Image Treatment of a 2D Vapor-Liquid Compound Droplet in a Linearized Steady Viscous Flow

2008 ◽  
Vol 130 (7) ◽  
Author(s):  
D. Palaniappan
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Extensional Flow ◽  
Flow Fields ◽  
Uniform Flow ◽  
Creeping Flow ◽  
Physical Parameters ◽  
Vertex Angle ◽  
Compound Droplet ◽  
Vapor Liquid

The classical method of images is used to construct closed form exact solutions for the two-dimensional (2D) perturbed flow fields in the presence of a 2D vapor-liquid compound droplet in the limit of low-Reynolds number. The geometry of the multiphase droplet is composed of two overlapping infinitely long cylinders Ca and Cb of radii a and b, respectively, intersecting at a vertex angle π∕2. The composite inclusion has the shape resembling a 2D snowman type of object with a vapor cylinder Ca partly protruded into the cylinder Cb filled with another fluid whose viscosity is different from that of the host fluid. The mathematical problem with this inclusion in the Stokes flow environment is formulated in terms of Stokes stream function with mixed boundary conditions at the boundary of the hybrid droplet. General expressions for the perturbed stream functions in the two phases are obtained in a straightforward fashion using Kelvin’s inversion together with shift and reflection properties of biharmonic functions. Application of our method to other related problems in creeping flow and possible further generalizations are also discussed. The general results are then exploited to derive singularity solutions for the hybrid droplet embedded in (i) a centered shear flow, (ii) a quadratic potential flow, and (iii) an extensional flow past the 2D vapor-liquid compound droplet. The image singularities in each case depend on the two radii of the cylinders, the center-to-center distance, and the viscosity ratio. The exact solutions are utilized to plot the flow streamlines and they show some interesting patterns. While the flow fields exterior to the droplet exhibit symmetrical topological structures, the interior flow fields show existence of free eddies—enclosed in a figure-eight separatrix—and stagnation points (hyperbolic points). The flow characteristics are influenced by the viscosity and radii ratios. Furthermore, the asymptotic analysis leads to a rather surprising conclusion that there is a (subdominant) uniform flow far away from the droplet in all cases. The existence of an origin, the natural center of the drop of the composite geometry, which neutralizes the uniform flow for a particular choice of the physical parameters, is illustrated. This reveals the sensitivity of the geometry in 2D Stokes flow. The present results may be of some interest in models involving a combination of stick and slip boundaries. Moreover, the method discussed here can be useful both as a teaching tool and as a building block for further calculations.

Viscous Flows Involving a Liquid-Vapor Compound Droplet

2001 ◽  
Author(s):  
D. Palaniappan
Keyword(s):  
Exact Solutions ◽  
Viscosity Ratio ◽  
Numerical Algorithms ◽  
Flow Fields ◽  
Continuous Phase ◽  
Boundary Integral ◽  
Liquid Volume ◽  
Drag Forces ◽  
Compound Droplet ◽  
Fluid Interactions

Abstract Exact analytical solutions for steady-state axisymmetric creeping flows in and around a compound multiphase droplet are presented. The solutions given here explain the droplet fluid interactions in uniform and nonuniform flow fields. The compound droplet has a two-sphere geometry with the two spherical surfaces (of unequal radii) intersecting orthogonally. The surface tension forces are assumed to be sufficiently large so that the interfaces have uniform curvature. The singularity solutions for the uniform and paraboloidal flows in the presence of a compound droplet are derived using the method of reflections. The exact solutions for the velocity and pressure fields in the continuous and dispersed phases are given in terms of the fundamental singularities (Green’s functions) and their derivatives. It is found that flow fields and the drag forces depend on two parameters namely, the viscosity ratio and the radii ratio. In the case of paraboloidal flows, a single or a pair of eddies is noticed in the continuous phase for various values of these parameters. The eddies changes their size and shape if the size of the droplet is altered. These observations may be useful in the study of hydrodynamic interactions of compound droplets in complex situations. It is found that the Stokes resistance is greater when the liquid volume is large compared to the vapor volume in uniform flow. It is also noticed that the maximum value of the drag in paraboloidal flow depends on the viscosity ratio and significantly on the liquid volume in the dispersed phase. The exact solutions presented here may be useful for boundary integral formulations that are based on special kernels and also in validating numerical algorithms and codes on multiphase flow and droplet-fluid interactions.

Exact solutions for Stokes flow in and around a sphere and between concentric spheres

2009 ◽  
Vol 631 ◽  
pp. 363-373 ◽  
Author(s):  
P. N. SHANKAR
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Fields ◽  
Concentric Sphere ◽  
Mixing Properties ◽  
Concentric Spheres ◽  
General Method

A general method is suggested for deriving exact solutions to the Stokes equations in spherical geometries. The method is applied to derive exact solutions for a class of flows in and around a sphere or between concentric spheres, which are generated by meridional driving on the spherical boundaries. The resulting flow fields consist of toroidal eddies or pairs of counter-rotating toroidal eddies. For the concentric sphere case the exact solution when the inner sphere is in instantaneous translation is also derived. Although these solutions are axisymmetric, they can be combined with swirl about a different axis to generate fully three-dimensional fields described exactly by simple formulae. Examples of such complex fields are given. The solutions given here should be useful for, among other things, studying the mixing properties of three-dimensional flows.

scholarly journals On the Perturbation of the Three-Dimensional Stokes Flow of Micropolar Fluids by a Constant Uniform Magnetic Field in a Circular Cylinder

2011 ◽  
Vol 2011 ◽  
pp. 1-41 ◽  
Author(s):  
Panayiotis Vafeas ◽  
Polycarpos K. Papadopoulos ◽  
Pavlos M. Hatzikonstantinou
Keyword(s):  
Magnetic Field ◽  
Circular Cylinder ◽  
Stokes Flow ◽  
Uniform Magnetic Field ◽  
Colloidal Suspension ◽  
Three Dimensional ◽  
Flow Fields ◽  
Creeping Flow ◽  
Spherical Particles ◽  
Circular Duct

Modern engineering technology involves the micropolar magnetohydrodynamic flow of magnetic fluids. Here, we consider a colloidal suspension of non-conductive ferromagnetic material, which consists of small spherical particles that behave as rigid magnetic dipoles, in a carrier liquid of approximately zero conductivity and low-Reynolds number properties. The interaction of a 3D constant uniform magnetic field with the three-dimensional steady creeping motion (Stokes flow) of a viscous incompressible micropolar fluid in a circular cylinder is investigated, where the magnetization of the ferrofluid has been taken into account and the magnetic Stokes partial differential equations have been presented. Our goal is to apply the proper boundary conditions, so as to obtain the flow fields in a closed analytical form via the potential representation theory, and to study several characteristics of the flow. In view of this aim, we make use of an improved new complete and unique differential representation of magnetic Stokes flow, valid for non-axisymmetric geometries, which provides the velocity and total pressure fields in terms of easy-to-find potentials. We use these results to simulate the creeping flow of a magnetic fluid inside a circular duct and to obtain the flow fields associated with this kind of flow.

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

1975 ◽  
Vol 67 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Allen T. Chwang ◽  
T. Yao-Tsu Wu
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Fundamental Solutions ◽  
The Body ◽  
Circular Cylinders ◽  
Geometrical Parameters ◽  
Flow Problems ◽  
Singularity Method ◽  
Wide Range ◽  
Body Shapes

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

Motion of slender bodies in unsteady Stokes flow

2011 ◽  
Vol 688 ◽  
pp. 66-87 ◽  
Author(s):  
Efrath Barta
Keyword(s):  
Stokes Flow ◽  
Slenderness Ratio ◽  
Creeping Flow ◽  
Slender Body Theory ◽  
Wide Range ◽  
Unsteady Stokes Flow ◽  
Slender Bodies ◽  
Set Up ◽  
Micro Organisms ◽  
Parameter Values

AbstractThe flow regime in the vicinity of oscillatory slender bodies, either an isolated one or a row of many bodies, immersed in viscous fluid (i.e. under creeping flow conditions) is studied. Applying the slender-body theory by distributing proper singularities on the bodies’ major axes yields reasonably accurate and easily computed solutions. The effect of the oscillations is revealed by comparisons with known Stokes flow solutions and is found to be most significant for motion along the normal direction. Streamline patterns associated with motion of a single body are characterized by formation and evolution of eddies. The motion of adjacent bodies results, with a reduction or an increase of the drag force exerted by each body depending on the direction of motion and the specific geometrical set-up. This dependence is demonstrated by parametric results for frequency of oscillations, number of bodies, their slenderness ratio and the spacing between them. Our method, being valid for a wide range of parameter values and for densely packed arrays of rods, enables simulation of realistic flapping of bristled wings of some tiny insects and of locomotion of flagella and ciliated micro-organisms, and might serve as an efficient tool in the design of minuscule vehicles. Its potency is demonstrated by a solution for the flapping of thrips.

Evolution and stationarity of liquid toroidal drop in compressional Stokes flow

2017 ◽  
Vol 835 ◽  
pp. 1-23 ◽  
Author(s):  
B. K. Ee ◽  
O. M. Lavrenteva ◽  
I. Smagin ◽  
A. Nir
Keyword(s):  
Cross Sections ◽  
Capillary Number ◽  
Initial Conditions ◽  
Extensional Flow ◽  
Building Blocks ◽  
Physical Parameters ◽  
Ambient Fluid ◽  
Medicinal Substances ◽  
Expansion Dynamic ◽  
Axisymmetric Disturbances

Dynamics of fluid tori in slow viscous flow is studied. Such tori are of interest as future carriers of biological and medicinal substances and are also viewed as potential building blocks towards more complex particles. In this study the immiscible ambient fluid is subject to a compressional flow (i.e., bi-extensional flow), and it comprises a generalization of our earlier report on the particular case with viscosity ratio$\unicode[STIX]{x1D706}=1$(see Zabarankinet al.,J. Fluid Mech., vol. 785, 2015, pp. 372–400), where$\unicode[STIX]{x1D706}$is the ratio between the torus viscosity and that of the ambient fluid. It is found that, for all viscosity ratios, the torus either collapses towards the axis of symmetry or expands indefinitely, depending on the initial conditions and the capillary number,Ca. During these dynamic patterns the cross-sections exhibit various forms of deformation. The collapse and expansion dynamic modes are separated by a limited deformation into a deformed stationary state which appears to exist in a finite interval of the capillary number,$0<Ca<Ca_{cr}(\unicode[STIX]{x1D706})$, and is unstable to axisymmetric disturbances, which eventually cause the torus either to collapse or to expand indefinitely. The characteristic dimensions and shapes of these unstable stationary tori and their dependence on the physical parametersCaand$\unicode[STIX]{x1D706}$are reported.

scholarly journals Experimental and an electrolyte non-random two-liquid model to predict the vapor–liquid equilibrium of CO2 in aqueous solutions of diethylenetriamine

2020 ◽  
pp. 174751982096417
Author(s):  
Ruilei Zhang ◽  
Yandong Tang ◽  
Weifeng Shan ◽  
Haijun Liu ◽  
Haijun Li ◽  
...  
Keyword(s):  
Aqueous Solutions ◽  
Saturated Vapor ◽  
Physical Parameters ◽  
Liquid Equilibrium ◽  
Absolute Deviation ◽  
Vapor Liquid Equilibrium ◽  
Liquid Theory ◽  
Equilibrium Data ◽  
Experimental Values ◽  
Vapor Liquid

The absorption and desorption data of CO2 in aqueous solutions with a mass fraction of 10% and 20% of diethylenetriamine are measured at 313.15, 343.15, 373.15, and 393.15 K. The electrolyte non-random two-liquid theory is developed using Aspen V9.0 to correlate and predict the vapor–liquid equilibrium of CO2 in aqueous diethylenetriamine solutions. The model predicted the heat capacity and saturated vapor pressure data of diethylenetriamine, the mixed heat of a diethylenetriamine–H2O binary system, and the vapor–liquid equilibrium data of a diethylenetriamine–H2O–CO2 ternary system. The physical parameters and the interaction parameters of the model system are calculated. The model predicted CO2 solubility showing a 10% average absolute deviation from experimental data. The calculated values of the model are basically consistent with the experimental values.

Non-Newtonian slender drops in a nonlinear extensional flow

2019 ◽  
Vol 877 ◽  
pp. 561-581 ◽  
Author(s):  
Moshe Favelukis
Keyword(s):  
Stability Analysis ◽  
Power Law ◽  
Analytical Solutions ◽  
Viscosity Ratio ◽  
Extensional Flow ◽  
Shear Thinning ◽  
Newtonian Liquid ◽  
Creeping Flow ◽  
Stationary States ◽  
Power Law Index

In this theoretical report we explore the deformation and stability of a power-law non-Newtonian slender drop embedded in a Newtonian liquid undergoing a nonlinear extensional creeping flow. The dimensionless parameters describing this problem are: the capillary number $(Ca\gg 1)$, the viscosity ratio $(\unicode[STIX]{x1D706}\ll 1)$, the power-law index $(n)$ and the nonlinear intensity of the flow $(|E|\ll 1)$. Asymptotic analytical solutions were obtained near the centre and close to the end of the drop suggesting that only Newtonian and shear thinning drops $(n\leqslant 1)$ with pointed ends are possible. We described the shape of the drop as a series expansion about the centre of the drop, and performed a stability analysis in order to distinguish between stable and unstable stationary states and to establish the breakup point. Our findings suggest: (i) shear thinning drops are less elongated than Newtonian drops, (ii) as non-Newtonian effects increase or as $n$ decreases, breakup becomes more difficult, and (iii) as the flow becomes more nonlinear, breakup is facilitated.

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