### Lagrangian representation for incompressible flow

Even Eulerian representation gives a field description of flow problems, it’s characterized and troubled by the convection terms. On the other hand, Lagrangian representation does not have this drawback, although the nonlinearity problem posed by convection term is replaced by nonlinearity of mapping between initial spacial cooridnates depending on particle index and spacial coordinates at some time later, as I am about to show.

Denote spacial coordinates, velocity, pressure in Eulerian representation as $x_i$, $u_i(x,t)$, $p(x,t)$ (symbol without subscript means the vector of proper dimension), one has the governing equations for inviscid incompressible flow:

$\partial u_i/\partial x_i=0$

$\partial u_i/\partial t+u_j\partial u_i/\partial x_j=-\partial p/\partial x_i+\nu \partial^2 u_i/\partial x_jx_j$

For some reference time $t^0$, and material coordinate (particle index) $\xi$, spacial position is described by a particle trajectory mapping from Lagrangian coordinate to Eulerian coordinate:

$x_i=\phi_i(\xi,t)$

Mapping $\phi$ between material coordinate $\xi$ and spacial coordinate $x$ is called deformation.

When material coordinate coincides with spacial coordinate at reference time $t^0$, we have

$x(\xi,t^0)=\phi(\xi,t^0)=\xi$

in which $x$, $\phi$ and $\xi$ are all generally in $\mathbb{R}^3$. Then the Lagrangian representation of unknowns are

$x=\phi(\xi,t)$

$u^t_i(\xi)=u_i(x,t)=u_i(\phi(\xi,t),t)$

$p^t(\xi)=p(\phi(\xi,t),t)$

Here symbols with superscript denotes the variables in Lagrangian representation.  The material derivative and spacial/reference derivative, for some variable $q$ are related by

$\partial q/\partial x_i=(\partial q/\partial \xi_j)(\partial \xi_j/\partial x_i)= (\partial q/\partial \xi_j)(\partial \phi_i(\xi,t)/\partial \xi_j)^{-1}=F_{ij}^{-1}\partial q/\partial \xi_j$

in which $F_{ij}$ is the Jacobi matrix of $\phi$, i.e. the deformation gradient:

$F=\nabla_{\xi}\phi=\frac{\partial \phi_i}{\partial \xi_j}$

A second order tensor’s reference coordinate form and material coordinate form is connected by Piola transform:

$T(\xi)=J\tau(x)F^{-t}\Longrightarrow \partial u_i/\partial \xi_j=J\partial u_i/\partial x_k F_{jk}^{-1}$

where $T$ is a second order tensor in reference configuration, and $\tau$ is second order tensor corresponding to $T$  in deformed configuration. The superscript $-t$ means “inverse” & “transpose”. Considering the divergence operation, we have:

$\nabla_{\xi}\cdot T=\frac{\partial T_{ij}}{\partial \xi_j}$

$\nabla_x\cdot \tau=\frac{\partial \tau_{ij}}{\partial x_j}$

The divergence theorem leads to

$\int_{\Omega_0}\nabla_{\xi}\cdot T=\int_{\Gamma_0}Tn_0$

$\int_{\Omega}\nabla_x\cdot \tau=\int_{\Gamma}\tau n$

Following theorem can be proved for the Piola transform:

$\nabla_{\xi}\cdot T(\xi)=J\nabla_x\cdot \tau(x),\forall x=\phi(\xi,t)$

$T(\xi)n_0dS_0=\tau(x)ndS,\forall x=\phi(\xi,t)$

and area derivatives $dS_0$ and $dS$ are related by

$dS=J|F^{-t}n^0|dS_0$

Apply results above to the viscosity term in momentum equation, we have

$\nu\frac{\partial^2 u_i}{\partial x_jx_j}=\nu J^{-1}\frac{\partial}{\partial \xi_j}(JF_{jk}^{-1}F_{kl}^{-1}\frac{\partial u_i}{\partial \xi_l})$

Finally, the Lagrangian formulation for momentum equation is:

$\frac{Du_i(\phi(\xi,t),t)}{Dt}=-F_{ij}^{-1}\frac{\partial p(\phi(\xi,t),t)}{\partial \xi_j}+\nu J^{-1}\frac{\partial}{\partial \xi_j}(JF_{jk}^{-1}F_{kl}^{-1}\frac{\partial u_i(\phi(\xi,t),t)}{\partial \xi_l})$

$J$ here bears the physical interpretation of volume change ratio, which is constantly one for incompressible flow. Depending on whether the reference time $t^0$ is taken as some time during the movement or always the initial time with initial configuration, the Lagrangian description can be divided into updated Lagrangian formulation or Total Lagrangian formulation.