Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 98

5.2.1.3 General Mass Conservation—Continuity Equation

The previous examples were solved using the steady one-dimensional form of the
mass conservation equation. In general cases where the flow varies in three dimen-
sions and also unsteady in time, a more general equation is needed. Following the
same principle of conservation, the rate of change of mass inside a control volume
is equal to the net mass of fluid flowing in or out of the control volume across its
boundary surface. Taking outflow of mass as positive we get

¶

 u

¶ ò

ò

r dV =- × r n ds

(5.4)

t 

s



V

net inflow of mass

rate of change of mass

This equation is also called the continuity equation. Using Gauss's theorem Eq. (5.4)
is rearranged to give the mass conservation in differential equation form as

∂ ρ

+∇ · ρ

u

= 0 (5.5)

∂ t

Expanding this we get

( ) ( ) ( )

∂ ρ ∂ ρ u ∂ ρ v ∂ ρ w

+ + + = 0 (5.6)

∂ t ∂ x ∂ y ∂ z

5.2.1.4 Steady Converging Pipe Example

Consider an incompressible 2D steady fluid flow through a converging pipe shown
in Fig. 5.5 .

Fig. 5.5 Schematic of fluid flow through a converging pipe. Section 1 of the pipe has the largest
cross-sectional area. Section 2 is the converging section of the pipe. Section 3 has the smallest
cross-sectional area