The Normal-Stress Balance and Capillary Flows

To obtain the normal component, which is generally referred to as the normal-stress balance, wetaketheinnerproductof(2-134) withn. Recalling that T = -pi + Tandf = -pi + t, this gives 

ptot and ptot represent the actual total pressure in the exterior and interior fluids, including both dynamic and hydrostatic contributions. In crossing an interface, we see that the normal component of the total stress undergoes a jump equal to y(V n). In the limiting case of no motion in the fluids, this implies that 

This equilibrium condition is known as the Young-Laplace equation. The physical significance of (2-136) is that the pressure inside a curved interface at equilibrium is larger than that outside by an amount that depends on the curvature V n and y. Now the curvature term V n can be expressed as the sum of the two principle radii of curvature of S at any point xs. on the interface, that is, 

We have obtained the Young Laplace relationship from normal-stress balance (2-135) by invoking the limiting case of no motion in the fluids. One might be tempted to suppose that the condition of no motion is independent of the interface, e.g., that it is determined by whether there is some source of fluid motion in the bulk-phase fluids away from the interface. However, this is not correct. In fact, if (V n) / constant on the interface (i.e., independent of position on S), the Young-Laplace equation cannot be satisfied, and there must be fluid motion so that the balance of normal stresses includes viscous contributions. Utilizing (2 137), we see that the condition (V n) = constant requires the sum of Rf1 and R2 1 to be constant on S. Examples of surfaces that satisfy this requirement are a sphere, where R = Ih = R (the radius) a circular cylinder, where R = R, R2 = 00 and a flat interface, where Ri = R2 = 00. 

We have said that the Young Laplace equation cannot be satisfied unless V n = constant. The proof is more or less trivial. Let us suppose that u = 0. Then, according to the equation of fluid statics, (2 61), 

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