Mechanical Engineering

The self-similar flow and free-surface shape induced by a flat plate entering an inviscid and incompressible liquid are investigated for arbitrary initial conditions. An analytical solution, which is based on two governing expressions, namely the complex velocity and the derivative of the complex potential, is obtained. These expressions are derived in an auxiliary parameter plane using integral formulae proposed for the determination of an analytical function from its modulus and argument given on the boundary of the parameter region. We derive a system of an integral and an integro-differential equation in terms of the velocity modulus and the velocity angle at the free surface, which are determined by the dynamic and kinematic boundary conditions. A numerical procedure for solving these equations is carefully validated by comparisons with results available in the literature. The results are presented in terms of the free surface shape, the angles at the tip of the splash jet, the contact angles at the intersection with the plate surface, pressure distribution and force coefficients. New features caused by the flow unsteadiness are found and discussed.

The self-similar flow and free-surface shape induced by a flat plate entering an inviscid and incompressible liquid are investigated for arbitrary initial conditions. An analytical solution, which is based on two governing expressions, namely the complex velocity and the derivative of the complex potential, is obtained. These expressions are derived in an auxiliary parameter plane using integral formulae proposed for the determination of an analytical function from its modulus and argument given on the boundary of the parameter region. We derive a system of an integral and an integro-differential equation in terms of the velocity modulus and the velocity angle at the free surface, which are determined by the dynamic and kinematic boundary conditions. A numerical procedure for solving these equations is carefully validated by comparisons with results available in the literature. The results are presented in terms of the free surface shape, the angles at the tip of the splash jet, the contact angles at the intersection with the plate surface, pressure distribution and force coefficients. New features caused by the flow unsteadiness are found and discussed.

The non-linear problem of cavity flow past a hydrofoil is considered with taking into account fluid viscosity in the cavity closure region and surface tension, which affect the cavity detachment. The theoretical model is based on the concept of viscous-inviscid interaction between the outer inviscid cavity flow and the inner turbulent separated flow downstream of the cavity. The outer inviscid flow is solved by constructing the complex flow potential, and the wake model is based on the method of integral relationships for separated turbulent flows. The obtained numerical results are compared with experimental data.

The limit combination of the initial parameters corresponding to flow separation from the wedge vertex during the initial stage of the oblique impact of a wedge on a liquid half-space is investigated on the basis of an analytical solution of the problem. The liquid is assumed to be ideal and incompressible; gravity and surface tension effects are ignored. The flow generated by the impact is two-dimensional and potential. The analysis revealed a limit combination of the initial parameters at which the pressure along the whole length of one side of the wedge becomes less than the atmospheric pressure. Such a pressure distribution results in the ventilation of the wedge side, which starts from the contact point on the free surface and extends suddenly along the whole length of the wedge side, thus leading to flow separation from the wedge vertex.

Cavity flow past a circular cylinder is considered accounting for the surface tension on the cavity boundary. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. The solution is based on two derived governing expressions, which are the complex velocity and the derivative of the complex potential defined in an auxiliary parameter region. An integral equation in the velocity magnitude along the free surface is derived from the dynamic boundary condition. The Brillouin-Villat criterion is employed to determine the location of the point of flow separation. The cases of zero surface tension and zero cavitation number are obtained as limiting cases of the solution. Numerical results concerning the effects of surface tension and cavitation development on the cavity detachment, the drag force and the geometry of the free boundaries are presented over a wide range of the Weber and the cavitation numbers.

A general similarity solution for water-entry problems of a wedge with its inner angle fixed and its sides in expansion is obtained with flow detachment, in which the speed of expansion is a free parameter. The known solutions for a wedge of a fixed length at the initial stage of water entry without flow detachment and at the final stage corresponding to Helmholtz flow are obtained as two special cases, at some finite and zero expansion speeds, respectively. An expanding horizontal plate impacting a flat free surface is considered as the special case of the general solution for a wedge inner angle equal to${\rm\pi}$. An initial impulse solution for a plate of a fixed length is obtained as the special case of the present formulation. The general solution is obtained in the form of integral equations using the integral hodograph method. The results are presented in terms of free-surface shapes, streamlines and pressure distributions.

The self-similar solution that characterizes the water impact, with a constant vertical velocity, of a wedge entering the free surface with an arbitrary orientation is derived analytically. The study is carried out by assuming the fluid to be ideal, weightless and with negligible surface tension effects. The solution is based on the complex analysis of nonlinear two-dimensional problems of unsteady free boundary flows and is written in terms of two governing functions, which are the complex velocity and the derivative of the complex potential defined in a parameter domain. The boundary value problem is reduced to the system of an integral and an integro-differential equation in terms of the velocity modulus and of the velocity angle to the free surface, both written as functions of a parameter variable. The system of equations is solved through a numerical procedure which is validated in the case of symmetric wedges. Comparisons with data available in literature are established for this purpose. Results are presented in terms of free surface shape, contact angles at the intersection with the wedge boundary, pressure distribution, force and moment coefficients. For a given wedge angle, the changes induced by the heel angle on the above quantities are discussed. A criterion is proposed to determine the limit conditions beyond which flow separation from the wedge apex occurs. Comparisons with experimental results available in literature are presented.

The hydrodynamic problem of impact between a solid wedge and a liquid wedge is analysed. The liquid is assumed to be ideal and incompressible; gravity and surface tension effects are ignored. The flow generated by the impact is assumed to be irrotational and therefore can be described by the velocity potential theory. The solution procedure is based on the analytical derivation of the complexvelocity potential in a parameter plane and the function mapping conformally the parameter plane onto the similarity plane. The mapping function is found as a combination of the derivatives of the complex potential in the similarity and parameter planes, through the integral equations for mixed and homogeneous boundary-value problems in terms of the velocity modulus and the velocity angle with the fluid boundary, together with the dynamic and kinematic boundary conditions. These equations are solved through a numerical method. The procedure is first verified through comparisons with some known results. Simulations are then made for a variety of cases, and detailed results are presented in terms of the free surface shape, streamlines, pressure distribution on the wetted solid surface, and contact angles between the free surface and the body surface.

The collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the...

The collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the...

Experimental studies of supercavitating models moving at speeds in the range from 400 m/s to 1000 m/s revealed a regime of bouncing motion, in which the rear part of an axisymmetric body periodically bounces against the free boundaries of the supercavity. The impulsive force generated by the impacts is the main concern in this paper. The analysis is performed in the approximation of two-dimensional potential flow of an ideal and incompressible liquid with negligible surface tension effects. The primary interest of the study is to determine the added mass taking into account the shape of the cavity. The theoretical study is based on the integral hodograph method, which makes it possible to obtain analytic expressions for the flow potential and for the complex velocity in an auxiliary parameter plane and obtain a parametric solution to the problem. The problem is reduced to a system of two integro-differential equations in two unknowns: the velocity magnitude on the cavity boundary an...

Cavity flow past an obstacle in the presence of an inflow vorticity is considered. The proposed approach to the solution of the problem is based on replacing the continuous vorticity with its discrete form in which the vorticity is concentrated along vortex lines coinciding with the streamlines. The flow between the streamlines is vortex free. The problem is to determine the shape of the streamlines and cavity boundary. The pressure on the cavity boundary is constant and equal to the vapour pressure of the liquid. The pressure is continuous across the streamlines. The theory of complex variables is used to determine the flow in the following subregions coupled via their boundary conditions: a flow in channels with curved walls, a cavity flow in a jet and an infinite flow along a curved wall. The numerical approach is based on the method of successive approximations. The numerical procedure is verified considering a body with a sharp edge, for which the point of cavity detachment is ...

It has been shown by experimental and numerical studies that various cavitation instabilities occur in inducers for rocket engines when the cavity length exceeds about 65% of the blade spacing. On the other hand, it has been pointed out by an experimental study that the cavitation instabilities occur when the pressure gradient near the throat becomes small to some degree. The present study is motivated to examine the latter criterion based on pressure gradient for cavitation instabilities from the viewpoint of theoretical analysis. For this purpose, analyses of steady flow and its stability were carried out for cavitating flow in cascades with circular arc and plano-convex blades by a singularity method based on closed cavity model. It was found that the criterion based on the cavity length for the occurrence of cavitation instabilities is more adequate than the criterion based on the pressure gradient. It was also found that the steady cavity length and the stability of the flow in...

A general similarity solution for the water-entry problem of a two-dimensional arbitrarily shaped body undergoing expansion in a prescribed manner is obtained. The free surface is assumed to detach from the curved body surface in the mathematical formulation of the problem. The point of detachment is determined based on the Brillouin-Villat criterion, which states that the curvature of the body and the curvature of the free surface must be the same at the point of flow detachment. In this extended abstract, results are presented for water entry of an expanding circular cylinder. The integral hodograph method [1,2] is used to derive analytical expressions for the complex potential and for the complex velocity, both defined in a parametric plane, for which the first quadrant is chosen. It enables the original partial differential equation with nonlinear boundary conditions on the unknown free surface to be reduced to a system of integro-differential equations along straight lines in t...

Microencapsulation of functional ingredients has been under development in the cosmetic, pharmaceutical, nutraceutical and food industries for several decades, using various preparation techniques such as spray-drying, liposome encapsulation and coacervation, where single or multiwall microcapsules, or matrix-type capsules can be made. The choice of an appropriate encapsulation technology involves consideration of many important aspects such as cost, number of steps involved, encapsulation efficiency and storage stability. Multifunctional microcapsules must be able not only to entrap and protect the additive(s) from the surrounding environment and during storage, but also to deliver and to release their contents at the target via some trigger mechanism. Typically, microcapsules are spherical, with diameters larger than a few microns. Therefore they are too big to be used for Pickering stabilisation and control of the product microstructure, where they can have an additional and unique functionality. For example, by using conventional emulsifiers, one can prepare emulsions with a multi-year shelf-life, but to make shelf-stable foams with lifetimes longer than a few days is a real challenge, unless the continuous phase of the product is gelled or solidified, which in many cases leads to undesired textural changes. One way to produce super-stable foams via pure interfacial stabilisation is to use particle dispersions, at the right conditions by tuning the particle wettability, inter-particle interactions and particle morphology. Many authors have demonstrated the potential of spherical (or roughly spherical or aggregated) nanoparticles as foam or emulsion stabilisers, while only recently, stabilisation by high-aspect-ratio anisotropic or 'rod-like' particles has been described. The anisotropic micro-rods are able to give extra stability due to their ability to intertwine at the bubble interface, thus increasing the surface coverage and interfacial rigidity. At the same time, the microrod length can be comparable with the diameter of the drops or bubbles they stabilise, rather than being much smaller as is the case with spherical nanoparticles. A relevant question is whether we can use the size and the structuring advantage of the rod-like microparticles and also make them perform the function of (edible) micro-capsules.

We report a generic technique to image and study the wettability of spherical nanoparticles adsorbed at liquid surfaces and demonstrate that nanoparticle monolayers can be imprinted at air-water and oil-water interfaces and their three-phase contact angle at the original liquid interface can be determined by an atomic force microscopy scan on a replica of the interface; the technique is tested using four different types of nanoparticles, the smallest one having a radius of 37 nm.

In this study we consider the stability of horizontal two-layered liquid film attached to a heated solid substrate. The film can contain surfactant that is soluble in both liquid phases. The evaporation of solvent from the upper film is also taken into account. Thus, the two-layered film can exhibit both thermocapillary and Marangoni instabilities coupled with the effect of solvent mass loss. The problem is solved in the framework of the lubrication approximation. We derive a system of partial differential equations describing the evolution of long-wave disturbances in the presence of surfactant and evaporation. Appropriate rescaling is proposed and a numerical analysis of the dimensionless groups for particular system of water-light oil film upon a horizontal PVC plate is performed. The model allows one to investigate the role of different factors on the film stability: the surfactant concentration and distribution coefficient, the critical concentrations of micellization, the surface viscosities, the adsorption isotherms of the surfactant at the liquid surfaces and the intensity of evaporation. Based on this model the full linear analysis of the stability is given in Part II [see Danov et al. (1997b) Chem. Engng Sci., submitted]. The non-linear effects are also taken into account in Part III [see Chem. Engng Sci. submitted], where these effects are studied numerically for a particular case of PVC/tetrachlorethane/water/vapour system.

A general method for calculating the spontaneous curvature and the bending modulus of surfactant monolayers or grafted polymer brushes is presented which is based on a density functional theory, where the bending parameters and the spontaneous curvature are analytically represented in terms of integral moments of the solution for the flat geometry. The novelty here is that the problem is solved for an arbitrary form of the free energy density function with the only requirement being that the energy functional is local. The difference compared to other approaches is that for the case of local density functional theories; the lateral pressure and its derivatives with respect of curvature could be expressed analytically in terms of the density profiles of the flat geometry. This general approach enables a variety of problems involving polymer brushes, surfactants, and amphiphilic diblock copolymers to be treated directly and has direct application in modeling the phase behavior of microemulsions and surfactant solutions. As particular examples we consider the cases of low-density arbitrary solvent, arbitrary density in ideal solvent and Θ solvent of a polymer brush and an amphiphilic diblock copolymer.

We study the effect of the particle wettability on the preferred type of emulsion stabilised solely by food colloid particles. We present results obtained with the recently developed gel trapping technique (GTT) for characterisation of wettability and surface structuring of individual food colloid particles adsorbed at air-water and oil-water interfaces. This method allows us to replicate a particle monolayer onto the surface of polydimethylsiloxane (PDMS) without altering the position of the particles. By observing the polymer surface with scanning electron microscopy (SEM), we are able to determine the contact angle of the individual particles at the initial liquid interface. We demonstrate that the GTT can be applied to fat crystal particles, calcium carbonate particles coated with stearic acid and spray-dried soy protein/calcium phosphate particles at air-water and oil-water interfaces. Subsequently, we prepare emulsions of decane and water stabilised by the same food colloid particles and correlate the wettability data obtained for these particles to the preferred type of emulsions they stabilise.