Viscosity, Newtonian fluids and the Navier-Stokes equations

Viscosity is internal friction: layers of fluid sliding past each other exchange momentum. The viscous stress \(\tau\) must vanish when the fluid moves rigidly, so it cannot depend on the velocity itself (a uniform translation) or on the antisymmetric part \(W\) of the velocity gradient (a rigid rotation). It can depend on the rate of strain \(S = \tfrac12(\nabla\mathbf{u} + \nabla\mathbf{u}^T)\). A Newtonian fluid is one in which the dependence is linear and the same in every direction. The most general such law is \[ \tau = 2\mu S + \lambda(\nabla\cdot\mathbf{u})\,I, \] and in incompressible flow the second term vanishes, leaving \(\tau = 2\mu S\), \(\tau_{ij} = \mu(\partial_j u_i + \partial_i u_j)\). The constant \(\mu\) is the dynamic viscosity. Water, air and oil are Newtonian to high accuracy; blood, paint and toothpaste are not.

Insert \(\sigma = -pI + 2\mu S\) into Cauchy's equation. The divergence of the viscous stress is \[ \partial_j\bigl(\mu(\partial_j u_i + \partial_i u_j)\bigr) = \mu\,\Delta u_i + \mu\,\partial_i(\nabla\cdot\mathbf{u}) = \mu\,\Delta u_i, \] using equality of mixed partial derivatives and incompressibility. Dividing by the constant density and writing \(\nu = \mu/\rho\), the kinematic viscosity, gives the incompressible Navier-Stokes equations: \[ \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \nu\Delta\mathbf{u} + \mathbf{f}, \qquad \nabla\cdot\mathbf{u} = 0, \] where \(p\) now denotes pressure divided by density. For water at 20 °C, \(\nu \approx 1.0\times10^{-6}\) m²/s; for air, \(\nu \approx 1.5\times10^{-5}\) m²/s.

Read each term as a statement about a fluid particle. \(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u}\) is its acceleration: local change plus the nonlinear advection. \(-\nabla p\) pushes it from high pressure to low. \(\nu\Delta\mathbf{u}\) is diffusion of momentum: a particle moving faster than the average of its neighbours is slowed down, exactly as the heat equation smooths temperature. \(\mathbf{f}\) is any external force per unit mass. The constraint \(\nabla\cdot\mathbf{u} = 0\) has no time derivative in it; the pressure is whatever it must be to keep it true, and taking the divergence gives \(-\Delta p = \partial_i u_j\,\partial_j u_i - \nabla\cdot\mathbf{f}\).

Viscosity also changes the boundary condition. At a solid wall a viscous fluid does not slip: \(\mathbf{u}\) equals the wall's velocity. This is an experimental fact, and it raises the order of the equations enough to impose it. The equations carry several names because they were found several times: Navier (1822) from a molecular model, Cauchy and Poisson in the 1820s and 1831, Saint-Venant (1843), and Stokes (1845), whose continuum derivation is the one given here.

The mathematical questions this course builds towards are posed for the equations on all of \(\mathbb{R}^3\), with velocity decaying at infinity, or on the periodic box \(\mathbb{T}^3 = \mathbb{R}^3/(2\pi\mathbb{Z})^3\): given a smooth divergence-free initial velocity \(\mathbf{u}_0\), find \(\mathbf{u}\) and \(p\) for \(t \gt 0\). The periodic case removes boundaries and infinity and keeps only the equations.

Picture it: a deck of cards pushed along the top card; each card drags the one below by friction, and the drag is proportional to how fast neighbouring cards slide past each other. That is \(\tau = \mu\,du/dy\) in one line. Think it: Navier-Stokes is the Euler equations plus a Laplacian: a nonlinear transport equation coupled to a heat equation and a divergence constraint. The Laplacian smooths; the nonlinearity can steepen; which wins is the central question.