I am trying to write the finite element formulation of the Cauchy momentum equation (Navier Stokes) in Lagrangian formalism:
$\begin{aligned} &\nabla\cdot\sigma+\rho\mathbf{g}=\rho\dfrac{\partial\mathbf{v}}{\partial t} \quad\text{with}\quad \begin{cases} ~\sigma=2\mu\mathbf{D}-p\mathbf{I}\\ ~\mathbf{D}=\frac{1}{2}\Big(\nabla\mathbf{v}+\nabla\mathbf{v}^\top\Big) \end{cases} \end{aligned}$
The weak form is quite straightforward and well documented (https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations). In the following, $\Omega$ is the current fluid domain and $\mathbf{w}$ is the test function that should disappear later with the FEM discretization:
$\displaystyle\int_\Omega\sigma:\nabla\mathbf{w}\,d\mathbf{x}+\int_\Omega \mathbf{w}^\top\rho\frac{\partial\mathbf{v}}{\partial t}\,d\mathbf{x}=\int_\Omega \mathbf{w}^\top\rho\mathbf{g}\,d\mathbf{x}+\int_{\partial\Omega}\mathbf{w}^\top\mathbf{t}\,d\mathbf{x} \quad\text{with}\quad \mathbf{t}=\sigma\cdot\mathbf{n}$
But now I would like to solve the system with a Newton-Raphson method, meaning that I must linearize with respect to a velocity/pressure increment $\delta\mathbf{v}$/$\delta\mathbf{p}$ (my new unknowns) and derive a tangent stiffness matrix $\mathbf{K}$. Then solve a discretized system:
$\mathbf{K}\,[\delta\mathbf{v}~~\delta\mathbf{p}]^\top=\text{Residual (until convergence)}.$
I know that in solid mechanics, one could write the system using the PK2 stress tensor and the Green-Lagrange strain tensor such that the integrals can be performed in the (constant) initial domain $\Omega_0$. However, I do not know how to do in the case of Navier-Stokes because my unknown is not a displacement field. I guess this approach is uncommon because I could not find clear documentation about it.