Page 309 - Process Modelling and Simulation With Finite Element Methods

296 Process Modelling and Simulation with Finite Element Methods

The governing equation for the fluid velocity and pressure can be written in
terms of the Navier-Stokes equations which is the equation of motion for
incompressible flow:
aU
p -- V .p (Vu + (Vu)')+ p (u.V)u + Vp = F ; (8.5)
at

where F is body force which includes gravitational force and, due to the level set
treatment of interfacial stresses, the surface tension term. The two components of
the F term can be represented as,

The delta function treats the surface tension term at the interface which is
determined by the position of the zero level set-which can be as many fluid-
fluid interfaces as necessary to demarcate the dispersed phase. The Heaviside
function, incorporated in order to describe the steep change in physical
properties, is represented in terms of level set function such as,

if$=O (8.9b)

if$>O (8.9~)
The density and viscosity are constant in each fluid and are represented in terms
of Heaviside function as,

p = H($)+&(l-
PC

(8.11)

We solve the above set of equations using FEMLAB. Smoothed approximants to
the Heaviside function are used to avoid Gibbs phenomena resulting in poor
convergence.