View source for Bingham fluid

A material for which, in uni-directional shear flow, the shear stress <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105501.png" /> and the shear rate (velocity gradient) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105502.png" /> satisfy the constitutive law

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105504.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105505.png" />, with the obvious extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105506.png" />. This can be written in terms of an  "effective viscosityeffective viscosity"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105507.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105508.png" />, taking the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b1105509.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b11055010.png" /></td> </tr></table>

This model is seen to incorporate a characteristic stress, the  "yield stress" , below which the material is regarded as absolutely rigid.

The Bingham model is a reasonable description of many pastes, slurries and gels; such materials include grease, paint, mud, lava, and many food and other consumer products.

The extension of the model to a fully three-dimensional description was derived by J.G. Oldroyd [[#References|[a1]]], who proposed that the unyielded material be regarded as a linear elastic solid and showed that the effective viscosity and yield criterion must be

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b11055011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b11055012.png" /> is the deviatoric stress tensor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110550/b11055013.png" /> is the rate of strain tensor.

A review of elementary flow problems can be found in [[#References|[a2]]]. The attempt to solve problems with more complex kinematics has led to theoretical and computational difficulties, stemming from over-idealized modelling (in (a1)) of the fluid below the yield point.

One should expect the flow field to contain yielded and unyielded (rigid) zones, separated by yield surfaces whose location is not known a priori. These surfaces would be difficult to track numerically and ad hoc smoothed out models have been proposed as a remedy in finite-element simulations [[#References|[a3]]], [[#References|[a4]]].

These models permit some viscous creep of the unyielded material, as a byproduct, so to speak; but there is a general theoretical need for some such relaxation of the model. For in asserting that the unyielded material is absolutely rigid, one is deprived of any means of determining the stress distribution within it, since the stress equilibrium equations are statically indeterminate, in general ([[#References|[a2]]] gives all the obvious exceptions). This indeterminacy makes it impossible to give a satisfactory discussion of the so-called squeeze flow paradox, for example [[#References|[a5]]]; here, Bingham material squeezed between parallel discs should be rigid on the central plane (because the shear stress vanishes) but must move radially outwards. In this context, the bi-viscosity model (in which the unyielded fluid is treated as a Newtonian liquid of large viscosity) has been used to obtain approximate analytical solutions [[#References|[a6]]], and so clear up some of the difficulties.

Some experimental justification for the use of smoothed-out models has been provided in [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.G. Oldroyd,   "A rational formulation of the equations of plastic flow for a Bingham solid"  ''Proc. Cambridge Philos. Soc.'' , '''43'''  (1947)  pp. 100–105</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.B. Bird,   G.C. Dai,   B.J. Yarusso,   "The rheology and flow of viscoplastic materials"  ''Rev. Chem. Eng.'' , '''1'''  (1983)  pp. 1–70</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Bercovier,   M. Engelman,   "A finite element method for incompressible non-Newtonian flows"  ''J. Comp. Phys.'' , '''36'''  (1980)  pp. 313–326</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.C. Papanastasiou,   "Flows of materials with yield"  ''J. Rheol.'' , '''31'''  (1987)  pp. 385–404</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.G. Lipscomb,   M.M. Denn,   "Flow of Bingham fluids in complex geometries"  ''J. Non-Newton. Fluid Mech.'' , '''14'''  (1984)  pp. 337–346</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S.D.R. Wilson,   "Squeezing flow of a Bingham material"  ''J. Non-Newton. Fluid Mech.'' , '''47'''  (1993)  pp. 211–219</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H.A. Barnes,   K. Walters,   "The yield stress myth"  ''Rheol. Acta'' , '''24'''  (1985)  pp. 323–326</TD></TR></table>