The momentum conservation equation for a Newtonian fluid is obtained by substituting \(\boxed{T_{ij} = -p\delta_{ij} + \tau_{ij} = -p\delta_{ij} + 2\mu \left(S_{ij} - \frac{1}{3}S_{mm}\delta_{ij} \right) + \mu_v S_{mm} \delta_{ij}}\) into Cauchy’s \(\boxed{\rho \frac{D u_j}{D t} = \rho g_j + \frac{\partial}{\partial x_i} \left( T_{ij} \right)}\) to obtain \[ \rho \left( \frac{\partial u_j}{\partial t} + u_i \frac{\partial u_j}{\partial x_i} \right) = - \frac{\partial p}{\partial x_j} + \rho g_j + \frac{\partial}{\partial x_i} \left[ \mu \left( \frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) + \left( \mu_v - \frac{2}{3} \mu \right) \frac{\partial u_m}{\partial x_m} \delta_{ij} \right] \] where we have used \( (\partial p/\partial x_i)\delta_{ij} = \partial p/\partial x_j \), material derivative \(\frac{D F}{D t} \equiv \frac{\partial F}{\partial t} + \mathbf{u} \cdot \nabla F, \text{or} \frac{D F}{D t} \equiv \frac{\partial F}{\partial t} + u_i \frac{\partial F}{\partial x_i}\) with \( F = u_j \), and \(S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)\) This is the Navier-Stokes momentum equation

When temperature differences are small within the flow, \(\mu\) and \(\mu_v\) can be taken outside the spatial derivative operating on the contents of the [..] brackets in the Navier-Stokes momentum equation, which then reduces to \[ \rho \frac{D u_j}{D t} = -\frac{\partial p}{\partial x_j} + \rho g_j + \mu \frac{\partial^2 u_j}{\partial x_i^2} + \left(\mu_v + \frac{1}{3} \mu\right) \frac{\partial}{\partial x_j} \frac{\partial u_m}{\partial x_m} \quad \text{(compressible)} \]

For incompressible fluids \(\boxed{\rho \frac{D u_j}{D t} = -\frac{\partial p}{\partial x_j} + \rho g_j + \mu \frac{\partial^2 u_j}{\partial x_i^2} + \left(\mu_v + \frac{1}{3} \mu\right) \frac{\partial}{\partial x_j} \frac{\partial u_m}{\partial x_m} \quad \text{(compressible)}}\) in vector notation reduces to \[ \rho \frac{D \mathbf{u}}{D t} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \quad \text{(incompressible)} \]

The net viscous force per unit volume in incompressible flow, the last term on the right in this equation = \(\mu \left( \partial^2 u_j / \partial x_i^2 \right)\), can be obtained from the divergence of the strain rate tensor or from the curl of the vorticity \[ \mu \frac{\partial^2 u_j}{\partial x_i^2} = 2\mu \frac{\partial S_{ij}}{\partial x_i} = \mu \frac{\partial}{\partial x_i} \left( \frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) = -\mu \varepsilon_{jik} \frac{\partial \omega_k}{\partial x_i}, text{ or } \mu \nabla^2 \mathbf{u} = -\mu \nabla \times \boldsymbol{\omega} \]

If viscous effects are negligible, which is commonly true in exterior flows away from solid boundaries, \(\boxed{\rho \frac{D u_j}{D t} = -\frac{\partial p}{\partial x_j} + \rho g_j + \mu \frac{\partial^2 u_j}{\partial x_i^2} + \left(\mu_v + \frac{1}{3} \mu\right) \frac{\partial}{\partial x_j} \frac{\partial u_m}{\partial x_m} \quad \text{(compressible)}}\) further simplifies to the Euler equation \[ \rho \frac{D \mathbf{u}}{D t} = -\nabla p + \rho \mathbf{g} \]