Chapter 8 Configuring Fluidity - The Applied Modelling and ...

14 Model equations
Using the equation of continuity (2.12) and Euler’s equation [Batchelor, 1967] (the force equation for
an inviscid fluid) in the form
gives
∂u
∂t
1
= −u · ∇u − ∇p, (2.14)
ρ
∂
(ρu) = −∇p − ∇ · (ρuu), (2.15)
∂t
where uu is a tensor which represents the dyadic product of vectors which, using index notation,
can be written as uiuj. Writing Π = pI + ρuu, (2.15) can be written as
∂
(ρu) + ∇ · Π = 0, (2.16)
∂t
where Π is clearly a symmetric tensor and is termed the momentum flux density tensor.
The effects of viscosity on the motion of a fluid are now considered. To express the equations of
motion governing a viscous fluid, some additional terms are required. The equation of continuity
(conservation of mass) is equally valid for any viscous as well as inviscid fluid. However, Euler’s
equation (2.14) and hence (2.16) require modification.
By adding −τ to the previously introduced momentum flux density tensor, Π, so that
Π = pI + ρuu − τ = −σ + ρuu, (2.17)
where σ = −pI + τ, the viscous transfer of momentum in the fluid can be taken into account. σ is
called the stress tensor and gives the part of the momentum flux which is not due to direct transfer
of momentum with the mass of the fluid. τ is named the deviatoric or viscous stress tensor. These
tensors and the forms which they may take are discussed in more detail in section 2.3.3. Thus, the
most general form of the momentum equation of a compressible viscous fluid may be written as
� �
∂u
ρ + u · ∇u = −∇ · σ + ρF , (2.18)
∂t
where F is a volume force per unit mass (e.g., gravity, astronomical forcing).
2.3.2.1 Compressible equations in conservative form
Using the conservation laws outlined above the following point-wise PDE system governing the
motion of a compressible fluid is obtained
∂ρ
+ ∇ · (ρu) = 0, (2.19a)
∂t
∂
(ρu) + ∇ · (ρuu − σ) = ρF , (2.19b)
∂t
∂
(ρE) + ∇ · (ρEu − σu + q) = ρF · u, (2.19c)
∂t
where E ≡ e + |u| 2 /2 is the total specific energy (in which e is the internal energy). (2.19a) is exactly
the conservative form of the continuity equation given in (2.12), (2.19b) is equivalent to equation
(2.18) and (2.19c) is obtained from making the substitutions w = e + p/ρ and E ≡ e + |u| 2 /2 in (2.18).