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Referring again to well‐recognized texts [1, 2], we will remind that the principle of conservation of mass, applied to an infinitesimally small control volume (CV), is mathematically expressed as

(1)

where ρ is the density and is the velocity vector. Equation (1) is often called the continuity equation. A mathematical expression of the conservation of momentum is

(2)

whose left‐hand side represents the time rate of change of momentum (i.e. mass times acceleration), and the right the forces acting upon the fluid by adjacent fluid particles (i.e. the divergence of the stress tensor, ) and other relevant objects through gravity or electromagnetically generated forces (i.e. a net body force per unit volume, ). Forces attributable to adjacent fluid particles are collectively represented by stress, which is another intensive property quantifying force per unit area.

An important requirement for mathematical modeling of fluid flows is to relate internal fluid stresses to characteristics of the fluid's motion. Such a relationship, which depends on the substance, is termed a constitutive relationship. There are many types of material behavior, requiring different constitutive models, but the most commonly used model relates shear stresses to strain rate (i.e. velocity gradient) in a linear manner. Fluids to which this linear relationship applies are known as Newtonian fluids. For example, a shear stress component τ_xy for a Newtonian fluid is characterized with

(3)

where μ is the dynamic viscosity and u is the x‐direction component of the velocity vector. For a Newtonian fluid, viscosity is independent of the strain rate. As long as it is homogeneous, glass is a Newtonian fluid although its viscosity is a very strong function of both composition and temperature (Chapter 4.1).

When the Newtonian constitutive model is substituted into the more general momentum Eq. (2) and the fluid is assumed to be incompressible, the result is the well‐known Navier–Stokes equation

(4)

where P is the fluid pressure. Not all fluids behave according to Eq. (3). Such non‐Newtonian fluids must be mathematically modeled with different constitutive relations substituted into Eq. (2) [5, 6].

Note that Eq. (4) is a vector equation, although it can be decomposed into vector component (scalar) equations, which is often done for the numerical application. In Cartesian coordinates, these component (scalar) equations are the following:

(5)

(6)

(7)

In fluid mechanics problems, the unknown field variables are the three components of velocity (e.g. u, v, and w) and fluid pressure, requiring four independent equations, which are the three momentum component Eqs. (5)–(7) and the continuity Eq. (1).

The equation for energy conservation is derived in a similar fashion. A common form of the energy equation is

(8)

where C_p and T are the specific heat and temperature, respectively, and k_t the thermal conductivity. In Eq. (8), the transient and advective transport terms are on the left side, whereas the right side has the conduction term and all other energy transfers accounted for as “sources.” An example of a “source” is Joule dissipation; that is, electrical energy converted to thermal energy by an electrical current passing through a resistive material (with an associated drop in electric potential).

In CFD, various transport phenomena are cast in the following general form known as the advection–diffusion equation [7],

(9)

where ϕ represents a generic field variable related to the property of interest, and Γ a generic diffusion coefficient. For example, temperature is the relevant field variable in the energy Eq. (8). (Likewise, ρc_p is substituted for ρ in the energy equation as it represents its “thermal inertia” per unit volume.) The first member on the left side of Eq. (9) is a transient term, which can be ignored for steady‐state problems. The second is the advection term, representing transport by fluid motion. The first member on the right side of Eq. (9) is the diffusion term, representing transport by atomic‐scale interactions, and all others are treated as “sources,” often because the model contains mathematical expressions which do not admit to the form of the one of the three standard terms.

Several commonly used mathematical expressions for the advective–diffusive transport of various field variables are listed in Table 2. Some are expressions of a fundamental principle, while others are consequences of further abstraction (e.g. turbulent kinetic energy), although still derived from first principles.

Table 2 List of commonly used transport equations in advection–diffusion form.

		ϕ	Transient	+	Advection	=	Diffusion	+	Source
A	Continuity	1		+		=		+	0
B	Momentum (x‐direction)	u		+		=	∇ ⋅ (μ ∇ u)	+
	Momentum (y‐direction)	v		+		=	∇ ⋅ (μ ∇ v)	+
	Momentum (z‐direction)	w		+		=	∇ ⋅ (μ ∇ w)	+
C	Energy	T		+		=	∇ ⋅ (k ∇ T)	+	S T
D	Turbulent kinetic energy	k		+		=	∇ ⋅ (μt, k ∇ k)	+	S k
E	Turbulence dissipation	ε		+		=	∇ ⋅ (μt, ɛ ∇ ε)	+	S ε
F	Electric charge/potential	E		+	0	=	∇ ⋅ (ke ∇ E)	+	0
G	Species A	mf A		+		=		+	S A

In Table 2, the rows D and E show the components of turbulence quantities for turbulent kinetic energy, k, and turbulence dissipation rate, ε. Glass flows are never turbulent, thanks to the stabilizing influence of glass extreme viscosity, in contrast to flows of air, gas, oxygen, combustion fumes, and other fluids that are part of glass processes.