On oscillatory convection with the Cattaneo–Christov hyperbolic heat-flow model

Abstract

Adoption of the hyperbolic Cattaneo–Christov heat-flow model in place of the more usual parabolic Fourier law is shown to raise the possibility of oscillatory convection in the classic Bénard problem of a Boussinesq fluid heated from below. By comparing the critical Rayleigh numbers for stationary and oscillatory convection, R_c and R_S respectively, oscillatory convection is found to represent the preferred form of instability whenever the Cattaneo number C exceeds a threshold value C_T≥8/27π²≈0.03. In the case of free boundaries, analytical approaches permit direct treatment of the role played by the Prandtl number ${\mathcal{P}}_1$, which—in contrast to the classical stationary scenario—can impact on oscillatory modes significantly owing to the non-zero frequency of convection. Numerical investigation indicates that the behaviour found analytically for free boundaries applies in a qualitatively similar fashion for fixed boundaries, while the threshold Cattaneo number C_T is computed as a function of ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$ for both boundary regimes.

1. Introduction

The notion of thermal transport obeying a hyperbolic rather than diffusive (Fourier-type) law of conduction goes back to Maxwell's early work on kinetic theory [1], and is a topic of considerable ongoing concern. In part, such interest derives from foundational problems in the Fourier theory of heat where—owing to its parabolic nature—thermal disturbances propagate with infinite speed [2,3]; hyperbolic modifications of Fourier's law overcome this problem because thermal signals may then travel as waves (heat waves) [4,5]. However, the significance of thermal waves is deep, and reaches beyond their theoretical implications: for instance, such waves have important applications in the study of energy transport [6] and thermal shocks in solids [7]; heat transport in nanomaterials and nanofluids [8–11]; biological tissues and surgical procedures [12], including skin burns [13] and radiofrequency heating [14,15]; convection in fluids and porous media [16–18]; alongside astrophysical contexts, particularly thermohaline convection [19,20]. In addition, work on hyperbolic theories of thermal transport has inspired related studies in other flux-based problems, for example, those involving advection–diffusion equations [21,22], and discontinuity waves [23,24].

Given such varied applications, and the clear need to improve theoretical understanding of thermal waves, it is the concern of this article to study the effects of hyperbolic heat-flow within a well-established context, namely the classic Rayleigh–Bénard convection problem of a Boussinesq fluid heated from below [25,26]. Indeed, by investigating hyperbolic heat-flow in such a canonical problem, we seek not only to further understanding of thermal waves per se, but also contribute to ongoing research into novel aspects of thermal convection and non-stationary heat conduction more generally [3–31]. To this end, we focus on the commonly encountered Maxwell–Cattaneo reformulation of Fourier's law in which the heat-flow vector Q_i is expressed in terms of both gradients in the local temperature T, and inertial effects owing to the relaxation time τ taken for such gradients to become established [2,32], that is,

$$\tau \frac{\mathrm{\partial }{Q}_i}{\mathrm{\partial }t}+Q_i=-\kappa \frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_i},$$ 1.1

where κ is the thermal conductivity, and standard indicial notion applies throughout. The thermal relaxation time τ is typically rather small [3,4], but can take relatively large values (approx. 100 s) in some contexts (e.g. biological tissues [13]). Curiously, while equation (1.1) is perhaps the most frequently encountered model for hyperbolic heat-flow (see, for example, the articles cited above), it is known to yield a paradoxical result for moving bodies whereby the process of heat conduction becomes dependent on the frame of motion [32]. To circumvent such difficulties, Christov and Jordan [32,33] proposed a modification to equation (1.1) by which Galilean invariance is restored after replacing the partial time derivative with a material derivative, i.e.

$$\tau (\frac{\mathrm{\partial }{Q}_i}{\mathrm{\partial }t}+v_j\frac{\mathrm{\partial }{Q}_i}{\mathrm{\partial }{x}_j})+Q_i=-\kappa \frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_i}.$$ 1.2

As we shall be working with a system in convective motion, we adopt this Cattaneo–Christov formulation throughout. Note that for problems such as ours, inertial effects introduced by the relaxation time are best discussed in terms of the dimensionless Cattaneo number C(τ), which accounts for combined effects, including system length scales (see equation (2.11)).

The problem of thermal convection with a hyperbolic heat-flow model seems to have been first investigated by Straughan & Franchi [16], who examined the instability of a system heated from above; however, it is Straughan's more recent work, in which he considers oscillatory convection with a Cattaneo–Christov model in a fluid layer heated from below [17], that is most relevant to our present discussion. In particular, if we denote the critical Rayleigh numbers for stationary and oscillatory convection by R_c and R_S, respectively, then for fluids confined by free boundaries Straughan derives an expression for R_S(C) which decreases with Cattaneo number, suggesting a transition from stationary to oscillatory convection as C is increased beyond a threshold value C_T for which R_S(C_T)=R_c; in the case of fixed boundaries, and for Prandtl number ${\mathcal{P}}_1=6$, Straughan then uses numerical methods to bound this value to the interval C_T∈(2.2×10⁻²,2.3×10⁻²) [17].

Straughan's preliminary investigation yields results which are valuable and compelling; however, Straughan's work raises a number of fundamental problems in need of serious consideration: for example, in the free boundary case it is of pressing importance to determine whether threshold behaviour is actually physical. In fact, by examining the frequency of oscillation γ one may show that oscillatory solutions are forbidden unless the Cattaneo number exceeds some bounding value C_B (see §4); for threshold behaviour to obtain generally, therefore, one must confirm the condition $C_{\mathrm{T}}({\mathcal{P}}_1)>C_{\mathrm{B}}({\mathcal{P}}_1)$ for arbitrary Prandtl number. Indeed, the importance of the Prandtl number itself is something Straughan does not address, possibly because ${\mathcal{P}}_1$ does not enter into calculations of the critical Rayleigh number R_c for stationary convection (when γ=0). However, for oscillatory convection (γ≠0), the role played by the Prandtl number becomes fundamental; in particular, the set of crucial parameters—the threshold Cattaneo number C_T, critical wavenumber a_S and oscillation frequency γ—all exhibit strong ${\mathcal{P}}_1$ dependence (see §§4–7).

In this article, therefore, we substantially develop Straughan's ideas [17] to form a comprehensive theory of thermal convection with the Cattaneo–Christov heat-flow model, paying particular attention to the foundational issues described above. Beginning with a short review of the classical theory [26] in §§2 and 3, we proceed to the theory of oscillatory convection in §4, where we generalize and simplify some of Straughan's initial work on the critical Rayleigh number R_S, deriving a number of new analytical results and asymptotic expressions in the process. In particular, we present calculations of both the bounding C_B and threshold Cattaneo number $C_{\mathrm{T}}({\mathcal{P}}_1)\ge 8/27{\pi }^2\approx 0.03$ for the first time, and thereby analyse ${\mathcal{P}}_1$-C parameter space generally into regions where either stationary or oscillatory convection is preferred (§5). Such an analysis allows us to show rigorously that the condition $C_{\mathrm{T}}({\mathcal{P}}_1)>C_{\mathrm{B}}({\mathcal{P}}_1)$ holds for arbitrary ${\mathcal{P}}_1$, and thence—in the case of free boundary conditions—that discontinuous transitions from stationary to oscillatory convection are permitted physically. For fixed boundary conditions, we analyse the system numerically (§6) by calculating the threshold parameters for Prandtl number in the range ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$, and—in a qualitative fashion at least—recover much of the behaviour observed analytically for free boundaries. These results, alongside some of their potential consequences for thermal convection more generally, are discussed further in §7.

2. Convection model

The basic fluid model adopted here comprises equations governing the conservation of mass and energy, alongside Christov's Galilean invariant formulation of the Cattaneo heat-flow law [2,26,32]. Thus, denoting the velocity, pressure, temperature and heat-flow as v_i, P, T and Q_i, respectively, the momentum, energy and heat-flow equations are (cf. Chandrasekar [26] and Straughan [17])

$$\begin{array}{rl}(\frac{\mathrm{\partial }{v}_i}{\mathrm{\partial }t}+v_j\frac{\mathrm{\partial }{v}_i}{\mathrm{\partial }{x}_j})& =-\frac{1}{{\rho }_0}\frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_i}+\nu {\mathrm{\nabla }}^2{v}_i-\frac{\rho }{{\rho }_0}g{\mathrm{\lambda }}_i,\end{array}$$ 2.1a

$$\begin{array}{rl}{\rho }_0{c}_{\mathrm{V}}(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+v_j\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_j})& =-\frac{\mathrm{\partial }{Q}_j}{\mathrm{\partial }{x}_j}\end{array}$$ 2.1b

$$\begin{array}{rl}\text{and}\tau (\frac{\mathrm{\partial }{Q}_i}{\mathrm{\partial }t}+v_j\frac{\mathrm{\partial }{Q}_i}{\mathrm{\partial }{x}_j})+Q_i& =-\kappa \frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_i}.\end{array}$$ 2.1c

Equations (2.1a–c) include constant coefficients for the fluid viscosity ν, gravitational acceleration g=|g|, specific heat at constant volume c_V, thermal relaxation time τ and thermal conductivity κ. Here, we assume a Cartesian (x,y,z) geometry in which gravity acts in the negative z-direction, i.e. λ_i is the unit vector λ_i=(0,0,1), while the Laplacian operator is ∇²≡∂²/∂x²+∂²/∂y²+∂²/∂z². We also employ the Boussinesq approximation in the buoyancy term, viz

$$\rho (T)={\rho }_0[1+\alpha (T_{\alpha }-T)],$$ 2.2

where ρ₀ is the fluid density when it is at temperature T=T_α, and α is a thermal expansion coefficient. Note that by asserting a material derivative in the Cattaneo heat-flow law, model (2.1) differs from that employed by Straughan, who used an upper convected Oldroyd time derivative [17,32,33]. However, because both models assume an incompressible equation of state, i.e.

$$\frac{\mathrm{\partial }{v}_i}{\mathrm{\partial }{x}_i}=0,$$ 2.3

our focus on a linearized system means that this difference is not important, and the more compact and intuitive form given by equation (2.1c) is preferred.

(a). Steady-state and linearized system

We now suppose that the fluid occupies a semi-infinite region $(x,y)\in {\mathbb{R}}^2$ confined between two parallel planes, one at z=0, and the other at z=d, where the upper and lower planes are held at constant temperatures T_u and T_l respectively, i.e.

$$T(0)=T_{\mathrm{l}}\text{and}T(d)=T_{\mathrm{u}},\hspace{0.17em}\text{with}\hspace{0.17em}{T}_{\mathrm{l}}>T_{\mathrm{u}}.$$ 2.4

In this way our system corresponds to the classic Bénard problem of a fluid heated from below, with the z-component of the velocity w vanishing on the boundaries, i.e. (cf. Chandrasekhar [26])

$$v_z=w=0\text{at}\hspace{0.17em}z=0,d.$$ 2.5

In steady-state, solutions to system (2.1) are given by

$$v_i=v_{i0}=0,T=T_0=T_{\mathrm{l}}-\beta {\mathrm{\lambda }}_j{x}_j,Q_i=Q_{i0}=\beta \kappa {\mathrm{\lambda }}_i\text{and}P=P_0(z),$$ 2.6

where β denotes the temperature gradient, and P₀ has a profile such that the buoyancy force is balanced by pressure gradients, that is,

$$\beta =\left|\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_i}\right|=\frac{T_{\mathrm{l}}-T_{\mathrm{u}}}{d}\text{and}\frac{\mathrm{\partial }{P}_0}{\mathrm{\partial }{x}_i}=-g\rho (T_0){\mathrm{\lambda }}_i.$$ 2.7

Our main purpose here is to consider the stability of these steady-state solutions once perturbed. However, before proceeding it is expedient to recast our system in a dimensionless form by defining the normalized quantities

$${\tilde{x}}_i=\frac{x_i}{d},\tilde{t}=\frac{\nu }{d^2}t,{\tilde{v}}_i=\frac{v_i}{\nu /d},\tilde{P}=\frac{P}{{\nu }^2{\rho }_0/d^2},\tilde{T}=\sqrt{\frac{\alpha g\kappa d^2}{\beta {\nu }^3{\rho }_0{c}_{\mathrm{V}}}}T,{\tilde{Q}}_i=\frac{d}{\kappa }\left(\frac{\tilde{T}}{T}\right)Q_i.$$ 2.8

Consequently, after adding small perturbations $\{{\tilde{u}}_i,\tilde{\theta },{\tilde{q}}_i,\tilde{p}\}$ to the steady-state solutions, that is,

$${\tilde{v}}_i={\tilde{v}}_{i0}+{\tilde{u}}_i={\tilde{u}}_i,\tilde{T}={\tilde{T}}_0+\tilde{\theta },{\tilde{Q}}_i={\tilde{Q}}_{i0}+{\tilde{q}}_i\text{and}\tilde{P}={\tilde{P}}_0+\tilde{p},$$ 2.9

system (2.1) may be linearized to give a set of normalized equations, viz

$$\begin{array}{rl}\frac{\mathrm{\partial }{\tilde{u}}_i}{\mathrm{\partial }\tilde{t}}& =-\frac{\mathrm{\partial }\tilde{p}}{\mathrm{\partial }{\tilde{x}}_i}+\widetilde{{\mathrm{\nabla }}^2}{\tilde{u}}_i+R\tilde{\theta }{\mathrm{\lambda }}_i,\end{array}$$ 2.10a

$$\begin{array}{rl}{\mathcal{P}}_1\frac{\mathrm{\partial }\tilde{\theta }}{\mathrm{\partial }\tilde{t}}& =R\tilde{w}-\frac{\mathrm{\partial }{\tilde{q}}_j}{\mathrm{\partial }{\tilde{x}}_j}\end{array}$$ 2.10b

$$\begin{array}{rl}\text{and}2{\mathcal{P}}_{1}C\frac{\mathrm{\partial }{\tilde{q}}_i}{\mathrm{\partial }\tilde{t}}& =-{\tilde{q}}_i-\frac{\mathrm{\partial }\tilde{\theta }}{\mathrm{\partial }{\tilde{x}}_i},\end{array}$$ 2.10c

where the dimensionless Prandtl number ${\mathcal{P}}_1$, Cattaneo number C and Rayleigh number R_a=R² are defined

$${\mathcal{P}}_1=\frac{\nu {\rho }_0{c}_{\mathrm{V}}}{\kappa },C=\frac{\tau \kappa }{2{\rho }_0{d}^2{c}_{\mathrm{V}}}\text{and}R=\sqrt{R_a}=\sqrt{\frac{\alpha g{d}^4\beta {\rho }_0{c}_{\mathrm{V}}}{\nu \kappa }}.$$ 2.11

Hence, denoting the divergence of the heat-flow

$$Q_q=\frac{\mathrm{\partial }{\tilde{q}}_i}{\mathrm{\partial }{\tilde{x}}_i},$$ 2.12

we take the curl of equation (2.10a) twice, the divergence of equation (2.10c) once, and the inner product any vector equations with λ_i, and thereby eliminate $\tilde{p}$ from the linearized system (2.10) to obtain (cf. Chandrasehkar [26])

$$\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }t}{\mathrm{\nabla }}^{2}w& =R(\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2})+{\mathrm{\nabla }}^{4}w,\end{array}$$ 2.13a

$$\begin{array}{rl}{\mathcal{P}}_1\frac{\mathrm{\partial }\theta }{\mathrm{\partial }t}& =Rw-Q_q\end{array}$$ 2.13b

$$\begin{array}{rl}\text{and}2C{\mathcal{P}}_1\frac{\mathrm{\partial }{Q}_q}{\mathrm{\partial }t}& =-Q_q-{\mathrm{\nabla }}^2\theta ,\end{array}$$ 2.13c

where w=u_z is the z-component of the velocity field, and the tilde notation has been dropped for brevity. These equations describe the evolution of perturbations to the conductive steady state (2.6) in a form conveniently reduced to three variables; we are thus well placed to investigate the stability of the system by further decomposing the problem into normal modes.

(b). Analysis into normal modes

Let us now write the perturbations in separable form assuming an exponential time dependence such that

$$w=W(z)f(x,y){\mathrm{e}}^{\sigma t},\theta =\mathit{\Theta }(z)f(x,y){\mathrm{e}}^{\sigma t}\text{and}{Q}_q=Q(z)f(x,y){\mathrm{e}}^{\sigma t},$$ 2.14

where σ is a constant frequency, W, Θ and Q are some eigenfunctions to be found, and f(x,y) is a plane tiling function satisfying

$${\mathrm{\nabla }}^{2}f(x,y)=-a^{2}f(x,y),$$ 2.15

with a as a characteristic wavenumber or inverse length scale. Then system (2.13) becomes (cf. Chandrasehkar [26] and Straughan [17])

$$\begin{array}{rl}\sigma (D^2-a^2)W& ={(D^2-a^2)}^{2}W-a^{2}R\mathit{\Theta },\end{array}$$ 2.16a

$$\begin{array}{rl}\sigma {\mathcal{P}}_1\mathit{\Theta }& =RW-Q\end{array}$$ 2.16b

$$\begin{array}{rl}\text{and}2\sigma {\mathcal{P}}_{1}CQ& =-Q-(D^2-a^2)\mathit{\Theta },\end{array}$$ 2.16c

where with Φ∈{W,Q,Θ} the operators D and D² are

$$D\mathrm{\Phi }=\frac{\mathrm{d}\mathrm{\Phi }}{\mathrm{d}z}\text{and}{D}^2\mathrm{\Phi }=\frac{{\mathrm{d}}^2\mathrm{\Phi }}{\mathrm{d}{z}^2}.$$ 2.17

Equations (2.16) represent the starting point for both the analytical and numerical work on oscillatory convection forming the main basis of this article. However, because we shall be comparing results for oscillatory convection with those of stationary convection, it is appropriate at this stage to briefly review some classical results.

3. Stationary convection

For stationary convection, instability sets in through the marginal state characterized by σ=0, and we can eliminate Θ and Q from equations (2.16) to give [26]

$${(D^2-a^2)}^{3}W=-a^2{R}^{2}W=-a^2{R}_{a}W.$$ 3.1

For free boundaries, we have that

$$W=0,D^{2}W=0,\mathit{\Theta }=0,\text{at}\hspace{0.17em}z=0,1,$$ 3.2

so by system (2.16) and equation (3.1), one deduces that all even derivatives of W vanish at the boundaries, and W(z) may be decomposed into odd Fourier modes, thus

$$W(z)=\sum _{n=1}^{\mathrm{\infty }}{W}_n,\text{with}\hspace{0.17em}{W}_n=A_n\sin (n\pi z)$$ 3.3

as the nth mode weighted by the constant coefficient A_n. Hence, following substitution of this expansion into equation (3.1), the orthogonality of the Fourier series gives a Rayleigh number R_a for the nth mode [26]

$$R_a(n)=\frac{{\mathrm{\Lambda }}_n^3}{a^2},\text{where}\hspace{0.17em}{\mathrm{\Lambda }}_n\equiv n^2{\pi }^2+a^2.$$ 3.4

Because we seek a minimum value for the onset of convection, differentiating R_a(n) with respect to a thus yields a critical Rayleigh number R_c(n)=R_a(a_c) for the nth mode corresponding to critical wavenumber a_c(n), where (∂R_a/∂a)_ac=0, such that

$$R_{\mathrm{c}}(n)=\frac{27n^4{\pi }^4}{4}\text{and}{a}_{\mathrm{c}}(n)=\frac{n\pi }{\sqrt{2}}.$$ 3.5

Naturally, because the sequence {R_a(n)} increases monotonically with n, the absolute critical Rayleigh number occurs when n=1, i.e. (cf. Chandrasekhar [26]),

$$R_{\mathrm{c}}=\frac{27{\pi }^4}{4}\approx 657.511\text{and}{a}_{\mathrm{c}}=\frac{\pi }{\sqrt{2}}\approx 2.221,$$ 3.6

where we adopt the convention that unless the mode number n is explicitly stated in a parameter's argument, expressions shall be quoted assuming n=1 throughout.

In the case of fixed boundary conditions, that is,

$$W=0,DW=0,\mathit{\Theta }=0,\text{at}\hspace{0.17em}z=0,1,$$ 3.7

system (2.16) may be solved numerically to obtain (cf. section 6 and Chandrasehkar [26])

$$R_{\mathrm{c}}\approx 1707.762\text{and}{a}_{\mathrm{c}}\approx \mathrm{3.116.}$$ 3.8

Note that because σ=0, in the case of stationary convection we have by system (2.16) that neither the Prandtl number ${\mathcal{P}}_1$ nor Cattaneo number C enter into calculations of critical values, so it is not surprising that we recover the classical results associated with the more usual Fourier law.

4. Oscillatory convection

Proceeding to the focus of the present work, we now consider the onset of instability via some kind of oscillatory mode. Some of the initial results in this section are equivalent to those first obtained by Straughan [17]; however, the theory described here substantially develops Straughan's ideas, and departs from his analysis in key areas. Crucially, for example, we establish rigorously that oscillatory modes are in fact physically permitted solutions, paying particular attention to the role played by Prandtl number ${\mathcal{P}}_1$ (which Straughan did not consider), and thereby derive a number of important analytical results, limiting behaviours, and threshold values (cf. §1).

For oscillatory solutions, we assume that σ may be complex and non-zero, and define γ such that [26]

$$\sigma =\mathrm{i}\gamma ,$$ 4.1

then eliminating Θ and Q from system (2.16), we have

$$-{(D^2-a^2)}^2[(D^2-a^2)-\sigma ]W+\sigma {\mathcal{P}}_1(2C{\mathcal{P}}_1\sigma +1)(D^2-a^2)W=(2C{\mathcal{P}}_1\sigma +1)a^2{R}_{a}W.$$ 4.2

Hence, for the free boundary conditions given in equation (3.2), one finds (as we had for stationary convection) that all even derivatives of the eigenfunction W(z) vanish at the boundaries, and our Fourier decomposition (3.3) gives

$$({\mathrm{\Lambda }}_n^3-2{\mathcal{P}}_1^{2}C{\mathrm{\Lambda }}_n^2{\gamma }^2-{\mathrm{\Lambda }}_n{\mathcal{P}}_1{\gamma }^2-a^2{R}_a)=(2C{\mathcal{P}}_1{a}^2{R}_a-{\mathrm{\Lambda }}_n^2({\mathcal{P}}_1+1)+2C{\mathcal{P}}_1^2{\mathrm{\Lambda }}_n{\gamma }^2)\sigma .$$ 4.3

From equation (4.3), we see that for an arbitrary γ the Rayleigh number will in general be complex. As observed by Chandrasekhar [26], the physical requirement that R_a be real thus places constraints on the relationship between the real and imaginary components of σ=iγ. We therefore study the onset of convection via a purely oscillatory mode by asserting γ to be real in which case comparison of real and imaginary parts in equation (4.3) yields (cf. Chandrasehkar [26] and Straughan [17])

$${\mathrm{\Lambda }}_n^3-2{\mathcal{P}}_1^{2}C{\mathrm{\Lambda }}_n^2{\gamma }^2-{\mathrm{\Lambda }}_n{\mathcal{P}}_1{\gamma }^2=a^2{R}_a$$ 4.4a

and

$${\mathrm{\Lambda }}_n^2({\mathcal{P}}_1+1)-2C{\mathcal{P}}_1^2{\mathrm{\Lambda }}_n{\gamma }^2=2C{\mathcal{P}}_1{a}^2{R}_a.$$ 4.4b

Straughan uses this system to eliminate γ² and obtain an expression for R_a [17]; however, as we shall discuss further in later sections, examination of the oscillation frequency is integral to establishing the physicality of solutions, and for this reason we instead begin by eliminating R_a to determine γ, viz

$${\gamma }^2(n)=\frac{1}{4C^2{\mathcal{P}}_1^3}[{\mathcal{P}}_1(2C{\mathrm{\Lambda }}_n-1)-1].$$ 4.5

Indeed, because we need γ²>0, for given n, a and ${\mathcal{P}}_1$ this expression places a restriction on the Cattaneo number for physical solutions to obtain; in particular, we require

$$C>\frac{({\mathcal{P}}_1+1)}{2(n^2{\pi }^2+a^2){\mathcal{P}}_1}.$$ 4.6

Assuming condition (4.6) to hold, we substitute for γ² in system (4.4) to give

$$R_a(n)=\frac{1}{4C^2{\mathcal{P}}_1^2{a}^2}[({\mathcal{P}}_1+1){\mathrm{\Lambda }}_n+2C{\mathcal{P}}_1^2{\mathrm{\Lambda }}_n^2],$$ 4.7

and then—as we did for stationary convection—solve (∂R_a/∂a)_aS=0 to determine the critical Rayleigh number R_S(n)=R_a(a_S) for the nth mode (cf. Straughan [17])

$$R_{\mathrm{S}}(n)=\frac{n^2{\pi }^2}{2C}{(1+\frac{a_S^2(n)}{n^2{\pi }^2})}^2,$$ 4.8

where a_S(n) is the corresponding critical wavenumber

$$a_S^2(n)=n^2{\pi }^2{[\frac{({\mathcal{P}}_1+1)}{2n^2{\pi }^{2}C{\mathcal{P}}_1^2}+1]}^{1/2}.$$ 4.9

Here that we have adopted an ‘S’ subscript notation for these critical values, both to emphasize that they represent Straughan's results generalized to the nth mode [17], and to distinguish them from the values for stationary convection outlined in §3. At this stage in the analysis it is appropriate to quote the generalized results, because physical solutions obtain more readily when n is large. Indeed, for the critical modes, that is, a=a_S, inequality (4.6) indicates the requirement

$$C>C_{\mathrm{B}}(n)\equiv \frac{({\mathcal{P}}_1+1)}{2n^2{\pi }^2(2{\mathcal{P}}_1+1)},\text{i.e.}\hspace{0.17em}\underset{n\to \mathrm{\infty }}{lim}{C}_{\mathrm{B}}(n)=0,$$ 4.10

where C_B∈(1/4n²π²,1/2n²π²) is defined here as the ‘bounding’ Cattaneo number.

Note again that the sequence of critical Rayleigh numbers {R_S(n)} increases monotonically with n, so that—seeking a minimum value—by equations (4.8) and (4.9) we have an absolute critical Rayleigh number when n=1, i.e. (figure 1)

$$R_{\mathrm{S}}=\frac{{\pi }^2}{2C}{[1+{(\frac{({\mathcal{P}}_1+1)}{2{\pi }^{2}C{\mathcal{P}}_1^2}+1)}^{1/2}]}^2,$$ 4.11

with corresponding wavenumber a_S and frequency γ determined by equations (4.9) and (4.5) after taking n=1.

Figure 1.

Critical Rayleigh numbers R_a (a), wavenumbers a (b), and asymptotic limits (dashed-dotted curves) plotted as a function of C for ${\mathcal{P}}_1=6$ in the case of free boundaries. Note that beyond the threshold value C_T≈0.0342 (circles) we have that R_S<R_c, in which case oscillatory convection (solid curves) is preferred over stationary convection (dashed curves), and the wavenumber shifts discontinuously (circles). Here, the oscillatory convection curves represent physical solutions only when inequality (4.14) is satisfied with ${\mathcal{P}}_1=6$, i.e. C>C_B(6)≈0.0273 (cf. figure 3).

The critical Rayleigh number given by equation (4.11) is equivalent to that derived by Straughan, though here we have expressed R_S in a far more compact form that makes its functional dependence on the Cattaneo number more apparent. Indeed, we see immediately that R_S is a strictly decreasing function of C, while in the asymptotic limits we have

$$R_{\mathrm{S}}(C)\to \{\begin{array}{ll}\frac{({\mathcal{P}}_1+1)}{4C^2{\mathcal{P}}_1^2}\to \mathrm{\infty },& \text{as}\hspace{0.17em}C\to 0,\\ \frac{2{\pi }^2}{C}\to 0,& \text{as}\hspace{0.17em}C\to \mathrm{\infty },\end{array}$$ 4.12a

with

$$a_S^2(C)\to \{\begin{array}{ll}\pi \sqrt{\frac{({\mathcal{P}}_1+1)}{2C{\mathcal{P}}_1^2}}\to \mathrm{\infty },& \text{as}\hspace{0.17em}C\to 0,\\ {\pi }^2,& \text{as}\hspace{0.17em}C\to \mathrm{\infty }.\end{array}$$ 4.12b

Similar limits with ${\mathcal{P}}_1$ may also be obtained for given Cattaneo number C. Thus, at fixed ${\mathcal{P}}_1,$ we find that for increasing $C:0\to \mathrm{\infty }$, then the Rayleigh number decreases like $R_{\mathrm{S}}(C):\mathrm{\infty }\to 0$.

Naively, therefore, one supposes there to exist a threshold Cattaneo number C_T defined such that R_S(C_T)=R_c, beyond which (C>C_T) the preferential form of instability switches from stationary to oscillatory convection (R_S<R_c). Furthermore, given our critical wavenumbers

$$a_S^2={\pi }^2{[\frac{({\mathcal{P}}_1+1)}{2{\pi }^{2}C{\mathcal{P}}_1^2}+1]}^{1/2}>a_{\mathrm{c}}^2=\frac{{\pi }^2}{2},$$ 4.13

one expects this change to be characterized by a discontinuous transition in the value of a (figure 1). Nevertheless, for the n=1 mode to yield physical solutions, inequality (4.14) demands that

$$C>C_{\mathrm{B}}({\mathcal{P}}_1)\equiv \frac{({\mathcal{P}}_1+1)}{2{\pi }^2(2{\mathcal{P}}_1+1)},$$ 4.14

so for a given Prandtl number ${\mathcal{P}}_1$, these kinds of transitions between forms of convection will only occur in the fashion described provided C_T>C_B. It is to this particular issue—the problem of determining whether the threshold behaviour is physical—that we now turn.

5. Threshold Cattaneo number

Because the threshold Cattaneo number C_T is defined such that R_S(C_T)=R_c, for a given Prandtl number ${\mathcal{P}}_1$ we have

$$R_{\mathrm{S}}(C_{\mathrm{T}})=\frac{{\pi }^2}{2C_{\mathrm{T}}}{[1+{(\frac{({\mathcal{P}}_1+1)}{2{\pi }^2{C}_{\mathrm{T}}{\mathcal{P}}_1^2}+1)}^{1/2}]}^2=R_{\mathrm{c}},\text{i.e.}\hspace{0.17em}{C}_T^{3/2}[C_T^{1/2}-\left(\frac{2{\pi }^2}{R_{\mathrm{c}}}\right)]=\frac{({\mathcal{P}}_1+1)}{4R_{\mathrm{c}}{\mathcal{P}}_1^2},$$ 5.1

an expression which may be solved numerically (figure 2). By equation (5.1), we see that C_T is a monotonically decreasing function of ${\mathcal{P}}_1$, obeying the asymptotic behaviour

$$C_{\mathrm{T}}({\mathcal{P}}_1)\to \{\begin{array}{ll}\frac{1}{3{\pi }^2{\mathcal{P}}_1\sqrt{3}}\to \mathrm{\infty },& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to 0,\\ \frac{2{\pi }^2}{R_{\mathrm{c}}}=\frac{8}{27{\pi }^2},& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to \mathrm{\infty }.\end{array}$$ 5.2

Assuming all values of C>C_T to represent physical solutions, therefore, the curve $C_{\mathrm{T}}({\mathcal{P}}_1)$ may be used to divide C-${\mathcal{P}}_1$ parameter space into regions where either stationary convection (R_c<R_S) or oscillatory convection (R_c>R_S) with mode n=1 is the preferred mechanism for instability (figure 2). We now show that this assumption is correct.

Figure 2.

The threshold Cattaneo number C_T (solid curve), bounding Cattaneo number C_B (dashed curve), and asymptotic limits (dashed-dotted curves) plotted as a function of ${\mathcal{P}}_1$. For C=C_T we have R_S=R_c, so the curve $C_{\mathrm{T}}({\mathcal{P}}_1)$ divides ${\mathcal{P}}_1$-C parameter space into regions where either oscillatory convection (C>C_T) or stationary convection (C<C_T) by the critical mode n=1 is preferred. Similarly, ${\mathcal{P}}_1$-C space may be divided by the curve $C_{\mathrm{B}}({\mathcal{P}}_1)$ into regions where oscillatory convection by the critical n=1 mode is either physically permitted (C>C_B), or forbidden (C<C_B). Oscillatory convection can occur in regions below the curve C_B, but will do so only for mode numbers n>1 or wavenumbers a>a_S (see inequality (4.6)). Note that for given C, one may also view the threshold curve in terms of the threshold Prandtl number ${\mathcal{P}}_T(C)$ as defined in equation (5.8).

Recall that for the lowest mode (n=1), physical solutions require C>C_B, where C_B is a lower bound on the Cattaneo number defined in equation (4.14). Then, since R_S is a strictly decreasing function of C, for a given Prandtl number ${\mathcal{P}}_1$ we may define

$$R_B\equiv R_{\mathrm{S}}(C_{\mathrm{B}})=\frac{{\pi }^4{(2{\mathcal{P}}_1+1)}^3}{{\mathcal{P}}_1^2({\mathcal{P}}_1+1)}$$ 5.3

as an upper bound on the critical Rayleigh number for oscillatory convection by the n=1 mode, and with limits

$$R_B({\mathcal{P}}_1)\to \{\begin{array}{ll}\frac{{\pi }^4}{{\mathcal{P}}_1^2}\to \mathrm{\infty },& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to 0,\\ 8{\pi }^4=\frac{32{\pi }^4}{4},& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to \mathrm{\infty }.\end{array}$$ 5.4

Furthermore, differentiation of R_B with respect to ${\mathcal{P}}_1$ shows that it is a strictly decreasing function of Prandtl number, viz

$$\frac{\mathrm{d}{R}_B}{\mathrm{d}{\mathcal{P}}_1}=-\frac{{\pi }^4({\mathcal{P}}_1+2){(2{\mathcal{P}}_1+1)}^2}{{\mathcal{P}}_1^3{({\mathcal{P}}_1+1)}^2}<0,\mathrm{\forall }\hspace{0.17em}{\mathcal{P}}_1\in {\mathbb{R}}^{+}.$$ 5.5

Hence, by our asymptotic limits (5.4), we have with R_S(C_B)=R_B and R_S(C_T)=R_c that

$$R_{\mathrm{S}}(C_{\mathrm{B}})>\frac{32{\pi }^4}{4}>\frac{27{\pi }^4}{4}=R_{\mathrm{S}}(C_{\mathrm{T}}),\mathrm{\forall }\hspace{0.17em}{\mathcal{P}}_1\in {\mathbb{R}}^{+},$$ 5.6

and, because R_S(C) is a strictly decreasing function of C, that

$$C_{\mathrm{T}}({\mathcal{P}}_1)>C_{\mathrm{B}}({\mathcal{P}}_1),\mathrm{\forall }\hspace{0.17em}{\mathcal{P}}_1\in {\mathbb{R}}^{+}.$$ 5.7

Thus, for any given value of ${\mathcal{P}}_1$, inequality (5.7) tells us that C>C_T also guarantees C>C_B, and the values R_S(C)<R_c will correspond to physical solutions. Hence, for free boundaries at least, when discussing the transition from stationary to oscillatory convection, it is sufficient—as we have done so far—to restrict our attention to critical values associated with the lowest mode.

Of course, one may express these thresholds equivalently in terms of the Prandtl number ${\mathcal{P}}_1$; indeed, because $\mathrm{\partial }{R}_{\mathrm{S}}/\mathrm{\partial }{\mathcal{P}}_1<0$, then for fixed Cattaneo number C we can define a threshold Prandtl number ${\mathcal{P}}_T$ such that $R_{\mathrm{S}}({\mathcal{P}}_1)<R_{\mathrm{c}}$ for all ${\mathcal{P}}_1>{\mathcal{P}}_T$ (figure 2). By equation (5.1), this value may be written in the closed form

$${\mathcal{P}}_T=\frac{1+\sqrt{1+16R_{\mathrm{c}}{C}^{3/2}(\sqrt{C}-\sqrt{2{\pi }^2/R_{\mathrm{c}}})}}{8R_{\mathrm{c}}{C}^{3/2}(\sqrt{C}-\sqrt{2{\pi }^2/R_{\mathrm{c}}})},$$ 5.8

with asymptotic behaviour corresponding to limits (5.2)

$${\mathcal{P}}_T(C)\to \{\begin{array}{ll}\mathrm{\infty },& \text{as}\hspace{0.17em}C\to \frac{2{\pi }^2}{R_{\mathrm{c}}},\\ \frac{1}{3{\pi }^{2}C\sqrt{3}},& \text{as}\hspace{0.17em}C\to \mathrm{\infty },\end{array}$$ 5.9

where $C\to 2{\pi }^2/R_{\mathrm{c}}$ is taken from above. As we found in our discussion of $C_{\mathrm{T}}({\mathcal{P}}_1)$, therefore, ${\mathcal{P}}_T(C)$ is thus a strictly decreasing function of C defined on $C\in (2{\pi }^2/R_{\mathrm{c}},\mathrm{\infty })$.

An important consequence of inequality (5.7) is that the transition from stationary to oscillatory convection occurs with finite frequency. Indeed, for a given Prandtl number ${\mathcal{P}}_1$, one may define a threshold wavenumber a_T=a_S(C_T) and frequency γ_T=γ(C_T) for the critical modes R_S(C_T)=R_c, where γ_T>0 and a_T>a_c. These threshold values, which have asymptotic limits

$$a_{\mathrm{T}}({\mathcal{P}}_1)\to \{\begin{array}{ll}\pi {\left(\frac{3\sqrt{3}}{2{\mathcal{P}}_1}\right)}^{1/4}\to \mathrm{\infty },& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to 0,\\ \pi ,& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to \mathrm{\infty }\end{array}$$ 5.10a

and

$${\gamma }_{\mathrm{T}}^2({\mathcal{P}}_1)\to \{\begin{array}{ll}\frac{9{\pi }^4}{2{\mathcal{P}}_1^{3/2}}{\left(\frac{3}{4}\right)}^{1/4}\to \mathrm{\infty },& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to 0,\\ \frac{135{\pi }^4}{256{\mathcal{P}}_1^2}\to 0,& \text{as}\hspace{0.17em}{\mathcal{P}}_1\to \mathrm{\infty },\end{array}$$ 5.10b

are discussed further in §7. Because a_S decreases with C, and because oscillatory solutions have C>C_T, the threshold wavenumber a_T represents an upper limit on values taken by a_S.

6. Numerical analysis

In the case of fixed boundary conditions, i.e. those given by equation (3.7), system (2.16) may be solved numerically using the Chebyshev tau-QZ method described by Dongerra et al. [34] and outlined in the context of thermal convection by Straughan [17]. Under this approach, one begins by transforming the z-coordinate such that our problem is defined on z∈(−1,1), and then proceeds to eliminate terms with derivatives higher than D² by introducing an auxiliary variable χ(z) such that χ(z)f(x,y) e^σt=∇²w. In this way, system (2.16) may be written in the form

$$\begin{array}{rl}(4D^2-a^2)W-\chi & =0,\end{array}$$ 6.1a

$$\begin{array}{rl}(4D^2-a^2)\chi -a^{2}R\mathit{\Theta }& =\sigma \chi ,\end{array}$$ 6.1b

$$\begin{array}{rl}(4D^2-a^2)\mathit{\Theta }+Q& =-2\sigma C{\mathcal{P}}_{1}Q\end{array}$$ 6.1c

$$\begin{array}{rl}\text{and}-RW+Q& =-\sigma {\mathcal{P}}_1\mathit{\Theta }.\end{array}$$ 6.1d

Thence, after expanding the quantities Φ∈{W,χ,Θ,Q} as Chebyshev polynomials T_n(z) weighted by constant coefficients ϕ_n, viz

$$\mathrm{\Phi }(z)=\sum _{n=0}^{N-1}{\varphi }_n{T}_n(z),\text{with }N\text{ even,}$$ 6.2

one may approximate the system to (in principle) arbitrary precision in terms of the eigenvalue problem

$$A_{ij}{\text{x}}_j=\sigma B_{ij}{\text{x}}_j,$$ 6.3

where x_j is a vector of length 4N comprising the ϕ_n, i.e.

$${\text{x}}_j={(w_0,\dots ,w_{N-1},{\chi }_0,\dots ,{\chi }_{N-1},{\theta }_0,\dots ,{\theta }_{N-1},q_0,\dots ,q_{N-1})}^{\text{T}},$$ 6.4

and A_ij≡A_ij(a,R) and $B_{ij}\equiv B_{ij}(C,{\mathcal{P}}_1)$ are 4N×4N matrices with constant coefficients inclusive of the expanded boundary conditions (cf. Straughan [17]).

Given R, C and ${\mathcal{P}}_1$, therefore, system (6.3) may be solved for the wavenumber a subject to the condition ℜ{σ}=0; critical values are then obtained by minimizing R_a(a)=R². In this way, both stationary (γ=0) and oscillatory (γ≠0) solutions may be found such that subsequent variation of C yields an intersection problem for the threshold Cattaneo number C_T (figure 3).

Figure 3.

Critical Rayleigh numbers R_a (a) and wavenumbers a (b) plotted as a function of C for ${\mathcal{P}}_1=6$ in the case of fixed boundaries (cf. figure 1). Beyond the threshold value C_T≈0.02223 (circles) we have R_S<R_c, in which case oscillatory convection (dashed curves) is preferred over stationary convection (solid curves), and the wavenumber shifts discontinuously (circles). Note that it has not been possible to compute R_S for Cattaneo number less than approximately 0.0165; presumably this is because—as we had in the case of free boundaries in figure 1, where C_B≈0.0273—there exists some bounding Cattaneo number C_B≈0.0165 below which solutions are non-physical. As elsewhere, data in these plots are given to ±0.1% uncertainty, and computed using N=40 polynomials (cf. table 1).

Truncation of the Chebyshev polynomial expansion means that the accuracy of the solutions given by our numerical method are dependent on N. For the range of parameters explored in this article we find—up to six significant figures at least—that changing the number of Chebyshev polynomials from N=40 to N=50 does not impact on computed values, and so all numerical data are quoted assuming N=40. A more important consideration turns out to be the resolution for the scan over wavenumber a; here, we have employed a grid such that the critical wavenumber a_S may be determined to within ±0.1%. The threshold Cattaneo number C_T then follows by examining R_S(C) and finding upper C_u and lower C_l bounds such that R_S(C_l)>R_c>R_S(C_u); again, by choosing a suitably fine grid for test values of C, this process allows us to resolve the threshold value C_T to within ±0.1% (and similarly for γ_T). Note that investigation of the upper a_u and lower a_l wavenumbers associated with the C_u and C_l bounds on C_T (the bounding wavenumbers around a_T) indicates a corresponding uncertainty on a_T to within the ±0.1% for which a_S is initially computed, i.e.

$$\frac{1}{2}\left|\frac{a_{\mathrm{u}}-a_{\mathrm{l}}}{a_{\mathrm{T}}}\right|<0.001,\text{where}\hspace{0.17em}{a}_{\mathrm{T}}=\frac{1}{2}(a_{\mathrm{l}}+a_{\mathrm{u}}).$$ 6.5

Data corresponding to the intersection problem when ${\mathcal{P}}_1=6$ are displayed graphically in figure 3, in which case we find a threshold Cattaneo number C_T=(2.223±0.0022)×10⁻², consistent with Straughan's lower precision result of C_T∈(2.2×10⁻²,2.3×10⁻²) [17]. Note from the logarithmic plot of the Rayleigh number that the oscillatory solution R_S appears to approach a power law R_S∝1/C as the Cattaneo number becomes large, and therefore—in a qualitative fashion at least—seems to obey similar behaviour to the free boundary solution discussed in §4 (cf. figure 1). Likewise, as C is increased beyond the threshold value C_T, the solution switches from stationary convection to oscillatory convection with a discontinuous transition to a larger wavenumber and narrower convection cells, that is, (a_c,a_S=a_T)≈(3.116,4.873) at the threshold. Again, this behaviour corresponds to that seen in the case of free boundaries, with further increases to C resulting in a reduction in the wavenumber (suggesting subsequent broadening of convection cells) as a_S tends towards some constant asymptotic limit.

Naturally, one may solve the intersection problem for different values of ${\mathcal{P}}_1$, and thereby investigate the effect of Prandtl number on those values of Cattaneo number C_T, wavenumber a_T and oscillation frequency γ_T at which the transition from stationary to oscillatory convection occurs. Indeed, as we did for free boundary conditions in §5, one may then divide our ${\mathcal{P}}_1$-C parameter space into regions where either stationary convection (R_c<R_S) or oscillatory convection (R_c>R_S) is preferred (figure 4). Here, we compute 401 values for Prandtl number in the range ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$ (table 1). In a similar fashion to the free boundary case, note that the threshold Cattaneo number appears to obey some kind of asymptotic behaviour, with $C_{\mathrm{T}}\propto 1/{\mathcal{P}}_1$ when the Prandtl number is small, tending towards a constant value $C_{\mathrm{T}}({\mathcal{P}}_1)\approx 0.02$ when ${\mathcal{P}}_1$ is large (cf. figure 2). This value is comparable to that obtained for free-boundaries, where we found C_T≥8/27π²≈0.03 (see §5).

Figure 4.

The threshold Cattaneo number C_T plotted as a function of ${\mathcal{P}}_1$ for fixed boundary conditions. As we found in the free boundary case, for C=C_T we have R_S=R_c, so the curve $C_{\mathrm{T}}({\mathcal{P}}_1)$ divides ${\mathcal{P}}_1$-C parameter space into regions where either oscillatory convection (C>C_T) or stationary convection (C<C_T) is preferred (cf. figure 2). A subset of the data used to produce this figure is given in table 1.

Table 1.

Numerically computed values for the threshold Cattaneo numbers C_T, wavenumbers a_T, and oscillation frequencies γ_T in the case of fixed boundary conditions. This table comprises a subset of the 401 values used to produce figure 5 for Prandtl numbers in the range ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$, with data quoted to within ±0.1% uncertainty.

${\mathcal{P}}_1$	CT	aT	γT	${\mathcal{P}}_1$	CT	aT	γT
10−2.0	1.234	5.621	212.5	10+0.0	0.03214	5.175	12.10
10−1.9	0.9894	5.786	199.1	10+0.1	0.02965	5.123	9.928
10−1.8	0.7935	5.954	186.5	10+0.2	0.02768	5.073	8.094
10−1.7	0.6368	6.110	173.5	10+0.3	0.02614	5.027	6.569
10−1.6	0.5116	6.232	159.6	10+0.4	0.02492	4.986	5.307
10−1.5	0.4116	6.295	144.0	10+0.5	0.02396	4.950	4.271
10−1.4	0.3318	6.292	127.3	10+0.6	0.02321	4.918	3.428
10−1.3	0.2681	6.228	110.4	10+0.7	0.02262	4.891	2.746
10−1.2	0.2173	6.127	94.52	10+0.8	0.02215	4.869	2.195
10−1.1	0.1768	6.008	80.24	10+0.9	0.02178	4.850	1.752
10−1.0	0.1445	5.891	67.94	10+1.0	0.02149	4.834	1.397
10−0.9	0.1189	5.783	57.54	10+1.1	0.02126	4.821	1.112
10−0.8	0.09842	5.687	48.81	10+1.2	0.02108	4.811	0.8860
10−0.7	0.08215	5.608	41.44	10+1.3	0.02093	4.802	0.7049
10−0.6	0.06924	5.535	35.18	10+1.4	0.02082	4.796	0.5608
10−0.5	0.05898	5.469	29.80	10+1.5	0.02073	4.790	0.4459
10−0.4	0.05085	5.407	25.16	10+1.6	0.02066	4.785	0.3544
10−0.3	0.04442	5.347	21.14	10+1.7	0.02060	4.781	0.2817
10−0.2	0.03933	5.288	17.66	10+1.8	0.02055	4.778	0.2238
10−0.1	0.03531	5.232	14.67	10+1.9	0.02052	4.776	0.1779
10−0.0	0.03214	5.175	12.10	10+2.0	0.02049	4.774	0.1414

7. Effect of Prandtl number on threshold values

Recall that in the case of free boundary conditions the wavenumber $a_{\mathrm{T}}^2\equiv a_S^2({\mathcal{P}}_1,C_{\mathrm{T}})$ and frequency $\gamma \equiv {\gamma }_{\mathrm{T}}({\mathcal{P}}_1,C_{\mathrm{T}})$ associated with the oscillatory solution at threshold Cattaneo number are defined by

$$a_{\mathrm{T}}^2\equiv {\pi }^2{[\frac{({\mathcal{P}}_1+1)}{2{\pi }^2{C}_{\mathrm{T}}{\mathcal{P}}_1^2}+1]}^{1/2}\text{and}{\gamma }_T^2\equiv \frac{1}{4C_{\mathrm{T}}^2{\mathcal{P}}_1^3}[{\mathcal{P}}_1(2C_{\mathrm{T}}({\pi }^2+a_{\mathrm{T}}^2)-1)-1]$$ 7.1

respectively (cf. equations (4.5) and (4.13)), whereas for fixed boundaries these threshold values are given numerically by the returned values of a_S(C_T) and γ(C_T) at the threshold solution R_c=R_S(C_T) (table 1). Given their explicit dependence on ${\mathcal{P}}_1$, alongside the ${\mathcal{P}}_1$ dependence of C_T itself, it is therefore clear that both a_T and γ_T will be influenced by the Prandtl number rather strongly (figure 5).

Figure 5.

Threshold wavenumbers $a_{\mathrm{T}}\equiv a_S({\mathcal{P}}_1,C_{\mathrm{T}})$ (a,b) and frequencies $\gamma \equiv {\gamma }_{\mathrm{T}}({\mathcal{P}}_1,C_{\mathrm{T}})$ (c,d) as a function of Prandtl number ${\mathcal{P}}_1$ for the case of both free (a,c) and fixed (b,d) boundary conditions. The analytical free boundary solutions (see equations (7.1)) have asymptotic limits with ${\mathcal{P}}_1$ (dashed curves) given by equations (5.10).

Plots of the threshold wavenumbers and frequencies as a function of Prandtl number in the range ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$ are given for both free and fixed boundary conditions in figure 5. Note in the free boundary case that a_T decreases with ${\mathcal{P}}_1$, with asymptotic behaviour $a_{\mathrm{T}}\propto {\mathcal{P}}_1^{-1/4}$ when ${\mathcal{P}}_1$ is small, converging to the value $a_{\mathrm{T}}({\mathcal{P}}_1\to \mathrm{\infty })=\pi$ when the Prandtl number is large (see the limiting behaviour expressed in equation (5.10a)). Because $a_{\mathrm{T}}>\pi >a_{\mathrm{c}}=\pi /\sqrt{2}$, this means that the transition from stationary to oscillatory convection is always marked initially by a discontinuous shift to smaller convection cells. Such an effect is to a certain extent replicated in the case of fixed boundary conditions; indeed, again we see what appears to be convergence towards a constant value a_T≈4.77 when the Prandtl number is large (table 1), while—on the domain ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$ at least—we have a_T>a_c≈3.116 (see equation (3.8)). In this second case, however, we find that the threshold wavenumber initially increases with ${\mathcal{P}}_1$, before decreasing, reaching a maximum value a_T≈6.3 at ${\mathcal{P}}_1\approx 0.035$.

In the absence of a full nonlinear analysis (which would reveal more detailed information about the geometry of convection cells, but is well beyond the scope of the present article), it is not immediately clear why this discrepancy between the boundary condition regimes should arise at low ${\mathcal{P}}_1$. What is more, our picture is further complicated by the highly coupled nature of the threshold system whereby both Cattaneo number and Prandtl number play a part, with C_T dependent on ${\mathcal{P}}_1$. However, some insights may be reached by returning to the definition of ${\mathcal{P}}_1$, viz

$${\mathcal{P}}_1=\frac{\nu {\rho }_0{c}_{\mathrm{V}}}{\kappa }$$ 7.2

(see equations (2.11)); thus, at fixed conductivity κ, we see that the low Prandtl number regime corresponds to one with vanishing viscosity.¹ In the case of fixed boundaries at low ${\mathcal{P}}_1$, therefore, there is a marked contrast between the vanishing ‘stiffness’ of the fluid between the planes of confinement ($\nu \to 0$), and the no-slip condition for the velocity of the fluid at the walls; boundary effects are thus far more significant. Compared with the free boundary case, when no such difference is present, it is then not so surprising that the behaviour at low Prandtl number should be distinct. Of course, why the value of a_T should reach a maximum when the boundaries are fixed, rather than simply converging to another constant value as ${\mathcal{P}}_1\to 0$, or indeed how ${\mathcal{P}}_1$ affects a_T more generally in either regime, are both questions in need of resolution. However, because the wavenumber will correspond to the geometry of the convection cells, it is the opinion of the author that satisfactory answers will require undertaking a more sophisticated nonlinear study than is appropriate to consider here.

On the other hand, a physical basis for the ${\mathcal{P}}_1$ dependence of the threshold oscillation frequency γ_T is more apparent. Indeed, in both boundary regimes, one expects that high viscosity (i.e. large Prandtl number) will impede the overall motion of the fluid, thereby curtailing oscillatory motion. Such behaviour, whereby γ_T always decreases with ${\mathcal{P}}_1$, is present for both free and fixed boundary conditions (figure 5), whereas the slightly ‘kinked’ dependence in the latter case seems in some way to reflect the corresponding variation in the wavenumber (cf. the γ_T dependence on a_T in the case of free boundaries indicated by equations (7.1)).

8. Conclusion

We have studied the canonical Rayleigh–Bénard convection problem of a Boussinesq fluid layer heated from below, using the hyperbolic Cattaneo–Christov heat-flow model in place of the more usual Fourier law [2,17,25,26,32,33], and in so doing developed a linear theory for thermal instability by oscillatory modes (which are forbidden classically, see §§1–4). In the case of free boundary conditions, critical Rayleigh numbers for both stationary and oscillatory convection, denoted R_c and R_S respectively, may be determined analytically, and thus allow for a comparison between preferred forms of instability. When considering stationary convection the perturbation time dependence has no component of gyration (σ=iγ=0), meaning that both Cattaneo number C and Prandtl number ${\mathcal{P}}_1$ effects vanish (i.e. those terms in σC and $\sigma {\mathcal{P}}_1$). Under these conditions, therefore, solutions are equivalent to the classical results obtained with the more usual Fourier law, and onset of instability occurs with critical Rayleigh number R_c=27π⁴/4≈657.511, corresponding to a critical wavenumber $a_{\mathrm{c}}=\pi /\sqrt{2}\approx 2.221$ [26]. For oscillatory convection, however, when γ≠0, the critical Rayleigh number R_S acquires both C and ${\mathcal{P}}_1$ dependence, so that by examining the necessary condition for physical solutions γ²>0 overstability is rendered possible provided C is larger than a lower bounding value $C_{\mathrm{B}}({\mathcal{P}}_1)$. In this case, the associated critical wavenumber a_S also acquires C and ${\mathcal{P}}_1$ dependence, with $a_S(C,{\mathcal{P}}_1)>a_{\mathrm{c}}$ for all $C,{\mathcal{P}}_1\in {\mathbb{R}}^{+}$, converging on the classical result $a_S\to a_{\mathrm{c}}$ as $C,{\mathcal{P}}_1\to \mathrm{\infty }$.

Crucially, for given Prandtl number, we have demonstrated that R_S(C) is a strictly decreasing function of C, obeying a dependence $R_{\mathrm{S}}(C)\propto 1/C^2\to 0<R_{\mathrm{c}}$ when C is large, whereas for relatively small values of the Cattaneo number R_S exceeds the classical (Fourier law) value with R_S(C)>R_c. Thus, provided C>C_B, there exists some threshold Cattaneo number C_T such that R_S(C_T)=R_c beyond which the preferred form of instability switches from stationary to oscillatory convection. Notably, such a transition is marked by a discontinuous shift from the stationary wavenumber $a_{\mathrm{c}}=\pi /\sqrt{2}$ to narrow convection cells in the oscillatory regime, with associated wavenumber a_S>a_c; for example, at Prandtl number ${\mathcal{P}}_1=6$, when C_T≈0.0342, we found a threshold oscillatory wavenumber a_T=a_S(C_T)≈3.347>a_c≈2.221 (figure 1).

Indeed, explicit calculations of the threshold Cattaneo number have been presented here for the first time, allowing us to divide ${\mathcal{P}}_1$-C parameter space in a general sense via the curve $C_{\mathrm{T}}({\mathcal{P}}_1)$ into regions where either stationary (C<C_T) or oscillatory (C>C_T) convection is preferred. When expressed at fixed C in terms of the threshold Prandtl number ${\mathcal{P}}_T$, this curve may also be written in closed form (§5). Such an analysis has enabled us to demonstrate rigorously that $C_{\mathrm{T}}({\mathcal{P}}_1)>C_{\mathrm{B}}({\mathcal{P}}_1)$ for arbitrary Prandtl number, and therefore that threshold behaviour associated with the critical convective mode (n=1) is always permitted physically.

For fixed boundary conditions, both stationary and oscillatory behaviour may be studied numerically using a Chebyshev-tau algorithm [34], after expanding the problem into Chebyshev polynomials and approximating the system in terms of a finite number of linear equations (§6). As expected, in the case of stationary convection one recovers the classical (Fourier law) results R_c≈1707.776 and a_c≈3.116 [26], whereas for oscillatory convection one finds that the overstable Rayleigh number R_S is less than R_c for C sufficiently large. In this way, we have studied the transition from stationary-to-oscillatory convection for Prandtl numbers in the range ${\mathcal{P}}_1\in [{10}^{-2},{10}^{+2}]$, paying special attention to the wavenumbers a_T and oscillation frequencies γ_T associated with the threshold solution R_S(C_T)=R_c. Note that in the fixed boundary case we find that $C_{\mathrm{T}}\gtrsim 0.02$, whereas for free boundaries this value is slightly higher, with C_T≥8/27π²≈0.03. By using N=40 polynomials, such an approach yields relatively precise values, to within ±0.1% uncertainty. As in the case of free boundaries, we find that the Prandtl number has important consequences for the threshold parameters; indeed, the effect of low ${\mathcal{P}}_1$ on the wavenumber a_T is particularly pronounced, possibly owing to the contrast between no-slip conditions at the boundary, and inviscid (or very low ν) motion within the fluid layer (§7).

Overall, our analysis has shown that hyperbolic heat-flow effects have profound consequences for thermal convection, and substantially lower instability thresholds when the Cattaneo number is relatively large, in which case the Rayleigh number for oscillatory modes scales as R_S∝1/C². Why this particular scaling should obtain is not immediately clear; however, the emergence of oscillatory solutions makes sense when we consider the relationship between our problem and overstability in classical systems. Indeed, Cattaneo terms impart a kind of elasticity to the fluid, and allow thermal disturbances to propagate as waves in a semi-analogous fashion to the way in which rotational effects and impressed magnetic fields permit wave propagation and overstability in classical convection [26]. Intuitively, therefore, one expects that increases to the Cattaneo number should make this effect more pronounced, meaning that the onset of oscillatory convection occurs at lower Rayleigh numbers when C is large (with ∂R_S/∂C<0), as we have observed.

As discussed in the Introduction, new technologies present heat transfer problems on ever reduced scales, and the phenomena we describe seem likely to be of particular relevance to such systems (where hyperbolic heat-flow effects are known to be especially pronounced), alongside biological contexts for which thermal relaxation times are relatively long [8–11,13–15]. Nevertheless, one of the pressing issues surrounding hyperbolic modifications to the usual Fourier law of heat conduction is to establish which kind of heat-flow model should be properly employed (the model used here is simply one more frequently encountered). Crucially, our discussion emphasizes that a Cattaneo–Christov heat-flow law has implications beyond simply the speed at which thermal signals propagate; rather, that Cattaneo terms lead to pronounced transitions in overall system behaviour: in our case, the emergence of discontinuous transitions from stationary to (otherwise forbidden) oscillatory convection, with marked shifts in wavenumber and gyration frequency. In terms of experimental application, such transitions have the potential to act as very clear signatures of governing dynamics, and thence—in some respects at least—a means of model validation. Indeed, it seems plausible that by studying the transition between forms of convection within an appropriate experimental context (e.g. small-scale system), one may determine various Cattaneo thresholds C_T, and thence relaxation times τ. In this respect, a key advantage of our analysis is the inclusion of effects owing to Prandtl number ${\mathcal{P}}_1$, which in some circumstances can be used as a control parameter [35]; for example, we have shown that the Cattaneo threshold $C_{\mathrm{T}}({\mathcal{P}}_1)$ can be conceived equivalently as a Prandtl threshold ${\mathcal{P}}_T(C)$, so that system bifurcations could potentially be triggered by varying ${\mathcal{P}}_1$.

The theory presented here substantial develops existing work on Cattaneo–Christov heat-flow effects in thermal convection [16,17]; however, a number of theoretical issues precluded by our linear approach remain, not least that of developing an understanding of how the Prandtl number impacts on convection cell geometry, and thus the physical basis for discontinuous transitions between stationary a_c and convective a_T=a_S(C_T) wavenumbers at the Cattaneo threshold. Indeed, having investigated Cattaneo–Christov heat-flow effects in the classic Rayleigh–Bénard system, it is important to consider its role in other thermal convection systems, especially those which are known to yield oscillatory or non-stationary instability under the more usual Fourier law, such as magnetized or rotating fluids [26,28]. The author plans to address some of these problems in future publications.

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Funding statement

J.J.B. is sponsored by a Leverhulme Trust Grant (Tipping Points Project, University of Durham).

Conflict of interests

The author reports no competing interests.