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One of the main objectives of kinetic theory is to describe the macroscopic properties of fluids from microscopic or molecular considerations like the mass, velocity, kinetic energy as well as internal degrees of freedom of molecules, and interaction forces between the molecules. The theory ultimately provides macroscopic conservation and constitutive equations that govern the flow and thermal field variables such as density, velocity, pressure, temperature, stress and heat flux. Explicit expressions of fluid properties such as viscosity and thermal conductivity can also be obtained. While conservation equations are straightforward to establish, and are valid for any fluid,

constitutive equations for stress and heat flux remain elusive for complex fluids and non- isothermal flow under non-ordinary conditions. The role of kinetic theory in establishing appropriate constitutive equations for gases and liquids will be reviewed.

2.2.1 Constitutive equation for heat

When the thermal relaxation time τ for a solid is comparable to the thermal process time, Fourier’s law ceases to be valid. In this case, the constitutive equation for the heat flux,

Q, is the Maxwell-Cattaneo or Cattaneo-Vernoti equation, which must be used, namely [24]: K T t ∂ τ = − ∇ ∂ Q +Q , (2.2.1)

where T is the temperature, K is the thermal conductivity and ∇ is the gradient operator. For a moving fluid, however, this equation needs to be reformulated and rendered objective (frame indifferent). This was achieved by Christov [14], and later revisited by Khayat & Ostoja-Starzewski [15]. The resulting equation is simply reproduced here as

K T t δ τ = − ∇ δ Q +Q , (2.2.2)

where the Jaumann or Li-type derivative is given by

( )

( )

_{( ) ( )}

_{( )}

t t V V V

δ ∂

≡ + ⋅∇ − ⋅∇ − ∇⋅

δ ∂ , (2.2.3)

where V is the velocity. It is not difficult to see that this equation relates the temperature gradient at a material point X and time t to the heat flux vector at the same point at time t + τ for a medium of thermal conductivity k. The relation reads:

(

, t+ τ = − ∇

)

K T

( )

, t

and equation (2.2.1) is recovered by Taylor expansion of (2.2.4) for small relaxation time. Note that equation (2.2.1) is hyperbolic. It is worth noting that other two-phase systems do reflect the phase-lagging character exhibited in (2.2.1). One such connection is the case of a hot jet penetrating a porous medium [25] or the flow through porous layers [26]. In this regard, equations (2.1a and b) of [25] are comparable to equations (4) and (5) of [20]. Also, in this case, the former equations switch from parabolic to hyperbolic for high jet penetration velocity.

2.2.2 Conservation and constitutive equations

Both conservation and constitutive equations can have their basis in kinetic theory. The theory is particularly well established for rarefied monatomic gases, but lately has been generalized to include dense gases and liquids. Its foundations were established in 1867 by James Clerk Maxwell (1831-1879) who proposed a general transport equation for arbitrary macroscopic quantities associated with mean values of microscopic quantities (Maxwell 1867). This equation of transport relates the time evolution of a macroscopic quantity with the motion of the molecules, collision between the molecules and action of external forces. The theory was valid for any molecular interaction potential, but the kinetic theory of gases gained a new impulse in 1872 with the work by Ludwig Eduard Boltzmann (1844-1906), who proposed an integro-differential equation – the Boltzmann equation – which represents the evolution of the velocity distribution function in the phase space spanned by the coordinates and velocities of the molecules [28]. Further advances were due to Sydney Chapman (1888-1970) and David Enskog (1884-1947) who calculated independently and by different methods the transport coefficients for gases whose molecules interact according to any kind of spherically symmetric potential function [29,30]. Another method derived from the Boltzmann equation was proposed in 1949 by Harold Grad (1923-1986) who expanded the distribution function in terms of tensorial Hermite polynomials and introduced balance equations corresponding to higher order moments of the distribution function [31].

The extension of kinetic theory to describe liquids involves daunting difficulties associated with multi-body molecular interactions and additional degrees of freedom. Nevertheless, with the proper development of collision integrals based on suitable interaction potential, Boltzmann equation was applied to liquids [32-35]. In this case, Grad’s 13-moment method can be applied by projecting the extended Boltzmann equation for liquids.

Before pursuing our discussion on non-Fourier effects, it is helpful to recall the governing equations for incompressible non-isothermal flow of a fluid of density ρ, viscosity µ and thermal diffusivity κ. The conservation and constitutive equations will be cast in dimensionless form, with D, U, D

U and U D

µ being the reference length, velocity, time and stress, respectively. The current paper is mainly concerned with closely coupled flow and heat transfer problems. In this case, a suitable velocity scale is taken as U

D κ

= . For a general fluid, the equations for conservation of linear momentum and energy read, respectively [36]:

(

)

1 t Pr− v + ⋅∇v v = −∇ + ∇ ⋅p σ, (2.2.5) t v j Ec :σ v θ + ⋅∇θ = − + ∇ , (2.2.6)

where j= ∇⋅q. The Prandtl and Eckert numbers are given by Pr and Ec D2 K T

ν µ

= =

κ ∆ ,

respectively, ∆T being a typical temperature differential of the process. Here ν is the kinematic viscosity and

p

K c κ =

ρ is the thermal diffusivity. In this work, Newtonian non- Fourier fluids will be considered. It is not unconceivable that most non-Fourier fluids may exhibit a non-Newtonian character. However, in order to keep the discussion tractable, the scope of the paper will be restricted to Newtonian fluids. In this case,

t

σ= ∇ ∇v+ v , and one recovers the Navier-Stokes equations. For an incompressible flow, adding the continuity equation, and equations (2.2.5) and (2.2.6) reduce to

0 v ∇ ⋅ = , (2.2.7)

(

)

1 t Pr− v + ⋅∇v v = −∇ + ∆p v, (2.2.8) t v j Ec v: v θ + ⋅∇θ = − + ∇ ∇ , (2.2.9)

As to the constitutive equation for heat, the formulation can be simplified by casting the constitutive equation for heat flux in terms of the scalar variable j. Thus, upon taking the divergence of equation (2.2.9), and recalling again the identity

(

a b

)

a: b a

(

b

)

∇⋅ ⋅∇ = ∇ ∇ + ⋅∇ ∇⋅ , one obtains the following constitutive equation for j:

(

t

)

C j + ⋅∇ + = −∆θv j j . (2.2.10) The following non-dimensional groups have been introduced, namely, the Cattaneo number, C, given as 2 C D τκ = . (2.2.11)

Depending on the boundary conditions, the heat flux vector, q, may or may not be obtained separately. If the boundary conditions are imposed on the temperature and j, then q can be calculated by solving

C t q v q q v q ∂ ⎛ _{+ ⋅∇ − ⋅∇ + = −∇θ}⎞ ⎜ _∂ ⎟ ⎝ ⎠ , (2.2.12)

once v, p, θ and j are determined. If, on other hand, the heat flux vector is imposed at the boundary, then equations must be solved simultaneously. The Fourier model is recovered upon setting C = 0 (zero relaxation time). In this case, q= −∇θ and j= −∆θ.

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