# Slip Conditions for flow between Parallel Plates

• SebastianRM
In summary, the problem states that for a slip condition, the fluid velocity at y = 0 is V-2l\gamma and at y = h it is V+2l\gamma.
SebastianRM
Homework Statement
A) Analyze velocity profile
B) Determine shear stress at both walls
Relevant Equations
Fluid flow between two parallel plates with slip of,
$$\delta u \approx l (du/dy)$$
The problem states:

Two parallel plates separated by distance h, the plate at the top moves with velocity V, while the one at the bottom remains stationary.

My initial approach was:

I considered, ##du/dy = V/h## and for the shear stress ##\tau = \mu \frac{\partial u}{\partial y}##

For $$\frac{\partial u}{\partial y} = \frac{U_{top}-U_{bottom}}{h}$$

Where I considered the fluid's velocity at the top plate to be, ##U_{top} = V - \delta u## , and at the bottom, ##U_{bot} = 0 + \delta u##.

In order to improve my understanding, I searched for a diagram that illustrated the phenomena, which shows (apologies for the low quality) :

This ##b## distance in the diagram is the ##l## of my equation at the top. So I am wondering if it should be included as part of the ##dy## term, as ##dy = h - l## or should it be part of the ##\partial y = h - l ## or if my original analysis was correct?

Thanks a lot guys, your time and help are very appreciated.

The velocity at the top is V and the velocity at the base is 0. Forget the diagram. The shear rate is V/h.

BvU
Chestermiller said:
The velocity at the top is V and the velocity at the base is 0. Forget the diagram. The shear rate is V/h.

That is for no-slip condition, this is Slip condition. The velocity at the top is not V, because there is slip. Same at the bottom, the fluid is not attached to the plate, as there is slip.

Can you render the actual problem statement in full?:

BvU said:
Can you render the actual problem statement in full?:
The diagram says *No-slip* but that is a typo (the teacher said so).

Delta2
I see. In that case your picture is sensible and something similar occurs at the top plate.

Effectively ##h## is increased by ##2b##. Arguing that ##\tau## is a one-side derivative is thin ice, but I suppose that's what is meant.

SebastianRM
BvU said:
I see. In that case your picture is sensible and something similar occurs at the top plate.

Effectively ##h## is increased by ##2b##. Arguing that ##\tau## is a one-side derivative is thin ice, but I suppose that's what is meant.
I see, that makes sense. Would you say du/dy is the usual slope V/h ? While ##\partial u / \partial y ## accounts for the velocity V affected by slip and the "new" height, such that ## (V - 2\delta u)/(h + 2b) ## ?

Not both.

I get a fluid velocity at y = 0 of ##u(0)=l\gamma##, where ##\gamma## is the shear rate. And, at y = h, I get a fluid velocity of ##u(h)=V-l\gamma##. So, for the shear rate, I get $$\gamma=\frac{u(h)-u(0)}{h}=\frac{(V-l\gamma)-l\gamma}{h}=\frac{V-2l\gamma}{h}$$Solving for ##\gamma## gives:$$\gamma=\frac{V/h}{\left(1+2\frac{l}{h}\right)}$$
What do you guys think?

SebastianRM
Chestermiller said:
I get a fluid velocity at y = 0 of ##u(0)=l\gamma##, where ##\gamma## is the shear rate. And, at y = h, I get a fluid velocity of ##u(h)=V-l\gamma##. So, for the shear rate, I get $$\gamma=\frac{u(h)-u(0)}{h}=\frac{(V-l\gamma)-l\gamma}{h}=\frac{V-2l\gamma}{h}$$Solving for ##\gamma## gives:$$\gamma=\frac{V/h}{\left(1+2\frac{l}{h}\right)}$$
What do you guys think?
That is correct Sir, I think. By doing ##\delta u = \ell du/dy = \ell V/(h+2l) ## we can arrive to the same form.
Thank you!

## 1. What are slip conditions for flow between parallel plates?

Slip conditions refer to the boundary conditions that describe how a fluid behaves at the interface between a solid surface and the fluid. In the case of flow between parallel plates, slip conditions refer to the behavior of the fluid at the walls of the channel.

## 2. What is the no-slip condition?

The no-slip condition is a commonly used slip condition for flow between parallel plates. It states that the fluid velocity at the wall is equal to the velocity of the wall itself. This means that the fluid directly in contact with the wall has zero velocity.

## 3. What is the slip length?

The slip length is a measure of the slipperiness of a surface. It is defined as the distance from the wall at which the fluid velocity is equal to the velocity of the wall. A larger slip length indicates a more slippery surface, while a smaller slip length indicates a more sticky surface.

## 4. How does slip affect the flow between parallel plates?

Slip conditions can significantly affect the flow between parallel plates. In the case of a no-slip condition, the fluid directly in contact with the walls will have zero velocity, causing a velocity gradient in the fluid. This can lead to changes in the flow patterns and velocity profiles within the channel. Slip conditions can also affect the overall flow rate and pressure drop in the system.

## 5. What are some applications of slip conditions for flow between parallel plates?

Slip conditions are important to consider in various engineering and scientific applications, such as microfluidics, lubrication, and drag reduction. They can also play a role in the design and optimization of devices such as heat exchangers, pumps, and valves.

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