Fluid Sheet Research Articles

Suction-induced magnetohydrodynamics of a viscoelastic fluid over a stretching surface within a porous medium

The magnetohydrodynamic (MHD) flow over a stretching sheet of a viscoelastic fluid immersed in a porous medium is studied analytically. The flow is induced by suction and also by an infinite elastic sheet which is stretched along its own plane. The stretching of the sheet is assumed to be proportional to the distance from the slit. The governing equations are reduced to a non-linear ordinary differential equation by means of similarity transformation. The resulting non-linear equation is solved analytically and the streamlines of the flow field are obtained. The effect of various quantities such as suction parameter, Chandrasekhar number and porous parameter on the velocity fields are studied. Results show that the flow field can be divided into a near-field region (boundary-layer region) and a far-field region (free stream region). Suction on the surface plays an important role in the flow development in the near-field whereas the far-field is influenced mainly by stretching. The electromagnetic effect plays exactly the same role as the porous medium, which reduces the horizontal flow velocity resulting from stretching. The flow pattern also exhibits a substantial change as the suction effect increases, and in such case the growth of the near-field region can extend far away from the stretching surface. These results have possible technological applications in liquid-based systems involving stretchable materials.

Preserving Fluid Sheets with Adaptively Sampled Anisotropic Particles

This paper presents a particle-based model for preserving fluid sheets of animated liquids with an adaptively sampled Fluid-Implicit-Particle (FLIP) method. In our method, we preserve fluid sheets by filling the breaking sheets with particle splitting in the thin regions, and by collapsing them in the deep water. To identify the critically thin parts, we compute the anisotropy of the particle neighborhoods, and use this information as a resampling criterion to reconstruct thin liquid surfaces. Unlike previous approaches, our method does not suffer from diffusive surfaces or complex remeshing operations, and robustly handles topology changes with the use of a meshless representation. We extend the underlying FLIP model with an anisotropic position correction to improve the particle spacing, and adaptive sampling to efficiently perform simulations of larger volumes. Due to the Lagrangian nature of our method, it can be easily implemented and efficiently parallelized. The results show that our method can produce visually complex liquid animations with thin structures and vivid motions.

On axisymmetric flow of Sisko fluid over a radially stretching sheet

This study presents an analysis of the axisymmetric flow of a non-Newtonian fluid over a radially stretching sheet. The momentum equations for two-dimensional flow are first modeled for Sisko fluid constitutive model, which is a combination of power-law and Newtonian fluids. The general momentum equations are then simplified by invoking the boundary layer analysis. Then a non-linear ordinary differential equation governing the axisymmetric boundary layer flow of Sisko fluid over a radially stretching sheet is obtained by introducing new suitable similarity transformations. The resulting non-linear ordinary differential equation is solved analytically via the homotopy analysis method (HAM). Closed form exact solution is then also obtained for the cases n=0 and 1. Analytical results are presented for the velocity profiles for some values of governing parameters such as power-law index, material parameter and stretching parameter. In addition, the local skin friction coefficient for several sets of the values of physical parameter is tabulated and analyzed. It is shown that the results presented in this study for the axisymmetric flow over a radially non-linear stretching sheet of Sisko fluid are quite general so that the corresponding results for the Newtonian fluid and the power-law fluid can be obtained as two limiting cases.

Viscoplastic sheets and threads

A theory is presented for the dynamics of slender sheets of viscoplastic fluid. First a model suitable for relatively small curvature is presented and applied to the fall of a liquid bridge supported at its two ends. Second, order-one curvatures are considered, along with the sagging of a beam with a free end that is either emplaced or extruded horizontally. We then present a theory combining both limits of curvature, and consider the buckling of a nearly vertical column. Analogous models for a circular viscoplastic thread are also discussed.

Dynamics of free subduction from 3‐D boundary element modeling

In order better to understand the physical mechanisms underlying free subduction, we perform three‐dimensional boundary‐element numerical simulations of a dense fluid sheet with thickness h and viscosity η2 sinking in an ‘ambient mantle’ with viscosity η1. The mantle layer is bounded above by a traction‐free surface, and is either (1) infinitely deep or (2) underlain by a rigid boundary at a finite depth H + d, similar to the typical geometry used in laboratory experiments. Instantaneous solutions in configuration (1) show that the sheet's dimensionless ‘stiffness’ S determines whether the slab's sinking speed is controlled by the viscosity of the ambient mantle (S < 1) or the viscosity of the sheet itself (S > 10). Time‐dependent solutions with tracers in configuration (2) demonstrate a partial return flow around the leading edge of a retreating slab and return flow around its sides. The extra ‘edge drag’ exerted by the flow around the sides causes transverse deformation of the slab, and makes the sinking speed of a 3‐D slab up to 40% less than that of a 2‐D slab. A systematic investigation of the slab's interaction with the bottom boundary as a function of η2/η1 and H/h delineates a rich regime diagram of different subduction modes (trench retreating, slab folding, trench advancing) and reveals a new ‘advancing‐folding’ mode in which slab folding is preceded by advancing trench motion. The solutions demonstrate that mode selection is controlled by the dip of the leading edge of the slab at the time when it first encounters the bottom boundary.

First-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrows

An average modeling methodology under the lubrication approach is used to formulate a set of three coupled nonlinear partial differential equations based on the Nusselt scales. This system, known as the energy-integral method in literature, simplifies the Navier-Stokes equation at the first order and analyzes the dynamics of a thin sheet of fluid flowing over a topography with sinusoidally varying longitudinal furrows. Limiting cases of the linear stability results are mathematically discussed and the complete linear system is numerically handled by means of finite differences to approximate the eigenfunctions and their derivatives in a periodic domain. In a geometry which resembles a vertical shift of a topography, with the amplitude being equal to the shift length, it is found that such a geometry stabilizes the flow compared to its counterpart with no shift, such that the wave characteristics get affected. To confirm the stability results, a numerical investigation is performed.

Thermally induced van der Waals rupture of thin viscous fluid sheets

We consider the dynamics of a thin symmetric fluid sheet subject to an initial temperature profile, where inertia, viscous stresses, disjoining pressures, capillarity, and thermocapillarity are important. We apply a long-wave analysis in the limit where deviations from the mean sheet velocity are small, but thermocapillary stresses and heat transfer from the sheet to the environment are significant and find a coupled system of partial differential equations that describe the sheet thickness, the mean sheet velocity, and the mean sheet temperature. From a linear stability analysis, we find that a stable thermal mode couples the velocity to the interfacial dynamics. This coupling can be utilized to delay the onset of rupture or to promote an earlier rupture event. In particular, rupture can be induced thermally even in cases when the heat transfer to the surrounding environment is significant, provided that the initial phase shift between the initial velocity and temperature disturbances is close to ϕ = π/2. These effects suggest a strategy that uses phase modulation in the initial temperature perturbation related to the initial velocity perturbation that assigns priority of the rupture events at particular sites over several spatial periods.

Asymptotic behavior of a retracting two-dimensional fluid sheet

Two-dimensional (2D) capillary retraction of a viscous liquid film is studied using numerical and analytical approaches for both diphasic and free surface flows. Full 2D Navier-Stokes equations are integrated numerically for the diphasic case, while one-dimensional (1D) free surface model equations are used for free surface flows. No pinch-off is observed in the film in any of these cases. By means of an asymptotic matching method on the 1D model, we derive an analytical expansion of the film profile for large times. Our analysis shows that three regions with different timescales can be identified during retraction: the rim, the film, and an intermediate domain connecting these two regions. The numerical simulations performed on both models show good agreement with the analytical results. Finally, we report the appearance of an instability in the diphasic retracting film for small Ohnesorge number. We understand this as a Kelvin-Helmholtz instability arising due to the formation of a shear layer in the neck region during the retraction.

The mechanism of a splash on a dry solid surface

AbstractFrom rain storms to ink jet printing, it is ubiquitous that a high-speed liquid droplet creates a splash when it impacts on a dry solid surface. Yet, the fluid mechanical mechanism causing this splash is unknown. About fifty years ago it was discovered that corona splashes are preceded by the ejection of a thin fluid sheet very near the vicinity of the contact point. Here we present a first-principles description of the mechanism for sheet formation, the initial stages of which occur before the droplet physically contacts the surface. We predict precisely when sheet formation occurs on a smooth surface as a function of experimental parameters, along with conditions on the roughness and other parameters for the validity of the predictions. The process of sheet formation provides a semi-quantitative framework for studying the subsequent events and the influence of liquid viscosity, gas pressure and surface roughness. The conclusions derived from this framework are in quantitative agreement with previous measurements of the splash threshold as a function of impact parameters (the size and velocity of the droplet) and in qualitative agreement with the dependence on physical properties (liquid viscosity, surface tension, ambient gas pressure, etc.) Our analysis predicts an as yet unobserved series of events within micrometres of the impact point and microseconds of the splash.

Hydroelastic waves on fluid sheets

AbstractNonlinear travelling waves on a two-dimensional inviscid fluid sheet are investigated when the sheet is bounded above and below by two thin elastic plates. Symmetric and antisymmetric solution branches are identified, together with a pair of bifurcation branches. It is shown that far along the branches the solutions approach limiting configurations that correspond to static solutions valid in the absence of fluid forcing.

At least three scales of convection in a mantle with strongly temperature-dependent viscosity

We studied the convective patterns developing at high Rayleigh numbers and intermediate viscosity ratios in a fluid with a strongly temperature-dependent viscosity. Within this sluggish lid regime, three different scales of convection develop. The largest convective scale is cellular, with cold downwelling sheets of viscous fluid encasing hotter, less viscous, parts of the tank. Within each of those cells develop several (typically 3–7) hot 3D upwelling plumes. Upon impinging under the cold thermal boundary layer, each plume in turn generates locally a small ring of cold material which does not reach the bottom of the tank. Applying those results to the Earth’s mantle, we suggest that the large-scale features of mantle convection and the co-existence of several scales of convection, i.e. slabs and plumes, are produced by thermal convection in a mantle with high Rayleigh number and a strongly temperature-dependent viscosity material. This generates a convective pattern in two large-scale cells : the Pacific and the Indo-Atlantic “boxes”. Our experiments further suggest that what has been named the two hot superplumes, i.e. the two seismically slow regions encased within the subduction rings, are in fact each constituted of several hot instabilities. Moreover, plumes impacts under the lithosphere should be surrounded by cold rings of small extent. The asthenosphere appears therefore as the graveyard of both small-scale convection and hot plumes generated in the system.

Effect of temperature-dependent consistency index on the exiting sheet thickness in the calendering of power-law fluids

In this work we treat theoretically the calendering process of inelastic (power-law fluid) sheets of finite initial thickness, taking into account that the consistency index of the fluid is a well-defined function of the temperature. We predict the influence of the temperature-dependent consistency index on the exiting sheet thickness in the calendering process. The mass, momentum and energy balance equations, based on the lubrication theory, were nondimensionalized and solved for the velocity, pressure and temperature fields by using perturbation and numerical techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. When the above variables were obtained, the exiting sheet thickness in the calendering process was determined, considering the influence of the temperature variations in the process. The mentioned governing equations contain four dimensionless parameters: the Graetz number, Gz, a geometrical aspect ratio, β, the power law index of the fluid, n, and a parameter that takes into account the effect of the variable consistency index as a function of the temperature and ϵ, defined as the ratio of the Nahme–Griffith number, Na, to the Graetz number. Using the limit of ϵ ≪ 1, the dimensionless exiting sheet thickness of the calendering process have been obtained as a function of the involved dimensionless parameters. The numerical results show that the inclusion of temperature-dependent consistency index effect modifies the dimensionless exiting sheet thickness in about 6.91%.

The energetics of the breakup of a sheet and of a rivulet on a vertical substrate in the presence of a uniform surface shear stress

We use the lubrication approximation to obtain a complete description of the energetics of the breakup both of a fluid sheet of uniform thickness into a periodic array of infinitely many identical thin rivulets and of a single thin rivulet into one or more identical sub-rivulets on a vertical substrate in the presence of a prescribed uniform longitudinal shear stress at the free surface of the fluid by comparing the total energies of the different states. For both problems the situation when the volume flux is positive is relatively straightforward (and, in particular, qualitatively the same as that in the case of no prescribed shear stress), but when the volume flux is negative it is more complicated. However, whatever the value of the prescribed shear stress, there is always a critical thickness below which it is energetically favourable for a sheet to break up into rivulets and a critical semi-width below which it is energetically favourable for a rivulet to remain as a single rivulet, and a critical thickness above which it is energetically favourable for a sheet to remain as a sheet and a critical semi-width above which it is energetically favourable for a rivulet to break up into sub-rivulets.

Stability behavior of three non-Newtonian magnetic fluids in porous media

The present study deals with stability properties of two-dimensional non-Newtonian fluid layers moving in porous media, under the influence of uniform magnetic field. The system is composed of a middle fluid embedded between two semi-infinite fluids. The limiting case of the stability of one interface between two viscoelastic fluids is discussed. The presented analysis takes into account the modified Darcy's law. The principle aim of this work is to investigate the influence of fluid elasticity and the porosity effect on the growth rate in the presence of a magnetic field. The stability analysis is performed theoretically and stability diagrams are obtained. The stability analysis shows that non-Newtonian (viscoelastic) fluid layers have a higher growth rate than Newtonian fluid layers, indicating that non-Newtonian fluid sheets are more unstable than Newtonian fluid sheets. It is observed that in fluid sheets the fluid elasticity tends to damp the stability, whereas the fluid viscosity results in an enhancement of stability. The phenomenon of the dual role is found to increase the porous parameter as well as the magnetic permeability ratio. It has been found that the increase of the viscosity coefficient damps the growth rate, while increasing the Reynolds number has the opposite effect.

Effects of General Rotation and a Horizontal Magnetic Field on the Rayleigh–Taylor Instability of Three-Fluid Layer in Porous Media

The Rayleigh–Taylor instability of superposed conducting fluids through porous media under the influence of a uniform magnetic field and general rotation was investigated. The system is composed of a middle fluid sheet of finite thickness embedded between two semi-infinite fluids. The fluids are assumed incompressible and perfectly conducting. A dispersion equation is derived by solving the linearized equations of motion using the boundary conditions. The behavior of the growth rate with respect to the horizontal magnetic field, the vertical and horizontal components of rotation, the viscosity, and the permeability of the porous medium are examined numerically. The results show that the system is divided into two modes (two growth rates). The horizontal magnetic field, and the vertical and horizontal components of rotation, viscosity and porosity have a stabilizing effect on the growth rate of an unstable configuration.

Experimental study of atomization patterns produced by the oblique collision of two viscoelastic liquid jets

Experimental observations and analysis are presented for the formation and atomization of the fluid sheet created by obliquely colliding jets of viscoelastic fluids. Solutions of mono-disperse polystyrene (PS) in diethylphthalate and of poly-disperse polyethylene oxide (PEO) in glycerol/water mixtures were used to investigate the effects of fluid elasticity on the break-up patterns generated by the impact of two jets ejected from nozzles with an internal diameter of 0.85 mm. Various regimes of behaviour were identified which depend on the jet speed. The structures observed for these elastic fluids differ somewhat from those previously reported for Newtonian viscous fluids, and also show different behaviours depending on the degree of viscoelasticity. This study focuses on the periodic atomization, the so-called fishbone pattern, which occurs when the impinging jets form a liquid sheet which breaks up into a regular succession of ligaments and droplets. High-speed flash photography reveals that low concentrations of polymers significantly affect the evolution of the sheet and its fragmentation, the shapes of the ligaments, and the final drop sizes. The maximum fishbone angle is defined and shown to be a useful tool to describe the variation of the atomization pattern with polymer concentration. For the PS solutions the variation of maximum fishbone angle with reduced polymer concentration ( c/c * ) follows a single master curve, but although the same is true for PEO with high molecular weights, the curves remain separate for low molecular weights. Observation of the fishbone patterns formed by the oblique impact of jets may provide a useful tool to observe and characterize inter-chain interaction in high speed extensional flow of polymer solutions.

Propagation of large-amplitude waves on dielectric liquid sheets in a tangential electric field: Exact solutions in three-dimensional geometry

Nonlinear waves on sheets of dielectric liquid in the presence of an external tangential electric field are studied theoretically. It is shown that waves of arbitrary shape in three-dimensional geometry can propagate along (or against) the electric field direction without distortion, i.e., the equations of motion admit a wide class of exact traveling wave solutions. This unusual situation occurs for nonconducting ideal liquids with high dielectric constants in the case of a sufficiently strong field strength. Governing equations for evolution of plane symmetric waves on fluid sheets are derived using conformal variables. A dispersion relation for the evolution of small perturbations of the traveling wave solutions is obtained. It follows from this relation that, regardless of the wave shape, the amplitudes of small-scale perturbations do not increase with time and, hence, the traveling waves are stable. We also study the interaction of counterpropagating symmetric waves with small but finite amplitudes. The corresponding solution of the equations of motion describes the nonlinear superposition of the oppositely directed waves. The results obtained are applicable for the description of long waves on fluid sheets in a horizontal magnetic field.

Thin film formation during splashing of viscous liquids

After impact onto a smooth dry surface, a drop of viscous liquid initially spreads in the form of a thick lamella. If the drop splashes, it first emits a thin fluid sheet that can ultimately break up into droplets causing the splash. Ambient gas is crucial for creating this thin sheet. The time for sheet ejection, t{ejt}, depends on impact velocity, liquid viscosity, gas pressure, and molecular weight. A central air bubble is trapped below the drop at pressures even below that necessary for this sheet formation. In addition, air bubbles are entrained underneath the spreading lamella when the ejected sheet is present. Air entrainment ceases at a lamella velocity that is independent of drop impact velocity as well as ambient gas pressure.

Newtonian pizza: spinning a viscous sheet

We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radiusR(t) tends to infinity in finite time. As timetapproaches the critical timet*, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thicknesshin the central region is described by a similarity solution of the second kind, withh∝ (t* −t)αwhere the exponent α satisfies a nonlinear eigenvalue problem. Finally, for non-zero surface tension, we find that the exponent increases rapidly to infinity at a critical value of the rotational Bond numberB= 1/4. ForB> 1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than to stretch, with the limiting behaviour described by a similarity solution of the first kind.

Events before droplet splashing on a solid surface

A high-velocity (≈1 ms−1) impact between a liquid droplet (≈1 mm) and a solid surface produces a splash. Classical observations traced the origin of this splash to a thin sheet of fluid ejected near the impact point, though the fluid mechanical mechanism leading to the sheet is not known. Mechanisms of sheet formation have heretofore relied on initial contact of the droplet and the surface. In this paper, we theoretically and numerically study the events within the time scale of about 1 μs over which the coupled dynamics between the gas and the droplet becomes important. The droplet initially tries to contact the substrate by either draining gas out of a thin layer or compressing it, with the local behaviour described by a self-similar solution of the governing equations. This similarity solution is not asymptotically consistent: forces that were initially negligible become relevant and dramatically change the behaviour. Depending on the radius and impact velocity of the droplet, we show that the solution is overtaken by initially subdominant physical effects such as the surface tension of the liquid–gas interface or viscous forces in the liquid. At low impact velocities surface tension stops the droplet from impacting the surface, whereas at higher velocities viscous forces become important before surface tension. The ultimate dynamics of the interface once droplet viscosity cannot be neglected is not yet known.