Analytic solutions for the morphology of the final state of the front.

This section demonstrates how to use the Shape_result class which allows to compute the shape of the flow front at the final step of the flow.

Coussot’s model

Coussot et al proposed a formula in 1996 to approximate the geometry of the frontal lobe of a flow at a final instant for rheological test on an inclined surface:

\[D = - H - \ln{(1 - H)}\]

with \(D\) and \(H\) being the normalized version of the distance of the front \(d\) and the fluid depth \(h\), obtained by computing:

\[D = \frac{\rho g d (\sin(\theta))^2}{\tau_c \cos(\theta)} \text{ and } H = \frac{\rho g h \sin(\theta)}{\tau_c}\]

where

• \(h\): fluid depth.

• \(x\): spatial dimension.

• \(g\): gravitational acceleration.

• \(\rho\): fluid density.

• \(\tau_c\): threshold constraint.

• \(\theta\): slope of the surface.

In a case where \(\theta = 0\), the equations are slightly different:

\[D^* = \frac{{H^*}^2}{2}\]

with:

\[D^* = \frac{\rho g d}{\tau_c} \text{ and } H^* = \frac{\rho g h}{\tau_c}\]

References

Coussot, P., Proust, S., & Ancey, C., 1996, Rheological interpretation of deposits of yield stress fluids, Journal of Non-Newtonian Fluid Mechanics, v. 66(1), p. 55–70, doi:10.1016/0377-0257(96)01474-7.

Examples

Coussot’s model example

Coussot's model example