Basics of Fluid Mechanics, 2014a

8.5. DERIVATIONS OF THE MOMENTUM EQUATION 253
or index notation
τ ij = −
(
P m + 2 ) (
3 μ∇·U
∂Ui
δ ij + μ + ∂U )
j
∂x j ∂x i
(8.99)
End Advance material
where δ ij is the Kronecker delta what is δ ij =1when i = j and δ ij =0otherwise. While
this expression has the advantage of compact writing, it does not add any additional
information. This expression suggests a new definition of the thermodynamical pressure
is
Thermodynamic Pressure
P = P m + 2 3 μ∇·U
(8.100)
Summary of The Stress Tensor
The above derivations were provided as a long mathematical explanation 19 . To
reduced one unknown (the shear stress) equation (8.61) the relationship between the
stress tensor and the velocity were to be established. First, connection between τ xy and
the deformation was built. Then the association between normal stress and perpendicular
stress was constructed. Using the coordinates transformation, this association was
established. The linkage between the stress in the rotated coordinates to the deformation
was established.
Second Viscosity Coefficient
The coefficient 2/3μ is experimental and relates to viscosity. However, if the
derivations before were to include additional terms, an additional correction will be
needed. This correction results in
P = P m + λ∇·U (8.101)
The value of λ is obtained experimentally. This coefficient is referred in the literature
by several terms such as the “expansion viscosity” “second coefficient of viscosity” and
“bulk viscosity.” Here the term bulk viscosity will be adapted. The dimension of the
bulk viscosity, λ, is similar to the viscosity μ.According to second law of thermodynamic
derivations (not shown here and are under construction) demonstrate that λ must be
positive. The thermodynamic pressure always tends to follow the mechanical pressure
during a change. The expansion rate of change and the fluid molecular structure through
λ control the difference. Equation (8.101) can be written in terms of the thermodynamic
pressure P ,as
[ ( ) ] ( 2
τ ij = − P +
3 μ − λ ∂Ui
∇·U δ ij + μ + ∂U )
j
(8.102)
∂x j ∂x i
19 Since the publishing the version 0.2.9.0 several people ask this author to summarize conceptually
the issues. With God help, it will be provide before version 0.3.1