(Solved): Flow of a Non-Newtonian Fluid. Fluid flows through a cylindrical pipe of constant crosssection and ...

Flow of a Non-Newtonian Fluid. Fluid flows through a cylindrical pipe of constant crosssection and radius $R$, under steady state conditions. The stress tensor for the fluid obeys $?=?p\mathbf{I}+2?\mathbf{e}+4?\mathbf{e}?\mathbf{e}$ where $\mathbf{I}$ is a tensor whose components are given by $I_{\mathrm{ij}}={?}_{\mathrm{ij}}$, where ${?}_{\mathrm{ij}}$ is the Kronecker delta, and $e=\frac{1}{2}\left(?v+?v^T\right)$ is the rate of strain tensor. The pressure gradient is constant along the axis of the pipe, $\frac{?p}{?z}=?K$ The parameters $?,?$, and $K$ are constant. This problem is analogous to the flow of a Newtonian fluid through a pipe, except that the fluid obeys the alternative expression for stress given above. There are no body force effects. a). Derive the velocity profile, $v_{\mathrm{z}}(r)$, for the flow. It may be helpful to recall that $\frac{df}{dr}+\frac{f}{r}=\frac{1}{r}\frac{d(rf)}{dr}$ where $f$ is a function of $r$, i.e. $f=f(r)$. In addition, the following information is given $\begin{array}{l}(???{)}_r=\frac{1}{r}\frac{?\left(r{?}_{rr}\right)}{?r}+\frac{1}{r}\frac{?{?}_{r?}}{??}?\frac{{?}_{??}}{r}+\frac{?{?}_{rz}}{?z}\\ (???{)}_{?}=\frac{1}{r}\frac{?{?}_{??}}{??}+\frac{?{?}_{r?}}{?r}+\frac{2{?}_{r?}}{r}+\frac{?{?}_{?z}}{?z}\\ (???{)}_z=\frac{1}{r}\frac{?\left(r{?}_{rz}\right)}{?r}+\frac{1}{r}\frac{?{?}_{?z}}{??}+\frac{?{?}_{zz}}{?z}\end{array}$ b). Derive the pressure distribution in the pipe to within an arbitrary constant. In doing this part, check dependence of the pressure on all coordinate directions, or write a good justification as to why the pressure should not depend on a particular coordinate variable. Show all work.