The Tangential-Stress Balance and Thermocapillary Flows

To this point, we have considered only the component of the stress balance (2-134), in the direction normal to the interface. There will be, in general, two tangential components of (2-134), which we obtain by taking the inner product with the two orthogonal unit tangent vectors that are normal to n. If we denote these unit vectors as t (with i = 1 or 2), the so-called shear-stress balances can be written symbolically in the form, 

We see that the shear- (i.e., tangential-) stress components are discontinuous across the interface whenever gradv y is nonzero. Now, the interfacial tension for a two-fluid system, made up of two pure bulk fluids, is a function of the local thermodynamic state - namely, the temperature and pressure. However, it is much more sensitive to the temperature than to the pressure, and it is generally assumed to be a function of temperature only. If the two-fluid system is a multicomponent system, it is often the case that there may be a preferential concentration of one or more of the components at the interface (for example, we may consider a system of pure A and pure B, which are immiscible, with a third solute component C that is soluble in A and/or B but that is preferentially attracted to the interface), and then the interfacial tension will also be a function of the (surface-excess) concentration of these solute components. Both the temperature and the concentrations of adsorbed species can be functions of position on the interface, thus leading to spatial gradients of y. 

If we consider, first, the case with grad vy = 0, we see that the tangential-stress balance requires continuity of the tangential stress. If the fluids are Newtonian, this condition can also be written in terms of the rate of strain, in the form 

The converse situation, however, is that the condition (2 141) absolutely requires that there be flow whenever gradsy 0. A system with gradvy / 0 at some initial instant, but 

Motion induced at the interface by interfacial tension gradients 

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