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Viscosity

A fluid’s resistance to flow. Viscosity plays a significant role in drag and boundary layer development in aerodynamics.
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The statement of the theorem

In the context of continuum mechanics, the viscous stress tensor τ\boldsymbol{\tau} for an incompressible Newtonian fluid is defined by the relationship between the shear stress and the rate of strain. Let u(x,t)\textbf{u}(\textbf{x}, t) be the velocity field, and ρ\rho be the fluid density. The rate of strain tensor S\textbf{S} is given by the symmetric part of the velocity gradient tensor ∇u\nabla \textbf{u}: \n\nS=12(∇u+(∇u)T)\textbf{S} = \frac{1}{2} \big( \nabla \textbf{u} + (\nabla \textbf{u})^T \big) \n\nFor a Newtonian fluid, the viscous stress tensor τ\boldsymbol{\tau} is linearly proportional to the rate of strain tensor S\textbf{S}, with the proportionality constant being the dynamic viscosity ν\nu (or 12ρν\frac{1}{2}\rho\nu depending on the specific definition used, but ν\nu is standard for the coefficient): \n\nτ=2νS=ν(∇u+(∇u)T)\boldsymbol{\tau} = 2\nu \textbf{S} = \nu \big( \nabla \textbf{u} + (\nabla \textbf{u})^T \big) \n\nAlternatively, considering the shear stress component τij\tau_{ij} in Cartesian coordinates, the definition is:\n\nτij=2νDev(u)ijDev(u)ij=ν(∂ui∂xj+∂uj∂xi)\tau_{ij} = 2\nu \frac{\text{Dev}(\textbf{u})_{ij}}{\text{Dev}(\textbf{u})_{ij}} = \nu \big( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \big) \n\nThis formulation quantifies the internal resistance to deformation, where ν\nu has units of Mass⋅Length−1⋅Time−1\text{Mass} \cdot \text{Length}^{-1} \cdot \text{Time}^{-1}.