Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 69

3.5 MOMENTUM EQUATION WITH ACCELERATION

In deriving Equation 3.33 , we made an assumption that the volume of interest (and the
fluid within the volume) had no acceleration at the instant in time that we were evaluating
the system. Therefore, this equation does not hold for a volume of interest or system of
interest that is accelerating. This is the case because typically when evaluating an accelerat-
ing system, you would use an inertial or fixed coordinate system (normally denoted as
XYZ ) instead of a reference (or relative) coordinate system, which follows the moving vol-
ume (normally denoted as xyz ). In the previous derivations, a relative coordinate system
was used for simplicity. Furthermore, Equation 3.33 does not hold for an inertial reference
frame because the relative momentum for each system is not the same:

d - XYZ
dt
In order to develop an equivalent formulation as Equation 3.33 , for an accelerating control
volume, a relationship between the inertial momentum ( - XYZ ) and the control volume
momentum ( - xyz ) must be found. To start, let us define Newton's second law in terms of
momentum and a system of interest:

d - xyz
dt

- 5 6¼

ð ð

d - XYZ
dt

d - XYZ
dt

d
dt

-

- XYZ dm

dm ð 3 : 35 Þ

5 5 5

system mass system mass

2 2

To define the inertial velocity component in terms of the system velocity, use the
following relationship:

- XYZ 5 - xyz 1 - r Þ

where - r is the velocity of the volume of interests reference frame relative to the inertial
reference frame. Making the assumption that the fluid is irrotational,
d - XYZ
dt

ð 3 : 36

d - xyz
dt

d - r
dt 5

- XYZ - xyz - r

ð 3 : 37 Þ

5 5 1 1

In Equation 3.37 , the first acceleration term ( - XYZ ) is the acceleration of the system rela-
tive to the inertial frame, the second acceleration term ( - xyz ) is the acceleration of the sys-
tem relative to the system reference frame, and the third acceleration term ( - r ) is the
acceleration of the system reference frame relative to the inertial reference frame. A rota-
tional system would have multiple accelerations terms (see discussion below). Substituting
the acceleration terms into Equation 3.35 ,

ð ð ð

d - XYZ
dt

d - xyz
dt

- - r dm dm dm

5 5 1

system 2 mass system 2 mass system 2 mass

ð ð

d - xyz
dt

d - xyz
dt

- - r dm dm ð 3 : 38 Þ

2 5 5

system mass system mass

2 2