Physlib.FluidDynamics.FluidFlow.Newtonian

Given a fluid flow in $d$ dimensions with velocity field $\mathbf{u}(t, \mathbf{x})$, the spatial velocity gradient is a function that maps a time $t$ and a spatial position $\mathbf{x} \in \mathbb{R}^d$ to a $d \times d$ matrix. The entry in the $i$-th row and $j$-th column of this matrix is the partial derivative of the $i$-th component of the velocity with respect to the $j$-th spatial coordinate, denoted as $\frac{\partial u_i}{\partial x_j}$ or $\partial_j u_i$.

Given a fluid flow in $d$ dimensions with velocity field $\mathbf{u}$, the Newtonian stress tensor $\mathbf{\sigma}$ is a field that maps a time $t$ and a spatial position $\mathbf{x}$ to a $d \times d$ matrix defined by the expression: $$\mathbf{\sigma}(t, \mathbf{x}) = -p(t, \mathbf{x}) \mathbf{I} + \mu(t, \mathbf{x}) \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^\intercal \right) + \lambda(t, \mathbf{x}) (\nabla \cdot \mathbf{u}) \mathbf{I}$$ where: - $p(t, \mathbf{x})$ is the pressure scalar field. - $\mu(t, \mathbf{x})$ is the shear viscosity. - $\lambda(t, \mathbf{x})$ is the second viscosity. - $\mathbf{I}$ is the $d \times d$ identity matrix. - $\nabla \mathbf{u}$ is the spatial velocity gradient matrix at $(t, \mathbf{x})$. - $\nabla \cdot \mathbf{u}$ is the divergence of the velocity field at $(t, \mathbf{x})$. - $(\nabla \mathbf{u})^\intercal$ denotes the transpose of the velocity gradient matrix.