Thermal Convection in a Cattaneo–Fox Porous Material with Guyer–Krumhansl Effects

Abstract

The Guyer–Krumhansl equation is coupled with the Cattaneo–Fox law for the temperature and heat flux fields to study thermal convection in a fluid-saturated Darcy porous material. In particular the effects of the Guyer–Krumhansl terms on oscillatory convection are studied. It is found that for a certain range of the Guyer–Krumhansl coefficient stationary convection occurs while changing the range results in oscillatory convection. Numerical results quantify this effect.

Appendices

Appendix 1: Cattaneo–Fox Law and Cattaneo–Christov Law

To clarify Cattaneo–Fox law and Cattaneo–Christov law we begin with the Maxwell–Cattaneo law of heat conduction Cattaneo (1948),

$$\begin{aligned} \tau Q_{i,t} + Q_i = - \kappa T_{,i}. \end{aligned}$$

(25)

When Eq. (25) is combined with the conservation of energy equation it leads to the hyperbolic telegraphist equation for the temperature field. Thus the temperature can propagate as a damped travelling. However, Eq. (25) is insufficient to describe heat transfer in a moving fluid (Dauby et al. 2002), or heat conduction in nanomaterials see, e.g. Wang et al. (2011) and references therein. As a result, the equation governing heat flux must involve an objective derivative.

Straughan and Franchi (1984) proposed the following modification on the Maxwell–Cattaneo law (25)

$$\begin{aligned} \tau \left( {\dot{Q}_i - \varepsilon _{ijk} w_j Q_k } \right) = - Q_i - \kappa T_{,i}, \end{aligned}$$

(26)

which has come to be known as the Cattaneo–Fox heat flux law. Here $\varvec{w}= \text {curl}\,\varvec{{v}}/2$, a superposed dot denotes the material time derivative. The derivative $\tau \left( {\dot{Q}_i - \varepsilon _{ijk} w_j Q_k } \right)$ is an objective (Jaumann) time derivative of Fox (1969) for the heat flux.

Christov (2009) proposed an appropriate objective derivative for the heat flux when dealing with a Cattaneo-type theory for a fluid. He suggests the following Lie derivative which is a frame indifferent objective rate,

$$\begin{aligned} \tau \left( {Q_{i,t} + v_j Q_{i,j} - Q_j v_{i,j} + v_{j,j} Q_i } \right) = - Q_i - \kappa T_{,i}, \end{aligned}$$

(27)

which has come to be known as The Cattaneo–Christov law.

The Cattaneo–Christov theory has been placed on a second thermodynamic basis by Morro (2010). Straughan (2010a) showed that the Cattaneo–Christov theory yields a well-defined thermo-acoustic theory for wave propagation in a gas.

Appendix 2: Guyer–Krumhansl Model

The generalization of Eq. (25) which follows from the solution of the linearized Boltzmann equation is a well-known Guyer–Krumhansl equation for heat flux (Guyer and Krumhansl 1964, 1966a, b). This equation has been analysed by Lebon and Dauby (1990) by means of variational argument in the context of extended thermodynamics, which has form

$$\begin{aligned} \tau Q_{i,t} + Q_i = - \kappa T_{,i} + \hat{\tau }\varDelta Q_i + 2\hat{\tau }Q_{k,ki}. \end{aligned}$$

(28)

Here $\hat{\tau }= \tau \tau _N c_s^2/5$ where $\tau _N$ is a relaxation time and $c_s$ is the mean speed of phonos.

Franchi and Straughan (1994) proposed modifying Eq. (26) by incorporating the Guyer–Krumhansl terms for heat flux. Then one would modify Eq. (26) to

$$\begin{aligned} \tau \left( {\dot{Q}_i - \varepsilon _{ijk} w_j Q_k } \right) = - Q_i - \kappa T_{,i}+\hat{\tau }(\varDelta Q_i + 2 Q_{k,ki}). \end{aligned}$$

(29)

Franchi and Straughan (1994) employed Eq. (29) to study problem of thermal convection.