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Derivation of the spatio-temporal model equations for the thermoacoustic resonator

We derive the model equations describing the thermoacoustic resonator, that is, an acoustical resonator containing a viscous medium inside. Previous studies on this system have addressed this sytem in the frame of the plane-wave approximation, we extend the previous model to by considering spatial effects in a large aperture resonator. This model exhibits pattern formation and localized structures scenario.


Sound propagation in a viscous medium
The propagation of sound in a viscous, heat-conducting medium is described by the equations of mechanics of continuum media [1], i.e. the continuity equation the momentum transfer equation and the heat transport equation where ρ and p represent, respectively, the density and acoustic pressure deviations from the equilibrium state (denoted by ρ 0 and p 0 ), v is the fluid particle velocity. The constants µ, µ B , κ and c p are the coefficients of shear and bulk viscosity, the thermal conductivity and the specific heat at constant pressure of the fluid, respectively. In Eq.(3), T denotes the perturbation of the medium temperature due to the acoustic wave, and x i the spatial coordinates. Equations (1)-(3) must be complemented with the equation of state relating the medium density with the other variables. For a linear medium (small amplitudes) it obeys the simple form p = c 2 0 ρ. In a highly absorbing medium, the thermal effects are dominant and the hydrodynamic nonlinearity can be neglected. Furthermore, the speed of sound depends on the average value (over one period) of the temperature T av = T 0 +T , where T 0 is the temperature at equilibrium, so the equation of state takes the form ρ = p/c 2 0 (T ). Expanding at leading order it follows where We assume that the fields can be described by quasi-plane waves, i.e. they take the form where a slowly changing temperatureT , responsible for the thermal self-action of the waves on v = v z , is added. Taking into account the slowly varying envelope condition and the linear impedance relation relation v = p/ρ 0 c 0 , the equation forT is obtained by averaging of Eq.(3) whereT represents the average temperature perturbation caused by the wave energy dissipation. In Eq.(7) the new parameter b is defined as b = (µ B + 4µ/3)/ρ 0 c p , and χ = κ/ρ 0 c p is the thermal diffusivity.
On the other side, substituting Eqs. (5) in Eqs. (1) and (2) it follows that, in the parabolic approximation, the pressure field is described by [1] where a = bω 2 /2ρ 0 c 3 0 and ∇ 2 ⊥ is a laplacian operator acting on transverse coordinates.

Modal equations
In a resonator, the reflection at the boundaries imply the presence of counterpropagating traveling waves,and the acoustic field p can be expressed in terms of forward and backward components (9) whose amplitudes are assumed to vary slowly in space and time (compared with the evolution described by the exponential factors). Substitution of expansion (9) into Eq. (7), leads to the following temperature where T 0 is the homogeneous component and T 1 is the amplitudes of the thermal grating induced by the acoustic wave. The acoustic resonator containing a viscous fluid medium inside (e.g. glycerine), has been previously addressed [2]in the frame of plane-wave approximation, we extend now the previous model by considering the sound diffraction and temerature diffusion. Projecting on the different longitudinal modes, the following equations for the amplitudes are obtained: where a = ω 2 (µ B + 4µ/3) /(2ρ 0 c 3 0 ).

Boundary Conditions.
We consider an acoustic resonator with length L, filled with a viscous medium, and bounded by reflecting surfaces, with intensity reflection coefficients R, located at z = 0 and z = L. We assume that one of the surfaces, vibrating at frequency ω, acts as ultrasonic source. In accordance with (9), boundary conditions satisfy where the detuning parameter δ has been defined as where m is an integer, selected in order to be ω c the cavity frequency that lies nearest to the driving frequency ω.

Mean Field Limit.
When the reflectivity of the boundary walls is high, R → 1, the resulting model can be greatly symplified adopting the mean field limit. Under this assumption, the two counterpropagating fields are approximately constant and equal along the cavity (i.e. there is no spatial modulation of the field envelopes), and one can assume thatP where the overbar means the spatial average over the longitudinal coordinate z, We also consider that the frequency detuning between the driving and a cavity mode is small, 2L (ω c − ω) /c << π, which allows to write the boundary condition Eq.(14) as Averaging Eqs.(11), with leads to they can be reduced just to one equation where the averaged temperature components as defined asT i = LT i and we have used the property (f g) =fḡ [3], valid in the mean field case. The transmitted pressure can be assumed to be and Eq.(19) is written as where the parameters have been introduced. As well, under the same assumptions, Eqs.(12) read

The original model
The mean-field model for the thermoacoustic resonator can be written as whereT g has already been assumed to be real. We note that the model is highly reminiscent of that for a coherently driven optical cavity filled with a non-instantaneous (and nonlocal) Kerr medium [4]. Unlike its optical analog, in this thermoacoustic model there are two nonlinearities (owed toT h and toT g ); notice however that for γ h = 4k 2 χ both nonlinearities behave as a single one (a single temperature fieldT h +T g can be defined so that its evolution and that of the pressure just depend on themselves in that case). Defining the following quantities the model equations, Eqs.(26) become We consider a resonator with high quality plates (T = 0.1), separated by L = 5 cm, driven at a frequency f = 2MHz and containing glycerine at 10 o C. Under these conditions the medium parameters are c 0 = 2 × 10 3 m s −1 , α 0 = 10 m −1 , ρ 0 = 1.2 × 10 3 kg m −3 , c p = 4 × 10 3 J kg −1 K −1 , σ = 10 −2 K −1 and κ = 0.5 W m −1 K −1 (χ = 10 −7 m 2 s −1 ). In this case t p = 2 × 10 −5 s, t g = 6 × 10 −2 s, and our length unit is l d = 2 mm. For a resonator with a large Fresnel number, the relaxation of the homogeneous the temperature is mainly due to the heat flux through the boundaries, and can be estimated from the Newton's colling law as t h ∼ 10 1 s. Then, under usual conditions we get the normalized decay times τ p ∼ 10 −6 , and τ g ∼ 10 −2 , and the diffusion constant D ∼ 10 0 .Note that for τ g = 1 a single temperature field (H + G) can be defined, whose evolution depends just on P and vice versa. This model exhibits a rich spatiotemporal dynamics and exhibits pattern formation and localized structure formed in the transverse cross-section of the cavity. Some of these results are studied in [5].