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## Homework Statement

I am trying to derive Poiseuille's equation using an infinitesimal volume element of the pipe in cylindrical coordinates as seen below. I am using x for the flow direction, u for the velocity in the flow direction, and μ for the viscosity constant.

https://www.physicsforums.com/attachment.php?attachmentid=69754&stc=1&d=1400000671

## Homework Equations

I know the viscous stress varies radially by τ = μ(du/dr)

This means that the viscous force on one face of my infinitesimal volume element is μ(du/dr)

_{r}r δ[itex]\phi[/itex] δx

And I think the viscous force on the opposite face is μ(du/dr)

_{r+δr}(r+δr) δ[itex]\phi[/itex] δx

Now I want to subtract one of these forces from the other to get the net viscous force on the infinitesimal element.

## The Attempt at a Solution

When I attempt to do this, I am left with two radial derivatives evaluated at different points. I get:

μ ( (du/dr)

_{r+δr}r δ[itex]\phi[/itex] δx + (du/dr)

_{r+δr}δr δ[itex]\phi[/itex] δx - (du/dr)

_{r}r δ[itex]\phi[/itex] δx)

Now I'm not exactly sure how to use Taylor's Theorem to get all of the derivatives evaluated at the same point and combine terms. Thanks.