Capillary Surface - The Stress Balance Equation - The Stress Tensor

The Stress Tensor

The stress tensor is related to velocity and pressure. Its actual form will depend on the specific fluid being dealt with, for the common case of incompressible Newtonian flow the stress tensor is given by

\begin{align} \sigma_{ij} &= -\begin{pmatrix} p&0&0\\ 0&p&0\\ 0&0&p \end{pmatrix} + \mu \begin{pmatrix} 2 \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} & \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \\ \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} & 2 \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \\ \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} & \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} & 2\frac{\partial w}{\partial z} \end{pmatrix} \\ &= -p I + \mu (\nabla \mathbf{v} + (\nabla \mathbf{v})^T) \end{align}

where is the pressure in the fluid, is the velocity, and is the viscosity.

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