On the surfactant mass balance at a deforming fluid interface

The amount of surfactants ~surface active agents! adsorbed onto a fluid interface affects its surface tension. Thus the distribution of surfactants must be determined to find the jump in the normal and tangential stresses across the interface. Scriven ~see also Aris, Slattery, and Edwards et al.! uses differential geometry to derive the correct surface balance equation for an arbitrary surface coordinate system. Also invoking differential geometry, Waxman develops a correct form in ~‘‘fixed’’! surface coordinates that advance only normal to the surface. To arrive at this balance without appealing to differential geometry, Stone presents a simple physical derivation which leads to a form of the mass balance which is easy to solve numerically. Unfortunately, Stone’s derivation leaves the nature of the unsteady time derivative ambiguous. Here we follow the spirit set in Stone to derive geometrically the surface balance in a way that keeps the nature of the time derivative explicit. We verify that in Stone’s form the time derivative must hold the fixed coordinates constant, as the numerical implementation of this form of the mass balance actually do. We also derive a new form valid in an arbitrary surface coordinate system. Consider a fixed point A on a fluid surface with local normal n as in Fig. 1. We locate any two perpendicular planes which intersect along n. The intersection of each of these planes with the surface near the point A define curves whose unit tangents are t1 and t2 . By construction ]t1/]s152~1/R1!n and ]t2/]s252~1/R2!n, where ds1 and ds2 are differential arcs and R1~.0! and R2~.0! are the radii of curvature of the curves. Geometrically, these differential arcs are ds15R1df1 and ds25R2df2 , where df1 and df2 are the differential angles in the figure, and ]t1/]f152n and ]t2/]f252n. Thus in this locally orthogonal system, the components of the surface metric tensor aab are: Aa115R1 , a1250, and Aa225R2 and the diagonal elements simply act as scale factors. These arcs define a patch of area dA5Aa11Aa22df1df25Aadf1df2 where a is the determinant of the metric tensor. The diagonal components of the curvature tensor bab are defined by @]ta /]fa#–n 5 baa /Aaaa ~no sum on a!; so b1152R1 and b2252R2 . The curvatures are negative because as drawn in Fig. 1 both arcs are concave down with respect to the normal. If U is the instantaneous material velocity vector at the fixed point, its components along $n,t1 ,t2% are U5Us(1)t11Us(2)t21Wn, where W is the normal component and Us(1) and Us(2) are the physical components tangent to the surface. The fixed point advances along the normal ~n! as shown in the Fig. 1 a distance WDt so that the patch perimeters have lengths (R11WDt)df1 and (R21WDt)df2 at the time t1Dt; thus the change in area of the patch is WDt(R11R2)df1df2 and the per unit area per unit time rate of change is