0=0`

Velocity at
`y=H=U`

Solving equation 'i' we get

`du=(\tau )/(\mu )dy\\\\\int du=\int (\tau )/(\mu )dy\\\\u(y)=(\tau )/(\mu )* y+c`

from the boundary conditions we obtain that c = 0 since
`v(0)=0`

Also we have

`U=(\tau )/(\mu )* H\\\\\therefore (\tau )/(\mu )=(U)/(H)\\\\\therefore u(y)=(U)/(H)* y`