Chapter 3 Newtonian Fluids
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Fluid Mechanics
FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 1. INTRODUCTION Mechanics is the oldest physical science that deals with both stationary and moving bodies under the influence of forces. Mechanics is divided into three groups: a) Mechanics of rigid bodies, b) Mechanics of deformable bodies, c) Fluid mechanics Fluid mechanics deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries (Fig.1.1.). Fluid mechanics is the branch of physics which involves the study of fluids (liquids, gases, and plasmas) and the forces on them. Fluid mechanics can be divided into two. a)Fluid Statics b)Fluid Dynamics Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of water is always flat and horizontal whatever the shape of its container. Fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow— the natural science of fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). -
Fluid Mechanics
cen72367_fm.qxd 11/23/04 11:22 AM Page i FLUID MECHANICS FUNDAMENTALS AND APPLICATIONS cen72367_fm.qxd 11/23/04 11:22 AM Page ii McGRAW-HILL SERIES IN MECHANICAL ENGINEERING Alciatore and Histand: Introduction to Mechatronics and Measurement Systems Anderson: Computational Fluid Dynamics: The Basics with Applications Anderson: Fundamentals of Aerodynamics Anderson: Introduction to Flight Anderson: Modern Compressible Flow Barber: Intermediate Mechanics of Materials Beer/Johnston: Vector Mechanics for Engineers Beer/Johnston/DeWolf: Mechanics of Materials Borman and Ragland: Combustion Engineering Budynas: Advanced Strength and Applied Stress Analysis Çengel and Boles: Thermodynamics: An Engineering Approach Çengel and Cimbala: Fluid Mechanics: Fundamentals and Applications Çengel and Turner: Fundamentals of Thermal-Fluid Sciences Çengel: Heat Transfer: A Practical Approach Crespo da Silva: Intermediate Dynamics Dieter: Engineering Design: A Materials & Processing Approach Dieter: Mechanical Metallurgy Doebelin: Measurement Systems: Application & Design Dunn: Measurement & Data Analysis for Engineering & Science EDS, Inc.: I-DEAS Student Guide Hamrock/Jacobson/Schmid: Fundamentals of Machine Elements Henkel and Pense: Structure and Properties of Engineering Material Heywood: Internal Combustion Engine Fundamentals Holman: Experimental Methods for Engineers Holman: Heat Transfer Hsu: MEMS & Microsystems: Manufacture & Design Hutton: Fundamentals of Finite Element Analysis Kays/Crawford/Weigand: Convective Heat and Mass Transfer Kelly: Fundamentals -
Derivation of Fluid Flow Equations
TPG4150 Reservoir Recovery Techniques 2017 1 Fluid Flow Equations DERIVATION OF FLUID FLOW EQUATIONS Review of basic steps Generally speaking, flow equations for flow in porous materials are based on a set of mass, momentum and energy conservation equations, and constitutive equations for the fluids and the porous material involved. For simplicity, we will in the following assume isothermal conditions, so that we not have to involve an energy conservation equation. However, in cases of changing reservoir temperature, such as in the case of cold water injection into a warmer reservoir, this may be of importance. Below, equations are initially described for single phase flow in linear, one- dimensional, horizontal systems, but are later on extended to multi-phase flow in two and three dimensions, and to other coordinate systems. Conservation of mass Consider the following one dimensional rod of porous material: Mass conservation may be formulated across a control element of the slab, with one fluid of density ρ is flowing through it at a velocity u: u ρ Δx The mass balance for the control element is then written as: ⎧Mass into the⎫ ⎧Mass out of the ⎫ ⎧ Rate of change of mass⎫ ⎨ ⎬ − ⎨ ⎬ = ⎨ ⎬ , ⎩element at x ⎭ ⎩element at x + Δx⎭ ⎩ inside the element ⎭ or ∂ {uρA} − {uρA} = {φAΔxρ}. x x+ Δx ∂t Dividing by Δx, and taking the limit as Δx approaches zero, we get the conservation of mass, or continuity equation: ∂ ∂ − (Aρu) = (Aφρ). ∂x ∂t For constant cross sectional area, the continuity equation simplifies to: ∂ ∂ − (ρu) = (φρ) . ∂x ∂t Next, we need to replace the velocity term by an equation relating it to pressure gradient and fluid and rock properties, and the density and porosity terms by appropriate pressure dependent functions. -
Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow Through Porous Media
Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Permalink https://escholarship.org/uc/item/6zv599hc Author Wu, Y.S. Publication Date 1990-02-01 eScholarship.org Powered by the California Digital Library University of California Lawrence Berkeley Laboratory e UNIVERSITY OF CALIFORNIA EARTH SCIENCES DlVlSlON Theoretical Studies of Non-Newtonian and Newtonian Fluid Flow through Porous Media Y.-S. Wu (Ph.D. Thesis) February 1990 TWO-WEEK LOAN COPY This is a Library Circulating Copy which may be borrowed for two weeks. r- +. .zn Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098. :0 DISCLAIMER I I This document was prepared as an account of work sponsored ' : by the United States Government. Neither the United States : ,Government nor any agency thereof, nor The Regents of the , I Univers~tyof California, nor any of their employees, makes any I warranty, express or implied, or assumes any legal liability or ~ : responsibility for the accuracy, completeness, or usefulness of t any ~nformation, apparatus, product, or process disclosed, or I represents that its use would not infringe privately owned rights. : Reference herein to any specific commercial products process, or I service by its trade name, trademark, manufacturer, or other- I wise, does not necessarily constitute or imply its endorsement, ' recommendation, or favoring by the United States Government , or any agency thereof, or The Regents of the University of Cali- , forma. The views and opinions of authors expressed herein do ' not necessarily state or reflect those of the United States : Government or any agency thereof or The Regents of the , Univers~tyof California and shall not be used for advertismg or I product endorsement purposes. -
Fluid Inertia and End Effects in Rheometer Flows
FLUID INERTIA AND END EFFECTS IN RHEOMETER FLOWS by JASON PETER HUGHES B.Sc. (Hons) A thesis submitted to the University of Plymouth in partial fulfilment for the degree of DOCTOR OF PHILOSOPHY School of Mathematics and Statistics Faculty of Technology University of Plymouth April 1998 REFERENCE ONLY ItorriNe. 9oo365d39i Data 2 h SEP 1998 Class No.- Corrtl.No. 90 0365439 1 ACKNOWLEDGEMENTS I would like to thank my supervisors Dr. J.M. Davies, Prof. T.E.R. Jones and Dr. K. Golden for their continued support and guidance throughout the course of my studies. I also gratefully acknowledge the receipt of a H.E.F.C.E research studentship during the period of my research. AUTHORS DECLARATION At no time during the registration for the degree of Doctor of Philosophy has the author been registered for any other University award. This study was financed with the aid of a H.E.F.C.E studentship and carried out in collaboration with T.A. Instruments Ltd. Publications: 1. J.P. Hughes, T.E.R Jones, J.M. Davies, *End effects in concentric cylinder rheometry', Proc. 12"^ Int. Congress on Rheology, (1996) 391. 2. J.P. Hughes, J.M. Davies, T.E.R. Jones, ^Concentric cylinder end effects and fluid inertia effects in controlled stress rheometry, Part I: Numerical simulation', accepted for publication in J.N.N.F.M. Signed ...^.^Ms>3.\^^. Date Ik.lp.^.m FLUH) INERTIA AND END EFFECTS IN RHEOMETER FLOWS Jason Peter Hughes Abstract This thesis is concerned with the characterisation of the flow behaviour of inelastic and viscoelastic fluids in steady shear and oscillatory shear flows on commercially available rheometers. -
Lecture 1: Introduction
Lecture 1: Introduction E. J. Hinch Non-Newtonian fluids occur commonly in our world. These fluids, such as toothpaste, saliva, oils, mud and lava, exhibit a number of behaviors that are different from Newtonian fluids and have a number of additional material properties. In general, these differences arise because the fluid has a microstructure that influences the flow. In section 2, we will present a collection of some of the interesting phenomena arising from flow nonlinearities, the inhibition of stretching, elastic effects and normal stresses. In section 3 we will discuss a variety of devices for measuring material properties, a process known as rheometry. 1 Fluid Mechanical Preliminaries The equations of motion for an incompressible fluid of unit density are (for details and derivation see any text on fluid mechanics, e.g. [1]) @u + (u · r) u = r · S + F (1) @t r · u = 0 (2) where u is the velocity, S is the total stress tensor and F are the body forces. It is customary to divide the total stress into an isotropic part and a deviatoric part as in S = −pI + σ (3) where tr σ = 0. These equations are closed only if we can relate the deviatoric stress to the velocity field (the pressure field satisfies the incompressibility condition). It is common to look for local models where the stress depends only on the local gradients of the flow: σ = σ (E) where E is the rate of strain tensor 1 E = ru + ruT ; (4) 2 the symmetric part of the the velocity gradient tensor. The trace-free requirement on σ and the physical requirement of symmetry σ = σT means that there are only 5 independent components of the deviatoric stress: 3 shear stresses (the off-diagonal elements) and 2 normal stress differences (the diagonal elements constrained to sum to 0). -
A Fluid Is Defined As a Substance That Deforms Continuously Under Application of a Shearing Stress, Regardless of How Small the Stress Is
FLUID MECHANICS & BIOTRIBOLOGY CHAPTER ONE FLUID STATICS & PROPERTIES Dr. ALI NASER Fluids Definition of fluid: A fluid is defined as a substance that deforms continuously under application of a shearing stress, regardless of how small the stress is. To study the behavior of materials that act as fluids, it is useful to define a number of important fluid properties, which include density, specific weight, specific gravity, and viscosity. Density is defined as the mass per unit volume of a substance and is denoted by the Greek character ρ (rho). The SI units for ρ are kg/m3. Specific weight is defined as the weight per unit volume of a substance. The SI units for specific weight are N/m3. Specific gravity S is the ratio of the weight of a liquid at a standard reference temperature to the o weight of water. For example, the specific gravity of mercury SHg = 13.6 at 20 C. Specific gravity is a unit-less parameter. Density and specific weight are measures of the “heaviness” of a fluid. Example: What is the specific gravity of human blood, if the density of blood is 1060 kg/m3? Solution: ⁄ ⁄ Viscosity, shearing stress and shearing strain Viscosity is a measure of a fluid's resistance to flow. It describes the internal friction of a moving fluid. A fluid with large viscosity resists motion because its molecular makeup gives it a lot of internal friction. A fluid with low viscosity flows easily because its molecular makeup results in very little friction when it is in motion. Gases also have viscosity, although it is a little harder to notice it in ordinary circumstances. -
Rheology of Petroleum Fluids
ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 20, 2012 Rheology of Petroleum Fluids Hans Petter Rønningsen, Statoil, Norway ABSTRACT NEWTONIAN FLUIDS Among the areas where rheology plays In gas reservoirs, the flow properties of an important role in the oil and gas industry, the simplest petroleum fluids, i.e. the focus of this paper is on crude oil hydrocarbons with less than five carbon rheology related to production. The paper atoms, play an essential role in production. gives an overview of the broad variety of It directly impacts the productivity. The rheological behaviour, and corresponding viscosity of single compounds are well techniques for investigation, encountered defined and mixture viscosity can relatively among petroleum fluids. easily be calculated. Most often reservoir gas viscosity is though measured at reservoir INTRODUCTION conditions as part of reservoir fluid studies. Rheology plays a very important role in The behaviour is always Newtonian. The the petroleum industry, in drilling as well as main challenge in terms of measurement and production. The focus of this paper is on modelling, is related to very high pressures crude oil rheology related to production. (>1000 bar) and/or high temperatures (170- Drilling and completion fluids are not 200°C) which is encountered both in the covered. North Sea and Gulf of Mexico. Petroleum fluids are immensely complex Hydrocarbon gases also exist dissolved mixtures of hydrocarbon compounds, in liquid reservoir oils and thereby impact ranging from the simplest gases, like the fluid viscosity and productivity of these methane, to large asphaltenic molecules reservoirs. Reservoir oils are also normally with molecular weights of thousands. -
Continuity Equation in Pressure Coordinates
Continuity Equation in Pressure Coordinates Here we will derive the continuity equation from the principle that mass is conserved for a parcel following the fluid motion (i.e., there is no flow across the boundaries of the parcel). This implies that δxδyδp δM = ρ δV = ρ δxδyδz = − g is conserved following the fluid motion: 1 d(δM ) = 0 δM dt 1 d()δM = 0 δM dt g d ⎛ δxδyδp ⎞ ⎜ ⎟ = 0 δxδyδp dt ⎝ g ⎠ 1 ⎛ d(δp) d(δy) d(δx)⎞ ⎜δxδy +δxδp +δyδp ⎟ = 0 δxδyδp ⎝ dt dt dt ⎠ 1 ⎛ dp ⎞ 1 ⎛ dy ⎞ 1 ⎛ dx ⎞ δ ⎜ ⎟ + δ ⎜ ⎟ + δ ⎜ ⎟ = 0 δp ⎝ dt ⎠ δy ⎝ dt ⎠ δx ⎝ dt ⎠ Taking the limit as δx, δy, δp → 0, ∂u ∂v ∂ω Continuity equation + + = 0 in pressure ∂x ∂y ∂p coordinates 1 Determining Vertical Velocities • Typical large-scale vertical motions in the atmosphere are of the order of 0. 01-01m/s0.1 m/s. • Such motions are very difficult, if not impossible, to measure directly. Typical observational errors for wind measurements are ~1 m/s. • Quantitative estimates of vertical velocity must be inferred from quantities that can be directly measured with sufficient accuracy. Vertical Velocity in P-Coordinates The equivalent of the vertical velocity in p-coordinates is: dp ∂p r ∂p ω = = +V ⋅∇p + w dt ∂t ∂z Based on a scaling of the three terms on the r.h.s., the last term is at least an order of magnitude larger than the other two. Making the hydrostatic approximation yields ∂p ω ≈ w = −ρgw ∂z Typical large-scale values: for w, 0.01 m/s = 1 cm/s for ω, 0.1 Pa/s = 1 μbar/s 2 The Kinematic Method By integrating the continuity equation in (x,y,p) coordinates, ω can be obtained from the mean divergence in a layer: ⎛ ∂u ∂v ⎞ ∂ω ⎜ + ⎟ + = 0 continuity equation in (x,y,p) coordinates ⎝ ∂x ∂y ⎠ p ∂p p2 p2 ⎛ ∂u ∂v ⎞ ∂ω = − ⎜ + ⎟ ∂p rearrange and integrate over the layer ∫p ∫ ⎜ ⎟ 1 ∂x ∂y p1⎝ ⎠ p ⎛ ∂u ∂v ⎞ ω(p )−ω(p ) = (p − p )⎜ + ⎟ overbar denotes pressure- 2 1 1 2 ⎜ ⎟ weighted vertical average ⎝ ∂x ∂y ⎠ p To determine vertical motion at a pressure level p2, assume that p1 = surface pressure and there is no vertical motion at the surface. -
Lecture 2: (Complex) Fluid Mechanics for Physicists
Application of granular jamming: robots! Cornell (Amend and Lipson groups) in collaboration with Univ of1 Chicago (Jaeger group) Lecture 2: (complex) fluid mechanics for physicists S-RSI Physics Lectures: Soft Condensed Matter Physics Jacinta C. Conrad University of Houston 2012 Note: I have added links addressing questions and topics from lectures at: http://conradlab.chee.uh.edu/srsi_links.html Email me questions/comments/suggestions! 2 Soft condensed matter physics • Lecture 1: statistical mechanics and phase transitions via colloids • Lecture 2: (complex) fluid mechanics for physicists • Lecture 3: physics of bacteria motility • Lecture 4: viscoelasticity and cell mechanics • Lecture 5: Dr. Conrad!s work 3 Big question for today!s lecture How does the fluid mechanics of complex fluids differ from that of simple fluids? Petroleum Food products Examples of complex fluids: Personal care products Ceramic precursors Paints and coatings 4 Topic 1: shear thickening 5 Forces and pressures A force causes an object to change velocity (either in magnitude or direction) or to deform (i.e. bend, stretch). � dp� Newton!s second law: � F = m�a = dt p� = m�v net force change in linear mass momentum over time d�v �a = acceleration dt A pressure is a force/unit area applied perpendicular to an object. Example: wind blowing on your hand. direction of pressure force �n : unit vector normal to the surface 6 Stress A stress is a force per unit area that is measured on an infinitely small area. Because forces have three directions and surfaces have three orientations, there are nine components of stress. z z Example of a normal stress: δA δAx x δFx τxx = lim τxy δFy δAx→0 δAx τxx δFx y y Example of a shear stress: δFy τxy = lim δAx→0 δAx x x Convention: first subscript indicates plane on which stress acts; second subscript indicates direction in which the stress acts. -
7. Fluid Mechanics
7. Fluid mechanics Introduction In this chapter we will study the mechanics of fluids. A non-viscous fluid, i.e. fluid with no inner friction is called an ideal fluid. The stress tensor for an ideal fluid is given by T1= −p , which is the most simplest form of the stress tensor used in fluid mechanics. A viscous material is a material for which the stress tensor depends on the rate of deformation D. If this relation is linear then we have linear viscous fluid or Newtonian fluid. The stress tensor for a linear viscous fluid is given by T = - p 1 + λ tr D 1 + 2μ D , where λ and μ are material constants. If we assume the fluid to be incompressible, i.e. tr D 1 = 0 and the stress tensor can be simplified to contain only one material coefficient μ . Most of the fluids can not be modelled by these two simple types of stress state assumption and they are called non-Newtonian fluids. There are different types of non-Newtonian fluids. We have constitutive models for fluids of differential, rate and integral types. Examples non-Newtonian fluids are Reiner-Rivlin fluids, Rivlin-Ericksen fluids and Maxwell fluids. In this chapter we will concentrate on ideal and Newtonian fluids and we will study some classical examples of hydrodynamics. 7.1 The general equations of motion for an ideal fluid For an arbitrary fluid (liquid or gas) the stress state can be describe by the following constitutive relation, T1= −p (7.1) where p= p (r ,t) is the fluid pressure and T is the stress. -
Chapter 15 - Fluid Mechanics Thursday, March 24Th
Chapter 15 - Fluid Mechanics Thursday, March 24th •Fluids – Static properties • Density and pressure • Hydrostatic equilibrium • Archimedes principle and buoyancy •Fluid Motion • The continuity equation • Bernoulli’s effect •Demonstration, iClicker and example problems Reading: pages 243 to 255 in text book (Chapter 15) Definitions: Density Pressure, ρ , is defined as force per unit area: Mass M ρ = = [Units – kg.m-3] Volume V Definition of mass – 1 kg is the mass of 1 liter (10-3 m3) of pure water. Therefore, density of water given by: Mass 1 kg 3 −3 ρH O = = 3 3 = 10 kg ⋅m 2 Volume 10− m Definitions: Pressure (p ) Pressure, p, is defined as force per unit area: Force F p = = [Units – N.m-2, or Pascal (Pa)] Area A Atmospheric pressure (1 atm.) is equal to 101325 N.m-2. 1 pound per square inch (1 psi) is equal to: 1 psi = 6944 Pa = 0.068 atm 1atm = 14.7 psi Definitions: Pressure (p ) Pressure, p, is defined as force per unit area: Force F p = = [Units – N.m-2, or Pascal (Pa)] Area A Pressure in Fluids Pressure, " p, is defined as force per unit area: # Force F p = = [Units – N.m-2, or Pascal (Pa)] " A8" rea A + $ In the presence of gravity, pressure in a static+ 8" fluid increases with depth. " – This allows an upward pressure force " to balance the downward gravitational force. + " $ – This condition is hydrostatic equilibrium. – Incompressible fluids like liquids have constant density; for them, pressure as a function of depth h is p p gh = 0+ρ p0 = pressure at surface " + Pressure in Fluids Pressure, p, is defined as force per unit area: Force F p = = [Units – N.m-2, or Pascal (Pa)] Area A In the presence of gravity, pressure in a static fluid increases with depth.