Often we are interested not just in the shear viscosity of a fluid, but the extensional viscosity of a fluid as well. Let's study an ideal uniaxial extensional flow shown in the following figure that occurs when a cylindrical fluid filament is stretch slowly in the absence of body forces such that the velocity profile becomes

$u_z={}^{}z$

Here, ${}^{}$ is the constant elongation rate imposed on the fluid

a$)$ Using continuity and arguments of symmetry, find the theta and $r$ components of the velocity profile, $u_r=?,u_{}=?$

b$)$ Determine all the terms in the shear rate tensor, ${}_{{?}_{?}}^{}.$

c$)$ Show further that for an incompressible Newtonian fluid in simple elongation flow that

${}_{rr}={}_{}=-{}^{}$

${}_{?}==2{}^{}$

d$)$ Next use the equations of motion to gain further information about the components of the stress tensor. For steady state flow with negligible inertial terms and no body forces, show that the components of the stress tensor $T=\frac{-}{p}$I $+$ are constant throughout the fluid.

e$)$ Show that the results from $($b$)$ lead to the conclusion that

$T_{?}=-p_0={}_{?}=-{}_{}=3(\frac{d{u}_z}{dz})=3{}^{}$