“…Therefore, instability of the P1-C0-S solution is similar to the case of pure thermal convection discussed by Serrano-Aguilera et al. (2021), who proved that the S-mode is the most unstable mode and the bifurcation is Hopf type, which is consistent with our conclusion. For the P2-C1-S solution, it shows an overstability with a critical angular frequency ω r = 12.07 at Ra c = 15 423 and the corresponding distribution of eigenfunctions θ and q are shown in figures 10( a 1) and 10( a 2).…”
Section: Resultssupporting
confidence: 92%
“…Our numerical results are in fairly good agreement with the data of Serrano-Aguilera et al. (2021) ( Ra c = 2438 by a spectral method), with a relative error of 1.39%. Figure 2.Base flows (first row) and their corresponding most unstable mode (second row).…”
Section: Validationsupporting
confidence: 89%
“…Our results have a rather small error compared with the result obtained by the spectral code (Serrano-Aguilera et al. 2021). Finally, the hydrostatic solution is selected to validate the solving of Poisson’s equation for the electric potential and charge conservation by LBM.…”
Section: Tablesupporting
confidence: 47%
“…Figure 17.Comparison of ( a ) radial temperature distribution and ( b ) local Nu between our numerical results and results of Serrano-Aguilera et al. (2021) at Pr = 0.706 and Ra = 4.7 × 10 4 .Figure 18.Comparison between the numerical and analytical solutions of hydrostatic state at Fe = 10 4 : ( a ) electric field E r and ( b ) charge density q . Three cases corresponding to radius ratio 0.1, 0.3 and 0.5, respectively, are considered.…”
Section: Tablementioning
confidence: 75%
“…Mizushima, Hayashi & Adachi (2001) researched the influence of A on the bifurcation type and clarified the origin of the dual solutions from the bifurcation analysis. Serrano-Aguilera, Blanco-Rodríguez & Parras (2021) conducted abundant numerical investigations on dual solutions for values of the Prandtl number between 0.01 and 1 and Rayleigh numbers between 10 2 and 5 × 10 6 . Steady flow transition to oscillation (Hopf bifurcation) was also studied systematically.…”
We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.
“…Therefore, instability of the P1-C0-S solution is similar to the case of pure thermal convection discussed by Serrano-Aguilera et al. (2021), who proved that the S-mode is the most unstable mode and the bifurcation is Hopf type, which is consistent with our conclusion. For the P2-C1-S solution, it shows an overstability with a critical angular frequency ω r = 12.07 at Ra c = 15 423 and the corresponding distribution of eigenfunctions θ and q are shown in figures 10( a 1) and 10( a 2).…”
Section: Resultssupporting
confidence: 92%
“…Our numerical results are in fairly good agreement with the data of Serrano-Aguilera et al. (2021) ( Ra c = 2438 by a spectral method), with a relative error of 1.39%. Figure 2.Base flows (first row) and their corresponding most unstable mode (second row).…”
Section: Validationsupporting
confidence: 89%
“…Our results have a rather small error compared with the result obtained by the spectral code (Serrano-Aguilera et al. 2021). Finally, the hydrostatic solution is selected to validate the solving of Poisson’s equation for the electric potential and charge conservation by LBM.…”
Section: Tablesupporting
confidence: 47%
“…Figure 17.Comparison of ( a ) radial temperature distribution and ( b ) local Nu between our numerical results and results of Serrano-Aguilera et al. (2021) at Pr = 0.706 and Ra = 4.7 × 10 4 .Figure 18.Comparison between the numerical and analytical solutions of hydrostatic state at Fe = 10 4 : ( a ) electric field E r and ( b ) charge density q . Three cases corresponding to radius ratio 0.1, 0.3 and 0.5, respectively, are considered.…”
Section: Tablementioning
confidence: 75%
“…Mizushima, Hayashi & Adachi (2001) researched the influence of A on the bifurcation type and clarified the origin of the dual solutions from the bifurcation analysis. Serrano-Aguilera, Blanco-Rodríguez & Parras (2021) conducted abundant numerical investigations on dual solutions for values of the Prandtl number between 0.01 and 1 and Rayleigh numbers between 10 2 and 5 × 10 6 . Steady flow transition to oscillation (Hopf bifurcation) was also studied systematically.…”
We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.
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