Motion of Particles in Fluids

Particle Technology

Motion of Particles in Fluids

Study of how solid particles move through fluids. Separation processes depend on particle behavior in moving fluids.

Drag Force

The resistance force exerted by a fluid on a moving particle

Drag in Non-Viscous Fluid

No drag occurs; velocity gradients are infinite at the surface. Occurs due to pressure differences around the particle; no shear stress contribution.

Pressure Distribution Around Cylinder

Pressure is highest at stagnation points (A, D) and lowest at maximum velocity points (B, C).

Drag in Viscous Fluid

Includes shear stress effects; thickness depends on pressure gradient, boundary layer formation and flow separation influence drag.

Boundary Layer

The thin fluid region near a solid surface where velocity is retarded due to viscosity, gradually increasing from zero at the surface to free-stream velocity away from it.

Flow Separation

Occurs when boundary layer detaches due to rising pressure.

Reynolds Number (Re)

Dimensionless ratio of inertial to viscous forces: 𝑅𝑒 = 𝜌𝑢𝑑/𝜇

Stokes’ Law

Drag force for creeping flow (Re < 1): 𝐹𝑑 = 3𝜋𝜇𝑢𝑑

Surface Friction Drag

Drag due to viscous shear at particle surface; contributes ~2/3 of total drag at low Re.

Form Drag

Drag due to pressure differences around the particle; contributes ~1/3 of total drag at low Re.

Region A: Stokes’ law applies, Region B: slope changes progressively, Region C: Newton’s law applies, Region D: Turbulent boundary separation.

Drag Coefficient Regions

Region A

Re < 0.2

Region B

0.2 < Re < 1000

Region C

500–1000 < Re < 2×10⁵

Region D

Re > 2×10⁵

Terminal Falling Velocity

Constant velocity reached when gravitational force equals drag force.

Equal Settling Velocity

Condition where particles of different sizes/densities settle at the same velocity.

Free Settling

Occurs in dilute suspensions (<0.1% solids); particles fall independently without interference.

Hindered Settling

Occurs in concentrated suspensions; particle interactions increase drag and slow settling.

Motion of Bubbles/Drops

Internal circulation reduces drag and increases velocity.

Correction Factor (Q)

Accounts for internal circulation effects in bubbles/drops; depends on viscosity ratio μi/μ.

Viscosity Ratio is Large

μi /μ is _____ ⟶ Q ≈ 1: behaves like a rigid particle (low circulation)

Viscosity Ratio is Small

μi /μ is _____⟶ Q ≈ 1.5:strong internal circulation, higher velocity.

Surface Tension

Small droplets remain spherical; large droplets deform, increasing resistance.

Internal Circulation Effect

Can increase falling velocity by up to 50% compared to rigid spheres.

Terminal Velocity in Shear-Thinning Fluids

Settling is slow compared to Newtonian Fluids.

Non-Newtonian Fluids

Fluids with non-linear stress-strain relationships; particle motion differs from Newtonian fluids.

Modified Stokes’ Law

Drag force depends on correction factor Y, which increases as the fluid becomes more shear-thinning (lower n). Adapted drag law for particles in non-Newtonian fluids.

Power-Law Fluids

Shear-thinning fluids where viscosity decreases with shear rate; particles may remain suspended depending on fluid consistency.

Yield-Stress Fluids

Fluids that require a minimum stress to initiate flow; while drag coefficient is solved iteratively.

Herschel–Bulkley Model

Describes yield-stress fluids combining yield stress and shear-thinning behavior.

Static Equilibrium Condition

In Herschel–Bulkley fluids, particles remain suspended if dimensionless parameter Z lies between 0.04–0.2.

Static Equilibrium Condition

Shear Stress Relation

Drag Coefficient (Creeping Flow)

Modified Stokesʼ Law

Terminal Settling Velocity for Shear-Thinning Fluids