An integration variable substitution in the form \( \sin \xi = x/\sqrt{2R_1 \zeta} \) allows the integral to be evaluated: \[ (\mathbf{F}_{st})_z = \mathbf{e}_z \cdot \mathbf{F}_{st} = \pi \sigma \sqrt{2R_1 \zeta} \sqrt{2R_2 \zeta} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \] For static equilibrium, \( \mathbf{F}_p + \mathbf{F}_{st} = 0 \), so the evaluated results of \(\boxed{\mathbf{F}_p = - \iint_A \Delta p \, \mathbf{n} \, dA = - \Delta p \int_{-\sqrt{2R_1 \zeta}}^{+\sqrt{2R_1 \zeta}} \left[ \int_{-\sqrt{2R_2 \zeta - x^2 R_2/R_1}}^{+\sqrt{2R_2 \zeta - x^2 R_2/R_1}} (-x/R_1, -y/R_2, 1) \, dy \right] dx}\) and \(\boxed{\mathbf{F}_{st} = \sigma \oint_C \mathbf{t} \times \mathbf{n} \, ds}\) require \( \Delta p = \sigma \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \), where the pressure is greater on the side of the surface with the centers of curvature of the interface. Thus in the absence of buoyant forces and fluid motion, a bubble in water will assume a spherical shape since that shape minimizes its radii of curvature, or equivalently, its surface area

Relation between the stresses on the two sides of a boundary between two fluids

For air bubbles in water, gravity is an important factor for bubbles of millimeter size. The hydrostatic pressure in a liquid is obtained from \( p_L = p_0 - \rho g z \), where \( z \) is measured positively upwards from the free surface and gravity acts downwards and \( p_0 \) is the pressure at \( z = 0 \). For a gas bubble beneath the free surface: \[ p_G = p_L + \sigma \left( \frac{1}{R_1} + \frac{1}{R_2} \right) = p_0 - \rho g z + \sigma \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]Gravity and surface tension forces are of the same order over a length scale \( (\sigma / \rho g)^{1/2} \). For an air bubble in water at 288 K, this length scale is \[ (\sigma / \rho g)^{1/2} = [7.35 \times 10^{-2} \, \text{N/m} / (9.81 \, \text{m/s}^2 \times 10^3 \, \text{kg/m}^3)]^{1/2} = 2.74 \times 10^{-3} \, \text{m} \]