Heated and Salted Below Porous Convection with Generalized Temperature and Solute Boundary Conditions
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Abstract
We address the problem of initiation of convective motion in the case of a fluid saturated porous layer, containing a salt in solution, which is heated and salted below. We amplify the very interesting recent results of Nield and Kuznetsov and examine in detail a whole range of temperature and salt boundary conditions allowing for a combination of prescribed heat flux and temperature. The behaviour of the transition from stationary to oscillatory convection is examined in detail as the boundary conditions vary from prescribed temperature and salt concentration toward those of prescribed heat flux and salt flux.
Keywords
Heated–salted below Stationary–oscillatory transition Double diffusive convection1 Introduction
Nield and Kuznetsov (2016) produced an inspiring article in which they address the behaviour of the onset of convective motion in a layer of porous material which is saturated by a fluid containing a dissolved salt. They consider both Brinkman and Darcy theory, and they are primarily interested in the case where the heat flux and salt flux are prescribed on the boundary. They do, however, also consider the case where general thermal and salt boundary conditions are employed which involve a combination of flux and prescribed temperature and salt. They develop an asymptotic and a numerical analysis to study how oscillatory convection behaves as boundary conditions of flux only are considered. It is well known that in the heated below–salted below situation there is a transition from stationary convection to oscillatory convection as the salt Rayleigh number increases. The current article is motivated entirely by the work of Nield and Kuznetsov (2016), and we analyse how the transition from stationary to oscillatory convection is affected as the boundary conditions change.
Double diffusive convection is a problem with many real-life applications and as such has attracted much attention in the research literature, see, e.g. Barletta and Nield (2011), Deepika (2018), Deepika and Narayana (2016), Harfash and Challoob (2018), Harfash and Hill (2014), Joseph (1970), Joseph (1976), Lombardo et al. (2001), Matta et al. (2017), Mulone (1994), Nield (1967, 1968), Nield and Kuznetsov (2016), Straughan (2011, 2014, 2015a, 2018, 2019) and Xu and Li (2019). Stability analyses in double diffusive convection were introduced in the fundamental articles of Nield (1967, 1968), and from an unconditional energy stability point of view by Joseph (1970, 1976). Research activity in this area has increased rapidly as is witnessed by the articles cited above and the references therein.
Another very interesting development in stability in thermal convection studies has been to consider boundary conditions which are more general than those of prescribing temperature and salt concentration. For example, isoflux conditions, isobaric conditions, or various combinations. In many cases, these boundary conditions lead to surprising and novel results, see, e.g. Barletta (2012), Barletta et al. (2010), Barletta and Rees (2012), Barletta and Celli (2018), Celli and Barletta (2019), Celli et al. (2016), Celli et al. (2013), Celli and Kuznetsov (2018), Falsaperla et al. (2010, 2011), Lagziri and Bezzazi (2019), McKibbin (1986), Mohammad and Rees (2017), Nield and Kuznetsov (2016), Rees and Barletta (2011), Rees and Mojtabi (2011), Rees and Mojtabi (2013), Salt (1988) and Webber (2006).
Given the interest in double diffusive convection, especially where the salt and temperature effects are in competition as in the heated and salted below case, and the attention to general boundary conditions where a combination of flux and prescribed temperature/salt is studied, we believe this work is noteworthy. We also corroborate some of the findings of Nield and Kuznetsov (2016) as boundary conditions of pure flux are approached.
2 Equations
3 Exact Theory
Values of \(R^*\) and \(C^*\) together with the critical value of a, namely, \(a_{\mathrm{cr}}\), and L
\({R^*}\) | \({C^*}\) | \({a_{\mathrm{{cr}}}}\) | L |
---|---|---|---|
\(4\pi ^2 \epsilon _1/(\epsilon _1-1)\approx \) 41.5562 | \(4\pi ^2/(\epsilon _1-1)\approx \) 2.0778 | \(\pi \approx \) 3.14159 | \(\infty \) |
(40.758, 40.759) | (2.038, 2.039) | 3.111 | 100 |
(35.642, 35.643) | (1.782, 1.783) | 2.879 | 10 |
(32.273, 32.274) | (1.614, 1.615) | 2.693 | 5 |
(28.400, 28.401) | (1.420, 1.421) | 2.442 | 2.5 |
(23.530, 23.531) | (1.177, 1.178) | 2.057 | 1 |
(16.201, 16.202) | (0.810, 0.811) | 1.201 | 0.1 |
(13.745, 13.746) | (0.687, 0.688) | 0.677 | 0.01 |
(12.742, 12.743) | (0.638, 0.639) | 0.214 | \(10^{-4}\) |
(12.642, 12.643) | (0.632, 0.633) | 0.06773 | \(10^{-6}\) |
(12.632, 12.633) | (0.632, 0.633) | 0.0213 | \(10^{-8}\) |
Values of L and \(\sigma _i\) representing the values of \(\sigma _i\) on the critical oscillatory convection curve when \(C=2.1\)
L | \(\sigma _i\) |
---|---|
100 | \(\pm \, 0.1672 \) |
10 | \(\pm \, 0.3209 \) |
5 | \(\pm \, 0.3426 \) |
2.5 | \(\pm \, 0.3266 \) |
1 | \(\pm \, 0.2632 \) |
0.1 | \(\pm \, 0.1034 \) |
0.01 | \(\pm \, 0.03428 \) |
\(10^{-4}\) | \(\pm \, 3.485\times 10^{-3}\) |
\(10^{-6}\) | \(\pm \, 3.494\times 10^{-4}\) |
Values of L and \(\sigma _i\) representing the values of \(\sigma _i\) on the critical oscillatory convection curve when \(C=1.7\)
L | \(\sigma _i\) |
---|---|
5 | \(\pm \, 0.1443 \) |
2.5 | \(\pm \, 0.2095 \) |
1 | \(\pm \, 0.1981 \) |
0.1 | \(\pm \, 0.08590 \) |
0.01 | \(\pm \, 0.02903 \) |
\(10^{-4}\) | \(\pm \, 2.970\times 10^{-3}\) |
\(10^{-6}\) | \(\pm \, 2.973\times 10^{-4}\) |
\(10^{-8}\) | \(\pm \, 2.950\times 10^{-5}\) |
\(10^{-10}\) | \(\pm \, 2.930\times 10^{-6}\) |
Values of L and \(\sigma _i\) representing the values of \(\sigma _i\) on the critical oscillatory convection curve when \(C=1\)
L | \(\sigma _i\) |
---|---|
\(10^{-1}\) | \(\pm \, 0.03968 \) |
\(10^{-2}\) | \(\pm \, 0.01613 \) |
\(10^{-4}\) | \(\pm \, 1.736\times 10^{-3}\) |
\(10^{-6}\) | \(\pm \, 1.754\times 10^{-4}\) |
\(10^{-8}\) | \(\pm \, 1.741\times 10^{-5}\) |
\(10^{-9}\) | \(\pm \, 5.508\times 10^{-6}\) |
\(10^{-10}\) | \(\pm \, 1.719\times 10^{-6}\) |
\(10^{-11}\) | \(\pm \, 5.602\times 10^{-7}\) |
\(10^{-12}\) | \(\pm \, 1.782\times 10^{-7}\) |
4 Numerical Methods
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=100\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
3.111 | 38.7205 | 0 | 0 |
3.111 | 39.1205 | 0.4 | 0 |
3.111 | 39.5205 | 0.8 | 0 |
3.111 | 39.9205 | 1.2 | 0 |
3.111 | 40.3205 | 1.6 | 0 |
3.111 | 40.7205 | 2.0 | 0 |
3.111 | 40.7305 | 2.01 | 0 |
3.111 | 40.7591 | 2.05 | \(\pm \, 0.07377\) |
3.111 | 40.7601 | 2.07 | \(\pm \, 0.1202 \) |
3.111 | 40.7616 | 2.1 | \(\pm \, 0.1672 \) |
3.111 | 40.7766 | 2.4 | \(\pm \, 0.4039 \) |
3.111 | 40.7916 | 2.7 | \(\pm \, 0.5462 \) |
3.111 | 40.8066 | 3.0 | \(\pm \, 0.6584 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=10\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
2.879 | 33.8601 | 0 | 0 |
2.879 | 34.3601 | 0.5 | 0 |
2.879 | 34.8601 | 1.0 | 0 |
2.879 | 35.3601 | 1.5 | 0 |
2.879 | 35.5601 | 1.7 | 0 |
2.879 | 35.6432 | 1.8 | \(\pm \, 0.07613\) |
2.879 | 35.6484 | 1.9 | \(\pm \, 0.1954 \) |
2.879 | 35.6537 | 2.0 | \(\pm \, 0.2657 \) |
2.879 | 35.6799 | 2.5 | \(\pm \, 0.4823 \) |
2.879 | 35.7061 | 3.0 | \(\pm \, 0.6282 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=5\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
2.693 | 30.6504 | 0 | 0 |
2.693 | 31.1594 | 0.5 | 0 |
2.693 | 31.6594 | 1.0 | 0 |
2.693 | 31.8594 | 1.2 | 0 |
2.693 | 32.0594 | 1.4 | 0 |
2.693 | 32.2780 | 1.7 | \(\pm \,0.1441 \) |
2.693 | 32.2836 | 1.8 | \(\pm \, 0.2120 \) |
2.693 | 32.2892 | 1.9 | \(\pm \,0.2628 \) |
2.693 | 32.2948 | 2.0 | \(\pm \,0.3053 \) |
2.693 | 32.3004 | 2.1 | \(\pm \,0.3426 \) |
2.693 | 32.3060 | 2.2 | \(\pm \,0.3762 \) |
2.693 | 32.3116 | 2.3 | \(\pm \,0.4070 \) |
2.693 | 32.3172 | 2.4 | \(\pm \,0.4357 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=2.5\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
2.442 | 26.9809 | 0 | 0 |
2.442 | 27.2809 | 0.3 | 0 |
2.442 | 27.5809 | 0.6 | 0 |
2.442 | 27.8809 | 0.9 | 0 |
2.442 | 28.4058 | 1.5 | \(\pm \, 0.1120 \) |
2.442 | 28.4119 | 1.6 | \(\pm \, 0.1678 \) |
2.442 | 28.4179 | 1.7 | \(\pm \, 0.2095 \) |
2.442 | 28.4240 | 1.8 | \(\pm \, 0.2441 \) |
2.442 | 28.4300 | 1.9 | \(\pm \, 0.2744 \) |
2.442 | 28.4361 | 2.0 | \(\pm \, 0.3016 \) |
2.442 | 28.4422 | 2.1 | \(\pm \, 0.3266 \) |
2.442 | 28.4482 | 2.2 | \(\pm \, 0.3495 \) |
2.442 | 28.4543 | 2.3 | \(\pm \, 0.3712 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=1\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
2.057 | 22.3523 | 0 | 0 |
2.057 | 22.8523 | 0.5 | 0 |
2.057 | 23.3523 | 1.0 | 0 |
2.057 | 23.4523 | 1.1 | 0 |
2.057 | 23.5302 | 1.2 | \(\pm \, 0.04205\) |
2.057 | 23.5366 | 1.3 | \(\pm \, 0.09629\) |
2.057 | 23.5429 | 1.4 | \(\pm \, 0.1295 \) |
2.057 | 23.5492 | 1.5 | \(\pm \, 0.1557 \) |
2.056 | 23.5556 | 1.6 | \(\pm \, 0.1782 \) |
2.056 | 23.5619 | 1.7 | \(\pm \, 0.1981 \) |
2.056 | 23.5682 | 1.8 | \(\pm \, 0.2162 \) |
2.056 | 23.5746 | 1.9 | \(\pm \, 0.2329 \) |
2.056 | 23.5809 | 2.0 | \(\pm \, 0.2485 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=0.1\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
1.201 | 15.3912 | 0 | 0 |
1.201 | 15.5912 | 0.2 | 0 |
1.201 | 15.7912 | 0.4 | 0 |
1.201 | 15.9912 | 0.6 | 0 |
1.201 | 16.1912 | 0.8 | 0 |
1.201 | 16.2063 | 0.9 | \( \pm \, 0.02731 \) |
1.201 | 16.2120 | 1.0 | \( \pm \, 0.03968 \) |
1.201 | 16.2177 | 1.1 | \( \pm \, 0.04903 \) |
1.201 | 16.2290 | 1.3 | \( \pm \, 0.06373 \) |
1.201 | 16.2404 | 1.5 | \( \pm \, 0.07563 \) |
1.201 | 16.2517 | 1.7 | \( \pm \, 0.08590 \) |
1.201 | 16.2631 | 1.9 | \( \pm \, 0.09506 \) |
1.201 | 16.2744 | 2.1 | \( \pm \, 0.1034 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=0.01\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
0.677 | 13.0586 | 0 | 0 |
0.677 | 13.2586 | 0.2 | 0 |
0.677 | 13.4586 | 0.4 | 0 |
0.677 | 13.6586 | 0.6 | 0 |
0.677 | 13.7518 | 0.8 | \( \pm \, 9.6837\times 10^{-3} \) |
0.677 | 13.7570 | 0.9 | \( \pm \, 0.01330 \) |
0.677 | 13.7623 | 1.0 | \( \pm \, 0.01613 \) |
0.677 | 13.7674 | 1.1 | \( \pm \, 0.01853 \) |
0.677 | 13.7779 | 1.3 | \( \pm \, 0.02258 \) |
0.677 | 13.7884 | 1.5 | \( \pm \, 0.02600 \) |
0.677 | 13.7988 | 1.7 | \( \pm \, 0.02903 \) |
0.677 | 13.8092 | 1.9 | \( \pm \, 0.03176 \) |
0.677 | 13.8197 | 2.1 | \( \pm \, 0.03428 \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=10^{-4}\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
0.214 | 12.1049 | 0 | 0 |
0.214 | 12.2549 | 0.15 | 0 |
0.214 | 12.4049 | 0.3 | 0 |
0.214 | 12.5549 | 0.45 | 0 |
0.214 | 12.6550 | 0.55 | 0 |
0.214 | 12.7451 | 0.7 | \( \pm \, 7.2263 \times 10^{-4} \) |
0.214 | 12.7501 | 0.8 | \( \pm \, 1.1629 \times 10^{-3} \) |
0.214 | 12.7552 | 0.9 | \( \pm \, 1.4773 \times 10^{-3} \) |
0.214 | 12.7602 | 1.0 | \( \pm \, 1.7357 \times 10^{-3} \) |
0.214 | 12.7652 | 1.1 | \( \pm \, 1.9603 \times 10^{-3} \) |
0.214 | 12.7753 | 1.3 | \( \pm \, 2.3459 \times 10^{-3} \) |
0.214 | 12.7853 | 1.5 | \( \pm \, 2.6765 \times 10^{-3} \) |
0.214 | 12.7953 | 1.7 | \( \pm \, 2.9705 \times 10^{-3} \) |
0.214 | 12.8053 | 1.9 | \( \pm \, 3.2379 \times 10^{-3} \) |
0.214 | 12.8154 | 2.1 | \( \pm \, 3.4849 \times 10^{-3} \) |
Values of \(a_{\mathrm{{cr}}},R,C\) and \(\sigma _i\) at criticality \(L=10^{-6}\)
\(a_{\mathrm{{cr}}}\) | R | C | \(\sigma _i\) |
---|---|---|---|
0.06773 | 12.0105 | 0 | 0 |
0.06773 | 12.1105 | 0.1 | 0 |
0.06773 | 12.3105 | 0.3 | 0 |
0.06773 | 12.5105 | 0.5 | 0 |
0.06773 | 12.6005 | 0.59 | 0 |
0.06773 | 12.6460 | 0.7 | \( \pm \, 7.5101 \times 10^{-5} \) |
0.06773 | 12.6510 | 0.8 | \( \pm \, 1.1801 \times 10^{-4} \) |
0.06773 | 12.6560 | 0.9 | \( \pm \, 1.4911 \times 10^{-4} \) |
0.06773 | 12.6610 | 1.0 | \( \pm \, 1.7438 \times 10^{-4} \) |
0.06773 | 12.6660 | 1.1 | \( \pm \, 1.9701 \times 10^{-4} \) |
0.06773 | 12.6760 | 1.3 | \( \pm \, 2.3552 \times 10^{-4} \) |
0.06773 | 12.6860 | 1.5 | \( \pm \, 2.6800 \times 10^{-4} \) |
0.06773 | 12.6960 | 1.7 | \( \pm \, 2.9798 \times 10^{-4} \) |
0.06773 | 12.7060 | 1.9 | \( \pm \, 3.2469 \times 10^{-4} \) |
0.06773 | 12.7160 | 2.1 | \( \pm \, 3.4936 \times 10^{-4} \) |
To find \((R^*,C^*)\) numerically is not trivial. We used two codes. One tracks along the stationary convection curve from \(C=0\) and then tracks along the oscillatory convection curve in the opposite direction. When we are in the vicinity of \((R^*,C^*)\), the two leading eigenvalues \(\sigma _1\) and \(\sigma _2\) one of which is real, whereas the other is complex, have real parts close to zero and close to each other; therefore, the code switches from one to the other and breaks down. Thus, we extrapolate from the stationary and oscillatory convection curves for a given value of L to find an approximate value for \(R^*\) and \(C^*\). We then employ a second code which compares \(\sigma _1\) and \(\sigma _2\) in the vicinity of the “crossing point” and varies over a suitable range of wavenumber a. In this way, we actually find where \(\sigma _1\) and \(\sigma _2\) swap places, to 3 decimal places of accuracy in R. Numerical results employing these procedures are presented next.
5 Numerical Results and Conclusions
The values of \(R^*\) and \(C^*\) as L varies are displayed in Table 1 and in Figs. 2 and 3. From Fig. 3, it appears that \((R^*,C^*)\) form approximately a straight line with L varying. The critical values of a are in agreement with the values in table 1 of Nield and Kuznetsov (2016), who give values of R and \(a_{\mathrm{cr}}\) as L varies for the problem of thermal convection (without a solute). We found \(R^*\in (12.631,12.632)\) and \(C^*\in (0.6315,0.6316)\) when \(L=10^{-10}\), and we found a similar value for \(C^*\) when \(L=10^{-12}\).
The critical wavenumber \(a_{\mathrm{{cr}}}\) varies with L, but we found no variation with C, as seen in Tables 5, 6, 7, 8, 9, 10, 11, 12 and 13. For \(L\le 0.1\), we found the graph of \(\log _{10}a\) against \(\log _{10}L\) yields approximately a straight line, as is seen in Fig. 4. This behaviour is already seen in the numbers of table 1 of Nield and Kuznetsov (2016). We actually found values of \(a_{\mathrm{{cr}}}\) when \(L=10^{-9},10^{-10},10^{-11}\) and \(10^{-12}\) to be \(a_{\mathrm{{cr}}}=0.01201,0.00671,0.00381\) and 0.00213, respectively.
From Tables 2–4, we see that the value of \(\sigma _i\) (the imaginary part of \(\sigma \)) on the oscillatory curve firstly increases as L decreases, reaches a maximum, and then decreases again with further decrease in L. Tables 2–4 correspond to values on the oscillatory branches of the instability curve shown in Fig. 2 for \(C=1,1.7\) and 2.1 (the dashed lines). We observe that as L becomes very small \(\sigma _i\) likewise becomes very small. This is in complete agreement with the findings of Nield and Kuznetsov (2016) who note in their conclusions (in our notation),...oscillatory instability can still occur as L tends to zero..., in practical situations it is likely that no oscillations will be observed. This behaviour is witnessed in Tables 5–13.
From Tables 5–13, we see that \(\sigma _i\) increases on the oscillatory curve as C increases, and this is in agreement with the exact case of prescribed temperature and concentration where we know the exact solution, cf. Eq. (12).
From Tables 5–13 and the exact solution when \(L=\infty \), we observe that the slope on the stationary convection curve is always 1. However, the slope on the oscillatory convection curve is 0.05 when \(L=\infty ,100\) and then increases to a maximum and then decreases again as \(L\rightarrow 0\). We found approximately, slopes of 0.0524, 0.0561, 0.0606, 0.0633, 0.0567, 0.0522, 0.0502 when \(L=10,5,2.5,1,0.1,0.01,10^{-4}\), respectively, and then for \(L=10^{-6}\) or smaller the slope is 0.05. It appears from the numerical results that the oscillatory curve is a straight line for all values of L, but we have no analysis to justify this. It is worth pointing out that while the oscillatory curve is close to a straight line for an anisotropic inertia coefficient it is not actually straight, see Straughan (2014). Also, the transition from stationary to oscillatory convection in other problems may involve curved stationary convection and oscillatory convection curves, see, e.g. the analysis of Straughan (2015b) when the heat flux is of Cattaneo–Christov type.
In conclusion, we have found the transition from stationary convection to oscillatory convection in the heated below–salted below situation when the Nield and Kuznetsov (2016) boundary conditions, (4), (5), are employed for various values of L. We have chosen for our numerical results the realistic value of \(\epsilon _1=\epsilon Le=20\), although I believe the behaviour found here is not simply restricted to this case. Our results are in agreement with those of Nield and Kuznetsov (2016), and we add further information to their findings.
Notes
Acknowledgements
This work was supported by an Emeritus Fellowship of the Leverhulme Trust, EM-2019-022/9. I am grateful to referees for constructive comments which have improved the manuscript.
Compliance with Ethical Standards
Conflicts of interest
There are no conflicts of interest with this work.
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