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# Bingham Fluid Flow in a Circular Tube

A Bingham fluid acts like a rigid body at low shear stress but flows like a viscous fluid at high shear.
Consider the laminar flow of a Bingham fluid in a long circular tube of radius and length . The fluid flows under the influence of a pressure difference . The fluid density is .
One can derive an analytical expression for the velocity, given by http://www.syvum.com/cgi/online/serve.cgi/eng/fluid/fluid807.html:
for ,
for ,
where .
The mass flow rate is given by:
, with .
Here is the zero–shear viscosity and is the yield stress. You can vary these two parameters.
For a Newtonian fluid, the mass flow rate is given by the Hagen–Poiseuille equation: .
This Demonstration plots the velocity profile (blue curve) and gives the value of the corresponding mass flow rate. In addition, the velocity profile for a Newtonian fluid is plotted (red curve). The viscosity, , of the Newtonian fluid is chosen such that both fluid mass flow rates are equal. The value of the viscosity, , is also shown.

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