scholarly journals Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number

2017 ◽  
Vol 822 ◽  
Author(s):  
Giovanni Fantuzzi ◽  
Andrew Wynn
Keyword(s):  
Prandtl Number ◽  
Energy Method ◽  
Marangoni Convection ◽  
Fluid Layer ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Energy Stability ◽  
Thermal Boundary Conditions ◽  
Set Up ◽  
The Stability

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

Finite amplitude cellular convection

1958 ◽  
Vol 4 (3) ◽  
pp. 225-260 ◽  
Author(s):  
W. V. R. Malkus ◽  
G. Veronis
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Heat Transport ◽  
Stability Problem ◽  
Dominant Role ◽  
Linear Equations ◽  
Finite Amplitude ◽  
Set Up ◽  
Rayleigh Numbers ◽  
Linear Stability Problem

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

scholarly journals On nonlinear overstable convection rolls in a rotating system

1984 ◽  
Vol 25 (3) ◽  
pp. 406-418 ◽  
Author(s):  
N. Riahi
Keyword(s):  
Prandtl Number ◽  
Travelling Waves ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Oscillatory Convection ◽  
Nonlinear Analyses ◽  
Rotating System ◽  
The Stability ◽  
Convection Rolls

AbstractFinite amplitude oscillatory convection rolls in the form of travelling waves are studied for a horizontal layer of a low Prandtl number fluid heated from below and rotating rapidly about a vertical axis. The results of the stability and nonlinear analyses indicate that there is no subcritical instability and that the oscillatory rolls are unstable for the ranges of the Prandtl number and the rotation rate considered in this paper.

Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity

1987 ◽  
Vol 178 ◽  
pp. 491-506 ◽  
Author(s):  
D. R. Jenkins
Keyword(s):  
Thermal Convection ◽  
Fluid Layer ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Temperature Dependent Viscosity ◽  
Weakly Nonlinear ◽  
Expansion Procedure ◽  
The Stability ◽  
Dependent Viscosity

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.

Oscillatory instabilities of convection rolls at intermediate Prandtl numbers

1986 ◽  
Vol 164 ◽  
pp. 469-485 ◽  
Author(s):  
E. W. Bolton ◽  
F. H. Busse ◽  
R. M. Clever
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Stability Boundary ◽  
Experimental Studies ◽  
Interesting Property ◽  
Number Range ◽  
Close Relationship ◽  
Prandtl Numbers ◽  
The Stability ◽  
Convection Rolls

The analysis of the instabilities of convection rolls in a fluid layer heated from below with no-slip boundaries exhibits a close competition between various oscillatory modes in the range 2 [lsim ] P [lsim ] 12 of the Prandtl number P. In addition to the even-oscillatory instability known from earlier work two new instabilities have been found, each of which is responsible for a small section of the stability boundary of steady rolls. The most interesting property of the new instabilities is their close relationship to the hot-blob oscillations known from experimental studies of convection. In the lower half of the Prandtl-number range considered the B02-mode dominates, which is characterized by two blobs each of slightly hotter and colder fluid circulating around in the convection roll in a spatially and time-periodic fashion. At higher Prandtl numbers the BE 1-mode dominates, which possesses one hot blob (and one cold blob) circulating with the convection velocity. Just outside the stability boundary there exist other growing modes exhibiting three or four blobs which may be observable in experiments.

Finite-amplitude stability of pipe flow

1971 ◽  
Vol 45 (4) ◽  
pp. 701-720 ◽  
Author(s):  
A. Davey ◽  
H. P. F. Nguyen
Keyword(s):  
Unit Length ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Wave Number ◽  
True Solution ◽  
Wide Range ◽  
Damping Rate ◽  
Real Value ◽  
Infinitesimal Disturbance ◽  
The Stability

In this paper we present some results concerning the stability of flow in a circular pipe to small but finite axisymmetric disturbances. The flow is unstable if the amplitude of a disturbance exceeds a critical value, the equilibrium amplitude, which we have calculated for a wide range of wave-numbers and Reynolds numbers. For large values of the Reynolds number, R, and for a real value of the wave-number, α, we indicate that the energy density of a critical disturbance is of order c2i, where −ααci is the damping rate of the associated infinitesimal disturbance. The energy, per unit length of the pipe, of a critical disturbance which is concentrated near the axis of the pipe is of order R−2, and the wave-number α is of order R1/3 For a critical disturbance which is concentrated near the wall of the pipe the energy is of order $R^{-\frac{3}{2}}$ and α is of order R½. This suggests that non-linear instability is most likely to be caused by a ‘centre’ mode rather than by a ‘wall’ mode. The wall mode solution is also essentially the solution for the problem of plane Couette flow when αR is large. We compare it with the true solution.In an appendix Dr A. E. Gill indicates how some of the results of this paper may be inferred from a simple scale analysis.

The transition from roll to square-cell solutions in Rayleigh–Bénard convection

1984 ◽  
Vol 139 ◽  
pp. 461-471 ◽  
Author(s):  
D. R. Jenkins ◽  
M. R. E. Proctor
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Finite Conductivity ◽  
Silicone Oils ◽  
High Prandtl Number ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Square Cell

We consider three-dimensional finite-amplitude thermal convection in a fluid layer with boundaries of finite conductivity. Busse & Riahi (1980) and Proctor (1981) showed that the preferred planform of convection in such a system is a square-cell tesselation provided that the boundaries are much poorer conductors than the fluid, in contrast to the roll solutions which are obtained for perfectly conducting boundaries. We determine here the conductivity of the boundaries at which the preferred planform changes from roll to square-cell type. We show that, for low-Prandtl-number fluids (e.g. mercury), square-cell solutions are realized only when the boundaries are almost insulating; while, for high-Prandtl-number fluids (e.g. silicone oils), square-cell solutions are stable when the boundaries have conductivity comparable to that of the fluid.

The Effect of Suction on the Stability of Fluid between Horizontal Plates at Differing Temperatures

1985 ◽  
Vol 199 (2) ◽  
pp. 145-151 ◽  
Author(s):  
M M Sorour ◽  
M A Hassab ◽  
F A Elewa
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Stability Criteria ◽  
Thermal Boundary Conditions ◽  
Wide Range ◽  
Fluid Layers ◽  
The Stability ◽  
Horizontal Fluid Layer

The linear stability theory is applied to study the effect of suction on the stability criteria of a horizontal fluid layer confined between two thin porous surfaces heated from below. This investigation covers a wide range of Reynolds number 0 ≥ Re ≥ 30, and Prandtl number 0.72 ≥ Pr ≥ 100. The results show that the critical Rayleigh number increases with Peclet number, and is independent of Pr as far as Re < 3. However, for Re > 3 the critical Rayleigh number is function of both Pr and Pe. In addition, the analysis is extended to study the effect of suction on the stability of two special superimposed fluid layers. The results in the latter case indicate a more stabilizing effect. Furthermore, the effect of thermal boundary conditions is also investigated.

Thermal Instability in Differentially Heated Inclined Fluid Layers

1972 ◽  
Vol 39 (1) ◽  
pp. 41-46 ◽  
Author(s):  
T. E. Unny
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Base Flow ◽  
Fluid Layers ◽  
Heated Fluid ◽  
The Stability ◽  
Horizontal Fluid Layer ◽  
Small Inclination

In an inclined adversely heated fluid layer confined between two rigid boundaries in a slot of large aspect ratio it is found that the unicellular base flow in the conduction regime becomes unstable with the formation of stationary secondary rolls with their axes along the line of inclination (x-rolls) for large Prandtl number fluids and axes perpendicular to the line of inclination (y-rolls) for small Prandtl number fluids. However, for angles near the vertical, the curve of the critical Rayleigh number versus inclination for x-rolls rises above that for y-rolls even for large Prandtl number fluids so that in a vertical fluid layer only cross rolls (y-rolls) could develop. The stability equations, as well as the results, reduce to those available for the horizontal fluid layer for which x-rolls are as likely to occur as y-rolls. It is seen that even a small inclination to the horizontal is enough to assign a definite direction for these two-dimensional cells, this direction depending on the Prandtl number. It is hoped that this basic information will be of help in the determination of the magnitude of the secondary cells in the postinstability regime and the heat transfer characteristics of the thin fluid layer.

scholarly journals Bounds on heat transfer for Bénard–Marangoni convection at infinite Prandtl number

2018 ◽  
Vol 837 ◽  
pp. 562-596 ◽  
Author(s):  
Giovanni Fantuzzi ◽  
Anton Pershin ◽  
Andrew Wynn
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Marangoni Convection ◽  
Fluid Layer ◽  
Piecewise Linear ◽  
Background Field ◽  
Background Fields ◽  
Method Analysis ◽  
Asymptotic Scaling

The vertical heat transfer in Bénard–Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number $Nu$ as a function of the Marangoni number $Ma$. Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that $Nu\leqslant 0.838Ma^{2/7}$. In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on $Nu$, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering’s formulation at a given $Ma$. Using a piecewise-linear, monotonically decreasing profile we then show that $Nu\leqslant 0.803Ma^{2/7}$, lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering’s original formulation. We subsequently utilise convex optimisation to optimise the bound on $Nu$ over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that $Nu\leqslant O(Ma^{2/7}(\ln Ma)^{-1/2})$ when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent $2/7$ is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.

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