On the surfactant mass balance at a deforming fluid interface
On the surfactant mass balance at a deforming fluid interface
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DOI：
10.1063/1.869098
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发表时间：
1996-11
影响因子：
4.6
通讯作者：
H. Wong;D. Rumschitzki;C. Maldarelli
中科院分区：
文献类型：
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作者：
H. Wong;D. Rumschitzki;C. Maldarelli
表面活性剂~表面活性剂的用量！吸附到流体界面上会影响其表面张力。因此，必须确定表面活性剂的分布，以找到跨越界面的法向应力和切向应力的跳跃。Scriven ~也见阿里斯，斯莱特里，爱德华兹等人！使用微分几何导出任意曲面坐标系的正确曲面平衡方程。同样调用微分几何，韦克斯曼开发了一个正确的形式在“固定”！仅垂直于曲面前进的曲面坐标。为了达到这种平衡，而不呼吁微分几何，斯通提出了一个简单的物理推导，导致一种形式的质量平衡，这是很容易解决的数字。不幸的是，斯通的推导使不稳定时间导数的性质模糊不清。在这里，我们遵循石头的精神，以几何方式导出表面平衡，保持时间导数的性质明确。我们验证了在斯通的形式中，时间导数必须保持固定坐标不变，就像这种形式的质量平衡的数值实现实际上所做的那样。我们还导出了一个新的形式有效的任意曲面坐标系。如图1所示，考虑流体表面上的固定点A，其局部法线为n。我们找出沿沿着n相交的任意两个垂直平面。这些平面中的每一个与点A附近的表面的相交定义了单位切线为t1和t2的曲线。通过构造]t1/]s152~1/R1！n和]t2/]s252~1/R2！n，其中ds 1和ds 2是微分弧，R1~.0！R2 =.0！是曲线的曲率半径。在几何上，这些微分弧是ds 15 R1 df 1和ds 25 R2 df 2，其中df 1和df 2是图中的微分角，以及]t1/] f152 n和]t2/] f252 n。因此，在这个局部正交系统中，表面度量张量aab的分量是：Aa 115 R1，a1250和Aa 225 R2，对角元素只是作为比例因子。这些弧定义了一个面积为dA 5Aa 11 Aa 22 df 1df 25 Aadf 1df 2的补丁，其中a是度量张量的行列式。曲率张量bab的对角分量由下式定义：b1152 R1和b2252 R2曲率是负的，因为如图1所示，两条弧相对于法线都是向下凹的。如果U是固定点处的瞬时物质速度矢量，则其沿沿着$n，t1，t2%的分量为U 5 Us（1）t11 Us（2）t21 Wn，其中W是法向分量，Us（1）和Us（2）是与表面相切的物理分量。定点沿着法线~n前进！如图1所示，距离WDt，使得在时间t1 Dt，斑块周长具有长度（R11 WDt）df 1和（R21 WDt）df 2;因此，斑块面积的变化为WDt（R11 R2）df 1df 2，并且每单位面积每单位时间的变化率为
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