Steady-state Shape Research Articles

Growth from initial to self-similar shape of an interface crack front

Initiation and growth towards a self-similar/final shape of an interface crack is analysed. Asymmetric double cantilever beam experiments are performed under prescribed displacement loading conditions. The onset of crack growth from an initially straight front is followed using digital microscope. It is observed that at first the crack grows from the longitudinal axis of the specimen symmetry. Subsequently, the crack spreads in both the longitudinal and the transverse directions finally forming a curvilinear front. From this stage, the crack grows in a self-similar shape and propagates along the interface. Such crack formation may result in a nonlinear force vs. displacement relationship which is otherwise related to constituent or interface constitutive nonlinearities. To study the consequences of the initial crack growth process a cohesive zone model is established. The numerical analysis starts with the assumption of a straight crack front configuration and continues up to the self-similar stage of propagation. A maximum cohesive zone stress is used as criterion to determine the position and the shape of the process zone front while critical crack opening is used to follow the crack front shape. It is concluded that the non-linear part of the force vs. displacement curve observed during loading can be at least partially interpreted as a crack front reformation from the straight to a steady-state quasi-parabolic shape.

Mechanisms Governing Protein Clustering and Shape Changes in Biological Membranes

A characteristic feature of eukaryotic cells is the variety of membrane bound organelles, distinguished by their unique morphology and chemical composition. Despite the differences in membrane composition across organelles, ramified, tubular or sheet-like shapes are generic large scale morphologies observed in internal membranes, which suggest involvement of common underlying principles. While there is detailed knowledge of the molecular processes involved in membrane remodelling at short scales, our understanding of the underlying physical principles governing large scale morphogenesis is still rudimentary. One common aspect is that membranes are subject to the action of curvature sensing and curvature generating proteins which modulate local membrane shape and lipid composition. Even at relatively small values of surface coverage these proteins can cluster through to a variety or processes and trigger morphological changes which are important for singling. Another important feature, specific to organelle membranes, especially those in the trafficking pathways, is that they are subject to and driven by a continuous flux of membrane bound materials, on time scales comparable to membrane relaxation times. Here we present Monte Carlo Simulation models that tests the effect of the above processes on the morphology of the membranes. We find that the steady state shapes obtained as a result of such active processes, bear a striking resemblance to the ramified morphologies of organelles in vivo, pointing to the relevance of nonequilibrium fission-fusion in organelle morphogenesis. We show that, due to membrane curvature and composition interactions, the curvature-inducing membrane-nematogens can aggregate spontaneously, even at low concentrations, leading to a variety of membrane morphologies such as tubular and the sheet conformations. Strong lipid-protein interaction can result in fast protein clustering indicating a route to a lipid mediated signal amplification.

Odd viscosity in two-dimensional incompressible fluids

In this work, we present observable consequences of parity-violating odd viscosity term in incompressible 2+1D hydrodynamics. For boundary conditions depending on the velocity field (flow) alone we show that: (i) The fluid flow quantified by the velocity field is independent of odd viscosity, (ii) The force acting on a closed contour is independent of odd viscosity, and (iii) The odd viscosity part of torque on a closed contour is proportional to the rate of change of area enclosed by the contour with the proportionality constant being twice the odd viscosity. The last statement allows us to define a measurement protocol of {\it odd viscostance} in analogy to Hall resistance measurements. We also consider {\it no-stress} boundary conditions which explicitly depend on odd viscosity. A classic hydrodynamics problem with no-stress boundary conditions is that of a bubble in a planar Stokes flow. We solve this problem exactly for shear and hyperbolic flows and show that the steady-state shape of the bubble in the shear flow depends explicitly on the value of odd viscosity.

Verification Firings of End-Burning Type Hybrid Rockets

The authors have previously proposed the concept of end-burning-type hybrid rockets, which would use cylindrical fuel grains consisting of an array of many small ports running in the axial direction, through which oxidizer gas would flow. Because of difficulty in manufacturing a fuel grain that satisfied requirements such as high volumetric filling rate (above 0.95) and microsized port intervals, the end-burning hybrid rocket had yet to be achieved. This paper reports the results of verification firing tests of a novel end-burning-type hybrid rocket made possible for the first time by recent progress in three-dimensional printing technology. The results clearly distinguish the initial transient and steady periods of the end-burning mode and prove that no oxidizer-to-fuel ratio shift occurs during firing. Because the initial transient is a period for the exit end face to attain a steady-state shape, an initial end-face shape being close to the steady-state shape can shorten this period. A firing test with fuel having tapered ports is shown to attain a steady-state shape in less than 1 s, which is much shorter than the nontapered case of about 6 s.

Equilibrium configuration of a bounded inextensible membrane subject to solar radiation pressure

The equilibrium shape of a thin inextensible membrane subject to solar radiation pressure under given boundary constraints is studied. The membrane is assumed to be insusceptible to elastic deformation and to have negligible bending resistance, and its steady-state shape is therefore described by a developable surface (i.e., a surface of zero Gaussian curvature), resulting from an equilibrium between radiation pressure and membrane tension forces. A quantitative understanding of the mechanics of such membranes is essential in characterizing the dynamics of solar sail spacecraft that use sail wing tip displacement as an attitude control mode. The analysis in this paper develops a theoretical foundation for the billowed wing shape. Under reasonable simplifying assumptions, the key result is that solar radiation pressure and a given wing tip displacement yield a billowed solar sail wing with the shape of a generalized cylinder (i.e., a developable ruled surface, whose rulings are all parallel, rather than a general developable with variable ruling directions). The base curve geometry for the solar sail is also determined as the solution to a boundary value problem. The results presented herein allow the shape of the billowed membrane to be computed to any desired precision, for any given tip displacement.

Investigation of the Evolution of Crystal Size and Shape during Temperature Cycling and in the Presence of a Polymeric Additive Using Combined Process Analytical Technologies.

Crystal size and shape can be manipulated to enhance the qualities of the final product. In this work the steady-state shape and size of succinic acid crystals, with and without a polymeric additive (Pluronic P123) at 350 mL, scale is reported. The effect of the amplitude of cycles as well as the heating/cooling rates is described, and convergent cycling (direct nucleation control) is compared to static cycling. The results show that the shape of succinic acid crystals changes from plate- to diamond-like after multiple cycling steps, and that the time required for this morphology change to occur is strongly related to the type of cycling. Addition of the polymer is shown to affect both the final shape of the crystals and the time needed to reach size and shape steady-state conditions. It is shown how this phenomenon can be used to improve the design of the crystallization step in order to achieve more efficient downstream operations and, in general, to help optimize the whole manufacturing process.

Understanding time scales of diffusive fluxes and the implication for steady state and steady shape conditions

AbstractThe diffusion equation is one of the most commonly used models for describing environmental problems involving heat, solute, and water transport. A diffusive system can be either transient or steady state. When a system is transient, the dependent variable (e.g., temperature, concentration, or hydraulic head) varies with time; whereas at steady state, the temporal variations are negligible. Here we consider an intermediate state, called steady shape, corresponding to the situation where temporal variations in diffusive fluxes are negligible but the dependent variable may remain transient. We present a general theoretical framework for identifying steady shape conditions and propose a novel method for evaluating the time scale needed for a diffusive system to approach both steady shape and steady state conditions.

Two-Dimensional Steady-State Boundary Shape Inversion of CGM-SPSO Algorithm on Temperature Information

Addressing the problem of two-dimensional steady-state thermal boundary recognition, a hybrid algorithm of conjugate gradient method and social particle swarm optimization (CGM-SPSO) algorithm is proposed. The global search ability of particle swarm optimization algorithm and local search ability of gradient algorithm are effectively combined, which overcomes the shortcoming that the conjugate gradient method tends to converge to the local solution and relies heavily on the initial approximation of the iterative process. The hybrid algorithm also avoids the problem that the particle swarm optimization algorithm requires a large number of iterative steps and a lot of time. The experimental results show that the proposed algorithm is feasible and effective in solving the problem of two-dimensional steady-state thermal boundary shape.

An investigation on the motion and deformation of viscoelastic drops descending in another viscoelastic media

In the present study, an investigation of the motion and shape deformation of drops is carried out in creeping flow to highlight the effect of viscoelastic properties on the problem. A perturbation method is employed to derive an analytical solution for the general case that both interior and exterior fluids are viscoelastic, both fluids obeying the Giesekus model. An experiment is also performed for the limiting case of an immiscible drop of a 0.03% (w/w) polyacrylamide in an 80:20 glycerol/water solution falling through a viscous Newtonian silicon oil (410 cP polydimethylsiloxane oil) in order to check the accuracy of the analytical solution. It is shown that the addition of elastic properties to the interior fluid may cause a decrease in the terminal velocity of the droplet while an increase in the elastic properties of the exterior fluid results in the opposite behavior and increases the terminal velocity. The well-known spherical shape of creeping drops for Newtonian fluids is modified by elasticity into either prolate or oblate shapes. Using the analytical solution, it is shown that normal stresses play a key role on the final steady-state shape of the drops. To keep the drops spherical in viscoelastic phases, it is shown that the effect of normal stresses on the interior and exterior media can cancel out under certain conditions. The results presented here may be of interest to industries dealing with petroleum and medicine processing, paint and power-plant related fields where knowledge of the shape and terminal velocity of descending droplets is of great importance.

Structural Damage Detection Using Slopes of Longitudinal Vibration Shapes

While structural damage detection based on flexural vibration shapes, such as mode shapes and steady-state response shapes under harmonic excitation, has been well developed, little attention is paid to that based on longitudinal vibration shapes that also contain damage information. This study originally formulates a slope vibration shape (SVS) for damage detection in bars using longitudinal vibration shapes. To enhance noise robustness of the method, an SVS is transformed to a multiscale slope vibration shape (MSVS) in a multiscale domain using wavelet transform, which has explicit physical implication, high damage sensitivity, and noise robustness. These advantages are demonstrated in numerical cases of damaged bars, and results show that MSVSs can be used for identifying and locating damage in a noisy environment. A three-dimensional (3D) scanning laser vibrometer (SLV) is used to measure the longitudinal steady-state response shape of an aluminum bar with damage due to reduced cross-sectional dimensions under harmonic excitation, and results show that the method can successfully identify and locate the damage. Slopes of longitudinal vibration shapes are shown to be suitable for damage detection in bars and have potential for applications in noisy environments.

Open water bells

A smooth circular moving liquid sheet is formed by the head-on impingement of two equal laminar water jets. We subject such a liquid sheet to uniform laminar air flow from one side such that the direction of air velocity is perpendicular to the liquid sheet. The pressure of the moving air deforms the liquid sheet giving rise to an open water bell. The water bell is symmetric suggesting that the gas flow around the bell is also symmetric and that the gravitational force is negligible. We have captured the shape of the water bells for varying air flow rates and for varying Weber numbers, and compared the measurements with theoretical predictions obtained from a force balance involving liquid inertia, surface tension, and pressure difference across the sheet. The pressure exerted by the gas phase on the front and the rear surface of the deformed liquid sheet is obtained from known results of flow past flat circular discs. The predicted steady state shapes match well with the measurements at low Weber numbers but differences are observed at high Weber numbers, where the sheet flaps and is no longer smooth. Interestingly, the shape predicted by assuming a constant pressure difference equal to the stagnation pressure over the whole of the front face of the sheet and free stream value over the whole of the rear face yields nearly identical results suggesting that an open water bell is similar to a closed water bell in that, to a good approximation, the pressure on either sides of the water bell is homogeneous.

An atomistic study of the correlation between the migration of planar and curved grain boundaries

Molecular dynamics simulations were performed to investigate the migration of curved and planar boundaries. The reduced mobilities of capillarity driven U-shaped half-loop twin boundaries in b.c.c iron (Fe) were computed between 800 and 1200K. To rationalize these results, simulations were also performed for planar twin boundaries of different inclination to determine their absolute mobilities and grain boundary energies. The variation of these properties with inclination was integrated into a continuum model which was found to produce the steady-state shapes of curved boundaries consistent with those from simulations. A further extension of the continuum model enabled estimations of the reduced mobility that were in good agreement with simulation results. It was also identified that atomistic events governing the migration of curved and planar boundaries shared a number of similarities. Overall, the analyses of the shapes, mobilities and atomic-scale migration mechanisms of curved and planar boundaries presented here provide a correlation between the migration of these types of twin boundaries.

Vibration analysis of a planetary gear system based on the transfer matrix method

This study models a 3-dimensional planetary gear system using the transfer matrix method. The local transfer matrices between each component of the planetary gear set were derived with consideration of the tooth width, and the transfer matrix of a planetary gear system corresponding to the inertial transfer matrix was determined. The eigenvalue analysis of the transfer matrix suggested an analysis method in the form of a lambda matrix, instead of the direct search method through a characteristic polynomial. The boundary conditions at the first and the last stations of the entire transfer matrix were partitioned into known and unknown values to generate a concentrated transfer matrix and a latent equation, and the eigenvalue problem in the lambda matrix was solved. The characteristics of the responses according to the phase state of the harmonic component of the transmission error were reviewed through the steady-state response and mode shape type.

CFD modeling of transient flow in pressurized pipes

The aim of this article is the analysis of the hydraulic transient flows in pressurized pipes using Computational Fluid Dynamics (CFD). The model uses a three-dimensional efficient mesh with a refined mesh in the viscous sublayer, corresponding to the best compromise between the maximum accuracy and the minimum computational effort. CFD results have been compared with collected data from an experimental pipe-rig and an excellent fitting was observed, providing that a hyperbolic time-domain function be used to describe the valve closure. Calculated velocity profiles have shown two regions with different behaviors: the wall region, dominated by the fluid viscosity, in which flow changes are faster and with sharp gradients; and the pipe core, strongly dependent on the fluid inertial forces, that tends to maintain its initial steady-state shape and to have memory of the past time history of the velocity distribution. Immediately after the valve closure, the flow is cancelled in the valve section, an invert flux is generated and a vortex sheet is formed (i.e., a cylindrical surface composed of vortices in the circumferential direction), propagating to the upstream end. The transient wall shear stress has shown a strong dependence on the time history of the local velocity variation.

Analytical solutions of one-way coupled magnetohydrodynamic free surface flow

We study the flow in a layer of conductive liquid under the influence of surface tension, gravity, and Lorentz forces due to imposed potential differences and transverse magnetic fields, as a function of the Hartmann number, the Bond number, the Reynolds number, the capillary number and the height-to-width ratio A. For aspect ratios A ≪ 1 and Reynolds numbers Re≤A, lubrication theory is applied to determine the steady state shape of the liquid surface to lowest order. Assuming low Hartmann (Ha ≤ O(1)), capillary (Ca ≤ O(A4)), Bond (Bo ≤ O(A2)) numbers and contact angles close to 90°, the flow details below the surface and the free surface elevation for the complete domain are determined analytically using the method of matched asymptotic expansions. The amplitude of the free surface deformation scales linearly with the capillary number and decreases with increasing Bond number, while the shape of the free surface depends on the Bond number and the contact angle condition. The strength of the flow scales linearly with the magnetic field gradient and applied potential difference and vanishes for high aspect ratio layers (A → 0). The analytical model results are confirmed by numerical simulations using a finite volume moving mesh interface tracking method, where the Lorentz force is calculated from the equation for the electric potential. It is shown that the analytical result for the free surface elevation is accurate within 0.4% from the numerical results for Ha2 ≤ 1, Ca ≤ A4, Bo ≤ A2, Re ≤ A and A ≤ 0.1 and within 2% for A=0.5. For A=0.1, the solution remains accurate within 1% of the numerical solution when either Ha2 is increased to 400, Ca to 200A4 or Bo to 100A2.

Competing effects of interface anisotropy and isotropic driving force on the growth of steady-state shape in phase-field modeling

It is well-known that the Wulff construction represents the steady-state shape during a shrink evolution in the anisotropic Allen–Cahn phase-field model (AC-PFM). Adding a constant driving force (ΔFmc) into the AC-PFM will result in the competition between the anisotropic shrink and the isotropic growth, which may lead to different phase shape at the steady state. Through numerical simulations and theoretical analyses, three types of evolution behaviors that depend on the strength of ΔFmc have been concluded: shrink resembling Wulff, growth resembling Wulff and growth deviating from Wulff. In other words, in the limit of ΔFmc→0, the steady-state shape predicted by PFM follows the Wulff shape, but deviates from Wulff when ΔFmc grows substantially, as the ΔFmc term represents the isotropic feature of the growth dynamics while the other parts denote the anisotropic feature in the Wulff shape. The equivalence of the steady-state shape to Wulff when ΔFmc→0 has been proven based on a numerically verified interface normal velocity model. The “critical” ΔFmc for marked deviation from Wulff estimated from order analysis of the system evolution equation, is very closed to the point where the normal velocity (Vn) changes its dependence on ΔFmc from Vn∝ΔFmc to Vn∝(ΔFmc)0.5.

Stratified thin-film flow in a rheometer

When two immiscible layered fluids are present in a rheometer, interfacial distortions driven by the centripetal pressure gradient can modify torque measurements and induce dewetting. In particular, we examine the steady-state interface shape of a thin film coating a stationary substrate beneath a second immiscible fluid that is driven by a rotating parallel-plate or cone. An asymptotic analysis of the interfacial distortion for the parallel-plate flow is compared with numerical solutions for both the parallel-plate and cone and plate configurations. We develop asymptotic criteria for dewetting of the thin film as a function of fluid and flow properties, and show that significant interfacial distortion and dewetting can occur due to secondary flow effects even at low Reynolds numbers. The distortion of the interface can result in increased or decreased torque measurements depending on the viscosity and density ratios between the two fluid layers. We relate these effects to recent experimental studies on liquid-infused rough media and discuss the stabilizing effect of surface microstructure.

Computational modeling of anodic current distribution and anode shape change in aluminium reduction cells

In aluminium reduction cells, the profile of a new carbon anode changes with time before reaching a steady state shape, since the anode consumption rate, depending on the current density normal to anode surfaces, varies from one region to another. In this paper, a two-dimension model based on Laplace equation and Tafel equation was built up to calculate the secondary current distribution, and the shift of anode shape with time was simulated with arbitrary Lagrangian-Eulerian method. The time it takes to reach the steady shape for the anode increases with the enlargement of the width of the channels between the anodes or between the anode and the sidewall. This time can be shortened by making a sloped bottom or cutting off the lower corners of the new anode. Forming two slots in the bottom surface increases the anodic current density at the underside of the anode, but leads to the enlargement of the current at the side of the anode.

Influence of Air Cooling Jets on the Steady-State Shape of Strips in Hot Dip Galvanizing Lines

The influence of air cooling jets on the steady-state shape of the strip in hot dip galvanizing lines is investigated. For this purpose, the force characteristic of a single nozzle is measured by a laboratory flow simulator. This approach differs from conventional methods, where typically, CFD simulations are used. Based on the air cooler force characteristic the stability of the equilibrium points is investigated and the influence of different types of boundary conditions, tensile loads, strip thicknesses, and cooler fan speeds is examined.

Vesicle dynamics in a confined Poiseuille flow: from steady state to chaos.

Red blood cells (RBCs) are the major component of blood, and the flow of blood is dictated by that of RBCs. We employ vesicles, which consist of closed bilayer membranes enclosing a fluid, as a model system to study the behavior of RBCs under a confined Poiseuille flow. We extensively explore two main parameters: (i) the degree of confinement of vesicles within the channel and (ii) the flow strength. Rich and complex dynamics for vesicles are revealed, ranging from steady-state shapes (in the form of parachute and slipper shapes) to chaotic dynamics of shape. Chaos occurs through a cascade of multiple periodic oscillations of the vesicle shape. We summarize our results in a phase diagram in the parameter plane (degree of confinement and flow strength). This finding highlights the level of complexity of a flowing vesicle in the small Reynolds number where the flow is laminar in the absence of vesicles and can be rendered turbulent due to elasticity of vesicles.