Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 49

The four remaining pressure forces (that act in the y- and z-directions) can be obtained
through a similar analysis method. By summing all of these six forces, the total surface
force acting on the fluid element can be obtained. It would be represented as

2 @

p dx

2

1 @

p
@

dx
2

2 @

p dy

2

dF - - - - p ð dydz Þð Þ 1 p ð dydz Þð 2 Þ 1 p ð dxdz Þð Þ

5

@ x x @ y

ð 3 : 2 Þ

1 @

p dy

2

2 @

p dz

2

1 @

p
@

dz
2

- - - p ð dxdz Þð 2 Þ 1 p ð dxdy Þð Þ 1 p ð dxdy Þð 2 Þ

1

@ y @ z z

Combining terms in the previous equations yields

2 @

p dx

2

2 @

p
@

dx
2

2 @

p dy

2

dF - - - - p ð dydz Þð Þ 12 p ð dydz Þð Þ 1 p ð dxdz Þð Þ

5

@ x x @ y

2 @

p dy

2

2 @

p
@

dz
2

2 @

p
@

dz
2

- - - p ð dxdz Þð 2 Þ 1 p ð dxdy Þð Þ 12 p ð dxdy Þð Þ

12

@ y z z

ð ð ð

ð 3 : 3 Þ

@ p dx

2

@ p dy

2

@ p dz

2

- - - 2 dydz Þð Þ 2 2 dxdz Þð Þ 2 2 dxdy Þð Þ

52

@ x @ y @ z

dxdydz

52 @

p

1 @

p

1 @

p

x - y - z - @ @ @

From a previous class in calculus, the final term in the parentheses (right-hand side of the
equation) of Equation 3.3 is the gradient (denoted as grad or

r

) of the pressure force in
Cartesian coordinates. Therefore, the surface forces acting on a differential fluid element
can be simplified to

dF -

- dxdydz

52 r

ð 3 : 4 Þ

Returning to Newton's second law, the sum of the forces acting on a differential fluid
element can then be represented as

dF - dF - d - - - dxdydz - - ρ dxdydz

2 r 5 ð

ρ 2 r Þ dxdydz ð : Þ

5 1 5 3 5

If one divides the summation of the force acting on a differential element of fluid by the
unit volume ( Equation 3.5 ), then one gets a relationship that holds for fluid particles, and
is in terms of density:

d -
dxdydz 5

d -
dV 5

- ρ 2 r - ð 3 : 6 Þ

- 5

For a static fluid flow case ð 0 Þ , Newton's second law of motion for a particle with a

finite volume simplifies to

d -
dV 5

- - -

ρ 2 r

5 ρ

0 ð 3 : 7 Þ

5