Surface traction in translating-sphere Stokes flow

On the sphere, pressure and the normal part of the viscous stress cancel, leaving uniform traction $-3\mu U/(2a)$.

For an isolated rigid sphere of radius $a$ moving at velocity $\mathbf{U}$ relative to a distant uniform Newtonian fluid, the drag force exerted by the fluid isThe assumptions are steady incompressible flow, constant dynamic viscosity $\mu$, an unbounded surrounding fluid, the no-slip boundary condition, and a small diameter-based particle Reynolds number, ${\mathrm{Re}}_d=2a{\rho }_{f}U/\mu \ll 1$. Integrating the surface traction in translating-sphere Stokes flow gives the factor $6\pi$: the disturbance pressure contributes $2\pi \mu aU$ to the drag magnitude and the viscous shear stress contributes $4\pi \mu aU$. See Richard Fitzpatrick, Axisymmetric Stokes Flow Around a Solid Sphere, equations 10.120–10.129.

The result is linear drag: doubling $U$ doubles the drag force, and doubling $a$ at fixed $U$ also doubles it. The required power is $6\pi \mu a{U}^2$, all supplied to viscous dissipation. Density enters the check of the particle Reynolds number, although it does not appear in the leading drag formula.

Nearby walls or other particles change the hydrodynamic resistance matrix and invalidate the isolated-sphere coefficient, even at very small Reynolds number. Finite Reynolds number introduces inertia corrections; the Oseen approximation resolves the distant region where the leading Stokes flow approximation ceases to be uniform. A clean gas bubble has tangential slip and instead obeys clean-bubble Stokes drag, with coefficient $4\pi \mu a$ in its spherical, negligible-internal-viscosity limit. See H. K. Moffatt, Six Lectures on General Fluid Dynamics, section 3.4 for the sphere solution and its small-Reynolds number limitation. The continuum mechanics and boundary assumptions must also remain valid as the particle becomes very small.

Balancing this Stokes drag law against gravity and buoyancy gives the Stokes settling velocity of an isolated rigid sphere.

Drag on a small particle when the gas mean free path is not negligible relative to its size. Continuum no-slip Stokes friction then needs a slip or kinetic correction depending on pressure and gas-surface interactions.

The exponential inertial relaxation time ${\tau }_p=m/\zeta$ for linear drag. A no-slip sphere has ${\tau }_p={\rho }_p{d}^2/(18\eta )$; particle density enters this time, whereas fluid density enters the fluid Reynolds number.

The dimensionless factor $C$ compares a corrected linear drag resistance with the continuum Stokes's law: dividing by $C>1$ reduces the resistance. The oil drop charge readout and calibration article explains why this matters for electric charge inference.

A spherical clean bubble with negligible internal dynamic viscosity has zero tangential traction and permits tangential slip. In the creeping-flow, negligible-deformation limit its drag is $-4\pi \mu a\mathbf{U}$, versus $-6\pi \mu a\mathbf{U}$ for a no-slip rigid sphere.

A spherical clean bubble with constant gas mass density and prescribed dissolution law $\dot{M}=-4\pi aD{\rho }_a$ obeys $\dot{a}=-D/a$. With negligible gas dynamic viscosity, instantaneous terminal velocity is $\dot{z}=\Delta \rho g{a}^2/(3\mu )$. Eliminating time yields the displayed radius-height relation and maximum rise $z_{\mathrm{diss}}=\Delta \rho g{a}_0^4/(12\mu D)$. This uses an instantaneous force balance; a literally constant speed during shrinkage instead gives $a^2$ linear in height. The Reynolds number must remain small for the Stokes flow drag model to apply.

For a clean inviscid bubble a distance $d\gg a$ below a gravity-restored approximately flat free surface, the surface displacement scales as $F/(\rho g{d}^2)=O(a^3/d^2)$. Zero normal velocity and zero tangential traction are imposed to leading order by an opposite normal image Stokeslet at distance $2d$ from the bubble. Its incident velocity is $-F/(8\pi \mu d)$, which gives the displayed correction at fixed buoyancy. The image can be represented by a same-radius inviscid drop of density $2\rho$ in liquid of density $\rho$. Its deformation-producing strain is $O(F/(\mu d^2))$, so the leading fractional capillary deformation is $O(\mathrm{Bo}(a/d{)}^2)$; small Bond number is the conventional stronger sufficient criterion.

A buoyantly rising inviscid bubble in otherwise quiescent unbounded liquid has an exterior Stokeslet with $F=4\pi \mu aU=4\pi \rho g{a}^3/3$. Its tangential traction vanishes on the sphere. The dynamic normal traction is $-3F\cos \theta /(4\pi a^2)$, while the hydrostatic normal-traction variation is $\rho ga\cos \theta$. Their dipoles cancel at the terminal velocity, leaving a constant total normal stress compatible with uniform gas pressure even when surface tension is zero. This is an exact spherical solution in the stated Stokes model, not a proof of shape stability under arbitrary perturbations.

The Stokes–Einstein relation connects translational diffusion and linear drag through $D=k_{B}T/\zeta$. For an unbounded spherical particle, $\zeta =6\pi \mu a$ and $D=k_{B}T/(6\pi \mu a)$.

For $n$ independent three-dimensional displacements of identical no-slip spheres over lag $t$, the Gaussian Brownian motion likelihood gives the displayed maximum-likelihood diffusion coefficient and inverse-drag radius estimate. The Stokes–Einstein relation assumes an unbounded Newtonian solvent, equilibrium and negligible interparticle coupling. Use squared distances, rather than squaring the mean distance. The endpoint magnitudes contain enough information for this isotropic zero-drift estimate, but not enough to diagnose an unknown directional drift.

Each Cartesian Brownian increment has variance $2Dt$. Summing their normalized squares gives a chi-squared distribution with $3n$ degrees of freedom. Hence $\hat{D}$ is unbiased and has relative standard deviation $\sqrt{2/(3n)}$. Its inverse is not unbiased: for $3n>2$, $\mathbb{E}[1/\hat{D}]=[3n/(3n-2)]/D$. Thus the direct inverse-diffusivity radius estimate has appreciable finite-sample bias when only a few trajectories are observed. Measurement noise or drift adds separate systematic error.

For dilute isolated spheres in a Newtonian solvent, $D\eta a/T=k_B/(6\pi )$. An experimental check independently measures temperature, viscosity and particle size, then obtains diffusion from a displacement variance or a scattering decay. Drift, walls, concentration effects, aggregation and a hydrodynamic radius different from a structural radius can defeat an unqualified comparison.