Viscoplastic squeeze flow between two identical infinite circular cylinders

Direct numerical simulations of closely interacting infinite circular cylinders in a Bingham fluid are presented, and results compared to asymptotic solutions based on lubrication theory in the gap. Unlike for a Newtonian fluid, the macroscopic flow outside of the gap between the cylinders is shown to have a large effect on the pressure profile within the gap and the resulting lubrication force on the cylinders. The presented results indicate that the asymptotic lubrication solution can be used to predict the lubrication pressure only if the surrounding viscoplastic matrix is yielded by a macroscopic flow. This has implications for the use of subgrid-scale lubrication models in simulations of noncolloidal particulate suspensions in viscoplastic fluids.

Figure 1

Schematic showing the problem geometry for the entire flow system (${\mathrm{\Omega }}_1$) investigated through numerical methods and the reduced system (${\mathrm{\Omega }}_2$) investigated with both analytical and numerical methods.

Figure 2

Velocity magnitude plots for increasing $\text{Bn}$ (left to right) with quiescent conditions (top row) and macroscopic shear rate $\dot{\gamma }=5$ (bottom row).

Figure 3

Pressure contours for the two-cylinder system in macroscopic shear flow (left) and quiescent (right) with $\text{Bn}=1000$.

Figure 4

Left: Ratio of center line pressure to surface pressure as a function of the local gap width for $\text{Bn}=0$ (circles), 50 (diamonds), 500 (crosses), 1000 (triangles), and 2000 (squares). Right: Pressure distributions over the entire cylinder surface for the limiting $\text{Bn}=0$ and $\text{Bn}=2000$ quiescent flow cases as a function of angle away from the gap center.

Figure 5

Pressure profile along the gap center line for quiescent (diamonds), sheared (circles), and reduced domain (square) systems with viscoplastic lubrication solution overlayed (solid line) with $\text{Bn}=50,500,1000,2000$ in plots (a) through (d), respectively.

Figure 6

Stacked area plots of the total drag force on a cylinder as a function of Bn for (a) quiescent and (b) shear flow background conditions. Overlaid are the predictions from viscoplastic lubrication theory (squares).

Figure 7

Local shear rates in the gap center for quiescent (squares) and sheared (diamonds) systems for $\text{Bn}=0,50,500,1000,2000$.

Figure 8

(a) Color map of ${log}_{10}\left(\dot{\gamma }\right)$ and (b) normalized second invariants of the velocity gradient tensor with unyielded areas masked in gray with $\text{Bn}=1000$ (quiescent case).

Figure 9

Left: rate of mechanical dissipation per unit $\mathrm{\Lambda }$. Right: stacked area plot showing the rate of work done by viscous dissipation in the shear and plastic regions, scaled by $\mathcal{W}=FV$, with markers indicating the power required to move the cylinders with the approach velocity (diamonds), the total viscous dissipation in the system (dashed line), the viscous dissipation in the gap region of the full system (squares), and the total viscous dissipation in the reduced system (triangles).