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I got an physical problem in paste extrusion that I am unable to solve:

A pasty material flows through a conical nozzle. The material follows Hurschel-Bulkley flow model (H-B model): $\tau=\tau_y+K(\dot\gamma)^n$, where $\tau_y$ is yield strength, $K$ is flow consistency and $n$ is flow index (they are all known values). The nozzle geometry is also known (inlet and outlet radius $R_0$ and $R_1$, half cone angle $\alpha$).
Find the analytic relation between inlet pressure $P$ and outlet flow rate $Q$.

Assumptions: (1) material is homogeneous, isotropic, incompressible; (2) laminar, steady state flow, no inertia; (3) No boundary slippery; (4) isothermal, the effects of heat generated from viscous flow can be neglected; (5) Von-Mises yield criterion.

! This is a 2D flow problem due to the symmetry of nozzle. Through literature I am aware of, people have been working on paste extrusion problems mainly via two ways:
One is originally from Snelling's paper dates back to 1960 (Snelling, G.R. and J.F. Lontz, 1960). In their work they used spherical coordinates and assumed the material in the conical nozzle all flows towards the apex, and that the material on a virtual spherical cap with same distance to the apex is flowing at the same speed (so velocity is only dependent on $r$ in spherical coordinates). The assumption was also used by A.P. Metzner (1970). And later, M.J. Adams and R.A. Basterfield (2005) used H-B model and Von-Mises yield criterion to derive the extrusion equation in conical nozzle. However, they seemed to neglect shear flow, and only considered elongational flow. And the assumption basically implies the material at the wall of nozzle flows at the same speed with that at centerline.
Another approach is by F.N. Cogswell (1972) who derived the conical flow using cylindrical coordinates where he considered both shear and elongational flow. But they neglected yield strength of the material which made the problem way easier to solve...

I am by no means an expert in mathematics. I tried to introduce H-B model into Cauchy momentum equation, while ended up with equations I am unable to solve. I am wondering if there is a way to get the analytic expression of flow rate $Q$ in this case?

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  • $\begingroup$ Are you willing to make the approximation that the included angle of the cone is small? $\endgroup$ – Chet Miller May 29 '18 at 3:27
  • $\begingroup$ Yes I would like to, if there is no trivial way around. The actual half-cone angle of the nozzle we use is ~5 degree. $\endgroup$ – Tao May 31 '18 at 14:35
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For a small convergence angle like 10 degrees, what you do is assume that, locally, the diameter is constant, and express the pressure gradient in terms of the volumetric flow rate and the diameter. The convergence is taken into account by integrating the pressure gradient axially then with respect to the axially varying diameter.

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  • $\begingroup$ Thank you Chester for your answer. The flowrate - pressure drop relation in a cylindrical nozzle is: $Q=\pi R^3(\frac{\bigtriangleup PR}{2KL})^{\frac{1}{n}}\frac{n}{n+1}\left [ (1-X)^{\frac{n+1}{n}}-\frac{2n}{2n+1}(1-X)^{\frac{2n+1}{n}}+\frac{2n^2}{(2n+1)(3n+1)}(1-X)^{\frac{3n+1}{n}} \right ]$, where $X=\frac{2τ_y L}{∆PR}$. Now I replace L with $dl$, and $R=R_0 - l tan \theta$, where $R_0$ and $\theta$ are the inlet radius and half-cone angle, both constants, and then integrate P wrt dl. The problem I encounter is the mathematical difficulty in intergrating this equation. $\endgroup$ – Tao Jun 4 '18 at 19:18
  • $\begingroup$ The equation should really be written in terms of dP/dL. You can solve this nonlinear equation of dP/dL at each value of R, and integrate it numerically. $\endgroup$ – Chet Miller Jun 4 '18 at 20:41

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