Smoluchowski Effective Viscosity Equation and Calculator

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Smoluchowski Effective Viscosity Equation and Calculator

Fluids Flow Design and Engineering
Hydraulic & Pneumatic Design Engineering

For dilute suspensions of charged spheres the Smoluchowski Equation states D is the dielectric constant of the suspending fluid, k, the specific electrical conductivity of the suspension, the electrokinetic potential of the particles, and R the particle radius. Surface charges are not uncommon in stable suspensions. Other, less well understood, surface forces are also important and frequently cause the particles to form loose aggregate. Here again, non-Newtonian behavior is encountered.

Eq. 1
µ_eff = µ_o · [ 1 + 2.5 · θ · ( 1 + ( ( D · ζ ) / ( 2 · π · R ) )² / ( µ_o · k_e ) ) ]

µ_o = Viscosity of the Suspension (cP)
θ = Volume Fraction of Spheres (dimensionless)
D = Dielectric Constant (A² · s²/cm³)
R = Particle Radius (cm)
ζ = Electro kinetic Potential of the Particles (J)
k_e = Specific Electrical Conductivity of the Suspension (ohm/cm)
µ_eff = Effective Viscosity (cP)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley & Sons, Chapter: 1, Page: 34.