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

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

I know that when deriving the velocity profile for flow between parallel plates, you need to use the [tex]\phi[/tex] and [tex]\rho[/tex] 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

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# Help with Hagen-Poiseuille Flow

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