CPA Flow Modeling with Navier-Stokes Equations

The Navier-Stokes equation is used to model CPA flow, while assuming that the inertial forces are negligible in comparison with the viscous forces (i.e., creeping flow) [36 (link)]:
$$\rho \frac{\partial \mathit{v}}{\partial t}=\nabla \cdot \left[-p\mathit{I}+\mu \left(\nabla \mathit{v}+{\left(\nabla \mathit{v}\right)}^{\prime }\right)-\frac{2}{3}\mu \left(\nabla \cdot v\right)\mathit{I}\right]+\rho \mathit{g}$$
where p is the pressure, I is the identity matrix, μ is the dynamic viscosity, g is the gravitational acceleration, the prime denotes matrix transposition. The conservation of mass is given by [36 (link)]:
$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \mathit{v}\right)=0$$
The coupling between the heat transfer and fluid mechanics models comes about in two ways: (i) by implementing temperature-dependent viscosity and density in Eqs (8)–(9), and (ii) by using the solution to the velocity field in heat transfer calculations based on Eq (6).
Special attention is paid to the free surface boundary condition, Fig 3(B). The normal stress at the free surface (i.e., at the CPA-air interface) is assumed zero, while surface tension is neglected [36 (link)]:
$$|\left[-p\mathit{I}+\mu \left(\nabla \mathit{v}+{\left(\nabla \mathit{v}\right)}^T\right)-\frac{2}{3}\mu \left(\nabla \cdot \mathit{v}\right)\mathit{I}\right]\cdot \widehat{\mathit{n}}=0$$
Lastly, a no slip boundary condition is assumed on all solid surfaces:
$${\mathit{v}}_{\mathit{w}}=0$$