/Thumb 301 0 R The Kelvin equation is based on the assumptions that the vapour behaves like an ideal gas, the liquid phase is incompressible, the pressure is constant inside the drop and the surface tension. The Young-Laplace equation gives the pressure difference across a curved surface and its most important application is in the derivation of the Kelvin equation. The exact Kelvin equation is deduced from the equilibrium condition of liquid drops. /Type /Annot 22 0 obj /MediaBox [0 0 544.252 742.677] From the former, the equilibrium contact angles of drops on an outer or inner surface of a cylindrical capillary have been calculated as a function of surface curvature, showing that the expressions for equilibrium contact angles vary for different geometries, in view of the difference in thickness of the film of uniform thickness with which the bulk liquid (drops or menisci) is at equilibrium. (equation.\(3.6\)) 117 0 R (equation.\(3.6a\)) 118 0 R (equation.\(3.6b\)) 119 0 R (equation.\(3.6c\)) 120 0 R (equation.\(3.7\)) 121 0 R << Derivation Short heuristic derivation Schrdinger's equation can be derived in the following short heuristic way. /Rect [102.255 97.5756 190.993 108.883] 5 0 obj >> /Dests 11 0 R /Filter /FlateDecode /Border [0 0 0] Appendix 2: Derivation of Young-Laplace and Kelvin Equations . /Thumb 271 0 R /Resources 233 0 R /Rotate 0 Derivation of Kelvin's Equation The vapor pressure of nitrogen has been measured in the liquid range below the normal The normal bOIling pOl.nt calculated from thIs equatIOn IS 77.364 K . /Rotate 0 endobj /CropBox [0 0 544.252 742.677] /S /URI /Type /Pages denoted in and/or out) the Gibbs free energy function is: the total Gibbs free energy function of a mixture, is the internal energy of the system. /CropBox [0 0 544.252 742.677] /Type /Annot /CropBox [0 0 544.252 742.677] /Last 12 0 R 1.15). /CropBox [0 0 544.252 742.677] Stratham Hill Stone . endobj << Various mathematical switches that depend on use of the Heaviside function. /Rect [397.83 627.555 504.382 635.094] Derivation of the Boltzmann integral follows in 4 steps. 1. Elliot University of Alberta Abstract The Kelvin equation quantifies the deviation in equilibrium vapor pressure above a curved surface from that which would exist above a plane surface at the same temperature. /Kids [33 0 R 34 0 R 35 0 R] Equation (22) is the equation of the reduced inversion curve. To browse Academia.edu and the wider internet faster and more securely, please take a few seconds toupgrade your browser. /Type /Page Derivation of the Kelvin Equation YAN Hong, WANG Xiao-Song, ZHU Ru-Zeng PDF 3527 Like Abstract Abstract: The exact Kelvin equation is deduced from the equilibrium condition of liquid. The Kelvin equation quantifies the deviation in equi librium vapor pressure above a curved surface from that which would exist above a plane surface at the same temperature. Instead, the disjoining pressure modifies the growth rate of the droplet height, which then follows a power law of 1/4. (1.33) into eqn. Kelvin-Voigt model. Course Hero is not sponsored or endorsed by any college or university. This paper will present the conditions for the stability of liquid films on and inside cylindrical solid substrates with nanometre scale characteristic dimensions. /MediaBox [0 0 544.252 742.677] endobj 40 0 obj Semantic Scholar extracted view of "The Kelvin equation." by A. Mitropoulos. /Parent 15 0 R 27 0 obj Academia.edu uses cookies to personalize content, tailor ads and improve the user experience. REVIEW OF THEORY Kelvin derived equation (1) by considering the equilibrium of a liquid contained in a capillary tube s/tuated in a sealed vessel as shown in Fig. /Type /Page /H /I endobj TU Guo-Yun; WANG Zheng-Wu; WANG Zhong-Ni; LIU Feng; XIAO Jian-Yi. /Contents 191 0 R Curvature radius. /Thumb 232 0 R 1 0 obj /CropBox [0 0 544.252 742.677] << The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. 19 0 obj We investigate the axisymmetric homogeneous growth of 10-100 nm water nanodroplets on a substrate surface. << /A << /H /I /Type /Page >> . 25 Issue (04): 640-644.doi: 10.3866/PKU.WHXB20090422, ARTICLE /Contents 255 0 R stream Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. endobj << 2.2 The Derivation of Maxwell Equations In this section we derive the Maxwell equations based of the differentiation form of a number of physical principles. /Type /Outlines stream << You can download the paper by clicking the button above. Role of Interfacial Properties, Effect of Oily Additives on Foamability and Foam Stability. 7 0 obj >> >> Problem 1: Under normal conditions . /A 174 0 R 2007-11-21T10:51:29+05:30 Consider a liquid droplet surrounded by its vapors (Fig. /First 12 0 R Physics. 9 0 obj /H /I /A << 3 0 obj /ModDate (D:20071121105129+05'30') /Subject () (5), and recognising from these equations that P c > > ( P v - P o). /MediaBox [0 0 544.252 742.677] (5), and recognising Based on an analysis internal to the tube, from these equations that Pc > > (Pv Po ). 10 0 obj Derivation of the Ideal Gas Equation Let us consider the pressure exerted by the gas to be ' p ' The volume of the gas be - ' v ' Temperature be - " T" "n" - be the number of moles of gas Universal gas constant - R According to Boyle's Law, At constant n & T, the volume bears an inverse relation with the pressure exerted by a gas. << Janet A.W. Assumptions 1. /Type /Page The final form is: Derivation of the Energy Equation: The energy equation is a generalized form of the first law of Thermodynamics (that you studied in ME3322 and AE 3004). /D (cor*) The purpose of this note is to outline this derivation for a gas in a spherical container. (equation.\(3.1\)) 112 0 R (equation.\(3.2\)) 113 0 R (equation.\(3.3\)) 114 0 R (equation.\(3.4\)) 115 0 R (equation.\(3.5\)) 116 0 R >> 24 0 obj >> << /Keywords () /H /I /Count 5 >> /Contents 214 0 R The Kelvin equation for the equilibrium vapour pressure outside a drop of liquid is widely quoted but its derivation is subject to three assumptions which are not obviously justified. 3 Properties of the Kelvin-Voigt Model The Kelvin-Voigt model is on the form (t)=G(t)+ d(t) dt, (1) where G is the material's shear storage modulus and is its viscosity. LIU Qing-Shan, YANG Miao, TAN Zhi-Cheng, WELZ-BIERMANN Urs. End of preview. /Contents 270 0 R endobj << (cite.b1) 77 0 R (cite.b10) 78 0 R (cite.b11) 79 0 R (cite.b12) 80 0 R (cite.b2) 81 0 R The main mechanism of droplet growth is attributed to the accumulation of laterally diffusing water monomers, formed by the absorption of water vapour in the environment onto the substrate. /Count 6 /Parent 13 0 R 2. For this purpose the general form of the Kelvin equation is derived. Ludwig Eduard Boltzmann (February 20, 1844 - September 5, 1906), an Austrian physicist famous for the invention of statistical mechanics. The Kelvin equation for a compressible liquid is also given while the compression coefficient and the Tolman length are introduced into the expression. /Border [0 0 0] pdfeTeX-1.304 /Author () 30 0 obj endobj YOU Yi, DENG Yong-Shu, LI Er-Jun, PEI Xiao-Mei, ZHAO Jian-Xi. << This video is related to Kelvin Equation derivation, applications of Kelvin Equation, Young and Laplace equation, effect of particle size on solubility, and . These calculations have been extended to the case of glass nanocapillaries and carbon nanotubes, finding good agreement with experimental results in the literature. For the bulk phases (i.e. endobj Concept Introduction: Ideal gas is the most usually used form of the ideal gas equation, which describes the relationship among the four variables P, V, n, and T. . << << (1.37). /Names 3 0 R /Length 745 /MediaBox [0 0 544.252 742.677] Acta Physico-chimica Sinica. /Resources 56 0 R By using our site, you agree to our collection of information through the use of cookies. >> << /MediaBox [0 0 544.252 742.677] Angular frequency [rad/s]. For the bulk phases (i.e. 41 0 obj /S /GoTo For given analytic data, Sulem, Sulem, Bardos, and Frisch >> 13 0 obj doi:10.1016/j.camwa.2007.03.008 Posted at 04:35h in examples of participant observation in psychology by cold imagery examples. the satl!ratlOn hoe are represented by log pT= 3.39858- 282.953j (T - 3.83). /, Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks, URL: /H /I >> /Border [0 0 0] Standard Linear Model. Elsevier /Thumb 238 0 R 01 Nov November 1, 2022. thermodynamic relations derivation. /Type /Page (section.2) 167 0 R (section.3) 168 0 R (section.4) 169 0 R (section.5) 170 0 R (title.0) 171 0 R] The European Physical Journal Special Topics. /Resources 267 0 R Ideal Gas Practice Problems. >> DOI: 10.1016/J.JCIS.2007.10.001; /Parent 12 0 R /Annots [268 0 R 269 0 R] Along with the Laplace equation, the Kelvin equation constitutes the fundamental relationship of surface chemistry [4], and its validity has been experimentally checked under a wide range of conditions [9-11]. << 21 0 obj The exact Kelvin equation is deduced from the equilibrium condition of liquid drops. /Rect [314.715 306.668 331.651 317.611] /Thumb 205 0 R /Contents 204 0 R endobj It predicts that undersaturated vapours will condense in channels of . Consequently some doubt has recently been thrown on the value of the equation for estimating the true vapour pressure. f The Kelvin Equation (Chapter 4, pp. (Section.1) 72 0 R (Section.2) 73 0 R (Section.3) 74 0 R (Section.4) 75 0 R (affn1) 76 0 R endobj 42 0 obj (cite.b3) 82 0 R (cite.b4) 83 0 R (cite.b5) 84 0 R (cite.b6) 85 0 R (cite.b7) 86 0 R Key words: 2 0 obj /CropBox [0 0 544.252 742.677] /D (cite.b1) i.e., u0001 u0002 Ph = Po + (Pv Po ) + Pc (2) M 2u0002 Pv = Po exp . Influence of the Disjoining Pressure on the Equilibrium Interfacial Profile in Transition Zone Between a Thin Film and a Capillary Meniscus, The Superspreading Effect of Trisiloxane Surfactant Solutions, Origin of line tension for a Lennard-Jones nanodroplet, Foamoil interaction in porous media: Implications for foam assisted enhanced oil recovery, IUPAC Technical Report, Recommendations for the Characterization of Porous Solids, Chemical physics of colloid systems and interfaces, The modified Thomson equation in the theory of heterogeneous vapor nucleation on charged solid particles, Adsorption and capillary condensation in porous media: Liquid retention and interfacial configurations in angular pores, The influence of spreading particles on the stability of thin liquid films, 9076Emulsions Foams and Suspensions - Fundamentals and Applications, INTERMOLECULAR AND SURFACE FORCES by Jacob N. Israelachvili, Thermal Conductivity and Operating Temperature Effect on the Interline Region in a Micro/Miniature Heat Pipe, Assessment of Hydrodynamic and Molecular-Kinetic Models Applied to the Motion of the Dewetting Contact Line between a Small Bubble and a Solid Surface, Barrat et al._Basic Concepts for simple and complex liquids, CAPILLARY LIQUID TRANSPORT IN POROUS MEDIA, ABSTRACT INSTABILITIES OF VOLATILE FILMS AND DROPS, The Integument of Water-walking Arthropods: Form and Function, Theoretical Models for Surface Forces and Adhesion and Their Measurement Using Atomic Force Microscopy, Effects of surfactant adsorption and surface forces on thinning and rupture of foam liquid films. If the surface of a liquid is curved, there must be a pressure difference: Young-Laplace Equation . /PageLayout /SinglePage >> endobj /Annots [287 0 R 288 0 R 289 0 R 290 0 R 291 0 R 292 0 R 293 0 R 294 0 R 295 0 R 296 0 R] endobj /Parent 13 0 R 37 0 obj /Parent 6 0 R that the equation is valid to a few per . Fahrenheit to Kelvin conversion is done by using a simple formula i.e. Fahrenheit and Kelvin are the two very commonly known and used scales for temperature. 18 0 obj Yanyan GONG,Haichun LIU,Duo ZHANG,Jing TONG. 3.4 The Vorticity Equation Kelvin's circulation theorem provides us with a constraint on the circulation around a material curve but it doesn't tell us what's happening to the circulation at a localised point. Mathematical Modelling of Natural Phenomena, Colloids and Surfaces A: Physicochemical and Engineering Aspects, Progress in Nuclear Magnetic Resonance , Colloids and Interface Science Communications, Advances in Colloid and Interface Science, ASME 2008 First International Conference on Micro/Nanoscale Heat Transfer, Parts A and B, International Journal of Mineral Processing, IHTC15-KN02 MULTI-SCALE INTERFACIAL PHENOMENA AND HEAT TRANSFER ENHANCEMENT, Equilibrium and Dynamics of Evaporating or Condensing Thin Fluid Domains: Thin Film Stability and Heterogeneous Nucleation, Thickness, stability and contact angle of liquid films on and inside nanofibres, nanotubes and nanochannels, Wetting in the nanoscale: A continuum mechanics approach, Deformation of Carbon Nanotubes by Exposure to Water Vapor, Numerical study of homogeneous nanodroplet growth, Computer Simulations of Evaporation of Pinned Sessile Droplets: Influence of Kinetic Effects, Evaporation of pinned sessile microdroplets of water on a highly heat-conductive substrate: Computer simulations, Prediction of self-desiccation in low water-to-cement ratio pastes based on pore structure evolution, Evaporation of Sessile Water Droplets in Presence of Contact Angle Hysteresis, Evaporation of sessile water droplets: Universal behaviour in presence of contact angle hysteresis, Stress from NaCl crystallisation by carbon dioxide injection in aquifers, A pore-level scenario for the development of mixed wettability in oil reservoirs, Selected Aspects of Atmospheric Ice and Salt Crystallisation, Effect of Oily Additives on Foamability and Foam Stability. It is a crucial part in chemical kinetics. 8`.pe$H+5&ttVKApQ )P^:aM^,zr})|?sCYcyxMUjC?$I%)e-i!x{nxY hX`Ie+nY*3oa_1XchZ]Jhe-qY1E6rj''qD /Type /Action 26 octubre octubre 16 0 obj /D (title.0) /Annots [258 0 R 259 0 R 260 0 R 261 0 R 262 0 R 263 0 R 264 0 R] endobj An equation similar to that of Kelvin can be derived for the solubility of small particles or droplets in a liquid, by means of the connection between vapour pressure and solubility, thus the Kelvin equation also applies to solids, to slightly soluble liquids, and their solutions if the partial pressure is replaced by the solubility of the solid (or a second liquid) at the given radius, , and (equation.-49) 142 0 R (equation.-52) 143 0 R (equation.-6) 144 0 R (equation.-61) 145 0 R (equation.-62) 146 0 R /CropBox [0 0 544.252 742.677] << YANG Bai-Qin, CHEN Kai, XING Hang, XIAO Jin-Xin. << /Annots [36 0 R 37 0 R 38 0 R 39 0 R 40 0 R 41 0 R 42 0 R 43 0 R 44 0 R] /Rect [293.809 246.892 300.783 257.836] Monday - Friday: 8am-5pm Saturday - Sunday: 8am-2pm The mathematical derivation for Kelvin waves can be found in Knauss. /CropBox [0 0 544.252 742.677] /Subtype /Link This equation was first introduced by Svente Arrhenius in 1889. /Type /Annot /Annots [240 0 R 241 0 R 242 0 R 243 0 R 244 0 R 245 0 R 246 0 R 247 0 R] /Parent 12 0 R The total energy E of a particle is 39 0 obj This is pdfeTeX, Version 3.141592-1.30.4-2.2 (Web2C 7.5.5) kpathsea version 3.5.5 endobj Kelvin Waves Next we will look at Kelvin waves, waves for which rotation is important and that depend on the existence of a boundary. endobj 20 0 obj 1. Equation (1.4) has been under active investigation for the last four decades. /Names [(Doc-Start) 57 0 R (Fig..1) 58 0 R (Fig..2) 59 0 R (Fig..3) 60 0 R (Proposition.1) 61 0 R /Type /Pages 11 0 obj >> (equation.-19) 132 0 R (equation.-21) 133 0 R (equation.-26) 134 0 R (equation.-3) 135 0 R (equation.-35) 136 0 R The Kelvin equation 1 relates the equilibrium vapour pressure of a liquid to the curvature of the liquid-vapour interface. /Rotate 0 /CropBox [0 0 544.252 742.677] << /Parent 6 0 R /S /D >> They are important in tidal wave propagation along boundaries, in wind-driven variability in the coastal ocean, and in El Nio. endobj The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. Droplets of water in from a /Resources 250 0 R W. Xiao. << /Parent 13 0 R /Length 3985 >> maxwell relations thermodynamics derivation pdf. >> This means that it retains its shape as it moves . Sign In Create Free Account. The Kelvin equation describes the change in vapour pressure due to a curved liquid-vapor interface, such as the surface of a droplet. /Contents 231 0 R /Title (Some mathematical aspects of the Kelvin equation) Kelvin equation, /Annots [273 0 R 274 0 R 275 0 R 276 0 R 277 0 R 278 0 R 279 0 R 280 0 R 281 0 R 282 0 R /Parent 14 0 R Kelvin. /Count 6 /S /URI 67-69 in Shaw) /FICL#3AEnfocus 2 0 R This is also sometimes called the . It helps to understand the impact of temperature on the rate of reaction. (Proposition.6) 67 0 R (Proposition.7) 68 0 R (Proposition.8) 69 0 R (Proposition.9) 70 0 R (Section.) These switches can be used to "cut-off" or start a /Type /Page /Parent 14 0 R /C [1 0 0] << endobj endobj Kelvin Plank Statement . >> /Border [0 0 0] /Thumb 198 0 R >> /Resources 299 0 R endobj /Type /Annot /Resources 216 0 R >> Enter the email address you signed up with and we'll email you a reset link. >> /Thumb 215 0 R /Length 542 Because of equal strain in both elememts, the model is also known as . X1+rfJJ^` $ /A 172 0 R /Thumb 228 0 R /Parent 15 0 R >> /C [0 1 1] /Contents 237 0 R >> /Resources 257 0 R Derivation of the Kelvin Equation[J].Acta Phys. /Resources 272 0 R endobj The usual expression for the Kelvin equation is obtained by incorporating P c = 2 / r into Eq. /Subtype /XML http://www.whxb.pku.edu.cn/EN/10.3866/PKU.WHXB20090422, http://www.whxb.pku.edu.cn/EN/Y2009/V25/I04/640, http://www.whxb.pku.edu.cn/EN/10.3866/PKU.WHXB20090422, Thermodynamic Properties of the Ether-Based Functionalized Ionic Liquids [MOEMIm]Cl and [EOEMIm]Cl, Physicochemical Properties of 1-Methoxyethyl-3-Methylimidazolium Glycine, Molar Surface Gibbs Free Energy of the Aqueous Solution of the Ionic Liquid [C, Thermodynamic Properties and Predicting the Surface Tension of Pyridinium-Based Ionic Liquids of [C, Volume and Surface Properties of Aqueous [C, Dynamic Surface Adsorption Properties of Sodium Dodecyl Sulfate Aqueous Solution, Hydrophobic Synergism between the Alkyl Tails of Gemini Surfactants during Adsorption and Aggregation, Estimation and Prediction of the Physicochemical Properties of Imidazolium-Based Ionic Liquids, Stability of Avermyctin Emulsion in Water, Perfluorobutyl-Based Fluorinated Surfactant with High Surface Activity, Adsorption of Photosensitive Quaternary Ammonium Gemini Surfactant a4-6-, Synthesis and Solution Properties of PDMS-, Theoretical Expression and Experiment on Surface Tension in Ideal Binary Mixtures of Surfactants, Thermodynamic Properties of Ionic Liquid BMIAlCl, Surfactivity and Micellization of Disodium Hexadecyl Diphenyl Ether Disulfonate in Aqueous Solution. /MediaBox [0 0 544.252 742.677] /CropBox [0 0 544.252 742.677] /Rotate 0 warframe infested colors; minnesota twins retired numbers; luxury party bus rental near tokyo 23 wards, tokyo; oklahoma weather almanac; walker edison annual revenue; titan bullies chalino; (page.52) 152 0 R (page.53) 153 0 R (page.54) 154 0 R (page.55) 155 0 R (page.56) 156 0 R /Contents 197 0 R /CropBox [0 0 544.252 742.677] 32 0 obj The vapor pressure of a curved surface is higher than that of a flat and non-curved surface. (page.62) 162 0 R (page.63) 163 0 R (page.64) 164 0 R (page.65) 165 0 R (section.1) 166 0 R >> /Title (doi:10.1016/j.camwa.2007.03.008) >> /Subtype /Link endobj Born . /S /GoTo /PTEX.Fullbanner (This is pdfeTeX, Version 3.141592-1.30.4-2.2 \(Web2C 7.5.5\) kpathsea version 3.5.5) In order to derive the Kelvin equation on thermodynamic grounds, consider the transfer of d moles of vapor in equilibrium with the bulk liquid at pressure Pq into a pore where the equilibrium pressure is P. This process consists of three steps evaporation from the bulk liquid, expansion of the vapor from Pq to P and condensation into the pore. /Metadata 5 0 R /Border [0 0 0] /Contents 248 0 R 25 0 obj endobj >> It will be convenient to dene the dual Kelvin transform K as adjoint operation, acting on measures on Rd 0: Rd 0 f(y)K(dy)= Rd 0 Kf(y)(dy)= Rd 0 |y| . 71 0 R /MediaBox [0 0 544.252 742.677] /C [0 1 1] << /MediaBox [0 0 544.252 742.677] By . /Rotate 0 /Parent 15 0 R Chapter 1: Interfacial Thermodynamics Part IIA, The general form of the Kelvin equation: Correlation between infinitesimal, The Young-Laplace equation (1.30) gives the equilibrium pressure difference across. The more rigorous equation (1.2) does not admit a simple solution in terms of elementary Vapour pressures, obtained from the Kelvin equation, were closely matched to the experimental outcomes even in extended confined channels (\(\sim 8 \,{\text{nm}}\)) making the equation applicable . << /Parent 14 0 R Posted on October 26, 2022 by . /CropBox [0 0 544.252 742.677] /Type /Pages /Annots [251 0 R 252 0 R 253 0 R 254 0 R] /Rotate 0 34 0 obj >> << >> Saturation vapor pressure, However, you can also think of the Voigt or Maxwell . First, we discuss the Ampere's law. /Type /Annot >> Entry Barriers, Observation and Modelling of Fluid Transport into Porous Paper Coating Structures, Effect of Surfactant on Interfacial Gas Transfer Studied by Axisymmetric Drop Shape AnalysisCaptive Bubble (ADSA-CB), A disjoining pressure valid for non-zero contact angles, A slope-dependent disjoining pressure for non-zero contact angles, Thickness measurements of nanoscale brine films on silica surfaces under geologic CO 2 sequestration conditions using synchrotron X-ray fluorescence, Droplets evaporation: Problems and solutions, Nano-and Microstructures for Thin-Film Evaporation-A Review, FUNDAMENTALS OF PETROLEUM RESERVOIR ENGINEERING, Combined Capillary Pressure and Resistivity Index Measurements in Tight Gas Sands Using Vapor Desorption Method. Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. The Kelvin equation describes the change in vapour pressure due to a curved liquidvapor interface, such as the surface of a droplet. /Type /Page /Thumb 249 0 R /Thumb 298 0 R D = S. /H /I /Type /Page /A 303 0 R >> endobj State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China, Editorial Office of Acta Physico-Chimica Sinica (, Address: School of Chemistry and Molecular Engineering, Peking University. In KHI, the velocity and density profiles are uniform in each fluid layer, but they are discontinuous at the (plane) interface between the two fluids. >> /Rect [246.062 554.204 253.183 560.793] endobj >> /Rotate 0 endobj (equation.\(2.17\)) 102 0 R (equation.\(2.18\)) 103 0 R (equation.\(2.2\)) 104 0 R (equation.\(2.3\)) 105 0 R (equation.\(2.4\)) 106 0 R 12 0 obj /Thumb 55 0 R >> /Resources 229 0 R model is a Maxwell model in parallel with an elastic element.