Bingham fluids: deformation and energy dissipation in triangular cross section tube flow

Abstract

Flow of Bingham plastics through straight, long tubes with non-circular cross-sections is studied by means of an analytical method that allows to model a wide spectrum of tube geometries. Shear stress normal to equal velocity lines, velocity field and plug zones are explored, in particular in a tube with equilateral triangular cross-section for small values of the Bingham number Bi, and they are compared with corresponding numerical solutions. We show that a circular plug is present at the center of the triangular tube cross-section, consistent with numerical simulations as well as with previous results in the literature, if calculations up to and including first order in the shape perturbation parameter $\epsilon$ are taken into account. However with the inclusion of the second order terms in the algorithm this structure is no longer present and no plug zone is predicted for the same pressure drop. We find that in that case normal shear stresses are always greater than the yield stress of the fluid. As a result, the central region becomes a pseudo-plug since it presents small but non-vanishing relative deformations and does not move as a rigid core. The energy dissipation function for the Bingham fluid flow is written in terms of natural coordinates. Its distribution depends only on the normal shear stress at any point with Bingham number as a parameter.

Appendix

Equation (27) is solved by decoupling variables in the form $w_2 (r,\theta )=R_2(r)+P_2(r)\cos {6\theta }$. The resulting equations for $R_2(r)\equiv R_{2a}/R_{2b}$ and $P_2(r)\equiv P_{2a}/P_{2b}$ were solved analytically using Wolfram Mathematica 10,

$$R_{2a}= {{Bi}} [{{Bi}}^4 (9 - 14 r) + 136 {{Bi}}^3 r + 48 {{Bi}}^2 (38 - 63 r) r 64 {{Bi}} r (90 r^2-29) + 320 r (2 - 15 r^3)]$$

(34)

$$\begin{aligned} R_{2b}& {}= 16A^2 r \end{aligned}$$

(35)

and

$$\begin{aligned} P_{2a}& {}= {{Bi}} \left\{ 9 ({{Bi}}-2) A\log (2 - {{Bi}}) (3 {{Bi}}^4 + 2504 {{Bi}}^3 r\right. \end{aligned}$$

(36)

$$\begin{aligned}&-\, 34524 {{Bi}}^2 r^2 + 123204 {{Bi}} r^3 - 128200 r^4 \nonumber \\&+\, 3 r \log {r}(230 {{Bi}}^3 - 2304 {{Bi}}^2 r+ 5760 {{Bi}} r^2 \nonumber \\&-\, 4000 r^3 + 27B (\log {r} - 2\log [1 -\frac{2r}{{{Bi}}}])))\nonumber \\&+\, 2 ({{Bi}} (-9 {{Bi}}^4 C + 4 {{Bi}}^2 (-485950 + {{Bi}} (354215 \nonumber \\&+\, {{Bi}} (5 {{Bi}} (34 + {{Bi}})-57723))) r + 12 {{Bi}} (1263470\nonumber \\&+\,{{Bi}} ({{Bi}} (3900 + (18299 - 13 {{Bi}}) {{Bi}})-696067)) r^2 \nonumber \\&+\,12 (-1943800 + {{Bi}} (19520+ {{Bi}} (677379\nonumber \\&+\, 5 {{Bi}} (4 {{Bi}}-23153)))) r^3 + 384600C r^4)\nonumber \\&-\, 243 {{Bi}} CrB\log ^2{r} -6 (-97190 + {{Bi}} (93379 \nonumber \\&+\, {{Bi}} ({{Bi}} (1912 + {{Bi}})-26193))) r B\log [2r-{{Bi}}] \nonumber \\&+\, 6 r \log {r} ({{Bi}}^3 (-97190+ {{Bi}} (72679 + {{Bi}} (-12738 \nonumber \\&+\, {{Bi}} (187 + {{Bi}})))) - 18 {{Bi}}^2 (-97190+ {{Bi}} (81859 \nonumber \\&+\, {{Bi}} ({{Bi}} (952 + {{Bi}})-18705))) r+ 72 {{Bi}} (-97190\nonumber \\&+\,{{Bi}} (86179 + {{Bi}} (-21513 + {{Bi}} (1312 + {{Bi}})))) r^2\nonumber \\&-\,80 (-97190+ {{Bi}} (88879 + {{Bi}} (-23268\nonumber \\&+\,{{Bi}} (1537 + {{Bi}})))) r^3+81 {{Bi}}CB \log [1 - \frac{2r}{{{Bi}}}]))\nonumber \\&-\,486 Ar({{Bi}}-2)(2 {{Bi}} (5 {{Bi}}^2 - 39 {{Bi}} r + 60 r^2)\nonumber \\&+\, 3B(\log {r} - \log [2r-{{Bi}}])) \text {Li}_2\left( \frac{2}{{{Bi}}}\right) \nonumber \\&\left. +\, 486 rB(2 {{Bi}}C - 3A({{Bi}}-2)\log [2 - {{Bi}}])\text {Li}_2\left( \frac{2r}{{{Bi}}}\right) \right\} \nonumber \\ P_{2b}=\, & {} [16A^2 r(3({{Bi}}-2)A\log [2-{{Bi}}]-2{{Bi}}C)] \end{aligned}$$

(37)

where

$$\begin{aligned} A=\, & {} 40 + ({{Bi}}-16) {{Bi}}\nonumber \\ B=\, & {} ({{Bi}} - 2 r) ({ {Bi}}^2 - 16 {{Bi}} r + 40 r^2)\nonumber \\ C=\, & {} 60 + {{Bi}} (5 {{Bi}}-39) \end{aligned}$$

(38)

In this, Li$_n(x)$ is the polylogarithm function.