Capillary Floating and the Billiard Ball Problem

In a study of capillary floating, Finn (J Math Fluid Mech 11:443?C458, 2009) described a procedure for determining cross-sections of non-circular, infinite convex cylinders that float horizontally on a liquid surface in every orientation with contact angle ??/2. Finn??s procedure yielded incomplete results for other contact angles; he raised the question as to whether an analogous construction would be feasible in that case. In the note, Finn (J Math Fluid Mech 11:464?C465, 2009) pointed out a connection with an independent problem on billiard caustics citing the unpublished work (Gutkin in Proceedings of the Workshop on Dynamics and Related Questions, PennState University, 1993) of the present author. Here we present a solution of the billiard problem in full detail, thus settling Finn??s question in a surprising way. In particular, we show that such floating cylinders exist if and only if the contact angle lies in a certain, explicitly described countably dense set. Moreover, for each element ?? in this set we exhibit a family of convex, non-circular cylinders that float in every orientation with contact angle ??. Our discussion contains other material of independent interest for the billiard ball problem.     

Steady-state solutions to Bentsen's equation

Based on experimentally observed phenomena and the physical requirement of a unique value of saturation at any location within a porous medium, a restrictive condition for a valid solution to Bentsen's equation is derived: ?² f/?S ²≤0. The steady-state solution to Bentsen's equation is shown to be identical to the Buckley-Leverett solution to the displacement equation, and the steady-state solution for the fractional flow is shown to be independent of the capillary number. It is proved that under steady-state conditions, the capillary term of the fractional flow equation in the frontal region does not depend on the capillary number. Therefore, the unrealistic triple-valued saturation profile of the original Buckley-Leverett solution resulted because the capillary term was in-appropriately neglected. The break-through recovery efficiency,Τ _bt , is shown to be a function of the capillary number. As the capillary number decreases, the break-through recovery efficiency increases and the maximum value ofΤ _bt can be obtained asN _c → 0. The Buckley-Leverett solution is the limiting solution asN _c → 0.     

Experimental investigation on flow modes of electrospinning

Electrospinning experiments are performed byusing a set of experimental apparatus,a stroboscopic systemis adopted for capturing instantaneous images of the conejet configuration.The cone and the jet of aqueous solutionsof polyethylene oxide(PEO) are formed from an orifice of acapillary tube under the electric field.The viscoelastic constitutive relationship of the PEO solution is measured anddiscussed.The phenomena owing to the jet instability aredescribed,five flow modes and corresponding structures areobtained with variations of the fluid flow rate Q,the electricpotential U and the distance h from the orifice of the capillary tube to the collector.The flow modes of the cone-jetconfiguration involves the steady bending mode,the rotating bending mode,the swinging rotating mode,the blurringbending mode and the branching mode.Regimes in the Q-Uplane of the flow modes are also obtained.These results mayprovide the fundamentals to predict the operating conditionsexpected in practical applications.     

The Effect of the Walls of an Arbitrary Tank in the Problem of Separation–Free Impact on a Floating Body

An algorithm for constructing an asymptotic power series for large depths is proposed. It allows one to use the well–known solution of the problem of impact on a rigid body floating on the surface of a fluid half–space to obtain an approximate solution of the impact problem for the same body floating on the surface of a fluid in a bounded basin. The case where the domain occupied by the fluid has two perpendicular planes of symmetry is considered. Asymptotic expressions are given for the velocity potential on the wetted part of the body surface and for the added mass. Examples of solutions are considered.     

New exact solution of Euler’s equations (rigid body dynamics) in the case of rotation over the fixed point

A new exact solution of Euler’s equations (rigid body dynamics) is presented here. All the components of angular velocity of rigid body for such a solution differ from both the cases of symmetric rigid rotor (which has two equal moments of inertia: Lagrange’s or Kovalevskaya’s case), and from the Euler’s case when all the applied torques are zero, or from other well-known particular cases. The key features are the next: the center of mass of rigid body is assumed to be located at meridional plane along the main principal axis of inertia of rigid body, besides, the principal moments of inertia are assumed to satisfy to a simple algebraic equality. Also, there is a restriction at choosing of initial conditions. Such a solution is also proved to satisfy to Euler–Poinsot equations, including invariants of motion and additional Euler’s invariant (square of the vector of angular momentum is a constant). So, such a solution is a generalization of Euler’s case.     

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kine- matics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their incre- ments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motion- deformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.     

INITIAL PRESSURE DISTRIBUTION DUE TO JET IMPACT ON A RIGID BODY

The analysis in this paper shows that, after an impulse due to a two-dimensional jet having velocityU and density ρ hitting a rigid body, the initial pressure distribution over the wall has the constant valueρU² relative to the ambient pressure. It also reveals that a discontinuity exists in the pressure at the intersection of the surface of the body and the surface of the jet. These results have been confirmed by a numerical solution based on a boundary element method.     

Residual mass and flow regimes for the immiscible liquid–liquid displacement in a plane channel

The motion of two immiscible liquids in a plane channel is analyzed for the case in which the flow conditions and the interactions between the liquids and the solid surface maintain the displaced fluid attached to the wall. The Galerkin Finite Element Method is used to compute the velocity field and the configuration of the interface between the two fluids. We compare the residual mass fraction left on the wall with its two counterparts in capillary tubes, namely residual mass fraction and dimensionless layer thickness of the displaced fluid. The main result of this comparison was that although there is a qualitative similarity concerning the layer thickness between the two cases, the residual fraction of mass presented an important difference, showing that when the aspect ratio of the capillary passage is large there is an increase in the displacement efficiency. The thickness of the displaced liquid film attached to the channel walls is a function of the capillary number (Ca) and the viscosity ratio (N_μ). A map of streamlines in the Cartesian space (Ca, N_μ) with the different flow regimes of the problem is presented. We also showed that we can adapt the available analytical results obtained for gas-displacement in capillary tubes to the plane channel case, for low values of Ca.     

Falling film flow along steep two-dimensional topography: The effect of inertia

Steady film flow along a vertical wall with isolated step changes is studied numerically for Reynolds numbers Re ∼ O(10⁻³–10²) and capillary numbers Ca ∼ O(10⁻²–10¹). The lengthscale of free surface capillary features upstream of a step-in or step-out decreases uniformly with Re and switches from a −1/3 to a −1/2 power-law dependence on Ca. The height of the capillary features first grows with Re, but eventually diminishes when inertia forces overpower capillary forces. Simultaneously, the key dynamics move from upstream to downstream of the step, and switch from capillary arrest to inertial re-directioning of the falling liquid. The latter mechanism involves a low-pressure region originating from the edge of the step. At a step-out, a new free surface feature appears with increasing Re, which is caused by liquid overshoot in the horizontal direction and is restrained initially by capillary and subsequently by inertial forces. Simple scaling arguments are shown to predict many of the above characteristics.     

An Analytical Solution for Capillary Gravity Drainage with Dominant Viscous Forces

The development of the capillary fringe during gravity drainage has a significant influence on saturation and pressure distributions in porous formations (Sarkarfarshi et al. in Int J Greenh Gas Control 23:61–71, 2014). This paper introduces an analytical solution for gravity drainage in an axisymmetric geometry with significant capillary pressure. The drainage process results from the injection of a lighter and less viscous injectant into a porous medium saturated with a heavier and more viscous pore fluid. If the viscous force dominates the capillary and the buoyancy forces, then the flow regime is approximated by differential equations and the admissible solution comprises a front shock wave and a trailing simple wave. In contrast to existing analytical solutions for capillary gravity drainage problems (e.g., Nordbotten and Dahle in 47(2) 2011; Golding et al. in J Fluid Mech 678:248–270 2011), this solution targets the saturation distribution during injection at an earlier point in time. Another contribution of this analytical solution is the incorporation of a completely drained flow regime close to the injection well. The analytical solution demonstrates the strong dependency of the saturation distribution upon relative permeability functions, gas entry capillary pressure, and residual saturation. The analytical results are compared to results from a commercial reservoir engineering software package (\(\hbox {CMG } \hbox {STARS}^{\mathrm{TM}}\)).     

Singular integral equations in elastic dielectrics

Galerkin representations for the displacement vector, polarization vector and the potential field are obtained by elementary matrix inversions of the equations of equilibrium. Matrices of fundamental solutions of an infinite elastic dielectric continuum subjected to a concentrated body force, an electric force, and a charge density, are constructed. Theorems are proved on the discontinuity of double layer potentials and R, M, M operators of single layer potentials. By means of these theorems, the solution of the two basic boundary value problems has been reduced to the solution of a system of seven singular integral equations.     

The sudden starting of a floating body in deep water

This paper applies Lagrangian method to discuss the sudden starting of a floating body in deep water and the analytical solutions are obtained. It is known from the numerical results that the disturbing domain extends and the dynamic pressure also increases when the breadth of the floating body keeps constant and its depth increases.     

Steady viscoelastic film flow over 2D topography: I. The effect of viscoelastic properties under creeping flow

Two-dimensional, steady flow of a viscoelastic film over a periodic topography under the action of a body force is studied. The exponential Phan-Thien and Tanner (ePTT) constitutive model is used. The conservation equations are solved via the usual mixed finite element method combined with a quasi-elliptic grid generation scheme in order to capture the large deformations of the free surface. The constitutive equation is weighted using the SUPG method and solved via the polymeric stress splitting EVSS-G technique. First, the code is validated by verifying that in isolated topographies the periodicity conditions result in fully developed viscoelastic film flow at the inflow/outflow boundaries and that its predictions for Newtonian fluids over 2D topography under creeping flow conditions coincide with those of previous works. Since the lubrication approximation is not invoked here, the topographical features can have wall segments that form any angle with the main flow, but only slight smoothing of the convex corners assists in reducing the stress singularity there. Thus, steady-state solutions are computed accurately up to high Deborah numbers, resulting in large deformations of the free surface. The magnitude of the capillary ridge in the film before the entrance to a step down of the substrate and of the capillary depression before a step up is increased as De increases up to ～0.7 due to increased fluid elasticity. Above this value they decrease, because increasing De increases also the shear and elongational thinning, which eventually affect them more. Increasing the ratio of solvent to polymer viscosities, β, the elongational parameter, ? and the molecular slip parameter, ξ, monotonically increases their magnitudes and especially that of the capillary ridge, but the mechanisms leading to these changes are different as explained in the text.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Multi-scale analysis of simulated capillary instability

This paper presents the 3-D multi-scale analysis of a cylindrical liquid ligament subjected to a capillary instability. This analysis aims to investigate the evolution of the ligament interface paying a specific attention to the physical mechanisms involved at small scales. The capillary instability behavior is obtained from direct numerical simulations. Calculations are performed for several wavenumbers of the initial sinusoidal perturbation. During the capillary instability, the scale space is divided in two regions: the small-scale region where a thinning mechanism is identified and the large-scale region where a thickening mechanism is observed. Although the characteristic scale d_max of the large-scale region displays a dynamics that agrees with the Rayleigh linear theory, this agreement is lost for the characteristic scale d₁ of the small scale region showing that the non-linear effects mainly concentrate on the small scales. The dynamics of the characteristic scale d₁ follows three successive regimes. The development of a simple model allows identifying the physical mechanisms related to these three regimes as well as their dependences with the wavenumber of the perturbation. Among other results it is found that the capillary contraction regime that develops when the breakup is approached is always preceded by an elongation mechanism whose effect is to increase the specific-surface-area of the ligament.     

A computational study of the influence of viscoelasticity on the interfacial dynamics of dip coating flow

The effect of viscoelasticity on the interfacial dynamics of air displacing a viscoelastic fluid under the presence of gravity, i.e., the dip coating flows is examined. A stabilized finite element method coupled with a pseudo-solid domain mapping technique is used to carry out the computations. The fluid is modeled by the Finitely Extensible Non-linear Elastic Chilton–Ralison (FENE-CR) constitutive equation. Simulations at various Ca and Bo are performed in order to determine the limiting condition for dip coating where the flow characteristics become independent of Bo. For all values of Ca and Bo studied, the flow is characterized by recirculation near the interface. To this end the film thickness scaled with the capillary length, as a function of Wi, at low Ca and high Bo collapses onto a single curve, and agrees with the analytical expression for the film thickness in the low Wi limit. As the value of Ca is increased, the corresponding value of Bo that is required to collapse the results onto a single curve, i.e., the dip coating flow limit, is correspondingly higher. For a fixed Ca and Wi, increasing Bo results in a decrease in the film thickness, an increase in the size of the recirculation region and an increase in the strain rates subsequently leading to an increase in the normal stresses. We show that the interfacial dynamics in the dip coating flow are qualitatively similar to those observed in the Hele-Shaw flow. Specifically, at low Wi, film thinning occurs and as the value of Wi is increased, the formation of normal elastic stress boundary layers in the capillary transition region is observed. This is accompanied by a sharp increase in the film thickness and a compression of the air–liquid interface in the capillary transition region.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio ( $M$ M ), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from $8.66\times 10^{-4}$ 8.66 × 10 - 4 to $4.33\times 10^{-3}$ 4.33 × 10 - 3 , which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for $Bo>1$ B o > 1 because the applied gravity has a stabilizing influence on the drainage process.     

Modeling of micropipette aspiration and optical tweezers stretching of erythrocytes with or without Malaria parasite

The erythrocytes play an important role in the human body. The healthy erythrocytes can undergo extremely large deformation while passing through small capillaries. Their infection by Malaria Plasmodium falcipurum (P.f.) will lead to capillary blockage and blood flow obstruction. Many experimental and computational methods have been applied to study the increase in stickiness and decrease in deformability of the Malaria (P.f.) infected erythrocytes. The novelty of this paper lies in the establishment of an multi-component model for investigating mechanical properties of Malaria (P.f.) infected erythrocytes, especially of their enclosed parasites. Finite element method was applied to simulate the erythrocytes’ deformation in micropipette aspiration and optical tweezers stretching using the computational software ABAQUS. The comparisons between simulations and experiments were able to quantitatively conclude the effects of stiffness and stickiness of the parasitophorous vacuole membrane on the cells’ deformation, which could not be obtained from experiments directly.     

EFFECTS OF A TRANSLATING LOAD ON A FLOATING PLATE—STRUCTURAL DRAG AND PLATE DEFORMATION

The elastic deformation of a structural plate floating on water caused by a translating three-dimensional load is investigated. The problem is akin to the landing and take-off of aircraft on a structural or ice sheet. The initial-boundary-value problem is solved analytically using a free-surface condition that incorporates the flexural rigidity of the plate. The three-dimensional load is modeled as an axisymmetric, translating pressure distribution. The time-dependent analytical solution is used to obtain the unsteady drag of this moving pressure, if it exists, as well as its asymptotic behavior at large time. The behavior of the transition of the drag near a critical speed related to the minimum celerity of the free waves of the hydroelastic system is examined. Asymptotic analysis shows that the drag attains a discontinuous but finite value as the translation speed approaches the critical speed, an essential difference from some existing two-dimensional results. The growth rate of the plate slope is found to be weakly singular, like log t, for large time. Comparisons with published experimental data for plate deformation are made for the case of an ice sheet. The agreement is very favorable. Implications on the operation of floating runways are discussed.