Patterns at the onset of electroconvection in freely suspended smectic films

We report the results of experiments on electrically driven convection that occurs in a thin, freely suspended film of smectic A liquid crystal when an electric field is applied in the plane of the film. Convection in a vortex pattern is found above a well-defined critical voltage. The film behaves as a two-dimensional isotropic liquid: neither its thickness nor the director field are modified by the flow. We present measurements of the critical voltage at the onset of convection in two experimental configurations—one which allows the injection of charges into the film from the electrodes, and one which does not. When injection is present, the critical voltage for the onset of flow increases monotonically with increasing frequency of applied field. With no injection, there is no instability at DC and the critical voltage diverges there. The nature of the flow pattern observed at onset changes with frequency. Below a certain frequency the film flows in vortices that extend over the width of the film; above this frequency the flow is confined to two lines of smaller vortices localized along the electrodes. We present a simple discussion of the mechanisms which drive the convection.     

Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection

We generalize an analogy between rotating and stratified shear flows. This analogy is summarized in Table 1. We use this analogy in the unstable case (centrifugally unstable flow vs. convection) to compute the torque in Taylor-Couette configuration, as a function of the Reynolds number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we predict that the non-dimensional torque G = T/ν² L, where L is the cylinder length, scales with Reynolds number R and gap width η, G = 1.46η^3/2(1 - η)^-7/4 R ^3/2. At larger Reynolds number, velocity fluctuations become non-negligible in the dissipation. In these regimes, there is no exact power law dependence the torque versus Reynolds. Instead, we obtain logarithmic corrections to the classical ultra-hard (exponent 2) regimes: G = 0.50 . These predictions are found to be in excellent agreement with avail-able experimental data. Predictions for scaling of velocity fluctuations are also provided. Received 7 June 2001 and Received in final form 7 December 2001     

The measurement of subcritical electroconvection

Under idealized conditions, electroconvection in a nematic liquid crystal sets in at a well-defined threshold of the driving ac voltage. Fluctuating convection rolls of a small amplitude below that threshold have been observed recently. The measurement of the amplitude of these fluctuations is described in detail in this paper. It is based on averaging the light intensity signals using the structure function and a quantitative analysis of the light deflection.     

Localized bifurcations and defect instabilities in the convection of a nematic liquid crystal

The stationary and the time-dependent homogeneous ordered states in convection may both become unstable against localized perturbations. Defects are then created and they may contribute to the disorganization of the homogeneous state. We present an experimental study of defects in some homogeneous stationary structures as well as in the traveling-wave states of convection of a nematic liquid crystal. We show that the core of the defects is a germ of the unstable state and it can become unstable under the external stress. Then, either fully homogeneous states with the symmetry of the core, or complex disordered states can develop from the local instability of defects in processes quite similar to displacive transitions in solids. Some of the main features are qualitatively similar to numerical simulations of an appropriate Landau-Ginzburg equation.     

A convection/reaction/switching system

We consider a spatially distributed hybrid system consisting of a convection/reaction system in which the reaction switches discontinuously in time between modes, independently at each spatial point on reaching “switching thresholds”. The model involves a novel formulation for evolution of the free boundary between the modal regions.     

Multiplicity of Solutions of the Weighted p-Laplacian with Isolated Singularity and Diffusion Suppressed by Convection

In this paper, the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and di?usion suppressed by convection ?div(|x|^α|?u|^p?2?u) + λ 1/|x|^β |?u|^p?2?u · x = |x|^γg(|x|) in B \ {0} subject to nonlinear Robin boundary value condition |x|^α|?u|^p?2?u · ?n = A ? ρu on ?B,where λ > 0, B ? R^N(N ≥ 2) is the unit ball centered at the origin, α > 0, p > 1, β ∈ R, γ > ?N, g ∈ C([0,1]) with g(0) > 0, A ∈ R, ρ > 0 and ?n is the unit outward normal. The same problem with di?usion promoted by convection, namely λ ≤ 0, has already been discussed by the last two authors (Song-Yin (2012)), where the existence, nonexistence and classi?cation of singularities for solutions are presented. Completely di?erent from [Song, H. J. and Yin, J. X., Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection, Math. Meth. Appl. Sci., 35, 2012, 1089–1100], in the present case λ > 0, namely the di?usion is suppressed by the convection, non-singular solutions are not only existent but also may be in?nite which vary according only to the values of solutions at the isolated singular point. At the same time, the singular solutions may exist only if the di?usion dominates the convection.     

Heat conduction, convection and radiolysis of the

Even if the variation of Nusselt number with Reynolds number has been observed before, little to no studies are concerning nanofluids. The present numerical data in the laminar regime were also found to correlate well with the Brinkman number for the individual sets and much better globally, by combining all the data sets for nanofluids and water.Thus, this work proposes a new correlation between heat transfer rate and Brinkman number which is a nondimensional number of viscosity, flow velocity and temperature. The results showed a good empirical equation that Nu/(Re^0.62 Pr^0.33) is dependent on a power law function to the Brinkman number in laminar flow regime. The proposed correlation can be applied both to water and nanofluid flow, with respect to the nanoparticle loading. It is expected that the equation suggested by this work can be useful to design heating/cooling devices.     

A study on Brinkman number variation on water based nanofluid heat transfer in partially heated tubes

Even if the variation of Nusselt number with Reynolds number has been observed before, little to no studies are concerning nanofluids. The present numerical data in the laminar regime were also found to correlate well with the Brinkman number for the individual sets and much better globally, by combining all the data sets for nanofluids and water.Thus, this work proposes a new correlation between heat transfer rate and Brinkman number which is a nondimensional number of viscosity, flow velocity and temperature. The results showed a good empirical equation that Nu/(Re^0.62 Pr^0.33) is dependent on a power law function to the Brinkman number in laminar flow regime. The proposed correlation can be applied both to water and nanofluid flow, with respect to the nanoparticle loading. It is expected that the equation suggested by this work can be useful to design heating/cooling devices.     

A monotone finite element scheme for convection-diffusion equations

A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an -matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.