Oscillatory Bifurcations in Porous Layers with Stratified Porosity, Driven by Each Coefficient of the Spectrum Equation

The onset of oscillatory bifurcations in a porous horizontal layer <i>L</i>, uniformly rotating about a vertical axis, with vertically stratified porosity, heated from below and salted from above and below, is investigated. Denoting by <inline-formula><math display="inline"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub><mo>,</mo><mspace width="0.166667em"></mspace><mrow><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the Prandtl numbers of the salt <inline-formula><math display="inline"><semantics><msub><mi>S</mi><mi>i</mi></msub></semantics></math></inline-formula> salting <i>L</i> from below <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and above <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>i</mi><mo>=</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> respectively, it is shown that: (i) in <i>L</i> the oscillatory bifurcations can occur only if one of the structural conditions <inline-formula><math display="inline"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>></mo><mn>1</mn><mo>,</mo><mspace width="4pt"></mspace><msub><mi>P</mi><mn>2</mn></msub><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>,</mo><mspace width="4pt"></mspace><msub><mi>P</mi><mn>2</mn></msub><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>></mo><mn>1</mn><mo>,</mo><mspace width="4pt"></mspace><msub><mi>P</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> is verified; (ii) exists a bound <inline-formula><math display="inline"><semantics><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>2</mn></msub></semantics></math></inline-formula> for the Rayleigh number <inline-formula><math display="inline"><semantics><msub><mi>R</mi><mn>2</mn></msub></semantics></math></inline-formula> of <inline-formula><math display="inline"><semantics><msub><mi>S</mi><mn>2</mn></msub></semantics></math></inline-formula> such that <inline-formula><math display="inline"><semantics><mrow><msub><mi>R</mi><mn>2</mn></msub><mo><</mo><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>2</mn></msub></mrow></semantics></math></inline-formula> guarantees the absence of cold convection; (iii) via a new approach based on the instability power of each coefficient of the spectrum equation, criteria of existence, location and frequency of oscillatory (Hopf) bifurcations are furnished for any porosity stratification law. These criteria, as far as we know are, for the case at stake, the first criteria of Hopf bifurcations appearing in literature. We are confident that, via experimental results, will be validated.