The stress tensor and Cauchy's momentum equation

Two kinds of force act on a piece of fluid. Body forces act on every particle, like gravity \(\rho\mathbf{g}\) per unit volume. Surface forces are exerted by the neighbouring fluid across the boundary: on a small surface patch with unit normal \(\mathbf{n}\) the fluid on the side \(\mathbf{n}\) points to pushes with a force \(\mathbf{t}(\mathbf{n})\,dS\), where \(\mathbf{t}\) is the traction. A priori \(\mathbf{t}\) could depend on \(\mathbf{n}\) in any way.

Cauchy's stress theorem (Cauchy, 1823) says the dependence is linear: there is a matrix \(\sigma(\mathbf{x},t)\), the stress tensor, with \[ \mathbf{t}(\mathbf{n}) = \sigma\,\mathbf{n}, \qquad t_i = \sigma_{ij}n_j. \] The argument balances forces on a tiny tetrahedron with three faces on the coordinate planes. Surface forces scale like the area, \(h^2\), while inertia and body forces scale like the volume, \(h^3\); letting \(h\to0\) forces the tractions on the four faces to balance, which is exactly linearity. The entry \(\sigma_{ij}\) is the \(i\)-component of force per area on a face whose normal points along \(j\).

Newton's second law for a material region \(V\) reads \(\frac{d}{dt}\int_V\rho\mathbf{u}\,dV = \int_V\rho\mathbf{f}\,dV + \oint_{\partial V}\sigma\mathbf{n}\,dS\). The divergence theorem and conservation of mass turn this into Cauchy's momentum equation \[ \rho\frac{D\mathbf{u}}{Dt} = \nabla\cdot\sigma + \rho\mathbf{f}, \qquad (\nabla\cdot\sigma)_i = \partial_j\sigma_{ij}. \] Balancing angular momentum in the same way (with no body torques) shows \(\sigma_{ij} = \sigma_{ji}\). A symmetric matrix has real eigenvalues and orthogonal eigenvectors, the principal stresses and principal axes: on planes normal to a principal axis the traction is purely normal.

In a fluid at rest there can be no shear stress, because a fluid yields to any shear. Every direction must then be a principal direction, which forces all eigenvalues equal: \(\sigma = -p\,I\), with \(p\) the pressure (the minus sign because pressure pushes inward). In a moving fluid we write \(\sigma = -pI + \tau\), where \(\tau\) is the viscous or deviatoric stress. Cauchy's equation has three components but \(\sigma\) has six independent entries, so it cannot be solved until a constitutive law says how \(\tau\) depends on the motion. Setting \(\tau = 0\) gives the Euler equations; a linear law in the rate of strain gives Navier-Stokes.

Picture it: cut the fluid with an imaginary knife in every direction through one point; the force per area on each cut is \(\sigma\) applied to the cut's normal, and the three principal cuts feel pure push or pull. Think it: \(\sigma\) is a symmetric linear map, and the mean normal stress \(-\tfrac13\operatorname{tr}\sigma\) is the part every direction agrees on: that is the mechanical definition of pressure.