Hydrostatics and pressure

If the fluid is at rest, \(\mathbf{u} = 0\), Cauchy's equation with \(\sigma = -pI\) and gravity \(\mathbf{g} = -g\,\mathbf{e}_z\) reduces to \[ \nabla p = \rho\mathbf{g}, \qquad \frac{\partial p}{\partial z} = -\rho g, \quad \frac{\partial p}{\partial x} = \frac{\partial p}{\partial y} = 0. \] Pressure depends only on height and grows with depth. For a liquid of constant density, a depth \(d\) below a free surface at atmospheric pressure \(p_0\) has \(p = p_0 + \rho g d\). Ten metres of water add \(1000\times9.81\times10 = 98\,100\) Pa, almost exactly one more atmosphere.

Forces on submerged surfaces are integrals of pressure. A vertical dam of width \(W\) holding water of depth \(H\) feels a gauge pressure \(\rho g(H-z)\) at height \(z\), so the total horizontal force is \[ F = \int_0^H \rho g (H - z)\,W\,dz = \tfrac12\rho g W H^2. \] It grows with the square of the depth, which is why dams are thick at the bottom. The force acts at one third of the depth from the bottom, not at the middle.

Archimedes' principle comes from the divergence theorem. The net pressure force on a submerged body occupying \(V\) is \[ -\oint_{\partial V} p\,\mathbf{n}\,dS = -\int_V \nabla p\,dV = -\int_V \rho_{\text{fluid}}\,\mathbf{g}\,dV = \rho_{\text{fluid}}\,g\,|V|\,\mathbf{e}_z, \] an upward force equal to the weight of the displaced fluid. A floating body sinks until the displaced weight matches its own, so wood of density 800 floats with four fifths of its volume under water.

For a gas the density depends on pressure. An ideal gas at constant temperature has \(\rho = pM/(RT)\), so \(dp/dz = -p/H\) with scale height \(H = RT/(Mg)\), and \[ p(z) = p_0\,e^{-z/H}. \] At \(288\) K, \(H \approx 8.4\) km: the pressure halves roughly every \(5.8\) km of altitude. Finally, taking the curl of \(\nabla p = \rho\mathbf{g}\) gives \(\nabla\rho\times\mathbf{g} = 0\): equilibrium is only possible when surfaces of constant density are horizontal. A tilted density layer cannot sit still; it must flow.

Picture it: a column of fluid above each point, whose weight per unit area is the pressure. Think it: hydrostatics is the statement that \(\rho\mathbf{g}\) is a gradient; when it is not (tilted density), no pressure can balance it and motion is forced. This is the seed of every buoyancy-driven flow.