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f p =
2 v is the viscous force. Equa-
tion (6.1) can be applied to these two quantities to interpolate them at any location
in the fluid.
We will first interpolate f p ( x ) in a straightforward manner, as stated by
p is the pressure force, and f v = μ
−∇
n
m j
ρ j p ( x )
f p ( x )=
−∇
p ( x )=
W ( x
x j ,h ) .
(6.5)
j =1
To calculate the per-particle force f i
,[Muller et al. 03] proposed an adapta-
tion to Equation (6.5) to obtain symmetric pressure forces between any pair of
particles i and j , by substituting ( p i + p j ) / 2 for p ( x ). Also, the acceleration of
the fluid, defined as a ( x )= f ( x ) ( x ), has to be matched by every fluid parti-
cle. Therefore, the pressure force f p ( x ) at an arbitrary location within the fluid
defined by Equation (6.5) is multiplied by m i
ρ i
to get the pressure force f i
acting
on a single particle i . The complete definition of f i
is given by
n
m i
ρ i
m j
ρ j
( p i + p j )
2
f i
−∇
p i =
W ( x i
x j ,h ) ,
=
(6.6)
j =1
where according to the ideal gas law p j = j , with k being a gas constant that
can be used to increase or decrease the compressibility of our simulated fluid. We
do not find it necessary to use a rest density as in [Muller et al. 03]. We choose
the smoothing kernel
W ( r ,h )=(45 /πh 4 )(1
/h ) 2
r
r
, which is the gra-
dient of the spiky kernel used for density computations. Note that
r
W ( r ,h ) is
undefined when r = 0 . Because a particle never exerts a force on itself, we define
W ( 0 ,h )=0.
Apart from Equation (6.6), there are other ways in which we can use Equa-
tion (6.1) to interpolate a gradient. According to [Colin et al. 06], some of them
yield more-accurate approximations of the gradient of a function. For instance, a
more-accurate approximation would be obtained by using the difference gradient
approximation formula (DGAF), as in [Colin et al. 06]. However, the inaccura-
cies of Equation (6.6) tend to decrease compressibility for areas with similar but
high pressure, while this is not the case for a DGAF approximation. Simulating
water or other incompressible fluids therefore benefits from using Equation (6.6).
The approximation of the viscous force f i
= μ∇
2 v is similar to the approxi-
mation of f i
, as the following demonstrates:
n
2 v = μ m i
ρ i
m j
ρ j ( v j
f i
2 W ( x i
= μ
x j ,h ) ,
v i )
(6.7)
j =1
where μ is the viscosity constant, used to increase or decrease the viscosity of the
simulated fluid. Here, we choose the smoothing kernel
2 W ( r ,h )=(45 /πh 6 )
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