Bingham flow
Bingham flow
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The pictures you see below represent the velocity profile and the module of the strain tensor of a Bingham flow through a tube of squared section.
For simplication, the following assumptions have been made on the Navier--Stokes--Bingham equations:
The fluid velocity along the solid boundary is zero;
There is no external distribution of forces such as the gravity or a magnetic field, instead the fluid is driven by a constant pressure drop C = 16;
At low Reynolds number, which we assume here, the viscous effects dominate the inertial effects;
In the first place, we are interested to the steady version.
Typically the relationship between the stress tensor and the strain tensor D(u) of a viscous fluid (eg. water) is linear. Along the boundary |D(u)| is at its maximum to maintain the no-slip boundary. As we move away from the boundary, the velocity increases gradually driven by the pressure drop until reaching the pic right in the middle of the tube. This explains the parabolic shape of the profile.Â
In the case of a viscoplastic fluid (here a Bingham fluid, e.g. Snow, Mud, Toothpaste), this relationship is nonlinear. The velocity is still increasing gradually as we move away from the boundary but the maximum is attained not in the middle but once |D(u)| drops below a certain value B called the strain threshold. Two regions take form in the tube, in the light region the fluid behaves like a viscous fluid and in the dark region it behaves like a rigid body. The dark areas increase with B and can cover the whole tube, which can eventually obstruct the circulation of the flow.
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