Stabilisation of adverse density gradients in binary mixtures via transverse confinement: comparison of theory, numerics and experiment

We investigate the effect of the transverse aspect ratio ( upper B <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>B</mml:mi> </mml:math> $B$ ) of a thermogravitational column (TGC) on the stability of a binary fluid mixture with a negative separation ratio ( psi <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ψ</mml:mi> </mml:math> $\psi$ ), which generates an adverse vertical density gradient. In extended systems, this configuration is unstable and limits separation. Here we show that confinement fundamentally alters this behaviour and can fully suppress the instability. Combining linear stability analysis of the three-dimensional basic state, three-dimensional simulations and experiments, we identify upper B <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>B</mml:mi> </mml:math> $B$ as the key control parameter. Linear stability analysis yields a correlation for the critical aspect ratio upper B Subscript italic c Baseline left parenthesis psi right parenthesis <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">c</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $B_{\textit{c}}(\psi )$ separating unstable and stable regimes. For upper B less than upper B Subscript italic c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>B</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">c</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $B \lt B_{\textit{c}}$ , the system remains in a quiescent state despite the presence of the adverse density stratification. Energy budget analysis shows that confinement enhances transverse diffusive transport, preventing the growth of long-wave instabilities that dominate in extended systems. Direct numerical simulations confirm the predicted transition and reveal the nonlinear evolution towards convection when upper B greater than upper B Subscript italic c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>B</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">c</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $B \gt B_{\textit{c}}$ . Experiments performed with different mixtures, Tol vertical bar <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> $|$ Ch 0.50 <jats:inline-graphic xmlns:xlink="http://www.w3.org/199