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.
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.