Basics of Fluid Mechanics, 2014a

242 CHAPTER 8. DIFFERENTIAL ANALYSIS
where δA y is the surface area of the tetrahedron
in the y direction, δA x is the surface
area of the tetrahedron in the x direction
and δA z is the surface area of the
tetrahedron in the z direction. The opposing
forces which acting on the slanted surface in the x direction are
)
F x = δA n
(τ nn ̂n · î − τ nl ̂l · î − τ nℵ̂ℵ·î
(8.43)
Where here ̂ℵ, ̂l and ̂n are the local unit coordinates on n surface the same can be
written in the x, and z directions. The transformation matrix is then
⎛ ⎞ ⎛
⎞
F x ̂n · î ̂l · î ̂ℵ·î
⎝ F y
⎠ ⎜
⎟
= ⎝ ̂n · ĵ ̂l · ĵ ̂ℵ·ĵ ⎠ δA n (8.44)
F x ̂n · ̂k ̂l · ̂k ̂ℵ·̂k
When the tetrahedron is shrunk to a point relationship of the stress on the two sides
can be expended by Taylor series and keeping the first derivative. If the first derivative
is neglected (tetrahedron is without acceleration) the two sides are related as
)
−τ yx δA y − τ xx δA x − τ zx δA z = δA n
(τ nn ̂n · î − τ nl ̂l · î − τ nℵ̂ℵ·î (8.45)
The same can be done for y and z directions. The areas are related to each other through
angles. These relationships provide the transformation for the different orientations
which depends only angles of the orientations. This matrix is referred to as stress
tensor and as it can be observed has nine terms.
The Symmetry of the Stress Tensor
A small liquid cubical has three possible rotation axes. Here only one will be discussed
the same conclusions can be drown on the other direction. The cubical rotation
can involve two parts: one distortion and one rotation 10 . A finite angular distortion of
infinitesimal cube requires an infinite shear which required for infinite moment. Hence,
the rotation of the infinitesimal fluid cube can be viewed as it is done almost as a solid
body rotation. Balance of momentum around the z direction shown in Figure 8.6 is
dθ
M z = I zz (8.46)
dt
Where M z is the cubic moment around the cubic center and I 11 zz is the moment of
inertia around that center. The momentum can be asserted by the shear stresses which
act on it. The shear stress at point x is τ xy . However, the shear stress at point x + dx
is
τ xy | x+dx
= τ xy + dτ xy
dx (8.47)
dx
10 For infinitesimal change the lines can be approximated as straight.
11 See for the derivations in Example 3.5 for moment of inertia.