Help with Hagen-Poiseuille Flow

[teeeeee]

Hi

I'm trying to derive the velocity profile for Hagen-Poiseuille flow through a pipe.
Using cylindrical coordinates (z direction horizontal), I began by applying the Navier-Stokes equations to each coordinate.

For z, I got: $$\frac{1}{\eta}$$ $$\frac{\partial p}{\partial z} = \frac{1}{\rho}$$ $$\frac{\partial}{\partial\rho}$$ $$(\rho \frac{\partial v_{z}}{\partial\rho})$$ and from this equation I got the result that $$v_{z} = \frac{1}{4\eta} \frac{\partial p}{\partial z} (\rho^{2} - R^{2})$$ where R is the radius of the pipe

The Navier-Stokes equations for the $$\rho$$ and $$\phi$$ directions give $$\frac{\partial p}{\partial \rho} = \mu g_{\rho}$$ and $$\frac{\partial p}{\partial \phi} = \mu g_{\phi}$$ respectively, where g is gravity in each direction and $$\mu$$ is the density of the fluid.

I know that when deriving the velocity profile for flow between parallel plates, you need to use the $$\phi$$ and $$\rho$$ equations to show what is a function of what, and what is independent of what.
However, I've managed to get the velocity profile without using any information from the other two coordinate equations. Have I missed something? Do I need to use these two equations for something?

Many thanks
teeeeee

 

[minger]

Let me walk you from the beginning. If we assume fully-developed flow, then the velocity becomes purely axial, and varies only with the lateral coordinates, that is:
$$\begin{equation} \begin{split} v=w=0 \\ u=u(y,z) \end{split} \end{equation}$$
The continuity and momentum equations can then reduce to:
$$\begin{equation} \begin{split} \frac{\partial u}{\partial x}=0 \\ -\frac{\partial \hat{p}}{\partial x} + \mu\left(\frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) = 0 \\ -\frac{\partial \hat{p}}{\partial y} = -\frac{\partial \hat{p}}{\partial y} \end{split} \end{equation}$$
These indicate that the total pressure is a function only of x. Since u does not vary with x, we can say that the gradient dp*/dx must be a negative constant. Then, we can combine to form the basic equation as:
$$\frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = \frac{1}{\mu}\frac{d\hat{p}}{dx} = \mbox{const}$$
We can then non-dimensionalize as:
$$\begin{equation} \begin{split} y* &= \frac{y}{h} \\ z* &= \frac{z}{h} \\ u* &= \frac{\mu u}{h^2(-d\hat{p}/dx)} \end{split} \end{equation}$$
Where h is a characteristic duct width. OK, now that we have those variables defined, we can rewrite the general equation as:
$$\nabla^{*2}(u*) = -1$$

OK, we're just about there. Now, for a Hagen-Poiseuille Fow, we have a circular duct, so the single variable is of course r. Non-dimensionally, we can write r*=r/r_o where r_o is the pipe radius. The Laplacian operator in cylindrical coordiantes reduces to:
$$\nabla^2 = \frac{1}{r}\frac{d}{dr}\left(r\frac{d}{dr}\right)$$

From here you can plug your operator into the general equation using the non-dimensional terms. If you need more help, come back with a good attempt and we can get you the rest of the way.