Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 91

3.5 The Conservation of Momentum could also include acceleration components. In Cartesian

component form this would be

ð ð ð

5 @
@

- bx 1 - sx 2 d - - rx ρ - xyz U dV u xyz ρ dV u xyz ρ

1

t

V

ð

V

ð

area

ð

5 @
@

- by 1 - sy 2 d - - ry ρ - xyz U dV v xyz ρ dV v xyz ρ

1

t

ð ð

area

ð

V V

5 @
@

- bz 1 - sz 2 d - - rz ρ - xyz U dV w xyz ρ dV w xyz ρ

1

t

V V area

3.6 It is common for heat exchange to occur within biological flows. To account for the conserva-

tion of energy within a fluid, the formulation would become

ð ð ð

ρ

ð

ρ

5 @
@

5 @
@

v 2
2 1

v 2
2 1

d - d - - -

Q

2 W

e ρ dV 1 e ρ U u 1 gz dV 1 u 1 gz U

t t

area area

V V

The rate of work in this equation would need to account for all of the various work terms
that may be applied to the fluid. In a simplified form, this would be

ρ

ð
area - U

ð ð

v 2
2 1

v 2
2 1

2 W shaft 2 W other 5 @
@

d -

Q

- dA - U u gz ρ dV u gz p

1 1 1 1 1

t

V area

The second law of thermodynamics can be applied to biofluids. It is represented as

ð ð

$ Q
T

@
@

d - - U s ρ dV s ρ

1

t

area

V

3.7 The Navier-Stokes equations are the solutions of Newton's Second Law of Motion applied to

fluid flow. For incompressible flows with a constant viscosity, these equations simplify to

2 u 2 u 2 u

@ u

@

u @

u

v @

u

w @

u

g x 2 @

p @

x 2 1 @ y 2 1 @

ρ

t 1 x 1 y 1

5 ρ

x 1 μ @ @ @ z @ @ @ @ z 2

@ v

@

u @

v

v @

v

w @

v

g y 2 @

p @

2 v

x 2 1 @

2 v

y 2 1 @

2 v

ρ

t 1 x 1 y 1

5 ρ

y 1 μ

@ @ @ z @ @ @ @ z 2

2 w
@

2 w 2 w

@ w

@

u @

w
@

v @

w
@

w @

w
@

g z 2 @

p @

x 2 1 @ y 2 1 @

ρ

5 ρ

z 1 μ

t 1 x 1 y 1 z @ @ @ z 2

in Cartesian coordinates and

v 2
r 1

2 v r
@θ

2 v r
@

@ v r

@

v r @

v r
@

v r @ v r

@θ 2

v z @

v r
@

g r 2 @

p @

@

1

r @ r 2 @

1

r 2 @

2

@θ 1 @

v θ

ρ

5 ρ

r 1 μ

rvðÞ

t 1 r 1

1 2

z @ r @ r

2

z 2

1

@ v z

@

v r @

v z
@

v r @ v z

@θ 1

v z @

v z
@

g z 2 @

p

r @

r @

v z
@ r 2 @

2 v z
@θ

1 @

2 v z
@

1 1

ρ

t 1 r 1

5 ρ

z 1 μ

2

z @ @ r r z 2

2 v θ
@θ

2 v θ
@

@ v θ

@

v r @

v θ
@

v r @ v θ

@θ 1

v r v r 1

v z @

v θ
@

g θ 2 @

p
@θ 1 μ

@
@

1

r @ r 2 @

1

r 2 @

2

@θ 1 @

v r

ρ

5 ρ

rvðÞ

t 1 r 1

1 2

z r @ r

2

z 2