Max Thickness and Velocity Profile of Paint on Vertical Wall | Bingham Plastic

[helppls]

Homework Statement

Consider a layer of paint on a vertical wall, with uniform thickness T and a yield stress of Tau_0. (Paint is a Bingham Plastic).

a) Derive an expression for the maximum thickness T0 that can be sustained without vertical flow.b) Derive an expression for the velocity profile when thickness is larger than T0.

Homework Equations

When shear stress is less than yield stress, du/dy=0

Tau= mu(dU/dy)+yield stress

The Attempt at a Solution

[/B]
For part A, I got (using momentum balance): T0= Tau_0/pg

Basically, the paint won't run if the shear stress does not exceed the yield stress, correct? So if the thickness is larger than the maximum thickness without vertical flow (T0), it can be assumed that the shear stress exceeds the yield stress. I'm confused at what to do from here? Because paint is a Bingham plastic and when the shear stress exceeds the yield stress:

Tau= mu(dU/dy)+yield stress

^Would I just solve this equation for U or does this question require T0 in the answer?

 

[Chestermiller]

Hi Liz,

Welcome to Physics Forums!

Let T be the total thickness of the paint film ($T>T_0$). Assume that the free surface is at y = 0, the wall is at y = T, and the z direction is downward. If I do a force balance on the portion of the film between y = 0 and arbitrary distance y through the film thickness, I obtain $$\tau_{yz}+\rho gy=0$$. Do you agree with this so far?

 

[helppls]

Yes, I do agree with that. It makes sense and I believe I got that from my force balance as well. I don't know where to go from here though

 

[Chestermiller]

Yes, I do agree with that. It makes sense and I believe I got that from my force balance as well. I don't know where to go from here though

So you have $$\tau_{yz}=-\rho g y\tag{1}$$And, for a Bingham plastic: $$\tau_{yz}(T_0)=-\rho g T_0=-\tau_0\tag{2}$$where $T_0$ is the value of y beyond which the absolute magnitude of the shear stress exceeds $\tau_0$
So, $$T_0=\frac{\tau_0}{\rho g}\tag{3}$$
Also, for a Bingham plastic,
For $|\tau_{yz}|\leq \tau_0$, $$\frac{du_z}{dy}=0\tag{4}$$
For $|\tau_{yz}|\geq \tau_0$ and $\tau_{yz}\leq 0$, $$-\tau_0+\mu\frac{du_z}{dy}=\tau_{yz}\tag{5}$$

If we combine Eqns. 2 and 5, we obtain: $$\mu\frac{du_z}{dy}=-\rho g (y-T_0)\tag{6}$$

Questions so far?

 

[helppls]

No, it makes sense to me. Would the next step be integration?

 

[Chestermiller]

No, it makes sense to me. Would the next step be integration?

Yes. If you integrate Eqn. 6 from an arbitrary value of y $(>T_0)$ to the wall y = T, what do you get (taking into account the no-slip boundary condition $u_z=0$ at y = T)?

 

[helppls]

I got this:

U=(−ρgy²/2μ)-T₀y/μ+ρgT²/2μ+T₀T/μ

After applying the boundary conditions, is this correct?

 

[Chestermiller]

I got this:

U=(−ρgy²/2μ)-T₀y+ρgT²/2μ+T₀T/μ

After applying the boundary conditions, is this correct?

No. The sign of $T_0y$ should be positive, not negative. Otherwise, OK. I wrote this equation in factored form:

$$u_z=\frac{\rho g}{2\mu}(T-y)(T+y-2T_0)$$

Now, what is the value of $u_z$ at $y=T_0$?

 

[helppls]

Oh, oops, you're right! Wouldn't it also be zero because T0 was defined as the max thickness without vertical flow?

 

[Chestermiller]

Oh, oops, you're right! Wouldn't it also be zero because T0 was defined as the max thickness without vertical flow?

There are two distinct regions in this problem:

1. The region between the free surface y = 0 and the critical distance $y=T_0$, that is $0<y<T_0$
2. The region between the critical distance $T_0$ and the wall $y = T$, that is $T_0<y<T$

Here are two multiple choice question for you:

In the region $T_0<y<T$, the downward velocity $u_z$ is
(a) Increasing with y
(b) Decreasing with y
(c) Constant

In the region $0<y<T_0$, the downward velocity $u_z$ is
(a) Increasing with y
(b) Decreasing with y
(c) Constant

Incidentally, for the region $T_0<y<T$, I've re-expressed the downward velocity in a nicer mathematical form:
$$u_z=\frac{\rho g}{2\mu}[(T-T_0)^2-(y-T_0)^2]$$
From this equation, what is the value of $u_z$ at $y=T_0$?
What is the value of $u_z$ at y = 0?