DOCSLIB.ORG

∫° Nonlinear Capillary Oscillation of a Charged Drop

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
∫° Nonlinear Capillary Oscillation of a Charged Drop Technical Physics, Vol. 45, No. 8, 2000, pp. 1001–1008. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 70, No. 8, 2000, pp. 45–52. Original Russian Text Copyright © 2000 by Belonozhko, Grigor’ev. GASES AND LIQUIDS Nonlinear Capillary Oscillation of a Charged Drop D. F. Belonozhko and A. I. Grigor’ev Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia E-mail: [email protected] Received May 18, 1999; in final form October 14, 1999 Abstract—In the quadratic approximation with respect to the amplitudes of capillary oscillation and velocity field of the liquid moving inside a charged drop of a perfectly conducting fluid, it is shown that the liquid drop oscillates about a weakly prolate form. This refines the result obtained in the linear theory developed by Lord Rayleigh, who predicted oscillation about a spherical form. The extent of elongation is proportional to the initial amplitude of the principal mode and increases with the intrinsic charge carried by the drop. An estimate is obtained for the characteristic time of instability development for a critically charged drop. © 2000 MAIK “Nauka/Interperiodica”. Studies of capillary oscillation and stability of a 2. Consider a drop of an inviscid, perfectly conduct- charged drop are motivated by numerous applications ing liquid of density ρ characterized by the surface ten- [1]. The problem was reviewed in [1–7] and references sion γ and carrying a charge Q. The linear theory of therein. However, most theoretical analyses are based capillary oscillation postulates that there exists a time on the linearized system of fluid-dynamics equations. moment that can be treated as t = 0, when the drop Only recent studies have captured the nonlinear nature geometry is described by the second mode of linear of the phenomenon and provided essentially new infor- capillary oscillation of a small finite amplitude ε and mation about the mechanism of instability of a highly the velocity field is zero. The initial drop volume is charged drop and its capillary oscillation [2–7]. equal to that of a spherical drop of radius R. The prob- lem is to determine the axially symmetric oscillations 1. A charged spherical drop with an intrinsic charge of the drop in the case of an axially symmetric potential Q greater than a certain critical value becomes unsta- velocity field in the drop. ble, because the electrical repulsive forces exceed the We use the dimensionless variables in which ρ ≡ 1, surface tension forces. At the close of the nineteenth γ ≡ ≡ century, Lord Rayleigh developed a linear model of this 1, and R 1. The mathematical model of the prob- instability [8]. Since then, studies of the instability of a lem is then written in spherical coordinates with the ori- highly charged drop and its generalizations have grown gin at the center of the drop as follows: into a broad area of research having important technical ∆ψ ==0; U — ⋅ ψ, (1) applications. ∆Φ ==0; E–—⋅Φ, (2) From the perspective of classical physics, the ther- mal motion of molecules gives rise to drop oscillations r ∞: |—Φ| 0, (3) of amplitude ~kT/γ about the equilibrium spherical r = 0: |—ψ| < ∞, (4) form, where k is Boltzmann’s constant, T is the absolute temperature, and γ is the surface tension of the liquid. r=1+ξΘ(),t Each mode is characterized by a specific critical value 1∂Φ –------ ∫------- dS =Q; Sr(),,Θϕ ≡0≤≤Θπ (5) of surface charge, and a supercritical charge leads to the 4π°∂n onset of instability. The stability of the drop with 0≤ϕ<2π, respect to its intrinsic charge Q is generally character- ≡ 2 πγ 3 ized by the so-called Rayleigh parameter W Q /4 R , r==1+ξΘ(),t: Φ const, (6) where R is the drop radius. The second mode is the most unstable, and the corresponding critical value of ∂ψ ∂ξ 1 ∂ξ ∂ψ ------- = ------ + ---- ------- ------- , (7) the Rayleigh parameter is W = 4. 2∂Θ∂Θ c ∂r∂tr In this paper, we solve Rayleigh’s problem in the 2 quadratic approximation with respect to the amplitudes ∂ψ 1 2 E ∆F– ------- –--- ()— ⋅ ψ +F==Fγ;F------ , (8) of the velocity field and surface perturbation and deter- ∂t 2 E E 8π mine the geometric form about which a subcritically ξ ε ()Θ charged liquid drop oscillates. t==0: r 1 ++* P2 cos , (9) 1063-7842/00/4508-1001 $20.00 © 2000 MAIK “Nauka/Interperiodica” 1002 BELONOZHKO, GRIGOR’EV In the zeroth approximation, the nontrivial relations 1 2 4 P (µ) = --(3µ – 1), dv = --π; are (2), (3), (5), and (8). They contain the electric 2 2 ∫ 3 potential, the capillary pressure, and the constant com- V (10) ponent of the difference of pressures inside and outside ≤ ≤ ε ε ( (Θ)) the drop. Problem (1)–(11) reduces to one for the static 0 r 1 + + P2 cos , Φ ψ ≡ ξ ≡ V(r, Θ, t) ≡ * field 0(r), with the known functions 0 0 and 0 0: 0 ≤ Θ ≤ π, 0 ≤ ϕ < 2π, 2Φ Φ d 0 d 0 ψ = 0. (11) ----------- = 0, E1 = –---------er, dr2 dr This is a boundary value problem with the free sur- face r = 1 + ξ(Θ, t) and three unknown functions: the 2π π Φ potential Φ = Φ(r, Θ, t) of the electric field outside the 1 d 0 (Θ) Θ ϕ ψ ψ Θ –----π-- ∫ ∫--------- sin d d = Q, (16) drop, the potential = (r, , t) of the velocity field 4 dr r = 1 inside the drop, and the surface deviation ξ = ξ(Θ, t) 0 0 from a regular sphere with r = 1. Φ ∞ d 0 The potentials Φ and ψ satisfy the Laplace equa- r : --------- 0, tions (1) and (2) under the boundary and initial condi- dr tions (3)–(8) and (9)–(11), respectively. where er is the unit radial vector. Both kinematic and dynamic boundary conditions Φ The solution of the problem is 0 = Q/r. Using this for the velocity field U, (7) and (8), contain nonlinear solution and formulas (1B), (2C), and (5D)–(10D) from terms. Condition (4) implies that U is bounded at the ∆F the Appendices, the first- and second order problems center of the drop. In (8), is the difference between are easily formulated. The first order problem is written the constant pressure components inside and outside as the drop, F is the electric pressure on the drop surface E ∆ψ ⋅ ψ [9, 10], and Fγ is the capillary pressure under the 1 = 0, U1 = — 1, (17) curved drop surface [11]. ∆Φ = 0, E = –— ⋅ Φ , (18) It follows from (3) that the electric field E must be 1 1 1 ∞ | Φ | bounded at infinity. The continuity of the tangential r : — 1 0, (19) components of E(r, t) at the charged surface of a con- | ψ | ∞ ductor means that the surface is equipotential (see (6)) r = 0: — 1 < , (20) [10]. The condition for the normal components of E(r, t) 2π π at the interface is written in (5) in an integral form, Φ 1 d 1 (Θ) Θ ϕ where n is the outward unit normal to the drop surface. r = 1: –----π-- ∫ ∫--------- sin d d = 0, (21) ξ 4 dr The quantity ∗ in (9) is determined by (10), which 0 0 means that the volumes of the initial drop and a spher- Φ – ξ Q = 0, (22) ical drop of unit radius are equal. 1 1 3. In dimensionless variables, the amplitude ε of an ∂ψ ∂ξ ---------1 = --------1, (23) initial deviation from spherical form is measured in ∂r ∂t units of the spherical drop radius. We treat ε as a small parameter and seek a solution to the problem as a series ∂ψ ∂Φ ε 1 Q 1 ξ ξ ∆ ξ expansion in integer powers of , omitting the terms of ---∂------ + ----π-- ---∂------ + 2 1Q = 2 1 + Ω 1, (24) order higher than two: t 4 r 3 ξ ε ( (Θ)) Φ Φ Φ Φ (ε ) t = 0: 1 = P2 cos ; = 0 + 1 + 2 + O , (12) 1 2 (25) ψ ψ ψ (ε3) P (cos(Θ)) = --(3cos (Θ) – 1), = 1 + 2 + O , (13) 2 2 ξ ξ ξ ξ (ε3) ψ r = 1 + , = 1 + 2 + O , (14) 1 = 0. (26) Φ ∼ ψ ∼ ξ ∼ O(εm); m = 1, 2; Φ = O(1). (15) The problem for the second-order quantities is writ- m m m 0 ten as follows: The functions Φ, ψ, and ξ are assumed to be of the ∆ψ ⋅ ψ same order as their partial derivatives. Under these 2 = 0, U2 = — 2, (27) assumptions, since the Laplace operator is linear, we ∆Φ ⋅ Φ can use the expansions obtained in the Appendices to 2 = 0, E2 = –— 2, (28) split the problem (1)–(11) into problems of zeroth, first, ∞ | Φ | Φ Φ r : — · 2 0, (29) and second order [11] for the seven functions 0, m, ψ ξ | ψ | ∞ m, and m, where m = 1 and 2. r = 0: — · 2 < , (30) TECHNICAL PHYSICS Vol. 45 No. 8 2000 NONLINEAR CAPILLARY OSCILLATION OF A CHARGED DROP 1003 π π − Θ 2 2 m(m + 1)Pm(cos )); and formulate the Cauchy prob- 1 dΦ ∂Φ ∂Φ r = 1: –------ ---------2 + 2ξ ---------1 + ξ ---------1 lem for a homogeneous system of ordinary differential π ∫ ∫ 1 ∂ 1 2 4 dr r ∂r equations for the unknown functions Cm(t), Fm(t), and 0 0 (31) Z (t). By finding these unknowns and substituting them ∂ξ ∂Φ m 1 1 Θ Θ ϕ into (37)–(39), we can write the stable solutions to the – -∂---Θ------∂---Θ---- sin d d = 0, first-order problem as ∂Φ ξ = εcos(ω t)P (cosΘ), (40) Φ ξ ξ 1 ξ2 1 2 2 2 – 2Q = – 1--------- + 1 Q, (32) ∂r εω ψ = –--------2 sin(ω t)P (cosΘ), (41) ∂ψ ∂2ψ ∂ξ ∂ξ ∂ψ 1 2 2 2 ---------2 + ξ -----------1 = --------2 + --------1 ---------1, (33) ∂r 1 2 ∂t ∂Θ ∂Θ Φ ε (ω ) ( Θ) ∂r 1 = Qcos 2t P2 cos , (42) ∂ψ ∂Φ 2 2 Q 2 ξ ξ ∆ ξ Q ---∂------ + ----π-- ---∂------ + 2 2Q = 2 2 + Ω 2 ω = 8 – 2W, W = ------ < 4.
Load more

Recommended publications
  • Polyelectrolyte Assisted Charge Titration Spectrometry: Applications to Latex and Oxide Nanoparticles
    Polyelectrolyte assisted charge titration spectrometry: applications to latex and oxide nanoparticles F. Mousseau1*, L. Vitorazi1, L. Herrmann1, S. Mornet2 and J.-F. Berret1* 1Matière et Systèmes Complexes, UMR 7057 CNRS Université Denis Diderot Paris-VII, Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, 75205 Paris, France. 2Institut de Chimie de la Matière Condensée de Bordeaux, UPR CNRS 9048, Université Bordeaux 1, 87 Avenue du Docteur A. Schweitzer, Pessac cedex F-33608, France. Abstract: The electrostatic charge density of particles is of paramount importance for the control of the dispersion stability. Conventional methods use potentiometric, conductometric or turbidity titration but require large amount of samples. Here we report a simple and cost-effective method called polyelectrolyte assisted charge titration spectrometry or PACTS. The technique takes advantage of the propensity of oppositely charged polymers and particles to assemble upon mixing, leading to aggregation or phase separation. The mixed dispersions exhibit a maximum in light scattering as a function of the volumetric ratio �, and the peak position �!"# is linked to the particle charge density according to � ~ �!�!"# where �! is the particle diameter. The PACTS is successfully applied to organic latex, aluminum and silicon oxide particles of positive or negative charge using poly(diallyldimethylammonium chloride) and poly(sodium 4-styrenesulfonate). The protocol is also optimized with respect to important parameters such as pH and concentration, and to the polyelectrolyte molecular weight. The advantages of the PACTS technique are that it requires minute amounts of sample and that it is suitable to a broad variety of charged nano-objects. Keywords: charge density – nanoparticles - light scattering – polyelectrolyte complex Corresponding authors: [email protected] [email protected] To appear in Journal of Colloid and Interface Science leading to their condensation and to the formation of the electrical double layer [1].
    [Show full text]
  • Influence of Polyelectrolyte Multilayer Properties on Bacterial Adhesion
    polymers Article Influence of Polyelectrolyte Multilayer Properties on Bacterial Adhesion Capacity Davor Kovaˇcevi´c 1, Rok Pratnekar 2, Karmen GodiˇcTorkar 2, Jasmina Salopek 1, Goran Draži´c 3,4,5, Anže Abram 3,4,5 and Klemen Bohinc 2,* 1 Department of Chemistry, Faculty of Science, University of Zagreb, Zagreb 10000, Croatia; [email protected] (D.K.); [email protected] (J.S.) 2 Faculty of Health Sciences, Ljubljana 1000, Slovenia; [email protected] (R.P.); [email protected] (K.G.T.) 3 Jožef Stefan Institute, Ljubljana 1000, Slovenia; [email protected] (G.D.); [email protected] (A.A.) 4 National Institute of Chemistry, Ljubljana 1000, Slovenia 5 Jožef Stefan International Postgraduate School, Ljubljana 1000, Slovenia * Correspondence: [email protected]; Tel.: +386-1300-1170 Academic Editor: Christine Wandrey Received: 16 August 2016; Accepted: 14 September 2016; Published: 26 September 2016 Abstract: Bacterial adhesion can be controlled by different material surface properties, such as surface charge, on which we concentrate in our study. We use a silica surface on which poly(allylamine hydrochloride)/sodium poly(4-styrenesulfonate) (PAH/PSS) polyelectrolyte multilayers were formed. The corresponding surface roughness and hydrophobicity were determined by atomic force microscopy and tensiometry. The surface charge was examined by the zeta potential measurements of silica particles covered with polyelectrolyte multilayers, whereby ionic strength and polyelectrolyte concentrations significantly influenced the build-up process. For adhesion experiments, we used the bacterium Pseudomonas aeruginosa. The extent of adhered bacteria on the surface was determined by scanning electron microscopy. The results showed that the extent of adhered bacteria mostly depends on the type of terminating polyelectrolyte layer, since relatively low differences in surface roughness and hydrophobicity were obtained.
    [Show full text]
  • Surface Polarization Effects in Confined Polyelectrolyte Solutions
    Surface polarization effects in confined polyelectrolyte solutions Debarshee Bagchia , Trung Dac Nguyenb , and Monica Olvera de la Cruza,b,c,1 aDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208; bDepartment of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208; and cDepartment of Physics and Astronomy, Northwestern University, Evanston, IL 60208 Contributed by Monica Olvera de la Cruz, June 24, 2020 (sent for review April 21, 2020; reviewed by Rene Messina and Jian Qin) Understanding nanoscale interactions at the interface between ary conditions (18, 19). However, for many biological settings two media with different dielectric constants is crucial for con- as well as in supercapacitor applications, molecular electrolytes trolling many environmental and biological processes, and for confined by dielectric materials, such as graphene, are of interest. improving the efficiency of energy storage devices. In this Recent studies on dielectric confinement of polyelectrolyte by a contributed paper, we show that polarization effects due to spherical cavity showed that dielectric mismatch leads to unex- such dielectric mismatch remarkably influence the double-layer pected symmetry-breaking conformations, as the surface charge structure of a polyelectrolyte solution confined between two density increases (20). The focus of the present study is the col- charged surfaces. Surprisingly, the electrostatic potential across lective effects of spatial confinement by two parallel surfaces
    [Show full text]
  • Semester II, 2015-16 Department of Physics, IIT Kanpur PHY103A: Lecture # 5 Anand Kumar
    Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 8 (Text Book: Intro to Electrodynamics by Griffiths, 3rd Ed.) Anand Kumar Jha 19-Jan-2018 Summary of Lecture # 7: • Electrostatic Boundary Conditions E E = Electric Field: = above below E − E = 00 above− below � 0 above − below Electric Potential: V V = 0 above − below • Basic Properties of Conductors (1) The electric field = 0 inside a conductor, always. = 0 = = 0 (2) The charge density inside a conductor. This is because . (3) Any net charge resides on the surface. Why? To minimize the energy. 0 ⋅ (4) A conductor is an equipotential. (5) is perpendicular to the surface, just outside the conductor. 2 Summary of Lecture # 7: Prob. 2.36 (Griffiths, 3rd Ed. ): - Surface charge ? = = − 2 - Surface charge ? 4 + - Surface charge ? = − 2 4 1 2 - ( ) ? = 4 out 2 1 0 � - ( ) ? = 4 out 2 10 �+ - ( ) ? =4 out out 2 � - Force on ? 0 40 - Force on ? 0 3 Summary of Lecture # 7: Prob. 2.36 (Griffiths, 3rd Ed. ): - Surface charge ? = Same ª = − 2 Same - Surface charge ? 4 ª + - Surface charge ? = − 2 Changes � 4 1 2 = - ( ) ? 4 Same ª out 2 1 0 � = 4 - ( ) ? Same ª out 2 � 10 + Changes � - ( ) ? =4 out out 2 - Force on ? 0 0 � 4 Same ª Bring in a third - Force on ? 0 Same ª charge 4 Surface Charge and the Force on a Conductor: What is the electrostatic force on the patch? Force per unit area on the patch is: = (? ) = other = + , above other patch above = + 2 other 0 � = + , below other patch below 1 = = + 2 2 other − � other above below 0 But, inside a metal, = , so = 0 below = = + = 2 2 other � above other � � 5 0 0 0 Surface Charge and the Force on a Conductor: What is the electrostatic force on the patch? Force per unit area on the patch is: = (? ) = other = = 2 other � above � Force per unit0 area on the patch is: 0 = = 2 2 other � Force per unit area is pressure.0 So, the electrostatic pressure is: = = 2 2 20 2 6 0 Capacitor: Two conductors with charge and .
    [Show full text]
  • Ion Current Rectification in Extra-Long Nanofunnels
    applied sciences Article Ion Current Rectification in Extra-Long Nanofunnels Diego Repetto, Elena Angeli * , Denise Pezzuoli, Patrizia Guida, Giuseppe Firpo and Luca Repetto Department of Physics, University of Genoa, via Dodecaneso 33, 16146 Genoa, Italy; [email protected] (D.R.); [email protected] (D.P.); [email protected] (P.G.); giuseppe.fi[email protected] (G.F.); [email protected] (L.R.) * Correspondence: [email protected] Received: 29 April 2020; Accepted: 25 May 2020; Published: 28 May 2020 Abstract: Nanofluidic systems offer new functionalities for the development of high sensitivity biosensors, but many of the interesting electrokinetic phenomena taking place inside or in the proximity of nanostructures are still not fully characterized. Here, to better understand the accumulation phenomena observed in fluidic systems with asymmetric nanostructures, we study the distribution of the ion concentration inside a long (more than 90 µm) micrometric funnel terminating with a nanochannel. We show numerical simulations, based on the finite element method, and analyze how the ion distribution changes depending on the average concentration of the working solutions. We also report on the effect of surface charge on the ion distribution inside a long funnel and analyze how the phenomena of ion current rectification depend on the applied voltage and on the working solution concentration. Our results can be used in the design and implementation of high-performance concentrators, which, if combined with high sensitivity detectors, could drive the development of a new class of miniaturized biosensors characterized by an improved sensitivity. Keywords: nanofunnel; FEM simulation; ionic current rectification; micro-nano structure interface 1.
    [Show full text]
  • Beryllium Desorption from Minerals and Organic Ligands Over Time
    Chemical Geology 439 (2016) 52–58 Contents lists available at ScienceDirect Chemical Geology journal homepage: www.elsevier.com/locate/chemgeo Beryllium desorption from minerals and organic ligands over time Vanessa Boschi ⁎,JaneK.Willenbring Department of Earth and Environmental Science, University of Pennsylvania, 251 Hayden Hall, 240 South 33rd Street, Philadelphia, PA 19104, USA article info abstract Article history: Beryllium isotopes sorbed to sediments have provided useful tools in the field of geochronology and geomor- Received 13 April 2016 phology over the last few decades. The use of beryllium isotopes relies on the premise that beryllium sorbed to Received in revised form 3 June 2016 sediments is unaltered over large timescales. Changes in the environmental chemistry, either in-situ or en Accepted 10 June 2016 route from soil to fluvial system, to the ocean, can cause beryllium desorption and may preclude some beryllium Available online 11 June 2016 isotopic applications. Keywords: Four mechanisms were tested to determine the relative desorption potential of beryllium including a reduction Beryllium in pH, an increase in ionic strength (NaCl) and complexation by soluble organic (malonic acid) and inorganic spe- Desorption cies (NaF). To assess the relative effect of each mechanism on beryllium desorption from both organic and min- Minerals eral fractions, we prepared separate solutions of beryllium bound to minerals and organic compounds and Organic ligands measured beryllium concentrations in solution before and after each chemical perturbation. We conclude a re- Inner and outer sphere complexation duction in pH resulted in the greatest amount of desorption among the four treatments, removing 97% and 75% of sorbed beryllium from illite and montmorillonite, respectively, and none from the organic ligands tested.
    [Show full text]
  • On the Computation of Foldings
    On the Computation of Foldings The process of determining the development (or net) of a polyhedron or of a developable surface is called unfolding and has a unique result, apart from the placement of different components in the plane. The reverse process called folding is much more complex. In the case of polyhedra it H. Stachel leads to a system of algebraic equations. A given development can Professor emeritus Institute of Discrete Mathematics correspond to several or even to infinitely many incongruent polyhedra. and Geometry The same holds also for smooth surfaces. In the paper two examples of Vienna University of Technology such foldings are presented. Austria In both cases the spatial realisations bound solids, for which mathe- matical models are required. In the first example, the cylinders with curved creases are given. In this case the involved curves can be exactly described. In the second example, even the ruling of the involved developable surface is unknown. Here, the obtained model is only an approximation. Keywords: folding, curved folding, developable surfaces, revolute surfaces of constant curvature. 1. UNFOLDING AND FOLDING surface Φ is unique, apart from the placement of the components in the plane. The unfolding induces an In Descriptive Geometry there are standard procedures isometry Φ → Φ 0 : each curve c on Φ has the same available for the construction of the development ( net length as its planar counterpart c ⊂ Φ . Hence, the or unfolding ) of polyhedra or piecewise linear surfaces, 0 0 i.e., polyhedral structures. The same holds for development shows in the plane the interior metric of developable smooth surfaces.
    [Show full text]
  • 6.007 Lecture 5: Electrostatics (Gauss's Law and Boundary
    Electrostatics (Free Space With Charges & Conductors) Reading - Shen and Kong – Ch. 9 Outline Maxwell’s Equations (In Free Space) Gauss’ Law & Faraday’s Law Applications of Gauss’ Law Electrostatic Boundary Conditions Electrostatic Energy Storage 1 Maxwell’s Equations (in Free Space with Electric Charges present) DIFFERENTIAL FORM INTEGRAL FORM E-Gauss: Faraday: H-Gauss: Ampere: Static arise when , and Maxwell’s Equations split into decoupled electrostatic and magnetostatic eqns. Electro-quasistatic and magneto-quasitatic systems arise when one (but not both) time derivative becomes important. Note that the Differential and Integral forms of Maxwell’s Equations are related through ’ ’ Stoke s Theorem and2 Gauss Theorem Charges and Currents Charge conservation and KCL for ideal nodes There can be a nonzero charge density in the absence of a current density . There can be a nonzero current density in the absence of a charge density . 3 Gauss’ Law Flux of through closed surface S = net charge inside V 4 Point Charge Example Apply Gauss’ Law in integral form making use of symmetry to find • Assume that the image charge is uniformly distributed at . Why is this important ? • Symmetry 5 Gauss’ Law Tells Us … … the electric charge can reside only on the surface of the conductor. [If charge was present inside a conductor, we can draw a Gaussian surface around that charge and the electric field in vicinity of that charge would be non-zero ! A non-zero field implies current flow through the conductor, which will transport the charge to the surface.] … there is no charge at all on the inner surface of a hollow conductor.
    [Show full text]
  • On the Motion of the Oloid Toy
    On the motion of the Oloid toy On the motion of the Oloid toy Alexander S. Kuleshov Mont Hubbard Dale L. Peterson Gilbert Gede [email protected] Abstract Analysis and simulation are performed for the commercially available toy known as the Oloid. While rolling on the fixed horizontal plane the Oloid moves very swinging but smooth: it never falls over its edges. The trajectories of points of contact of the Oloid with the supporting plane are found analytically. 1 Introduction Let us consider the motion of the Oloid on a fixed horizontal plane. The Oloid is a developable surface comprises of two circles of radius R whose planes of symmetry make a right angle between each other with the distance between the centers of the circles equals to their radius R. The resulting convex hull is called Oloid. The Oloid have been constructed for the first time by Paul Schatz [2,3]. The geometric properties of the surface of the Oloid have been discussed in the paper [1]. The Oloid is also used for technical applications. Special mixing-machines are constructed using such bodies [4]. In our paper we make the complete kinematical analysis of motion of this object on the horizontal plane. Further we briefly describe basic facts from Kinematics and Differential Geometry which we will use in our investigation. The Frenet - Serret formulas. Consider a particle which moves along a continuous differentiable curve in three - dimensional Euclidean Space 3. We can introduce the R following coordinate system: the origin of this system is in the moving particle, τ is the unit vector tangent to the curve, pointing in the direction of motion, ν is the derivative of τ with respect to the arc-length parameter of the curve, divided by its length and β is the cross product of τ and ν: β = [τ ν].
    [Show full text]
  • The Polycons: the Sphericon (Or Tetracon) Has Found Its Family
    The polycons: the sphericon (or tetracon) has found its family David Hirscha and Katherine A. Seatonb a Nachalat Binyamin Arts and Crafts Fair, Tel Aviv, Israel; b Department of Mathematics and Statistics, La Trobe University VIC 3086, Australia ARTICLE HISTORY Compiled December 23, 2019 ABSTRACT This paper introduces a new family of solids, which we call polycons, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable rollers such as the oloid and particular polysphericons. The paper concludes with a discussion of the polycons as stationary and kinetic works of art. KEYWORDS sphericon; polycons; tetracon; ruled surface; developable roller 1. Introduction In 1980 inventor David Hirsch, one of the authors of this paper, patented `a device for generating a meander motion' [9], describing the object that is now known as the sphericon. This discovery was independent of that of woodturner Colin Roberts [22], which came to public attention through the writings of Stewart [28], P¨oppe [21] and Phillips [19] almost twenty years later. The object was named for how it rolls | overall in a line (like a sphere), but with turns about its vertices and developing its whole surface (like a cone). It was realised both by members of the woodturning [17, 26] and mathematical [16, 20] communities that the sphericon could be generalised to a series of objects, called sometimes polysphericons or, when precision is required and as will be elucidated in Section 4, the (N; k)-icons. These objects are for the most part constructed from frusta of a number of cones of differing apex angle and height.
    [Show full text]
  • Chapter 5 Capacitance and Dielectrics
    Chapter 5 Capacitance and Dielectrics 5.1 Introduction...........................................................................................................5-3 5.2 Calculation of Capacitance ...................................................................................5-4 Example 5.1: Parallel-Plate Capacitor ....................................................................5-4 Interactive Simulation 5.1: Parallel-Plate Capacitor ...........................................5-6 Example 5.2: Cylindrical Capacitor........................................................................5-6 Example 5.3: Spherical Capacitor...........................................................................5-8 5.3 Capacitors in Electric Circuits ..............................................................................5-9 5.3.1 Parallel Connection......................................................................................5-10 5.3.2 Series Connection ........................................................................................5-11 Example 5.4: Equivalent Capacitance ..................................................................5-12 5.4 Storing Energy in a Capacitor.............................................................................5-13 5.4.1 Energy Density of the Electric Field............................................................5-14 Interactive Simulation 5.2: Charge Placed between Capacitor Plates..............5-14 Example 5.5: Electric Energy Density of Dry Air................................................5-15
    [Show full text]
  • Electrical Double Layer Interactions with Surface Charge Heterogeneities
    Electrical double layer interactions with surface charge heterogeneities by Christian Pick A dissertation submitted to Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy Baltimore, Maryland October 2015 © 2015 Christian Pick All rights reserved Abstract Particle deposition at solid-liquid interfaces is a critical process in a diverse number of technological systems. The surface forces governing particle deposition are typically treated within the framework of the well-known DLVO (Derjaguin-Landau- Verwey-Overbeek) theory. DLVO theory assumes of a uniform surface charge density but real surfaces often contain chemical heterogeneities that can introduce variations in surface charge density. While numerous studies have revealed a great deal on the role of charge heterogeneities in particle deposition, direct force measurement of heterogeneously charged surfaces has remained a largely unexplored area of research. Force measurements would allow for systematic investigation into the effects of charge heterogeneities on surface forces. A significant challenge with employing force measurements of heterogeneously charged surfaces is the size of the interaction area, referred to in literature as the electrostatic zone of influence. For microparticles, the size of the zone of influence is, at most, a few hundred nanometers across. Creating a surface with well-defined patterned heterogeneities within this area is out of reach of most conventional photolithographic techniques. Here, we present a means of simultaneously scaling up the electrostatic zone of influence and performing direct force measurements with micropatterned heterogeneously charged surfaces by employing the surface forces apparatus (SFA). A technique is developed here based on the vapor deposition of an aminosilane (3- aminopropyltriethoxysilane, APTES) through elastomeric membranes to create surfaces for force measurement experiments.
    [Show full text]