Fluid Mechanics (Shear Stress)

[lfwake2wake]

1. Question
A large plate moves with speed (v.o) over a stationary plate on a layer of oil of thickness (d) and viscosity (u). If the velocity profile is that of a parabola, with the oil at the plates having the same velocity as the plates, what is the shear stress on the moving plate from the oil? If a linear profile is assumed, what is the shear stress on the moving plate? (Answers: u*v.o/(2d) and u*v.o/d)

Homework Equations

t: shear stress
t=u*(dv)/(dy)
v=md^2+c

The Attempt at a Solution

So I have the answer, but I cannot figure out the solution. If the velocity changes parabolically, then you have the equation v=md^2+c. dv=2md. So when d=0, v=0 and t=u*(2md-0)/(d-0). This is my solution, but obviously is not correct with the above answers. Any help would be greatly appreciated.
Thanks

 

[Hao]

Parabolically can either mean $$v = a y^2 + b$$ or $$v = a \sqrt{y} + b$$, where a and c are constants to be determined. Think carefully about which is the correct profile.

Furthermore, avoid expressing v immediately as a function where y = d. The velocity profile is $$v(y)$$. The velocity at the moving plate is $$v(d)$$. You will make mistakes with your derivatives otherwise.

 

[Mene]

Dear student,

Thank you for your question. In order to solve this problem, we need to use the Navier-Stokes equation for fluid flow, which states that the shear stress (t) is equal to the product of the fluid's dynamic viscosity (u) and the velocity gradient (dv/dy).

In this case, we are dealing with a two-dimensional flow, where the velocity (v) is dependent on the distance from the stationary plate (y). Since the velocity profile is given as a parabola, we can assume that the velocity gradient (dv/dy) is constant and equal to 2v.o/d, where v.o is the velocity of the moving plate and d is the thickness of the oil layer.

Using the Navier-Stokes equation, we can write:

t=u*(dv/dy) = u*(2v.o/d)

Therefore, the shear stress on the moving plate from the oil is equal to u*v.o/d.

If we assume a linear velocity profile instead, we can use the equation v=md+c, where m is the velocity gradient and c is a constant. In this case, we can solve for m by using the boundary condition that the velocity at the stationary plate is equal to v.o:

v.o = m*0+c
c = v.o

Now, using the same Navier-Stokes equation, we get:

t=u*(dv/dy) = u*m

Since we know that m=v.o/d (from the linear velocity profile equation), we can substitute this value into the equation above and get:

t=u*v.o/d

I hope this helps you understand the solution better. Let me know if you have any further questions.

Best regards,