# Bingham Fluid Flow in a Circular Tube

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 .

R=0.01m

L=1m

ΔP=100N/

2

m

ρ=0.75kg/l

One can derive an analytical expression for the velocity, given by http://www.syvum.com/cgi/online/serve.cgi/eng/fluid/fluid807.html:

v

z

ΔP

2

R

4L

μ

0

2

r

R

τ

0

μ

0

r

R

R⩾r⩾

r

0

v

z

ΔP

2

R

4L

μ

0

2

r

0

R

r⩽

r

0

where =.

r

0

2L

τ

0

ΔP

The mass flow rate is given by:

w=2πρrdr=1-+

R

∫

0

v

z

ΔPρπ

4

R

8L

μ

0

4

τ

0

3

τ

R

1

3

4

τ

0

τ

R

τ

R

ΔPR

2L

Here is the zero–shear viscosity and is the yield stress. You can vary these two parameters.

μ

0

τ

0

For a Newtonian fluid, the mass flow rate is given by the Hagen–Poiseuille equation: .

w=

ΔPρπ

4

R

8μL

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.

μ

μ