Fluids: Liquids and Gases - Computer Animation: Algorithms and Techniques - page 273

( Fx , Fy , Fz ), is div F¼rF¼ @F x

@x þ @F y @y þ @F z

sink at a location; the divergence for a field, F ¼ .Ifno

@z

mass is created or destroyed inside the element, then it is reasonable that the total divergence within the
element must be equal to the flow differences across the element at the x , y , and z boundaries. This is
stated by the Divergence Theorem : the integral of the flow field's divergence over the volume of the
element is the same as the integral of the flow field over the element's boundary. So, instead of wor-
rying about all the sources and sinks inside of the element, all that has to be computed is the flow at the
boundaries.
For now, for a given three-dimensional element, consider only flow in the x -direction. To make the
discussion a bit more concrete, assume that the fluid is flowing left to right, as x increases, so that the
mass flows into a element on the left, at x , and out of the element on the right, at xþdx . In the following
equations, r is density, p is pressure, A is the area of the face perpendicular to the x -direction, V is the
volume of the element, and the element dimensions are dx by dy by dz . To compute the mass flowing
across a surface, multiply the density of the fluid times area of the surface times the velocity component
normal to the surface, v x in this case. The conservation of mass equation states that the rate at which
mass changes is equal to difference between mass flowing into the element and the mass flowing out
of the element (Eq. 8.29 ). The notation for a value at a given location is <value>( location or, when

appropriate, <value>( location surface .

@ðrVÞ
@t

¼ðrv x AÞj x þ dx ðrv x AÞj x (8.29)

Replace the volume, V , and area, A , by their definitions in terms of the element dimensions
(Eq. 8.30 ).

dðrdxdydzÞ
dt

¼ðrv x dydzÞj x þ dx ðrv x dydzÞj x (8.30)

Divide through by dxdydz (Eq. 8.31 ).

@r

@t ¼ rv x dx j x þ dx rv x dx j x

(8.31)

Replace the finite differences by their corresponding differential forms (Eq. 8.32 ).

@r

@t ¼ @ðrv x Þ

(8.32)

@x

Finally, include flow in all three directions (Eq. 8.33 ).

@r

@t ¼ @ðrv x Þ

þ @ðrv y Þ
@y

þ @ðrv z Þ
@z

(8.33)

@x

then a common notation is to define the

rF ¼ @F

@x ; @F @y ; @F

If r is used as the gradient operator,

@z

divergence operator as in Equation 8.34 .

¼ @F x

@x þ @F y @y þ @F z

r F (8.34)

@z