Navier–Stokes stress and viscosity

Write the Cauchy stress as pressure plus a viscous contribution. The traction on a surface with outward unit normal n is the stress tensor multiplied by n; pressure supplies a compressive normal traction, whereas viscosity can supply tangential traction. For an ordinary isotropic Newtonian fluid without couple stresses, the viscous contribution is linear in the symmetric velocity gradient. The antisymmetric part describes local rigid rotation, which changes orientation without changing the shape of an infinitesimal material element.

The displayed form uses dynamic shear viscosity mu and a second viscosity coefficient lambda. The bulk viscosity is zeta = lambda + 2 mu/3, so lambda itself need not be positive. Nonnegative shear and bulk viscosities give nonnegative viscous dissipation in this model. Neither incompressibility nor isotropy proves the extra assumption often called the Stokes hypothesis, zeta = 0. A material whose stress depends nonlinearly on deformation rate or on its deformation history requires a different constitutive description.

Apply momentum balance to a small material volume: inertia equals the divergence of stress plus the body force per unit volume. For spatially constant viscosity coefficients, differentiating the constitutive law gives the momentum equation below. The acceleration is the material derivative, including advection. Here b denotes force per unit mass, such as gravitational acceleration. If viscosity changes substantially with temperature or position, keep the divergence of the full viscous stress; pulling coefficients through derivatives silently removes additional terms.

For an incompressible flow, theta vanishes and the constant-mu viscous contribution reduces to mu times the vector Laplacian of velocity. Dividing by density introduces the kinematic viscosity nu = mu/rho, with units of square meters per second. It controls the diffusion time across a length scale. Pressure remains essential even when density is fixed: it adjusts to enforce the velocity constraint and boundary conditions. The continuity equation still accompanies momentum conservation; one vector equation cannot replace the complete fluid model.

A shear stress transports momentum across an internal surface. Only its spatial variation supplies a net force to a small volume, so uniform shear stress does not imply local acceleration. With constant mu, a linear shear profile has nonzero stress but zero velocity Laplacian. Mechanical work can still enter through the moving boundaries and become internal energy. The dissipation density for an incompressible Newtonian fluid is twice mu times the sum of squared deformation-tensor components; a complete thermal calculation accounts for that heating.

At an ordinary impermeable no-slip wall, the fluid velocity equals the wall velocity. These conditions are modeling assumptions supported in many continuum-scale experiments, with exceptions such as rarefied gases or measurable interfacial slip. Initial velocity and suitable pressure or boundary data complete the problem. Before solving, check the continuum scale, constitutive range, and thermal assumptions. A compact equation is not permission to omit the mechanisms that determine its coefficients or boundary behavior.