Bingham¶

Module: fluid

Category: materialprop

Type string: "Bingham"

Parameters¶

Name	Description	Default	Units
`mu`	shear viscosity	0	[]
`tauy`	yield stress	0	[]
`n`	exponent	1	[]

Description¶

The material type for a Bingham fluid ¹ is Bingham.

The viscous shear stress for this material model is

\[ \boldsymbol{\tau}=2\mu\mathbf{D} \]

where

\[ \mu=\mu_{\infty}+\frac{\tau_{y}}{\dot{\gamma}}\left(1-e^{-n\dot{\gamma}}\right) \]

Here, \(\dot{\gamma}=\sqrt{2\mathbf{D}:\mathbf{D}}\) is the engineering shear rate. In the limit as \(\dot{\gamma}\to0\) the viscosity is given by \(\mu=\mu_{\infty}+n\tau_{y}\). If we define the scalar shear stress \(\tau=\sqrt{\boldsymbol{\tau}:\boldsymbol{\tau}/2}\), if follows that \(\tau=\mu_{\infty}\dot{\gamma}+\tau_{y}\left(1-e^{-n\dot{\gamma}}\right)\). Effectively, this constitutive model represents a bilinear response for \(\tau\) versus \(\dot{\gamma}\), with a slope of \(\mu_{\infty}+n\tau_{y}\) when \(\tau<\tau_{y}\) and \(\mu=\mu_{\infty}\) when \(\tau>\tau_{y}\). The exponential function rounds the corner at the intersection of these two lines. For an ideal Bingham fluid one would need to let \(n\to\infty\). In practice, values of \(n\) between 5 and 50 work well, with the lower end of this range producing faster convergence of the finite element solution.

Example:

<viscous type="Bingham">
    <mu>1</mu>
    <tauy>40</tauy>
    <n>40</n>
</viscous>

Papanastasiou, Tasos C, "Flows of materials with yield", Journal of Rheology 31, 5 (1987), pp. 385--404. ↩