Graphical Analysis of Fluid Flow Through Polymeric Complex Structures Using Multi-scale Simulations - Selected Topics of Computational and Experimental Fluid Mechanics - page 283

where the computational space is subdivided into a finite number of cells of volume
ʔ

V . These volume elements, which have thermodynamical and dynamical proper-
ties, are defined as “particles”. For numerical work, A

( r , t ) is approximated by the

interpolant summation over the field at particle positions:

m j
ˁ j

A ( r ) = A j W ( | r − r j | , h ), (1)

j

where m j , ˁ j , and A j are the mass, density, and field function, respectively, at the

position of particle j . The quantity m j /ˁ j is the inverse of the number density at
particle j and is effectively the fluid volume associated with particle j . The summa-
tion is over all particles within the region of compact support of the kernel function.
In the following we will write, for simplicity, W ij instead of W

.
The kernel function monotonically decreases with distance and behaves as a delta
function as the smoothing length, h , tends to zero. We refer the reader to Benz
( 1990 ), Monaghan ( 1992 ) and Liu and Liu ( 2003 ) for a detailed discussion on the
kernel functions. Usually, numerical codes employ different kernel functions
(Gesteira et al. 2010 ). The Gaussian kernel is one of the most widely used (Mon-
aghan 1992 ). Other commonly used functions are the cubic, the quartic, and the
quintic polynomials (see Gesteira et al. ( 2010 ) for their mathematical expressions
and details). The tensile correction is automatically activated when using kernel
functions with first derivatives that go to zero with decreasing inter-particle spacing
(Monaghan 2000 ).
We now write down the SPH equations. The momentum conservation equation
for a continuum field is written in SPH form as

( | r i − r j | , h )

P j
ˁ

d v i
dt =−

P i
ˁ

m j

j + i + ʠ ij

∇ i W ij + g , (2)

2 2

j

where P k and ˁ k denote the pressure and density evaluated at the position of particle

k , respectively. The viscosity term, ʠ ij , is given by

− ʱ c ij μ ij
ˁ ij

; v ij · r ij < 0

ʠ ij = (3)

0 ; v ij · r ij > 0

r ij + ʷ

2

where μ ij = h v ij · r ij /( ) , r ij = r i − r j , v ij = v i − v j , with r k and v k being

2

01 h 2 , and the position and velocity of particle k , c ij = ( is a

free parameter that can be changed according to the problem under consideration.
The particles' positions are evolved using the equation (Monaghan 1989 )

c i + c j )/ 2, ʷ = 0 . ʱ

m j
ˁ ij v ij W ij ,

d r i
dt =

v i + ʵ

(4)

j