Fluid Sheet Research Articles

Mass Flux Distribution Measurements and Visualizations of a Fluid Sheet Generated by a Partially Immersed Dip-Lubricated Gear

A simplified model of gearbox lubrication flow is investigated. A metric gear with module size 4 mm, tooth number 38, tip radius 80.25 mm and width 40 mm, is partially immersed and rotated between 0–12 1/s frequencies in room temperature distilled water. Flash and laser sheet visualizations of the global flow field and the boundary layer are presented. Resolved mass flow distributions in the fluid sheet above the water surface are obtained with in-situ liquid trapping (bucketing). The relation between the circumferential velocity and integrated mass flux of the fluid sheet is found to be linear above a critical speed, for which an analytical formula is given. Using dimensional analysis, we show that the relation between the mass flux and gear speed signifies a Reynolds-number independent behavior of the boundary layer around the immersed part of the gear.

Transverse instabilities of ascending planar jets formed by wave impacts on vertical walls

When a steep breaking wave hits a vertical sea wall, in shallow water, a rapidly ascending planar jet forms. This jet is ejected with high acceleration due to pressure created by the violent wave impact on the wall, creating a so-called ‘flip-through’ event. Previous studies have focused on the impulsive pressures on, and within, the wall and on the velocity of the jet. Here, in contrast, we consider the formation and break-up of the jet itself. Experiments show that during flip-through a fluid sheet, bounded by a rim, forms. This sheet has unstable transitional behaviours and organizing jets; undulations in the thickness of the fluid sheet are rapidly amplified and ruptured into an array of vertical ligaments. Lateral undulations of the rim lead to the formation of finger-jets, which subsequently break up to form droplets and spray. We present a linear stability analysis of the rim–sheet systems that highlights the contributions of rim retraction and sheet stretching to the break-up process. The mechanisms for the sequential surface deformations in the rim–sheet system are also described. Multiple, distinct, instability modes are identified during the rim deceleration, sheet stretch attenuation and rim retraction processes. The wavenumbers (and deformation length scales) associated with these instability modes are shown to lead to the characteristic double peak spectrum of surface displacement observed in the experiments. These mechanisms help to explain the columnar structures often seen in photographs of violent wave impacts on harbour walls.

Magneto-hydrodynamics of coupled fluid–sheet interface with mass suction and blowing

There are large number of studies which prescribe the kinematics of the sheet and ignore the sheet's mechanics. However, the current boundary layer analysis investigates the mechanics of both the electrically conducting fluid and a permeable sheet, which makes it distinct from the other studies in the literature. One of the objectives of the current study is to (i) examine the behaviour of magnetic field effect for both the surface and the electrically conducting fluid (ii) investigate the heat and mass transfer between a permeable sheet and the surrounding electrically conducting fluid across the hydro, thermal and mass boundary layers. Self-similar solutions are obtained by considering the RK45 technique. Analytical solution is also found for the stretching sheet case. The skin friction dual solutions are presented for various types of sheet. The influence of pertinent parameters on the dimensionless velocity, shear stress, temperature, mass concentration, heat and mass transfer rates on the fluid–sheet interface is presented graphically as well as numerically. The obtained results are of potential benefit for studying the electrically conducting flow over various soft surfaces such as synthetic plastics, soft silicone sheet and soft synthetic rubber sheet. These surfaces are easily deformed by thermal fluctuations or thermal stresses.

Universal Scaling Law for the Collapse of Viscous Nanopores.

Below a threshold size, a small pore nucleated in a fluid sheet will contract to minimize the surface energy. Such behavior plays a key role in nature and technology, from nanopores in biological membranes to nanopores in sensors for rapid DNA and RNA sequencing. Here we show that nanopores nucleated in viscous fluid sheets collapse following a universal scaling law for the pore radius. High-fidelity numerical simulations reveal that the scaling is largely independent of the initial conditions, including the size, shape, and thickness of the original nanopore. Results further show that the scaling law yields a constant speed of collapse as observed in recent experiments. Nanopores in fluid sheets of moderate viscosity also attain this constant terminal speed provided that they are sufficiently close to the singularity.

Codimensional non-Newtonian fluids

We present a novel method to simulate codimensional non-Newtonian fluids on simplicial complexes. Our method extends previous work for codimensional incompressible flow to various types of non-Newtonian fluids including both shear thinning and thickening, Bingham plastics, and elastoplastics. We propose a novel time integration scheme for semi-implicitly treating elasticity, which when combined with a semi-implicit method for variable viscosity alleviates the need for small time steps. Furthermore, we propose an improved treatment of viscosity on the rims of thin fluid sheets that allows us to capture their elusive, visually appealing twisting motion. In order to simulate complex phenomena such as the mixing of colored paint, we adopt a multiple level set framework and propose a discretization on simplicial complexes that facilitates the tracking of material interfaces across codimensions. We demonstrate the efficacy of our approach by simulating a wide variety of non-Newtonian fluid phenomena exhibiting various codimensional features.

Dressler’s theory for curved topography flows: iterative derivation, transcritical flow solutions and higher-order wave-type equations

The Dressler equations are a system of two non-linear partial differential equations for shallow fluid flows over curved topography. The theory originated from an asymptotic stretching method formulating the equations of motion in terrain-fitted curvilinear coordinates. Apparently, these equations failed to produce a transcritical flow profile changing from sub- to supercritical flow conditions. Further, wave-like motions over a flat bottom are excluded because the bed-normal velocity component is not accounted for. However, the theory was found relevant for several environmental flow problems including density currents over mountains and valleys, seepage flow in hillslope hydrology, the development of antidunes, the formation of geological deposits from hyper-concentrated flows, and shallow-water flow modeling in hydraulics. In this work, Dressler’s theory is developed in an alternative way by a systematic iteration of the stream and potential functions in terrain-fitted coordinates. The first iteration was found to be the former Dressler’s theory, whereas a second iteration of the governing equations results in velocity components generalizing Dressler’s theory to wave-like motion. Dressler’s first-order theory produces a transcritical flow solution over topography only if the total head is fixed by a minimum value of the specific energy at the transition point. However, the theory deviates from measurements under subcritical flow conditions, given that the bed-normal velocity component is significant. A second iteration to the velocity field was used to produce a second-order differential equation that resembles the cnoidal-wave theory. It accurately produces flow over an obstacle including the critical point and the minimum specific energy as part of the numerical solution. The new cnoidal-wave model compares well with the theory of a Cosserat surface for directed fluid sheets, whereas the Saint-Venant theory appears to be poor.

Thin-sheet flow between coalescing bubbles

When two spherical bubbles touch, a hole is formed in the fluid sheet between them, and capillary pressure acting on its tightly curved edge drives an outward radial flow which widens the hole joining the bubbles. Recent images of the early stages of this process (Paulsen et al., Nat. Commun., vol. 5, 2014) show that the radius of the hole $r_{\!E}$ at time $t$ grows proportional to $t^{1/2}$, and that the rate is dependent on the fluid viscosity. Here, we explain this behaviour in terms of similarity solutions to a third-order system of radial extensional-flow equations for the thickness and velocity of the sheet of fluid between the bubbles, and determine the growth rate as a function of the Ohnesorge number $\mathit{Oh}$. The initially quadratic sheet profile allows the ratio of viscous and inertial effects to be independent of time. We show that the sheet is slender for $r_{\!E}\ll a$ if $\mathit{Oh}\gg 1$, where $a$ is the bubble radius, but only slender for $r_{\!E}\ll \mathit{Oh}^{2}a$ if $\mathit{Oh}\ll 1$ due to a compressional boundary layer of length $L\propto \mathit{Oh}\,r_{\!E}$, after which there is a change in the structure but not the speed of the retracting sheet. For $\mathit{Oh}\ll 1$, the detailed analysis justifies a simple momentum-balance argument, which gives the analytic prediction $r_{\!E}\sim (32a{\it\gamma}/3{\it\rho})^{1/4}t^{1/2}$, where ${\it\gamma}$ is the surface tension and ${\it\rho}$ is the density.

Free-surface dynamics of small pores

When the size of a pore nucleated in a fluid sheet is sufficiently small, the pore will contract and close driven by its large radial curvature. The dynamics of small contracting pores are relevant to a number of natural systems and practical applications, from fission pores in cell membranes to the fabrication of nanopores in sensors for DNA sequencing. Here, we report high-fidelity numerical simulations that provide detailed insight into the mechanisms of pore contraction and collapse in fluid sheets of low viscosity. Results uncover a scaling law that predicts the radius of a closing pore as a function of the time to collapse without free parameters. Simulations also show that contracting pores do not always proceed to collapse. Instead, some contracting pores reverse the direction of motion and expand.

Weakly nonlinear instability of planar viscoelastic sheets

A second-order weakly nonlinear analysis has been made of the temporal instability for the linear sinuous mode of two-dimensional planar viscoelastic liquid sheets moving in an inviscid gas. The convected Jeffreys models including the corotational Jeffreys model, Oldroyd A model, and the Oldroyd B model are considered as the rheology model of the viscoelastic fluid of the sheet. The solution for the second-order gas-to-liquid interface displacement has been derived, and the temporal evolution leading to the breakup has been shown. The second-order interface displacement of the linear sinuous mode is varicose, which causes the sheet to fragment into ligaments. First-order constitutive relations of the three rheology models become identical after linearization, so the linear instability results are also the same. For the second-order weakly nonlinear instability, the second-order constitutive relation varies among the corotational Jeffreys model, Oldroyd A model, and the Oldroyd B model, but although they have different disturbance pressures, their disturbance velocities and interface displacements are the same, and therefore, the sheets of the corotational Jeffreys fluid, Oldroyd A fluid, and the Oldroyd B fluid have the same instability behavior characterized by the wave profile and breakup time. The reason for the identical instability behavior is that the effect of different codeformations of the corotational frame, covariant frame, and the contravariant frame is counteracted by the corresponding change in the second-order disturbance pressure, leaving no influence on the second-order velocity. At wavenumbers with maximum instabilities, an increase in the elasticity, or a reduction of the deformation retardation time, leads to a larger linear temporal growth rate, greater second-order disturbance amplitude, and shorter breakup time, thereby enhancing instability. The mechanism of linear instability has been examined using an energy approach, which shows that the main cause of instability is the aerodynamic force.

EFFECT OF ELASTICITY OF BOGER FLUIDS ON THE ATOMIZATION BEHAVIOR OF AN IMPINGING JET INJECTOR

Gelled propellants are of interest for rocket and ramjet propulsion systems because of their safety and performance benefits. During spray tests with doublets, such as on impinging jet injectors, it was observed that the breakup of the fluid sheet of several gels did not lead to the formation of droplets; instead, threadlike structures were formed. Such structures may cause a reduction in combustion efficiency and the accumulation of unburned material in the combustion chamber. The goal of the work presented here is to determine which physical properties influence thread formation. Spray tests were conducted with Boger fluids under atmospheric conditions concerning pressure and temperature with a like-on-like impinging jet injector. The range of injection velocities from 5 to over 70 m/s was studied. Images of the sprays were obtained with a shadowgraph technique. In order to obtain quantitative results from shadowgraph images, a code was implemented to determine a characteristic dimension of the threads. A clear correlation between the relaxation time, which is an indicator of the intensity of elastic effects, and the dimension of the obtained threads was observed. These results suggest that thread formation is caused by nonlinear elastic effects, while shear viscosity and surface tension do not seem to influence the formation of these structures.

Influence of viscosity on the impingement of laminar liquid jets

The impingement of low-viscosity liquid jets has been studied extensively for over a century due to their fundamental scientific interest and their practical importance in spray and atomization technologies. However, the role of the fluid viscosity in the impingement dynamics is largely unknown despite the fact that viscous liquids are common in spray and atomization processes ranging from spray drying in the food industry to the atomization of gelled propellants in rocket engines. Here, we report direct numerical simulations that enable a detailed analysis of the influence of viscosity on the impingement dynamics. The simulations solve the complete Navier–Stokes system governing the free-surface dynamics, and so fully account for the interplay of inertia, viscous and capillary forces. Results show that the liquid viscosity profoundly affects the impingement dynamics. The collision of viscous liquid jets generates a fluid sheet that thins at a rate r−1 with the distance r from the impact point at intermediate viscosities, in contrast to the inertial case in which the sheet thins at a faster rate r−2. As the viscosity increases, the fluid sheets become thicker and more uniform, and contrary to the inertial case, the velocity of the sheets is lower than the velocity of the jets. Results further reveal that due to viscous stresses the impact pressure generated by the collision of viscous liquid jets scales as Re−1, where Re is the jet Reynolds number.

Codimensional surface tension flow on simplicial complexes

Many visually interesting natural phenomena are characterized by thin liquid sheets, long filaments, and droplets. We present a new Lagrangian-based numerical method to simulate these codimensional surface tension driven phenomena using non-manifold simplicial complexes. Tetrahedra, triangles, segments, and points are used to model the fluid volume, thin films, filaments, and droplets, respectively. We present a new method for enforcing fluid incompressibility on simplicial complexes along with a physically-guided meshing algorithm to provide temporally consistent information for interparticle forces. Our method naturally allows for transitions between codimensions, either from tetrahedra to triangles to segments to points or vice versa, regardless of the simulation resolution. We demonstrate the efficacy of this method by simulating various natural phenomena that are characterized by thin fluid sheets, filaments, and surface tension effects.

A Free-Surface Model of a Jet Impinging On a Sprinkler Head

Understanding the atomization of fire sprinkler sprays fills a critical gap in the modeling of fire suppression systems. Previous research by the authors has shown an instability model coupled with a stochastic transport model can paint most of the sprinkler spray picture, but requires input in the form of thickness and velocity of unstable fluid sheets. The model outlined describes a water jet impinging on a perforated deflector plate as a velocity potential. The free surface separating the jet from the surrounding air takes the form of a vortex sheet with the air assumed to be at rest. Through the use of the Green’s function, the fluid velocity potential can be posed as a boundary value problem. Any solution obtained is an exact solution to the inviscid flow equations and the interior flow a solution to the Navier-Stokes equations. The resulting model allows for the determination of the complete flow field over a sprinkler head of arbitrary geometry and input conditions. Knowledge of this flow field provides insight into the impact of sprinkler head geometry and fluid velocity as well as providing the above mentioned inputs for a complete model of fire sprinkler sprays.

Mathematical Modelling of Radiative Hydromagnetic Thermosolutal Nanofluid Convection Slip Flow in Saturated Porous Media

High temperature thermal processing of nanomaterials is an active area of research. Many techniques are being investigated to manipulate properties of nanomaterials for medical implementation. In this paper, we investigate thermal radiation processing of a nanomaterial fluid sheet extruded in porous media. A mathematical model is developed using a Darcy drag force model. Instead of using linear radiative heat flux, the nonlinear radiative heat flux in the Rosseland approximation is taken into account which makes the present study more meaningful and practically useful. Velocity slip and thermal and mass convective boundary conditions are incorporated in the model. The Buongiornio nanofluid model is adopted wherein Brownian motion and thermophoresis effects are present. The boundary layer conservation equations are transformed using appropriate similarity variables and the resulting nonlinear boundary value problem is solved usingMaple 14which uses the Runge-Kutta-Fehlberg fourth fifth order numerical method. Solutions are validated with previous nonmagnetic and nonradiative computations from the literature, demonstrating excellent agreement. The influence of Darcy number, magnetic field parameter, hydrodynamic slip parameter, convection-conduction parameter, convection-diffusion parameter, and conduction-radiation parameter on the dimensionless velocity, temperature, and nanoparticle concentration fields is examined in detail. Interesting patterns of relevance are observed to improve manufacturing of nanofluids.

Analytical studies of sloshing in half-cylindrical tanks

The technically important problem of two-dimensional inviscid standing-wave motion of a fluid in a semi-cylindrical vessel is examined using a variety of the Boussinesq wave equation as well as the directed fluid sheet technique. The associated eigenvalue problems are dealt with by employing perturbative procedures and series expansions. The predictions were compared with the outcome of laboratory tank experiments and also with ‘exact' results. The Boussinesq equation perturbative scheme based on an approximate representation of the vessel geometry not only gave the best results, but was also the most straightforward to apply. Furthermore, it yielded practically useful explicit formulae for the frequencies at which sloshing may be initiated. A limitation is, however, that this formalism is only applicable to systems with comparatively small fill ratios.

Viscous interfaces as source for material creep: A continuum micromechanics approach

Abstract It is generally agreed upon that fluids may play a major role in the creep behavior of materials comprising heterogeneous microstructures and fluid-filled porosities at small length scales. In more detail, nanoconfined fluid-filled interfaces are typically considered to act as a lubricants, once electrically charged solid surfaces start to glide along fluid sheets, with the fluid being typically in a liquid crystal state, which refers to an “adsorbed”, “ice-like”, or “glassy” structure of fluid molecules. Here, we aim at translating this interface behavior into apparent creep laws at the continuum scale of materials consisting of one non-creeping solid matrix with embedded fluid-filled interfaces. To this end, we consider a linear relationship between (i) average interface dislocations and (ii) corresponding interface tractions, with an interface viscosity as the proportionality constant. Homogenization schemes for eigenstressed heterogeneous materials are used to upscale this interface behavior to the much larger observation scale of a matrix-inclusion composite comprising an isotropic and linear elastic solid matrix, as well as interacting parallel interfaces of circular shape, which are embedded in the aforementioned matrix. This results in exponentially decaying macroscopic viscoelastic phenomena, with both creep and relaxation times increasing with increasing interface size and viscosity, as well as with decreasing elastic stiffness of the solid matrix; while only the relaxation time decreases with increasing interface density. Accordingly, non-asymptotic creep of hydrated (quasi-) crystalline materials at higher load intensities may be readily explained through non-stationarity, i.e. spreading, of liquid crystal interfaces throughout solid elastic matrices.

Instability regimes in the primary breakup region of planar coflowing sheets

AbstractThis article investigates the appearance of instabilities in two planar coflowing fluid sheets with different densities and viscosities via experiments, numerical simulation and linear stability analysis. At low dynamic pressure ratios a convective instability is shown to appear for which the frequency of the waves in the primary atomization region is influenced by both liquid and gas velocities. For large dynamic pressure ratios an asymptotic regime is obtained in which frequency is solely controlled by gas velocity and the instability becomes absolute. The transition from convective to absolute is shown to be influenced by the velocity defect induced by the presence of the separator plate. We show that in this regime the splitter plate thickness can also affect the nature of the instability if it is larger than the gas vorticity thickness. Computational and experimental results are in agreement with the predictions of a spatio-temporal stability analysis.

Stability of a moving radial liquid sheet: Time-dependent equations

We study the stability of a radial liquid sheet produced by head-on impingement of two equal laminar liquid jets. Linear stability equations are derived from the inviscid flow equations for a radially expanding sheet that govern the time-dependent evolution of the two liquid interfaces. The analysis accounts for the varying liquid sheet thickness while the inertial effects due to the surrounding gas phase are ignored. The analysis results in stability equations for the sinuous and the varicose modes of sheet deformation that are decoupled at the lowest order of approximation. When the sheet is excited at a fixed frequency, a small sinuous displacement introduced at the point of impingement grows as it is convected downstream suggesting that the sheet is unstable at all Weber numbers (We ≡ ρlU2h/σ) in the absence of the gas phase. Here, ρl is the density of the liquid, U is the speed of the liquid jet, h is the local sheet thickness, and σ is the surface tension. The sinuous disturbance diverges at We = 2 which sets the size of the sheet, in agreement with the results of Taylor [“The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets,” Proc. R. Soc. London, Ser. A 253, 313 (1959)]. Asymptotic analysis of the sinuous mode for all frequencies shows that the disturbance amplitude diverges inversely with the distance from the edge of the sheet. The varicose waves, on the other hand, are neutrally stable at all frequencies and are convected at the speed of the liquid jet.

On self-similar thermal rupture of thin liquid sheets

We consider the dynamics of a symmetrically heated thin incompressible viscous fluid sheet. We take surface tension to be temperature dependent and consequently the streamwise momentum equation includes the effects of thermocapillarity, inertia, viscous stresses, and capillarity. Energy transport to the surrounding environment is also included. We use a long-wave analysis to derive a single nondimensional system which, with appropriate choices of Reynolds number, recovers two previously studied cases. In both cases, we find conditions under which sufficiently large-amplitude initial temperature profiles induce film rupture in finite time, notably without the inclusion of disjoining pressures from van der Waals effects. When the Reynolds number is large, the similarity solution is governed by a balance of inertia and capillarity near the rupture location, analogous to the isothermal case. When the Reynolds number is small, the thermocapillary transients induce the same similarity solution over intermediate times that is found for the drainage of lamellae in foams. For O(1) Reynolds numbers, the dynamics are governed initially by the large Reynolds number evolution, and then a transition over several orders of magnitude in the sheet thickness needs to take place before the small Reynolds number similarity solution is observed.

Effect of pressure-dependent viscosity on the exiting sheet thickness in the calendering of Newtonian fluids

An analysis of calendering for an incompressible Newtonian fluid flow, with pressure-dependent viscosity is studied theoretically under assumptions of isothermal conditions. We predict the influence of pressure-dependent viscosity on the exiting sheet thickness of the sheet of fluid from the gap. The dimensionless mass and momentum balance equations, which are based on lubrication theory, were solved for the velocity and pressure fields by using perturbation techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. When the above variables were obtained, the dimensionless exiting sheet thickness was determined by considering the influence of the pressure variations in the calendering process. Moreover, quantities of engineering interest are also calculated, which include the cylinder-separating force and power required to drive both cylinders in terms of the geometrical and kinematical parameters of the system. The results show that the inclusion of pressure-dependent viscosity effect increases the leave-off distance and consequently the dimensionless exiting sheet thickness in comparison with the case of pressure-independent viscosity.