## 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}≥8/27*π*^{2}≈0.03. In the case of free boundaries, analytical approaches permit direct treatment of the role played by the Prandtl number *C*_{T} is computed as a function of

## 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,
*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.
*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 *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 *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

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* 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

## 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])
*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}/∂

*x*

^{2}+∂

^{2}/∂

*y*

^{2}+∂

^{2}/∂

*z*

^{2}. We also employ the Boussinesq approximation in the buoyancy term, viz

*ρ*

_{0}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.

### (a) Steady-state and linearized system

We now suppose that the fluid occupies a semi-infinite region *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.
*z*-component of the velocity *w* vanishing on the boundaries, i.e. (cf. Chandrasekhar [26])

In steady-state, solutions to system (2.1) are given by
*β* denotes the temperature gradient, and *P*_{0} has a profile such that the buoyancy force is balanced by pressure gradients, that is,
*C* and Rayleigh number *R*_{a}=*R*^{2} are defined
_{i}, and thereby eliminate *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
*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
*a* as a characteristic wavenumber or inverse length scale. Then system (2.13) becomes (cf. Chandrasehkar [26] and Straughan [17])
*Φ*∈{*W*,*Q*,*Θ*} the operators *D* and *D*^{2} are

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]
*W* vanish at the boundaries, and *W*(*z*) may be decomposed into odd Fourier modes, thus
*n*th 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 *n*th mode [26]
*R*_{a}(*n*) with respect to *a* thus yields a critical Rayleigh number *R*_{c}(*n*)=*R*_{a}(*a*_{c}) for the *n*th mode corresponding to critical wavenumber *a*_{c}(*n*), where (∂*R*_{a}/∂*a*)_{ac}=0, such that
*R*_{a}(*n*)} increases monotonically with *n*, the absolute critical Rayleigh number occurs when *n*=1, i.e. (cf. Chandrasekhar [26]),
*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,
*σ*=0, in the case of stationary convection we have by system (2.16) that neither the Prandtl 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

For oscillatory solutions, we assume that *σ* may be complex and non-zero, and define *γ* such that [26]
*Θ* and *Q* from system (2.16), we have
*W*(*z*) vanish at the boundaries, and our Fourier decomposition (3.3) gives
*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])
*γ*^{2} 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
*γ*^{2}>0, for given *n*, *a* and

Assuming condition (4.6) to hold, we substitute for *γ*^{2} in system (4.4) to give
*R*_{a}/∂*a*)_{aS}=0 to determine the critical Rayleigh number *R*_{S}(*n*)=*R*_{a}(*a*_{S}) for the *n*th mode (cf. Straughan [17])
*a*_{S}(*n*) is the corresponding critical wavenumber

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 *n*th 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*_{B}∈(1/4*n*^{2}*π*^{2},1/2*n*^{2}*π*^{2}) 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)
*a*_{S} and frequency *γ* determined by equations (4.9) and (4.5) after taking *n*=1.

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
*C*. Thus, at fixed

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* (figure 1). Nevertheless, for the *n*=1 mode to yield physical solutions, inequality (4.14) demands that
*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 *C*_{T} is a monotonically decreasing function of *C*>*C*_{T} to represent physical solutions, therefore, the curve *C*-*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.

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 *n*=1 mode, and with limits
*R*_{B} with respect to *R*_{S}(*C*_{B})=*R*_{B} and *R*_{S}(*C*_{T})=*R*_{c} that
*R*_{S}(*C*) is a strictly decreasing function of *C*, 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 *C* we can define a threshold Prandtl number *C* defined on

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 *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*_{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 *D*^{2} by introducing an auxiliary variable *χ*(*z*) such that *χ*(*z*)*f*(*x*,*y*) *e*^{σt}=∇^{2}*w*. In this way, system (2.16) may be written in the form
*Φ*∈{*W*,*χ*,*Θ*,*Q*} as Chebyshev polynomials *T*_{n}(*z*) weighted by constant coefficients *ϕ*_{n}, viz
_{j} is a vector of length 4*N* comprising the *ϕ*_{n}, i.e.
*A*_{ij}≡*A*_{ij}(*a*,*R*) and *N*×4*N* matrices with constant coefficients inclusive of the expanded boundary conditions (cf. Straughan [17]).

Given *R*, *C* and *a* subject to the condition ℜ{*σ*}=0; critical values are then obtained by minimizing *R*_{a}(*a*)=*R*^{2}. 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).

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.

Data corresponding to the intersection problem when *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 *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 *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 *C*_{T}≥8/27*π*^{2}≈0.03 (see §5).

## 7. Effect of Prandtl number on threshold values

Recall that in the case of free boundary conditions the wavenumber *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 *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).

Plots of the threshold wavenumbers and frequencies as a function of Prandtl number in the range *a*_{T} decreases with *a*_{T}≈4.77 when the Prandtl number is large (table 1), while—on the domain *a*_{T}>*a*_{c}≈3.116 (see equation (3.8)). In this second case, however, we find that the threshold wavenumber initially increases with *a*_{T}≈6.3 at

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 *C*_{T} dependent on *κ*, we see that the low Prandtl number regime corresponds to one with vanishing viscosity.^{1} In the case of fixed boundaries at low *a*_{T} should reach a maximum when the boundaries are fixed, rather than simply converging to another constant value as *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 *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 *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 *σC* and *R*_{c}=27*π*^{4}/4≈657.511, corresponding to a critical wavenumber *γ*≠0, the critical Rayleigh number *R*_{S} acquires both *C* and *γ*^{2}>0 overstability is rendered possible provided *C* is larger than a lower bounding value *a*_{S} also acquires *C* and

Crucially, for given Prandtl number, we have demonstrated that *R*_{S}(*C*) is a strictly decreasing function of *C*, obeying a dependence *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*_{S}>*a*_{c}; for example, at Prandtl number *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 *C* parameter space in a general sense via the curve *C*<*C*_{T}) or oscillatory (*C*>*C*_{T}) convection is preferred. When expressed at fixed *C* in terms of the threshold Prandtl number *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 *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*_{T}≥8/27*π*^{2}≈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 *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/*C*^{2}. 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

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.

## Data accessibility

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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.

## Acknowledgements

The author thanks Professor B. Straughan for many stimulating discussions, and two anonymous referees for insightful comments on the original manuscript which have helped to clarify its argument.

## Footnotes

- Received October 31, 2014.
- Accepted January 21, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.