Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel
The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximati...
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2025-04-01
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| author | Natalja Budkina Valentina Koliskina Andrei Kolyshkin Inta Volodko |
| author_facet | Natalja Budkina Valentina Koliskina Andrei Kolyshkin Inta Volodko |
| author_sort | Natalja Budkina |
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| description | The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. |
| format | Article |
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| institution | Kabale University |
| issn | 2311-5521 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
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| series | Fluids |
| spelling | doaj-art-bf0154f5773a4e1886806f8aab65225b2025-08-20T03:47:58ZengMDPI AGFluids2311-55212025-04-0110511110.3390/fluids10050111Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical ChannelNatalja Budkina0Valentina Koliskina1Andrei Kolyshkin2Inta Volodko3Institute of Applied Mathematics, Riga Technical University, Zunda Embankment 10, LV 1048 Riga, LatviaInstitute of Applied Mathematics, Riga Technical University, Zunda Embankment 10, LV 1048 Riga, LatviaInstitute of Applied Mathematics, Riga Technical University, Zunda Embankment 10, LV 1048 Riga, LatviaInstitute of Applied Mathematics, Riga Technical University, Zunda Embankment 10, LV 1048 Riga, LatviaThe weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data.https://www.mdpi.com/2311-5521/10/5/111weakly nonlinear instabilityconvective flowcollocation method |
| spellingShingle | Natalja Budkina Valentina Koliskina Andrei Kolyshkin Inta Volodko Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel Fluids weakly nonlinear instability convective flow collocation method |
| title | Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel |
| title_full | Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel |
| title_fullStr | Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel |
| title_full_unstemmed | Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel |
| title_short | Weakly Nonlinear Instability of a Convective Flow in a Plane Vertical Channel |
| title_sort | weakly nonlinear instability of a convective flow in a plane vertical channel |
| topic | weakly nonlinear instability convective flow collocation method |
| url | https://www.mdpi.com/2311-5521/10/5/111 |
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