Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 81

For this formulation, gravity does not need to align with a particular Cartesian direc-
tion. Substituting Equation 3.57 into Equation 3.55 and writing it in terms of vector compo-
nents gives us

dxdydz

1 @σ xx
@ x 1 @τ y 1 @τ zx

dm @

u
@

u @

u

v @

u
@

w @

u

yx

ρ g x

5

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

dxdydz

1 @τ

x 1 @σ y 1 @τ

dm @

v
@

u @

v
@

v @

v

w @

v

xy yy zy

ρ g y

5

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

dxdydz

x 1 @τ yz

1 @τ

y 1 @σ

dm @

w
@

u @

w
@

v @

w
@

w @

w
@

xz zz

ρ g z 5

t 1 x 1 y 1 @ @ @ z z

Using the relationship that dm

5 ρ

dV

5 ρ

dxdydz , the equations of motion become

g x 1 @σ

x 1 @τ yx y 1 @τ

@ u

@

u @

u

v @

u

w @

u
@

Du
Dt

xx zx

ρ

5 ρ 5 ρ

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

g y 1 @τ xy
@ x 1 @σ yy y 1 @τ zy

@ v

@

u @

v

v @

v

w @

v Dv

Dt

ρ

5 ρ 5 ρ

ð 3 : 58 Þ

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

g z 1 @τ xz
@ x 1 @τ y 1 @σ zz

@ w

@

u @

w
@

v @

w
@

w @

w
@

Dw
Dt

yz

ρ

5 ρ 5 ρ

t 1 x 1 y 1 @ @ z z

Equation 3.58 is the differential equations of motion, which are valid for any fluid that
is a continuum and for any fluid that has the force of gravity as the only body force. In
Chapter 2, we defined the normal stress as a function of hydrostatic pressure and the
shear stresses as a function of viscosity and shear rate (which is a function of velocity).
The following definitions apply:

2
3 μr U

μ @

u

-

σ p 2

52 2 1

xx

@ x

2
3 μr U

μ @

v

- σ p 2

52 2 1

yy

@ x

2
3 μr U

μ @

w
@

- σ zz 52 p 2

2 1

x

ð 3 : 59 Þ

@ u

y 1 @

v

τ xy 5 τ yx 5 μ

@ @ x

@ u

@ z 1 @

w
@

τ xz 5 τ zx 5 μ

x

@ v

z 1 @

w
@

τ

5 τ 5 μ yz zy

@ y

where r is the gradient operator and is defined as

5 @

f

1 @

f

1 @

f

x - y - z -

r f

@ @ @

where f is any function in the Cartesian coordinate system (note that the gradient function
can be calculated in any coordinate system for any function; we are just highlighting the
Cartesian coordinate system for this analysis).