Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 61

Substituting the appropriate values for mass into Equation 3.22 ,

ð
V ρ

ð
area ρ

dt system 5 @

dm

d - - dV 0 ð 3 : 24 Þ

1 U 5

@ t

The previous equation ( Equation 3.24 ) describes the changes in mass within a system of
interest. The first term (right-hand side of Equation 3.24 ), describes the time rate of change
of the mass within the volume of interest. This includes any possible change in density
within the volume or changes within the volume itself. The second term (right-hand side),
describes the mass flux into/out of the surfaces of interest. Mass that is entering into the
volume of interest would be considered a negative flux (because the velocity vector acts in
an opposite direction to the area vector), whereas mass leaving the volume of interest
would be a positive flux (the velocity and the area vectors are acting in the same direction).
By the conservation of mass principle, the time rate of change of mass within the volume of
interest has to be balanced by the flux of mass into/out of the volume of interest.
Equation 3.24 can be simplified in specific fluid cases. For an incompressible flow, there
is no change in density with time/space. This simplifies Equation 3.24 to

ð

dt system 5 ρ @

dm V

@

d - -

t 1 ρ

ð : Þ

U 5 0 3 25

area

because the volume integral of dV is simply the volume of interest. By canceling out the
density terms and making a further assumption that the volume of interest does not
change with time, Equation 3.25 becomes
dm
dt

system 5

ð

d - - ð : Þ

U 5 0 3 26

area

A volume that does not change with time would be considered non-deformable. This is
not always a good assumption in biofluids because blood vessels change shape when the
heart's pressure pulse is passed through it. Also, the lungs use a shape change to drive the
flow of air into or out of the system. However, in some biofluid cases, it might be
acceptable to make this assumption. In this textbook, we will assume that our volume of
interest is non-deformable unless stated otherwise. Equation 3.26 does not make an assump-
tion on the flow rate (i.e., is it steady or does it change with time), so this equation is valid
for any incompressible flow through a non-deformable volume. Although, remember that
by definition, steady flows can have no fluid property that changes with time. Therefore,
the first integral term in Equation 3.24 would be equal to zero. So for a general compressible
steady flow situation, Equation 3.24 would simplify to the mass flux equation:
ð
area ρ

d - - 0 ð 3 : 27 Þ

U 5

In fluid mechanics, the integral represented in Equation 3.26 is commonly referred to as
the volume (or volumetric) flow rate, Q . For an incompressible flow through a non-
deformable volume, the volume flow rate into the volume must be balanced by the flow