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Consider a nearly spherical bubble of radius $a$ translating with velocity $\mathbf{U}$ through a liquid of dynamic viscosity $\mu$. Neglect internal gas viscosity and use exterior Stokes flow. Let the surfactant concentration be $C=C_0+C^{\prime }$ and linearize surface tension as $\gamma ={\gamma }_0-{\gamma }_1{C}^{\prime }$ with ${\gamma }_1>0$. Surface transport has surface diffusion coefficient $D_s$ and exchange-relaxation rate $k$.

In the bubble's translating frame, write its tangential surface velocity aswhere $\mathbf{n}$ points out of the bubble. The clean-bubble value is $A=-1/2$; a completely immobile interface has $A=0$. The dipolar surfactant distribution is $C^{\prime }=B\mathbf{U}\cdot \mathbf{n}$, withCombining that transport solution with the tangential Marangoni stress balance givesIncreasing $\lambda$ suppresses interfacial motion, whereas stronger diffusion or exchange reduces the concentration contrast and restores mobility. The linearization separately requires $|C^{\prime }|/C_0\ll 1$, or $2|A||\mathbf{U}|a/(k{a}^2+2D_s)\ll 1$. Small Reynolds number and capillary number are also needed for Stokes flow and a nearly spherical shape; a large immobilization parameter alone does not establish those assumptions.

In the common SDS-PAGE version, protein denaturation and binding of an anionic surfactant reduce differences due to folded shape and intrinsic electric charge. An electric field drives the resulting protein complexes through the gel; smaller chains generally migrate farther. A lane of reference proteins supplies positions for estimating apparent mass. This estimate depends on the separation conditions: it is not a direct weighing of a molecule, and different proteins can migrate together.