Basics of Fluid Mechanics, 2014a

8.5. DERIVATIONS OF THE MOMENTUM EQUATION 245
τ zz + ∂τzz
∂z dz
(
τ yy + ∂τyy
∂y
)
dy
Z
τ yy
τ xz τ xy
(
τ xx
y
τ zz
x
τ xz + ∂τxz
)
∂x dx
(
τ xy + ∂τxy
)
∂x dx
(
τ xx + ∂τxx
∂x dx )
Fig. -8.8. The shear stress at different surfaces. All shear stress shown in surface x and x+dx.
cubic are surface forces, gravitation forces (body forces), and internal forces. The body
force that acting on infinitesimal cubic in x direction is
î · f B = f Bx dx dy dz (8.56)
The dot product yields a force in the directing of x. The surface forces in x direction
on the x surface on are
dA x
dA x
{ }} { { }} {
f xx = τ xx | x+dx
× dy dz − τ xx | x
× dy dz (8.57)
The surface forces in x direction on the y surface on are
dA y
dA y
{ }} { { }} {
f xy = τ yx | y+dy
× dx dz − τ yx | y
× dx dz (8.58)
The same can be written for the z direction. The shear stresses can be expanded into
Taylor series as
τ ix | i+di
= τ ix + ∂ (τ ix)
∂i ∣ di + ··· (8.59)
i
where i in this case is x, y, orz. Hence, the total net surface force results from the
shear stress in the x direction is
(
∂τxx
f x =
∂x
+ ∂τ yx
∂y
+ ∂τ )
zx
dx dy dz (8.60)
∂z