Proof with Explanation:
From the Newton's law of viscosity we have
[tex]\tau =\mu \frac{du}{dy}.........(i)[/tex]
where [tex]\tau [/tex] is the wall shear stress
[tex]\mu [/tex] is the coefficient of dynamic viscosity
[tex]\frac{du}{dy} [/tex] 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 [tex]y=0=0 [/tex]
Velocity at [tex]y=H=U [/tex]
Solving equation 'i' we get
[tex]du=\frac{\tau }{\mu }dy\\\\\int du=\int \frac{\tau }{\mu }dy\\\\u(y)=\frac{\tau }{\mu }\times y+c[/tex]
from the boundary conditions we obtain that c = 0 since [tex]v(0)=0[/tex]
Also we have
[tex]U=\frac{\tau }{\mu }\times H\\\\\therefore \frac{\tau }{\mu }=\frac{U}{H}\\\\\therefore u(y)=\frac{U}{H}\times y[/tex]
