Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 82

Substituting Equations 3.59 into Equations 3.58 , the equations of motion become

0
@

1
A

-

2

μ @ u
@

@ u

y 1 @ v

@ 2 p

3 μr U

2 @ μ @ μ

@ u

1 @ w
@

2 1

Du
Dt 5 ρ

x @ @ x

@ z x

ρ g x 1 1 1

@ x @ y @ z

h i

- 2 @ u

@

@ u

1 @ v
@

@ μ

2
3

μr U @ μ @ μ

@ u

1 @ w
@

2

2 @

p

x @ y x

@ z x

5 ρ g x

x 1

1 1

@ @ x @ y @ z

0
@

1
A

-

@ μ

@ u

1 @ v
@

@ 2 p

2
3

μr U μ @ v

@

@ μ

@ v

1 @ w
@

2 1 2

Dv
Dt 5 ρ

@ y x x @ z y

ρ g y

1 1 1

@ x @ y @ z

ð 3 : 60 Þ

h i

- 2 @ v

@

2 @ u

y 1 @ v

@ v

z 1 @ w

@ μ

3 μr U

@ μ @ μ

x 2

2 @

p

@ @ x @ @ y

5 ρ

g y

y 1

1 1

@ @ y @ x @ z

0 1

-

@ μ

@ u

z 1 @ w

@ v

1 @ w
@

2
3

μ @ w
@

@ μ @ 2 p μr U 2

2 1

Dw
Dt 5 ρ

@ @ x @ z y x

@ A

ρ g z 1

1 1

@ x @ y @ z

h i

- 2 @ w

@

2
3

@ μ

@ u

z 1 @ w

@ v

1 @ w
@

@ μ μr U @ μ

2

g z 2 @

p
@

x @ @ x @ z y

5 ρ

z 1

1 1

y
Equations 3.60 are the full Navier-Stokes equations that are valid for any fluid. If we
assume that the fluid is incompressible and the viscosity is uniform and constant, the
equations simplify to

@ z @ x @

2 u
@

2 u 2 u

@ u

@

u @

u
@

v @

u

w @

u
@

2 @

p @

x 2 1 @ y 2 1 @

ρ

t 1 x 1 y 1

5 ρ g x

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

5 ρ

y 1 μ

ð 3 : 61 Þ

t 1 x 1 y 1 @ @ @ 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

Equation 3.61 is the form of the Navier-Stokes equations that will be used often in this
textbook. In many biofluid mechanics examples, it is however more useful to solve the
Navier-Stokes equations in a cylindrical coordinate system. The Navier-Stokes equations in
cylindrical coordinates are as follows for incompressible fluids with a constant viscosity:

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 2

z r @ r z 2

1

2 v z
@θ

2 v z
@

@ v z

@

v r @

v z
@

v

r @

v z
@θ 1

v z @

v z
@

2 @

p
@

1

r @

r @

v z
@ r 2 @

1

2 1 @

ρ

5 ρ

g z

z 1 μ t 1 r 1

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 θ
r

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 1 2

z r @ r

2

z 2

ð 3 : 62 Þ