arXiv subjects

Stephen W. Morris

Publications and source records attributed to Stephen W. Morris.

34 records · Page 2Linked to original sources

arXiv2004

Codimension-two points in annular electroconvection as a function of aspect ratio

We rigorously derive from first principles the generic Landau amplitude equation that describes the primary bifurcation in electrically driven convection. Our model accurately represents the experimental system: a weakly conducting, submicron thick liquid crystal film suspended between concentric circular electrodes and driven by an applied voltage between its inner and outer edges. We explicitly calculate the coefficient $g$ of the leading cubic nonlinearity and systematically study its dependence on the system's geometrical and material parameters. The radius ratio $α$ quantifies the film's geometry while a dimensionless group ${\cal P}$, similar to the Prandtl number, fixes the ratio of the fluid's electrical and viscous relaxation times. Our calculations show that for fixed $α$, $g$ is a decreasing function of ${\cal P}$, as ${\cal P}$ becomes smaller, and is nearly constant for ${\cal P} \gtrsim 1$. As ${\cal P} \to 0$, $g \to \infty$. We find that $g$ is a nontrivial and discontinuous function of $α$. We show that the discontinuities occur at codimension-two points that are accessed by varying $α$.

nlin.PS↗

arXiv2004

Subdiffusive axial transport of granular materials in a long drum mixer

Granular mixtures rapidly segregate radially by size when tumbled in a partially filled horizontal drum. The smaller component moves toward the axis of rotation and forms a buried core, which then splits into axial bands. Models have generally assumed that the axial segregation is opposed by diffusion. Using narrow pulses of the smaller component as initial conditions, we have characterized axial transport in the core. We find that the axial advance of the segregated core is well described by a self-similar concentration profile whose width scales as $t^α$, with $α\sim 0.3 < 1/2$. Thus, the process is subdiffusive rather than diffusive as previously assumed. We find that $α$ is nearly independent of the grain type and drum rotation rate within the smoothly streaming regime. We compare our results to two one-dimensional PDE models which contain self-similarity and subdiffusion; a linear fractional diffusion model and the nonlinear porous medium equation.

cond-mat.soft↗

arXiv2003

Aspect ratio dependence of charge transport in turbulent electroconvection

We present measurements of the normalized charge transport or Nusselt number $\rm Nu$ as a function of the aspect ratio $Γ$ for turbulent convection in an electrically driven film. In analogy with turbulent Rayleigh-B{é}nard convection, we develop the relevant theoretical framework in which we discuss the local-power-law-scaling of $\rm Nu$ with a dimensionless electrical forcing parameter $\cal R$. For these experiments where $10^4 \lesssim {\cal R} \lesssim 2 \times 10^5$ we find that ${\rm Nu} \sim {\rm F}(Γ){\cal R}^γ$ with $γ$ either $ = 0.26 (\pm 0.02)$ or $0.20 (\pm 0.03)$, in excellent agreement with the theoretical predictions of $γ= 1/4$ and 1/5. Our measurements of the aspect ratio-dependence of ${\rm Nu}$ for $0.3 \leq Γ\leq 17$ compares favorably with the function ${\rm F}(Γ)$ from the scaling theory.

cond-mat↗

arXiv2001

Sequential Bifurcations in Sheared Annular Electroconvection

A sequence of bifurcations is studied in a one-dimensional pattern forming system subject to the variation of two experimental control parameters: a dimensionless electrical forcing number ${\cal R}$ and a shear Reynolds number ${\rm Re}$. The pattern is an azimuthally periodic array of traveling vortices with integer mode number $m$. Varying ${\cal R}$ and ${\rm Re}$ permits the passage through several codimension-two points. We find that the coefficients of the nonlinear terms in a generic Landau equation for the primary bifurcation are discontinuous at the codimension-two points. Further, we map the stability boundaries in the space of the two parameters by studying the subcritical secondary bifurcations in which $m \to m+1$ when ${\cal R}$ is increased at constant ${\rm Re}$.

nlin.PS↗

arXiv2001

Bifurcations in annular electroconvection with an imposed shear

We report an experimental study of the primary bifurcation in electrically-driven convection in a freely suspended film. A weakly conducting, submicron thick smectic liquid crystal film was supported by concentric circular electrodes. It electroconvected when a sufficiently large voltage $V$ was applied between its inner and outer edges. The film could sustain rapid flows and yet remain strictly two-dimensional. By rotation of the inner electrode, a circular Couette shear could be independently imposed. The control parameters were a dimensionless number ${\cal R}$, analogous to the Rayleigh number, which is $\propto V^2$ and the Reynolds number ${\cal R}e$ of the azimuthal shear flow. The geometrical and material properties of the film were characterized by the radius ratio $α$, and a Prandtl-like number ${\cal P}$. Using measurements of current-voltage characteristics of a large number of films, we examined the onset of electroconvection over a broad range of $α$, ${\cal P}$ and ${\cal R}e$. We compared this data quantitatively to the results of linear stability theory. This could be done with essentially no adjustable parameters. The current-voltage data above onset were then used to infer the amplitude of electroconvection in the weakly nonlinear regime by fitting them to a steady-state amplitude equation of the Landau form. We show how the primary bifurcation can be tuned between supercritical and subcritical by changing $α$ and ${\cal R}e$.

nlin.PS↗

arXiv2000

Electrically driven convection in a thin annular film undergoing circular Couette flow

We investigate the linear stability of a thin, suspended, annular film of conducting fluid with a voltage difference applied between its inner and outer edges. For a sufficiently large voltage, such a film is unstable to radially-driven electroconvection due to charges which develop on its free surfaces. The film can also be subjected to a Couette shear by rotating its inner edge. This combination is experimentally realized using films of smectic A liquid crystals. In the absence of shear, the convective flow consists of a stationary, azimuthally one-dimensional pattern of symmetric, counter-rotating vortex pairs. When Couette flow is applied, an azimuthally traveling pattern results. When viewed in a co-rotating frame, the traveling pattern consists of pairs of asymmetric vortices. We calculate the neutral stability boundary for arbitrary radius ratio $α$ and Reynolds number ${{\cal R} e}$ of the shear flow, and obtain the critical control parameter ${\cal R}_c (α, {{\cal R} e})$ and the critical azimuthal mode number ${m_c (α, {{\cal R} e})}$. The Couette flow suppresses the onset of electroconvection, so that ${\cal R}_c (α, {{\cal R} e}) > {\cal R}_c (α,0)$. The calculated suppression is compared with experiments performed at $α= 0.56 $ and $0 \leq {{\cal R} e} \leq 0.22 $.

patt-sol↗

arXiv1999

Development and geometry of isotropic and directional shrinkage crack patterns

We have studied shrinkage crack patterns which form when a thin layer of an alumina/water slurry dries. Both isotropic and directional drying were studied. The dynamics of the pattern formation process and the geometric properties of the isotropic crack patterns are similar to what is expected from recent models, assuming weak disorder. There is some evidence for a gradual increase in disorder as the drying layer become thinner, but no sudden transition, in contrast to what has been seen in previous experiments. The morphology of the crack patterns is influenced by drying gradients and front propagation effects, with sharp gradients having a strong orienting and ordering effect.

patt-sol↗

arXiv1998

Benard-Marangoni Convection in Two Layered Liquids

We describe experiments on B{é}nard-Marangoni convection in horizontal layers of two immiscible liquids. Unlike previous experiments, which used gases as the upper fluid, we find a square planform close to onset which undergoes a secondary bifurcation to rolls at higher temperature differences. The scale of the convection pattern is that of the thinner lower fluid layer for which buoyancy and surface tension forces are comparable. The wavenumber of the pattern near onset agrees with the prediction of the linear stability analysis for the full two-layer problem. The square planform is in qualitative agreement with recent one- and two-layer nonlinear theories, which fail however to predict the transition to rolls.

patt-sol↗

arXiv1998

The Dynamics of Granular Segregation Patterns in a Long Drum Mixer

We have studied the early time evolution of granular segregation patterns in a horizontal rotating cylinder partially filled with a sand/salt mixture. The growth of concentration fluctuations starting from premixed initial conditions was analyzed using Fourier techniques. In one mixture, we observed generally merging dynamics in the segregated bands up to the onset of saturation. At the threshold of saturation, we found a spectrum of wavelengths with a broad peak. The peak position was nearly independent of the tube rotation rate. The overall growth rate of Fourier modes had a maximum at a particular value of the angular rotation frequency. In a slightly different mixture, we observed transient traveling waves when the larger grains were in the majority. We measured the wave speed as a function of several parameters using presegregated initial conditions to launch waves of various wavelengths.

patt-sol↗

arXiv1997

Annular electroconvection with shear

We report experiments on convection driven by a radial electrical force in suspended annular smectic A liquid crystal films. In the absence of an externally imposed azimuthal shear, a stationary one-dimensional (1D) pattern consisting of symmetric vortex pairs is formed via a supercritical transition at the onset of convection. Shearing reduces the symmetries of the base state and produces a traveling 1D pattern whose basic periodic unit is a pair of asymmetric vortices. For a sufficiently large shear, the primary bifurcation changes from supercritical to subcritical. We describe measurements of the resulting hysteresis as a function of the shear at radius ratio $η\sim 0.8$. This simple pattern forming system has an unusual combination of symmetries and control parameters and should be amenable to quantitative theoretical analysis.

patt-sol↗

arXiv1997

Traveling Granular Segregation Patterns in a Long Drum Mixer

Mixtures of granular media often exhibit size segregation along the axis of a partially-filled, horizontal, rotating cylinder. Previous experiments have observed axial bands of segregation that grow from concentration fluctuations and merge in a manner analogous to spinodal decomposition. We have observed that a new dynamical state precedes this effect in certain mixtures: bi-directional traveling waves. By preparing initial conditions, we found that the wave speed decreased with wavelength. Such waves appear to be inconsistent with simple PDE models which are first order in time.

cond-mat.soft↗

arXiv1997

Weakly Nonlinear Analysis of Electroconvection in a Suspended Fluid Film

It has been experimentally observed that weakly conducting suspended films of smectic liquid crystals undergo electroconvection when subjected to a large enough potential difference. The resulting counter-rotating vortices form a very simple convection pattern and exhibit a variety of interesting nonlinear effects. The linear stability problem for this system has recently been solved. The convection mechanism, which involves charge separation at the free surfaces of the film, is applicable to any sufficiently two-dimensional fluid. In this paper, we derive an amplitude equation which describes the weakly nonlinear regime, by starting from the basic electrohydrodynamic equations. This regime has been the subject of several recent experimental studies. The lowest order amplitude equation we derive is of the Ginzburg-Landau form, and describes a forward bifurcation as is observed experimentally. The coefficients of the amplitude equation are calculated and compared with the values independently deduced from the linear stability calculation.

patt-sol↗

arXiv1996

Electroconvection in a Suspended Fluid Film: A Linear Stability Analysis

A suspended fluid film with two free surfaces convects when a sufficiently large voltage is applied across it. We present a linear stability analysis for this system. The forces driving convection are due to the interaction of the applied electric field with space charge which develops near the free surfaces. Our analysis is similar to that for the two-dimensional Bénard problem, but with important differences due to coupling between the charge distribution and the field. We find the neutral stability boundary of a dimensionless control parameter ${\cal R}$ as a function of the dimensionless wave number $κ$. ${\cal R}$, which is proportional to the square of the applied voltage, is analogous to the Rayleigh number. The critical values ${{\cal R}_c}$ and ${κ_c}$ are found from the minimum of the stability boundary, and its curvature at the minimum gives the correlation length ${ξ_0}$. The characteristic time scale ${τ_0}$, which depends on a second dimensionless parameter ${\cal P}$, analogous to the Prandtl number, is determined from the linear growth rate near onset. ${ξ_0}$ and ${τ_0}$ are coefficients in the Ginzburg-Landau amplitude equation which describes the flow pattern near onset in this system. We compare our results to recent experiments.

patt-sol↗

arXiv1996

The Spatio-Temporal Structure of Spiral-Defect Chaos

We present a study of the recently discovered spatially-extended chaotic state known as spiral-defect chaos, which occurs in low-Prandtl-number, large-aspect-ratio Rayleigh-Benard convection. We employ the modulus squared of the space-time Fourier transform of time series of two-dimensional shadowgraph images to construct the structure factor ${S}({\vec k},ω)$. This analysis is used to characterize the average spatial and temporal scales of the chaotic state. We find that the correlation length and time can be described by power-law dependences on the reduced Rayleigh number $ε$. These power laws have as yet no theoretical explanation.

chao-dyn↗

arXiv1996

Convective Fingering of an Autocatalytic Reaction Front

We report experimental observations of the convection-driven fingering instability of an iodate-arsenous acid chemical reaction front. The front propagated upward in a vertical slab; the thickness of the slab was varied to control the degree of instability. We observed the onset and subsequent nonlinear evolution of the fingers, which were made visible by a {\it p}H indicator. We measured the spacing of the fingers during their initial stages and compared this to the wavelength of the fastest growing linear mode predicted by the stability analysis of Huang {\it et. al.} [{\it Phys. Rev. E}, {\bf 48}, 4378 (1993), and unpublished]. We find agreement with the thickness dependence predicted by the theory.

patt-sol↗

arXiv1993

Spiral Defect Chaos in Large Aspect Ratio Rayleigh-Benard Convection

We report experiments on convection patterns in a cylindrical cell with a large aspect ratio. The fluid had a Prandtl number of approximately 1. We observed a chaotic pattern consisting of many rotating spirals and other defects in the parameter range where theory predicts that steady straight rolls should be stable. The correlation length of the pattern decreased rapidly with increasing control parameter so that the size of a correlated area became much smaller than the area of the cell. This suggests that the chaotic behavior is intrinsic to large aspect ratio geometries.

patt-sol↗