Lagrangian governing equations of incompressible fluid

The Particle finite element method (PFEM) is based on the Lagrangian form of the Navier-Stokes equation for incompressible Newtonian fluids. Assuming the density does not change in time for an incompressible fluid, the continuity equation reduces to zero requirements for the divergence of the velocity. The Navier-Stokes equations take the form

$$\rho \mbox{$\displaystyle\frac{\partial \,{\mbox{\boldmath$u$}}}{\partial \,{t}}$} = \rho \mbox{\boldmath$b$} + \nabla\cdot \mbox{\boldmath$\sigma$} \;,$$ (2.30)

$$\nabla\cdot \mbox{\boldmath$u$} = 0\;.$$ (2.31)

For the deviatoric stress in Newtonian fluids a linear dependency of stress tensor and strain rate tensor is adopted and for the Newtonian fluids. Considering the incompressibility of the fluid, the Cauchy stress reads

$$\mbox{\boldmath$\sigma$} =-p \mbox{\boldmath$I$} + 2\mu \nabla^s \mbox{\boldmath$u$} \;.$$ (2.32)

This equation is known as Stokes' law and its Cartesian form writes

$$\sigma_{ij}=-p\delta_{ij}+\mu\left(\mbox{$\displaystyle\frac{\par... ...j}}$}+\mbox{$\displaystyle\frac{\partial \,{u_j}}{\partial \,{x_i}}$}\right)\;.$$ (2.33)

Substituting the expression of Cauchy stress from Stokes' law (2.32) into the momentum equation (2.30) and rewriting gives

$$\rho \mbox{$\displaystyle\frac{\partial \,{\mbox{\boldmath$u$}}}{\partial \,{t}}$} = \rho \mbox{\boldmath$b$} + \mu\nabla^2 \mbox{\boldmath$u$} - \nabla p\;.$$ (2.34)

The governing equations of the mass (2.31) and momentum conservation (2.34) form can be written in the Cartesian form for the individual component $i$ using Einstein summation convention

$$\rho \mbox{$\displaystyle\frac{\partial \,{u_i}}{\partial \,{t}}$} = - \frac{\partial}{\partial x_i}p+\mu\frac{\partial}{\partial x_j}\left(\frac{\partial u_i}{\partial x_j}\right)+\rho b_i \;,$$ (2.35)

$$\mbox{$\displaystyle\frac{\partial \,{u_i}}{\partial \,{x_i}}$} = 0\;.$$ (2.36)

The equations are accompanied by a set of standard boundary conditions imposed on the complementary parts of the domain boundary

$$\tau_{ij}\nu_j - p\nu_i = \bar \sigma_{ni} \Gamma_{\sigma}\;,$$ (2.37)

$$u_i\nu_i = \bar u_n \qquad\; \Gamma_n\;,$$ (2.38)

$$u_i\zeta_i = \bar u_t \qquad\;\, \Gamma_t\;,$$ (2.39)

where $\nu$ or $n$ denotes the normal direction to the boundary and $\zeta$ or $t$ the tangential one. The bar sign over a quantity $\bar x$ stands for its prescribed value.