$($v$)$

rAs

$V=2rrz$

PAc

$\frac{{?}^{2r}[r{|}_r-r{|}_{r+r}]}{2z}\frac{2rr[P{|}_z-P{|}_{z+z}]}{2+rZ}$

$\frac{1}{r}\frac{d}{dr}(r)+\frac{dP}{dz}=0$

Define a small, finite control volume $($CV$)$ in the material of interest. The control volume must be very thin $()$ in the direction of transport.

Apply the conservation law to the CV$.$ For the conserved quantity of interest define the flow $($flux $x$ area$)$ in and out, everything generated or consumed, and everything accumulated within the CV$.$

Normalize by the volume of the CV$.$ Divide the entire equation by the finite volume defined in step $1.$

Take the limit of the resulting equation as the finite length scales approach zero $().$

Replace the flux in the differential with one of the four flux laws: Newton's, Fourier's, Fick's, or Ohm's. A direct substitution into the first order differential should yield a second order differential which can be solved for the desired profiles.

Solve the differential equation and apply appropriate boundary conditions to yield the solution.

Please include pictures with your solutions and any helpful advice to solve the problem.