Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 95

• the conservation of momentum (Newton's second law, the rate of change of mo-
mentum equals the sum of forces acting on the fluid) and;
• the conservation of energy (first law of thermodynamics, the rate of change of
energy equals the sum of rate of heat addition to and the rate of work done on the
fluid).

5.2.1 Mass Conservation

The mass flow rate through a surface of a control volume is

.

ò  u

m =- × r n ds (5.1)

s

where s is the surface of a control volume, and u × n is the velocity component acting
perpendicular to the surface area, A� The sign of the normal component of velocity
determines whether the fluid flows in or out of the control volume, and a negative
sign is added to denote that a net outflow of mass produces a positive value (Fig. 5.1 ).
For a one-dimensional steady flow the mass flow through a surface is given as

.



∫ r

m =− u n ⋅ = ds A r (5.2)

s

The amount of mass entering a control volume surface is equal to the mass leaving
the control volume surface and this is the principle of mass conservation, sum-
marised as (Fig. 5.2 )

∑∑ ∑ ∑

mm − =→ − 0 ( ρ uA ) ( ρ u A ) = 0

(5.3)

in out in in out out

Fig. 5.1 a Mass flow rate passing through an arbitrary control surface. Flow streamlines that enter
the control volume are accounted by its vector component normal to the surface face denoted by
u × n