Fluid Mechanics

Problem $\backslash$#$02$ Navier$-$Stokes Equations

A steady, axisymmetric, incompressible fluid of density p Flows radially $($in the r$-$direction$)$ outward owing to a pressure difference between two fixed porous, coaxial cylindrical shells of radii $\backslash ($ k R $\backslash )$ and $\backslash ($ R $\backslash ).$ Note that the velocity is not zero at the solid surfaces. Assume long cylinders along the z$-$axis normal to the plane with negligible end effects $($i$.$e$.\backslash (\backslash$frac$\{\backslash$partial$\}\{\backslash$partial z$\}\backslash )).$

$($A$)$ Simplify the equation of continuity to show that $\backslash (\backslash$Gamma$\_\{$v$\_\{$r$\}\}=\backslash )$ constant.

$($B$)$ Simplify the equations of motion for a Newtonian fluid of viscosity H $($I$.$e$.$ navier$-$stokes equations$).$