The Effect of Slip on the Discharge From Partially Filled Circular and Fully Filled Lens and Figure 8 Shaped Pipes

Abstract

In this work, we examine the effect of wall slip for a gravity-driven flow of a Newtonian fluid in a partially filled circular pipe. An analytical solution is available for the no-slip case, while a numerical method is used for the case of flow with wall slip. We note that the partially filled circular pipe flow contains a free surface. The solution to the Navier–Stokes equations in such a case is a symmetry of a pipe flow (with no free surface) with the free surface as the symmetry plane. Therefore, we note that the analytical solution for the partially filled case is also the exact solution for fully filled lens and figure 8 shaped pipes, depending on the fill level. We find that the presence of wall slip increases the optimal fill height for maximum volumetric flow rate, brings the “velocity dip” closer to the free surface, and increases the overall flow rate for any fill. The applications of the work are twofold; the analytical solution may be used to verify numerical schemes for flows with a free surface in partially filled circular pipes, or for pipe flows in lens and figure 8 shaped pipes. Second, the work suggests that flows in pipes, particularly shallow filled pipes, can be greatly enhanced in the presence of wall slip, and optimal fill levels must account for the slip phenomenon when present.

Channel flow in an inclined circular tube filled with a porous medium

“…1.4 0.0463 0.0453 0.0462 0.0454 0.0429 0.0353 0.01617 0.00578 (0.0056) 1.5 0.0264 0.0264 0.0264 0.0260 0.0250 0.0215 0.00111 0.00423 (0.0041) 1.6 0.0130 0.0130 0.0130 0.0129 0.0125 0.0113 0.00667 0.00282 (0.0026) 1.7 0.0051 0.0051 0.0051 0.0051 0.0050 0.0047 0.00325 0.00160 (0.0014) 1.8 0.0013 0.0013 0.0013 0.0013 0.0013 0.0013 0.00105 0.00064 (0.0003) 1.9 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00012 0.00010 (0.0000) 2.0 0 0 0 0 0 0 0 0 Greenwell and Wang (1980). Value with asterisk is from Irvine and Fullard (2016). Value with hat is from Wang (2000).…”

“…The pure viscous flow in a partially filled tube has been studied before, but only three sources considered the maximum flow rate, i.e. Greenwell and Wang (1980), Wang (2000) and Irvine and Fullard (2016). These limited reports are compared in table 3.…”

“…Conformal mapping was used by Duck (1979) and Greenwell and Wang (1980). Bipolar coordinates and various forms of series or integrals were used by Chaudhury (1964), Buffham (1968), Guo and Meroney (2013), Fullard and Wake (2015), Irvine and Fullard (2016). Internal matching was used by Wang (2000).…”