Fluid Flow

Bingham Plastic Fluid

Bingham plastic is a material that behaves as rigid body at low stresses but flows as a viscous fluid at high stress. This behaviour is exhibited by slurries, suspensions of solids in liquids, paints, emulsions, foams, etc.

Bingham model is described by following relation.

τ = τo + μp γ

where, τ is shear stress, γ is shear rate, τo is called minimum yield stress and μp is called plastic viscosity.

Reynolds number for Bingham plastic fluid is defined as

Re = D V ρ / μp

where, D is pipe inside diameter, V is fluid velocity and ρ is fluid density.

For Laminar flow, friction factor is provided by Buckingham Reiner equation.

Friction factor laminar flow Buckingham Reiner

where, He is Hedstrom number and is calculated as

He = D²ρτo / μp²

For turbulent flow, an empirical relationship was developed by Darby and Melson.

Friction factor turbulent flow Darby and Melson

The friction factor for a Bingham plastic can be calculated for any Reynolds number from the equation.

f = ( fLm + fTm ) 1/m

where, fL is laminar flow friction factor and fT is turbulent flow friction factor. Factor m is calculated from following equation.

m = 1.7 + 40,000 / Re

Pressure drop is calculated as

ΔP = 2fρV² (L'/D)
L' = L + Le

where, L is the pipe length and Le is equivalent length due to loss in pipe fittings and is calculated as

Le = kD / 4f



  1. Chemical Engineering Fluid Mechanics, Ron Darby, 2nd Edition