Convection in a compressible fluid with infinite Prandtl number

1980 ◽  
Vol 96 (3) ◽  
pp. 515-583 ◽  
Author(s):  
Gary T. Jarvis ◽  
Dan P. Mckenzie
Keyword(s):  
Prandtl Number ◽  
Viscous Dissipation ◽  
Coefficient Of Thermal Expansion ◽  
Frictional Heating ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Adiabatic Temperature ◽  
Time Dependent ◽  
Dependent Manner ◽  
Rayleigh Numbers

An approximate set of equations is derived for a compressible liquid of infinite Prandtl number. These are referred to as the anelastic-liquid equations. The approximation requires the product of absolute temperature and volume coefficient of thermal expansion to be small compared to one. A single parameter defined as the ratio of the depth of the convecting layer,d, to the temperature scale height of the liquid,HT, governs the importance of the non-Boussinesq effects of compressibility, viscous dissipation, variable adiabatic temperature gradients and non-hydrostatic pressure gradients. Whend/HT[Lt ] 1 the Boussinesq equations result, but whend/HTisO(1) the non-Boussinesq terms become important. Using a time-dependent numerical model, the anelastic-liquid equations are solved in two dimensions and a systematic investigation of compressible convection is presented in whichd/HTis varied from 0·1 to 1·5. Both marginal stability and finite-amplitude convection are studied. Ford/HT[les ] 1·0 the effect of density variations is primarily geometric; descending parcels of liquid contract and ascending parcels expand, resulting in an increase in vorticity with depth. Whend/HT> 1·0 the density stratification significantly stabilizes the lower regions of the marginal state solutions. At all values ofd/HT[ges ] 0·25, an adiabatic temperature gradient proportional to temperature has a noticeable stabilizing effect on the lower regions. Ford/HT[ges ] 0·5, marginal solutions are completely stabilized at the bottom of the layer and penetrative convection occurs for a finite range of supercritical Rayleigh numbers. In the finite-amplitude solutions adiabatic heating and cooling produces an isentropic central region. Viscous dissipation acts to redistribute buoyancy sources and intense frictional heating influences flow solutions locally in a time-dependent manner. The ratio of the total viscous heating in the convecting system, ϕ, to the heat flux across the upper surface,Fu, has an upper limit equal tod/HT. This limit is achieved at high Rayleigh numbers, when heating is entirely from below, and, for sufficiently large values ofd/HT, Φ/Fuis greater than 1·00.

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.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

scholarly journals Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

2013 ◽  
Vol 716 ◽  
pp. 203-227 ◽  
Author(s):  
Andrew J. Wells ◽  
J. S. Wettlaufer ◽  
Steven A. Orszag
Keyword(s):  
Rayleigh Number ◽  
Scaling Laws ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Solute Flux ◽  
Scaling Analysis ◽  
Mushy Layer ◽  
Similar System ◽  
Solute Fluxes ◽  
Rayleigh Numbers

AbstractWe model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

Influence of viscous dissipation on Bénard convection

1974 ◽  
Vol 64 (2) ◽  
pp. 369-374 ◽  
Author(s):  
D. L. Turcotte ◽  
A. T. Hsui ◽  
K. E. Torrance ◽  
G. Schubert
Keyword(s):  
Temperature Gradient ◽  
Viscous Dissipation ◽  
Dimensionless Parameter ◽  
Finite Amplitude ◽  
Adiabatic Temperature ◽  
Numerical Calculations ◽  
Bénard Convection ◽  
Benard Convection ◽  
Adiabatic Temperature Gradient ◽  
Flow Velocities

The approximations implicit in Bénard convection have been modified to include viscous dissipation. It is shown that both the influence of an adiabatic temperature gradient and of viscous dissipation are governed by the same dimensionless parameter Di = αgh/cp. Numerical calculations of finite amplitude convection are given for finite values of Di. It is found that increasing Di decreases flow velocities and finally stabilizes the flow.

Finite-amplitude convection in the presence of finitely conducting boundaries

2001 ◽  
Vol 432 ◽  
pp. 351-367 ◽  
Author(s):  
M. WESTERBURG ◽  
F. H. BUSSE
Keyword(s):  
Thermal Conductivity ◽  
Prandtl Number ◽  
Fluid Layer ◽  
Finite Amplitude ◽  
Onset Of Convection ◽  
Theoretical Predictions ◽  
Square Cells ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

Finite-amplitude convection in the form of rolls and their stability with respect to infinitesimal disturbances is investigated in the case of boundaries of the horizontal fluid layer which exhibit a thermal conductivity comparable to that of the fluid. It is found that even when rolls represent the preferred mode at the onset of convection a transition to square cells may occur at slightly supercritical Rayleigh numbers. The phenomenon of inertial convection in low Prandtl number fluids appears to become more pronounced as the conductivity of the boundaries is reduced. Modulated convection rolls have also been found as solutions of the problem. But they appear to be unstable in general. Experimental observations have been made and are found in general agreement with the theoretical predictions.

scholarly journals Finite-amplitude convection in rotating spherical fluid shells

1997 ◽  
Vol 332 ◽  
pp. 359-376 ◽  
Author(s):  
A. Tilgner ◽  
F. H. Busse
Keyword(s):  
Heat Transport ◽  
Critical Value ◽  
Finite Amplitude ◽  
Time Dependent ◽  
Polar Regions ◽  
Onset Of Convection ◽  
Prandtl Numbers ◽  
Spherical Fluid ◽  
Rayleigh Numbers

Finite-amplitude convection in rotating spherical fluid shells is considered for a variety of Prandtl numbers P and Rayleigh numbers Ra up to about 10 times the critical value. Convection at low Rayleigh numbers in the form of azimuthally periodic or weakly aperiodic drifting waves is characterized by relatively low heat transport, especially for P ≲ 1. The transition to strongly time-dependent convection leads to a rapid increase of the heat transport with increasing Rayleigh numbers. Onset of convection in the polar regions is delayed, but contributes a disproportionate fraction of the heat transport at high Rayleigh number. The differential rotation generated by convection, the distributions of helicity, and the role of asymmetry with respect to the equatorial plane are also studied.

The Natural Flavonoid Naringenin Inhibits the Cell Growth of WilmsTumorinChildren by Suppressing TLR4/NF-κB Signaling

2020 ◽  
Vol 20 ◽  
Author(s):  
Hongtao Li ◽  
Peng Chen ◽  
Lei Chen ◽  
Xinning Wang
Keyword(s):  
Cell Growth ◽  
Western Blot ◽  
Wilms Tumor ◽  
Critical Role ◽  
Time Dependent ◽  
Luciferase Assay ◽  
Dependent Manner ◽  
Tlr4 Expression ◽  
Protein Levels

Background: Nuclear factor kappa B (NF-κB) is usually activated in Wilms tumor (WT) cells and plays a critical role in WT development. Objective: The study purpose was to screen a NF-κB inhibitor from natural product library and explore its effects on WT development. Methods: Luciferase assay was employed to assess the effects of natural chemical son NF-κB activity. CCK-8 assay was conducted to assess cell growth in response to naringenin. WT xenograft model was established to analyze the effect of naringenin in vivo. Quantitative real-time PCR and Western blot were performed to examine the mRNA and protein levels of relative genes, respectively. Results: Naringenin displayed significant inhibitory effect on NF-κB activation in SK-NEP-1 cells. In SK-NEP-1 and G-401 cells, naringenin inhibited p65 phosphorylation. Moreover, naringenin suppressed TNF-α-induced p65 phosphorylation in WT cells. Naringenin inhibited TLR4 expression at both mRNA and protein levels in WT cells. CCK-8 staining showed that naringenin inhibited cell growth of the two above WT cells in dose-and time-dependent manner, whereas Toll-like receptor 4 (TLR4) over expression partially reversed the above phenomena. Besides, naringenin suppressed WT tumor growth in dose-and time-dependent manner in vivo. Western blot found that naringenin inhibited TLR4 expression and p65 phosphorylation in WT xenograft tumors. Conclusion: Naringenin inhibits WT development viasuppressing TLR4/NF-κB signaling

scholarly journals Synthesis and Evaluation of the Antitumor Activity of Novel 1-(4-Substituted phenyl)-2-ethyl Imidazole Apoptosis Inducers In Vitro

Molecules ◽  
2020 ◽  
Vol 25 (18) ◽  
pp. 4293
Author(s):  
Zhen-Wang Li ◽  
Chun-Yan Zhong ◽  
Xiao-Ran Wang ◽  
Shi-Nian Li ◽  
Chun-Yuan Pan ◽  
...  
Keyword(s):  
Antitumor Activity ◽  
Tumor Cells ◽  
Antiproliferative Activity ◽  
Antitumor Agent ◽  
Positive Control ◽  
Selectivity Index ◽  
Time Dependent ◽  
Potential Candidate ◽  
Dependent Manner

Novel imidazole derivatives were designed, prepared, and evaluated in vitro for antitumor activity. The majority of the tested derivatives showed improved antiproliferative activity compared to the positive control drugs 5-FU and MTX. Among them, compound 4f exhibited outstanding antiproliferative activity against three cancer cell lines and was considerably more potent than both 5-FU and MTX. In particular, the selectivity index indicated that the tolerance of normal L-02 cells to 4f was 23–46-fold higher than that of tumor cells. This selectivity was significantly higher than that exhibited by the positive control drugs. Furthermore, compound 4f induced cell apoptosis by increasing the protein expression levels of Bax and decreasing those of Bcl-2 in a time-dependent manner. Therefore, 4f could be a potential candidate for the development of a novel antitumor agent.

scholarly journals Moxifloxacin Activates the SOS Response in Mycobacterium tuberculosis in a Dose- and Time-Dependent Manner

2021 ◽  
Vol 9 (2) ◽  
pp. 255
Author(s):  
Angelo Iacobino ◽  
Giovanni Piccaro ◽  
Manuela Pardini ◽  
Lanfranco Fattorini ◽  
Federico Giannoni
Keyword(s):  
Gene Expression ◽  
Mycobacterium Tuberculosis ◽  
Physiological State ◽  
Transcriptional Response ◽  
Sos Response ◽  
Resistant Mutant ◽  
Time Dependent ◽  
Dependent Manner ◽  
Gyra Gene ◽  
Drug Concentrations

Previous studies on Escherichia coli demonstrated that sub-minimum inhibitory concentration (MIC) of fluoroquinolones induced the SOS response, increasing drug tolerance. We characterized the transcriptional response to moxifloxacin in Mycobacterium tuberculosis. Reference strain H37Rv was treated with moxifloxacin and gene expression studied by qRT-PCR. Five SOS regulon genes, recA, lexA, dnaE2, Rv3074 and Rv3776, were induced in a dose- and time-dependent manner. A range of moxifloxacin concentrations induced recA, with a peak observed at 2 × MIC (0.25 μg/mL) after 16 h. Another seven SOS responses and three DNA repair genes were significantly induced by moxifloxacin. Induction of recA by moxifloxacin was higher in log-phase than in early- and stationary-phase cells, and absent in dormant bacilli. Furthermore, in an H37Rv fluoroquinolone-resistant mutant carrying the D94G mutation in the gyrA gene, the SOS response was induced at drug concentrations higher than the mutant MIC value. The 2 × MIC of moxifloxacin determined no significant changes in gene expression in a panel of 32 genes, except for up-regulation of the relK toxin and of Rv3290c and Rv2517c, two persistence-related genes. Overall, our data show that activation of the SOS response by moxifloxacin, a likely link to increased mutation rate and persister formation, is time, dose, physiological state and, possibly, MIC dependent.

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