Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 80

∂τ yx dy
∂ y 2

τ yx +

∂τ zx dz
∂ z 2

τ zx −

Y

dy

∂σ xx dx
∂ x 2

p

∂σ xx dx
∂ x 2

σ xx +

σ xx −

∂τ zx dz
∂ z 2

τ zx +

dz

∂τ yx dy
∂ y 2

τ yx −

dx

X

Z

FIGURE 3.19

The normal and shear stresses acting in the x-direction on a differential fluid element. The stres-
ses that act in the other Cartesian directions can be derived in a similar manner. Recall that only six of these stress
values are independent for momentum conservation.

Quantifying the stresses in the x-direction as a component of the total surface forces,

dydz dydz dxdz

1 @τ

1 @σ xx
@

dx
2

2 @σ xx
@

dx
2

dy
2

yx

dF sx

5 σ 2 σ 1 τ

xx xx yx

x x @ y

dxdz dxdy dxdy

2 @τ yx
@

dy
2

1 @τ

dz
2

1 @τ

dz
2

zx zx

2 τ 1 τ 2 τ ð 3 : 56 Þ

yx zx zx

y @ z @ z

dxdydz

x 1 @τ yx

@σ

y 1 @τ

xx zx

5

@ @ @ z

Using a similar analysis for each of the remaining two directions and assuming that the
gravitational force is the only body force, the total force in each direction becomes

dxdydz

x 1 @τ yx

1 @σ

y 1 @τ

xx zx

dF x dF bx dF sx

5 ρ

g x

5 1

@ @ @ z

dxdydz

1 @τ xy
@ x 1 @σ yy y 1 @τ zy

dF y dF by dF sy

5 ρ

g y

5 1 ð 3 : 57 Þ

@ @ z

dxdydz

x 1 @τ yz

1 @τ

y 1 @σ

xz zz

dF z dF bz dF sz

5 ρ

g z

5 1

@ @ @ z