Rheology Bulletin 2010, 79(2)
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Chapter 5 Dimensional Analysis and Similarity
Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences. -
An Investigation Into the Effects of Viscoelasticity on Cavitation Bubble Dynamics with Applications to Biomedicine
An Investigation into the Effects of Viscoelasticity on Cavitation Bubble Dynamics with Applications to Biomedicine Michael Jason Walters School of Mathematics Cardiff University A thesis submitted for the degree of Doctor of Philosophy 9th April 2015 Summary In this thesis, the dynamics of microbubbles in viscoelastic fluids are investigated nu- merically. By neglecting the bulk viscosity of the fluid, the viscoelastic effects can be introduced through a boundary condition at the bubble surface thus alleviating the need to calculate stresses within the fluid. Assuming the surrounding fluid is incompressible and irrotational, the Rayleigh-Plesset equation is solved to give the motion of a spherically symmetric bubble. For a freely oscillating spherical bubble, the fluid viscosity is shown to dampen oscillations for both a linear Jeffreys and an Oldroyd-B fluid. This model is also modified to consider a spherical encapsulated microbubble (EMB). The fluid rheology affects an EMB in a similar manner to a cavitation bubble, albeit on a smaller scale. To model a cavity near a rigid wall, a new, non-singular formulation of the boundary element method is presented. The non-singular formulation is shown to be significantly more stable than the standard formulation. It is found that the fluid rheology often inhibits the formation of a liquid jet but that the dynamics are governed by a compe- tition between viscous, elastic and inertial forces as well as surface tension. Interesting behaviour such as cusping is observed in some cases. The non-singular boundary element method is also extended to model the bubble tran- sitioning to a toroidal form. -
RHEOLOGY #2: Anelasicity
RHEOLOGY #2: Anelas2city (aenuaon and modulus dispersion) of rocks, an organic, and maybe some ice Chris2ne McCarthy Lamont-Doherty Earth Observatory …but first, cheese Team Havar2 Team Gouda Team Jack Stress and Strain Stress σ(MPa)=F(N)/A(m2) 1 kg = 9.8N 1 Pa= N/m2 or kg/(m s2) Strain ε = Δl/l0 = (l0-l)/l0 l0 Cheese results vs. idealized curve. Not that far off! Cheese results σ n ⎛ −E + PV ⎞ ε = A exp A d p ⎝⎜ RT ⎠⎟ n=1 Newtonian! σ Pa viscosity η = ε s-1 Havarti,Jack η=3*107 Pa s Gouda η=2*108 Pa s Muenster η=6*108 Pa s How do we compare with previous studies? Havarti,Jack η=3*107 Pa s Gouda η=2*108 Pa s Muenster η=6*108 Pa s Despite significant error, not far off published results Viscoelas2city: Deformaon at a range of 2me scales Viscoelas2city: Deformaon at a range of 2me scales Viscoelas2city Elas2c behavior is Viscous behavior; strain rate is instantaneous elas2city and propor2onal to stress: instantaneous recovery. σ = ηε Follows Hooke’s Law: σ = E ε Steady-state viscosity Elas1c Modulus k or E ηSS Simplest form of viscoelas2city is the Maxwell model: t 1 J(t) = + ηSS kE SS kE Viscoelas2city How do we measure viscosity and elascity in the lab? Steady-state viscosity Elas1c Modulus k or EU ηSS σ σ η = η = effective [Fujisawa & Takei, 2009] ε ε1 Viscoelas2city: in between the two extremes? Viscoelas2city: in between the two extremes? Icy satellites velocity (at grounding line) tidal signal glaciers velocity (m per day) (m per velocity Vertical position (m) Vertical Day of year 2000 Anelas2c behavior in Earth and Planetary science -
Computational Study of Jeffreyв€™S Non-Newtonian Fluid Past a Semi
Ain Shams Engineering Journal (2016) xxx, xxx–xxx Ain Shams University Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com ENGINEERING PHYSICS AND MATHEMATICS Computational study of Jeffrey’s non-Newtonian fluid past a semi-infinite vertical plate with thermal radiation and heat generation/absorption S. Abdul Gaffar a,*, V. Ramachandra Prasad b, E. Keshava Reddy a a Department of Mathematics, Jawaharlal Nehru Technological University Anantapur, Anantapuramu, India b Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle 517325, India Received 7 October 2015; revised 13 August 2016; accepted 2 September 2016 KEYWORDS Abstract The nonlinear, steady state boundary layer flow, heat and mass transfer of an incom- Non-Newtonian Jeffrey’s pressible non-Newtonian Jeffrey’s fluid past a semi-infinite vertical plate is examined in this article. fluid model; The transformed conservation equations are solved numerically subject to physically appropriate Semi-infinite vertical plate; boundary conditions using a versatile, implicit finite-difference Keller box technique. The influence Deborah number; of a number of emerging non-dimensional parameters, namely Deborah number (De), ratio of Heat generation; relaxation to retardation times (k), Buoyancy ratio parameter (N), suction/injection parameter Thermal radiation; (fw), Radiation parameter (F), Prandtl number (Pr), Schmidt number (Sc), heat generation/absorp- Retardation time tion parameter (D) and dimensionless tangential coordinate (n) on velocity, temperature and con- centration evolution in the boundary layer regime is examined in detail. Also, the effects of these parameters on surface heat transfer rate, mass transfer rate and local skin friction are investigated. This model finds applications in metallurgical materials processing, chemical engineering flow con- trol, etc. -
Similitude and Theory of Models - Washington Braga
EXPERIMENTAL MECHANICS - Similitude And Theory Of Models - Washington Braga SIMILITUDE AND THEORY OF MODELS Washington Braga Mechanical Engineering Department, Pontifical Catholic University, Rio de Janeiro, RJ, Brazil Keywords: similarity, dimensional analysis, similarity variables, scaling laws. Contents 1. Introduction 2. Dimensional Analysis 2.1. Application 2.2 Typical Dimensionless Numbers 3. Models 4. Similarity – a formal definition 4.1 Similarity Variables 5. Scaling Analysis 6. Conclusion Glossary Bibliography Biographical Sketch Summary The concepts of Similitude, Dimensional Analysis and Theory of Models are presented and used in this chapter. They constitute important theoretical tools that allow scientists from many different areas to go further on their studies prior to actual experiments or using small scale models. The applications discussed herein are focused on thermal sciences (Heat Transfer and Fluid Mechanics). Using a formal approach based on Buckingham’s π -theorem, the paper offers an overview of the use of Dimensional Analysis to help plan experiments and consolidate data. Furthermore, it discusses dimensionless numbers and the Theory of Models, and presents a brief introduction to Scaling Laws. UNESCO – EOLSS 1. Introduction Generally speaking, similitude is recognized through some sort of comparison: observing someSAMPLE relationship (called similarity CHAPTERS) among persons (for instance, relatives), things (for instance, large commercial jets and small executive ones) or the physical phenomena we are interested. -
The Effect of Induced Magnetic Field and Convective Boundary Condition
Propulsion and Power Research 2016;5(2):164–175 HOSTED BY http://ppr.buaa.edu.cn/ Propulsion and Power Research www.sciencedirect.com ORIGINAL ARTICLE The effect of induced magnetic field and convective boundary condition on MHD stagnation point flow and heat transfer of upper-convected Maxwell fluid in the presence of nanoparticle past a stretching sheet Wubshet Ibrahimn Department of Mathematics, Ambo University, P.O. Box 19, Ambo, Ethiopia Received 5 February 2015; accepted 17 July 2015 Available online 30 May 2016 KEYWORDS Abstract The present study examines the effect of induced magnetic field and convective fl fl boundary condition on magnetohydrodynamic (MHD) stagnation point ow and heat transfer due Nano uid; fl Stagnation point flow; to upper-convected Maxwell uid over a stretching sheet in the presence of nanoparticles. fi Heat transfer; Boundary layer theory is used to simplify the equation of motion, induced magnetic eld, energy Convective boundary and concentration which results in four coupled non-linear ordinary differential equations. The condition; study takes into account the effect of Brownian motion and thermophoresis parameters. The Induced magnetic governing equations and their associated boundary conditions are initially cast into dimensionless field; form by similarity variables. The resulting system of equations is then solved numerically using Upper-convected Max- fourth order Runge-Kutta-Fehlberg method along with shooting technique. The solution for the well fluid governing equations depends on parameters such as, magnetic, velocity ratio parameter B,Biot number Bi, Prandtl number Pr, Lewis number Le, Brownian motion Nb, reciprocal of magnetic Prandtl number A, the thermophoresis parameter Nt, and Maxwell parameter β. -
Chapter 8 Dimensional Analysis and Similitude
Chapter 8 Dimensional Analysis and Similitude Ahmad Sana Department of Civil and Architectural Engineering Sultan Qaboos University Sultanate of Oman Email: [email protected] Webpage: http://ahmadsana.tripod.com Significant learning outcomes Conceptual Knowledge State the Buckingham Π theorem. Identify and explain the significance of the common π-groups. Distinguish between model and prototype. Explain the concepts of dynamic and geometric similitude. Procedural Knowledge Apply the Buckingham Π theorem to determine number of dimensionless variables. Apply the step-by-step procedure to determine the dimensionless π- groups. Apply the exponent method to determine the dimensionless π-groups. Distinguish the significant π-groups for a given a flow problem. Applications (typical) Drag force on a blimp from model testing. Ship model tests to evaluate wave and friction drag. Pressure drop in a prototype nozzle from model measurements. CIVL 4046 Fluid Mechanics 2 8.1 Need for dimensional analysis • Experimental studies in fluid problems • Model and prototype • Example: Flow through inverted nozzle CIVL 4046 Fluid Mechanics 3 Pressure drop through the nozzle can shown as: p p d V d 1 2 f 0 , 1 0 2 V / 2 d1 p p d For higher Reynolds numbers 1 2 f 0 V 2 / 2 d 1 CIVL 4046 Fluid Mechanics 4 8.2 Buckingham pi theorem In 1915 Buckingham showed that the number of independent dimensionless groups of variables (dimensionless parameters) needed to correlate the variables in a given process is equal to n - m, where n is the number of variables involved and m is the number of basic dimensions included in the variables. -
On Dimensionless Numbers
chemical engineering research and design 8 6 (2008) 835–868 Contents lists available at ScienceDirect Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd Review On dimensionless numbers M.C. Ruzicka ∗ Department of Multiphase Reactors, Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502 Prague, Czech Republic This contribution is dedicated to Kamil Admiral´ Wichterle, a professor of chemical engineering, who admitted to feel a bit lost in the jungle of the dimensionless numbers, in our seminar at “Za Plıhalovic´ ohradou” abstract The goal is to provide a little review on dimensionless numbers, commonly encountered in chemical engineering. Both their sources are considered: dimensional analysis and scaling of governing equations with boundary con- ditions. The numbers produced by scaling of equation are presented for transport of momentum, heat and mass. Momentum transport is considered in both single-phase and multi-phase flows. The numbers obtained are assigned the physical meaning, and their mutual relations are highlighted. Certain drawbacks of building correlations based on dimensionless numbers are pointed out. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Dimensionless numbers; Dimensional analysis; Scaling of equations; Scaling of boundary conditions; Single-phase flow; Multi-phase flow; Correlations Contents 1. Introduction ................................................................................................................. -
Dimensional Analysis and Similitude Lecture 39: Geomteric and Dynamic Similarities, Examples
Objectives_template Module 11: Dimensional analysis and similitude Lecture 39: Geomteric and dynamic similarities, examples Dimensional analysis and similitude–continued Similitude: file:///D|/Web%20Course/Dr.%20Nishith%20Verma/local%20server/fluid_mechanics/lecture39/39_1.htm[5/9/2012 3:44:14 PM] Objectives_template Module 11: Dimensional analysis and similitude Lecture 39: Geomteric and dynamic similarities, examples Dimensional analysis and similitude–continued Example 2: pressure–drop in pipe–flow depends on length, inside diameter, velocity, density and viscosity of the fluid. If the roughness-effects are ignored, determine a symbolic expression for the pressure–drop using dimensional analysis. Answer: We will apply Buckingham Pi-theorem Variables: Primary dimensions: No of dimensionless (independent) group: file:///D|/Web%20Course/Dr.%20Nishith%20Verma/local%20server/fluid_mechanics/lecture39/39_2.htm[5/9/2012 3:44:14 PM] Objectives_template Module 11: Dimensional analysis and similitude Lecture 39: Geomteric and dynamic similarities, examples Similitude: To scale–up or down a model to the prototype, two types of similarities are required from the perspective of fluid dynamics: (1) geometrical similarity (2) dynamic similarity 1. Geometric similarity: The model and the prototype must be similar in shape. (Fig. 39a) This is essential because one can use a constant scale factor to relate the dimensions of model and prototype. 2. Dynamic similarity: The flow conditions in two cases are such that all forces (pressure viscous, surface tension, etc) must be parallel and may also be scaled by a constant scaled factor at all corresponding points. Such requirement is restrictive and may be difficult to implement under certain experiential conditions. Dimensional analysis can be used to identify the dimensional groups to achieve dynamic similarity between geometrically similar flows. -
Rheology and Mixing of Suspension and Pastes
RHEOLOGY AND MIXING OF SUSPENSION AND PASTES Pr Ange NZIHOU, EMAC France USACH, March 2006 PLAN 1- Rheology and Reactors Reactor performance problems caused by rheological behaviors of suspensions et pastes 2- Rheology of complex fluids Definition Classification of mixtures Non-Newtonian behaviors Behavior laws of viscoplastic fluids Thixotropy Viscosity equations Rheological measurements 3-Factors influencing the rheological behavior of fluids 4- Mixing of pastes in agitated vessels Agitator and utilization Geometric parameters Dimensional numbers Dimensionless numbers 1- Rheology and Reactor DESIGN OF REACTOR FOR SCALE UP Flux Production Flux input output Accumulation Mass balance: ⎛⎞Aj ⎛⎞AAj,,in ⎛j out ⎞⎛Aj ⎞ ⎜⎟+=⎜⎟⎜ ⎟+⎜ ⎟ Flux ⎜⎟Flux Accumulation ⎝⎠⎝⎠Production ⎝ ⎠⎝ ⎠ 1 DIMENSIONS OF REACTOR IN VIEW OF SCALE CHANGE PERFORMANCE OF REACTOR: Thermodynamic and kinetic Hydrodynamic of the reaction Operating parameters: Composition Nature of reagents Conversion rate Pressure, temperature RTD Concentrations R Output Flow In Out Residence time Mass and heat transfer Geometric of reactor SIMILARITY PRINCIPLE: Geometric similitude Energetic similitude Kinematic similitude Thermal similitude 2 ENCOUNTERED PROBLEMS WITH REACTOR Existence of dead matter and recirculation: Stagnant fluid R Recirculation Presence of preferred passages R OBJECTIVE: Correct the flows or take it into consideration while designing the reactor 3 Ribbon impellers (agitators) for mixing Complex fluids 4 anchor Helicoidal ribbon Archemedian ribbon impeller 5 6 PLAN -
An Introduction to Dimensional Analysis David Dureisseix
An introduction to dimensional analysis David Dureisseix To cite this version: David Dureisseix. An introduction to dimensional analysis. Engineering school. Lyon, France. 2016, pp.20. cel-01380149v3 HAL Id: cel-01380149 https://cel.archives-ouvertes.fr/cel-01380149v3 Submitted on 12 Apr 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International License An introduction to dimensional analysis David Dureisseix D´epartement G´enieM´ecanique, INSA de Lyon April 12, 2019 This document is a short (and hopefully concise) introduction to dimensional analysis and is not expected to be printed. Indeed, it relies on URL links (in colored text) to refer to information sources and complementary studies, so it does not provide a large bibliography, nor many pictures. It has been realized with the kind help of Ton Lubrecht and Marie-Pierre Noutary. Photography by KoS, 2008, distributed under a CC BY-SA 3.0 license 1 Contents 1 Goals of dimensional analysis3 2 Physical quantities and -
Dimensional Analysis and Similitude
Fluid Mechanics Chapter 8 Dimensional Analysis and Similitude Dr. Amer Khalil Ababneh Introduction Because of the complexity of fluid mechanics, the design of many fluid systems relies heavily on experimental results. Tests are typically carried out on a subscale model, and the results are extrapolated to the full-scale system (prototype). The principles underlying the correspondence between the model and the prototype are addressed in this chapter. Dimensional analysis is the process of grouping of variables into significant dimensionless groups, thus reducing problem complexity. Similitude (Similarity) is the process by which geometric and dynamic parameters are selected for the subscale model so that meaningful correspondence can be made to the full size prototype. 8.2 Buckingham Π Theorem In 1915 Buckingham showed that the number of independent dimensionless groups (dimensionless parameters) can be reduced from a set of variables in a given process is n - m, where n is the number of variables involved and m is the number of basic dimensions included in the variables. Buckingham referred to the dimensionless groups as Π, which is the reason the theorem is called the Π theorem. Henceforth dimensionless groups will be referred to as π-groups. If the equation describing a physical system has n dimensional variables and is expressed as then it can be rearranged and expressed in terms of (n - m) π- groups as 1 ( 2 , 3 ,..., nm ) Example If there are five variables (F, V, ρ, μ, and D) to describe the drag on a sphere and three basic dimensions (L, M, and T) are involved. By the Buckingham Π theorem there will be 5 - 3 = 2 π-groups that can be used to correlate experimental results in the form F= f(V, r, m, D) 8.3 Dimensional Analysis Dimensional analysis is the process used to obtain the π-groups.