DEL BARRIO MINUTO DE DIOS A LA ORGANIZACIÓN MINUTO DE DIOS

In this chapter, the results are obtained for the case of convection of a non-Fourier fluid confined between two vertical walls. Since analytical results could not be obtained for the vertical slot configuration, the stability of conduction state is examined numerically for a fluid with a low Prandtl number (such as liquid helium II) and a fluid with an intermediate Prandtl number (such as nanofluids). In both cases, as the Cattaneo number increases the critical Grashof number shows a decreasing trend (Figures 4.6 and 4.8). In addition, when the Cattaneo number increases, an oscillatory branch is observed in the stability picture, which tries to lower the critical point. In general, it can be concluded that if the onset of convection occurs at a lower Grashof number (Rayleigh number) for the vertical slot (Rayleigh-Benard) configuration compared to that of for a Fourier fluid, it signals the presence of non-Fourier effects. In addition, if the conduction is lost to oscillatory convection, this can be another possible situation where non-Fourier effects are important.

In addition to liquid helium at very low temperatures (see section 4.2) that has small Prandtl number (Pr < 1) (T < 2.17 K), nanofluids have also non-Fourier character (see Chapter 2) (Pr ≈ 7). The relevance of the DPL model to nanofluids (NFs) has recently been recognized in the literature, and is emphasized in Chapter 2. The equivalence between the two-phase and DPL models allows the expression of the relaxation time in terms of the nanoparticle (NP) concentration [28]. Two non-dimensional groups are introduced for NFs in Chapter 2, the Cattaneo number, C ( Q₂

h τ κ

= ), and dimensionless retardation time, S ( CKF

K

= ). C and S are both related directly to the NP concentration. The retardation-to-relaxation time ratio is found to be equal to the NF solution-to-solvent thermal conductivity ratio, γ. Since S KF C

K

= , S must always be smaller than C for a NF. In this paper, only SPL (S = 0) fluids are studied. This limit case has multiple advantages over a fluid with retardation. It serves as reference case for a fluid with strong non-

Fourier character, is mathematically more manageable, and can, in the limit, reflect the behaviour of a NF with very high NP concentration.

As mentioned previously, if the onset of convection occurs at a lower Rayleigh number (Grashof number) for the Rayleigh-Benard configuration (vertical slot) compared to that of for a Fourier fluid, it signals the presence of non-Fourier effects. This phenomenon is observed experimentally [61], where the onset of convection for a nanofluid comprised of water and Al2O3 (Φ = 2.72%) and confined between two vertical plates occurs at a lower

Rayleigh number compared to that of a pure water (Figure 3 in [61] and θ=90°). From Figure 3 in [61], the ratio of critical Rayleigh number for NF over the critical Rayleigh number for pure water can be read as 0.75. Using Figure 4.6, it can be seen that for C ≈

0.0045 the ratio of critical Grashof number for a non-Fourier fluid over the critical Grashof number for a Fourier fluid is 0.75. Recall that if the conduction is lost to oscillatory convection, this can also signal the importance of non-Fourier effects. In this case, transient oscillatory convection is observed [62] at the onset for a layer of nanofluid confined between two horizontal plates (Rayleigh-Benard convection). In a recent experimental study, oscillatory convection is observed [63] for nanofluids (between two horizontal planes) with high thermophilic nanoparticles. For nanofluids, the oscillatory convection is resulted from the competition between two opposing factors: buoyancy (destabilizing) and thermophoresis (stabilizing).

The mechanism of the instability can also be studied through an examination of the disturbance-energy equation. The kinetic energy of the perturbations can be defined as:

(

)

∫

where Ω is the cell volume which is defined as Ω: -1/2 < x < +1/2, 0 < y < 2π/q. The energy balance rate is then obtained by taking the derivative of (4.6.1) with respect to time. Multiplying the linearized disturbance momentum equation by the perturbation velocity, vi′, the kinetic energy rate reads as

i i v dKE u v v dv u v dv dt _Ω t _Ω t t ′ ′ ′ ∂ ⎛ ∂ ∂ ⎞ ′ ′ ′ = = _⎜ + _⎟ ∂ ⎝ ∂ ∂ ⎠

∫

∫

Substituting u t∂ ∂′ and v∂ ∂′ t from (4.4.1b-c) into (4.6.2), the equation for the kinetic energy rate of the perturbations is

2 2 2 2

x x y x y

dKE

Gr v u v v u u v v

dt = − ′ ′ + ′ ′θ − ′ + ′ + ′ + ′ , (4.6.3)

with brackets defining the average over one spatial cycle. Unlike the Rayleigh-Benard problem, where the equation for the kinetic energy rate (equation 4.4 in [30]) consists of two terms (buoyancy and dissipation), in (4.6.3) three terms are present. The first term on the RHS, −Gr v u v_x ′ ′ , reflects the effects of the base flow interaction with the perturbations and is called ‘transfer’ [42]. The second term, v′ ′θ , represents the ‘buoyancy’ and the last term, − u′_x2+u′_y2 +v′_x2+v′_y2 shows the ‘dissipation’. Some points can be noted about (4.6.3). First, at the critical Grashof number, the kinetic energy rate dKE dt is equal to zero. Second, the dissipative term has always negative value, which emphasizes the fact that the viscosity always dissipates energy. In contrast, the values of buoyancy and transfer terms depend on v_x, u′, v′and θ′, which cannot be predicted easily. Third, it can be seen from (4.6.3) that for higher Grashof numbers, the transfer term plays a more important role. It was seen that the critical Grashof number decreases sharply for higher Cattaneo numbers (Figures 4.4 and 4.5). Therefore, as C increases, it is expected from equation (4.6.3) that the dominance of the transfer term will be replaced by the buoyancy term. The stationary (oscillatory) mode in Figure 4.5 signals the dominance of the transfer (buoyancy) term in (4.6.3).