Basics of Fluid Mechanics, 2014a

246 CHAPTER 8. DIFFERENTIAL ANALYSIS
after rearrangement equations such as (8.57) and (8.58) transformed into
internal forces
{ }} {
DU x
Dt ρ ✚✚dx dy ✚dz =
surface forces
{( }} {
∂τxx
∂x
+ ∂τ yx
∂y
+ ∂τ )
zx
✚✚dx
∂z dy ✚dz +
body forces
{ }} {
f Gx ρ✚✚dx dy ✚dz (8.61)
equivalent equation (8.61) fory coordinate is
ρ DU y
Dt
The same can be obtained for the z component
ρ DU z
Dt
=
=
(
∂τxy
∂x + ∂τ yy
∂y
+ ∂τ )
zy
+ ρf Gy (8.62)
∂z
(
∂τxz
∂x + ∂τ yz
∂y
+ ∂τ )
zz
+ ρf Gz (8.63)
∂z
Advance material can be skipped
Generally the component momentum equation is as
ρ DU i
Dt
=
(
∂τii
∂i
+ ∂τ ji
∂j + ∂τ )
ki
+ ρf Gi (8.64)
∂j
End Advance material
Where i is the balance direction and j and k are two other coordinates. Equation
(8.64) can be written in a vector form which combined all three components into one
equation. The advantage of the vector from allows the usage of the different coordinates.
The vector form is
ρ DU
Dt = ∇·τ (i) + ρf f G (8.65)
where here
τ (i) = τ ix î + τ iy ĵ + τ iẑk
is part of the shear stress tensor and i can be any of the x, y, or z.
Or in index (Einstein) notation as
ρ DU i
Dt
= ∂τ ji
∂x i
+ ρf Gi (8.66)
End Advance material
Equations (8.61) or(8.62) or(8.63) requires that the stress tensor be defined
in term of the velocity/deformation. The relationship between the stress tensor and
deformation depends on the classes of materials the stresses acts on. Additionally, the
deformation can be viewed as a function of the velocity field. As engineers do in general,
the simplest model is assumed which referred as the solid continuum model. In this
model the relationship between the (shear) stresses and rate of strains are assumed to be