I have a physics problem that stuck me for a while because I cannot solve the mathematical equations below. Hopefully I can get some insights here:) Problem is basically a paste flows through a conical nozzle, and find the relation between inlet pressure ($P_0$) and output flow rate ($Q$). Equations have been simplified to the following ($r$ is the radical component in spherical coordinates):
- $$ 0=-\frac{\partial p}{\partial r}+\frac{3\tau_{rr}}{r}+\frac{\partial\tau_{rr}}{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta r}}{\partial\theta}+\frac{\tau_{\theta r}}{r}\cot{\theta} $$
- $$ 0=\frac{3\tau_{\theta r}}{r}+\frac{\partial\tau_{\theta r}}{\partial r} $$
- $$ \sqrt{\frac{3}{4}\tau_{rr}^2+\tau_{r\theta}^2} = \tau_y+K_s\left[\sqrt{3\left(\frac{u_r}{r}\right)^2+\frac{1}{4}\ \left(\frac{1}{r}\ \frac{\partial u_r}{\partial\theta}\right)^2}\right]^{n_s} $$
- $$ Q=-\int_{0}^{\alpha}{u_r\bullet2\pi r\sin{\theta}}rd\theta $$
Boundary conditions: $ \left.u_r=0\right|_{\theta=\alpha}$, $\left.\frac{\partial\tau_{\theta r}}{\partial\theta}=0\right|_{\theta=0}$, and $P=\left.P_0\right|_{r=R_0}, P=\left.0\right|_{r=R_1}$.
$u_r$ is the velocity of a unit volume element and it's a function of both $r$ and $\theta$. Nozzle geometry are constants ($R_0, R_1,\alpha$): inlet, outlet radius and half cone angle. And material properties are also constants ($\tau_y,K_s,n_s$): yield strength, flow consistency and flow index. The components of deviatoric stress tensor ($\tau_{rr}, \tau_{\theta r}$) are unknown.
Any comment is appreciated, or if you have concerns about the correctness of the equations, feel free to leave a message as well. Many thanks!!