Hydromechanics E-Book

Emmanuil G. Sinaiski

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 776

Veröffentlichungsjahr: 2011

Beschreibung

Written by an experienced author with a strong background in applications of this field, this monograph provides a comprehensive and detailed account of the theory behind hydromechanics. He includes numerous appendices with mathematical tools, backed by extensive illustrations. The result is a must-have for all those needing to apply the methods in their research, be it in industry or academia.

Leseprobe

Contents

Cover

Related Titles

Title page

Copyright page

Dedication

Preface

List of Symbols

Chapter 1: Introduction

1.1 Goals and Methods of Continuum Mechanics

1.2 The Main Hypotheses of Continuum Mechanics

Chapter 2: Kinematics of the Deformed Continuum

2.1 Dynamics of the Continuum in the Lagrangian Perspective

2.2 Dynamics of the Continuum in the Eulerian Perspective

2.3 Scalar and Vector Fields and Their Characteristics

2.4 Theory of Strains

2.5 The Tensor of Strain Velocities

2.6 The Distribution of Velocities in an Infinitesimal Continuum Particle

2.7 Properties of Vector Fields. Theorems of Stokes and Gauss

Chapter 3: Dynamic Equations of Continuum Mechanics

3.1 Equation of Continuity

3.2 Equations of Motion

3.3 Equation of Motion for the Angular Momentum

Chapter 4: Closed Systems of Mechanical Equations for the Simplest Continuum Models

4.1 Ideal Fluid and Gas

4.2 Linear Elastic Body and Linear Viscous Fluid

4.3 Equations in Curvilinear Coordinates

Chapter 5: Foundations and Main Equations of Thermodynamics

5.1 Theorem of the Living Forces

5.2 Law of Conservation of Energy and First Law of Thermodynamics

5.3 Thermodynamic Equilibrium, Reversible and Irreversible Processes

5.4 Two Parameter Media and Ideal Gas

5.5 The Second Law of Thermodynamics and the Concept of Entropy

5.6 Thermodynamic Potentials of Two-Parameter Media

5.7 Examples of Ideal and Viscous Media, and Their Thermodynamic Properties, Heat Conduction

5.8 First and Second Law of Thermodynamics for a Finite Continuum Volume

5.9 Generalized Thermodynamic Forces and Currents, Onsager’s Reciprocity Relations

Chapter 6: Problems Posed in Continuum Mechanics

6.1 Initial Conditions and Boundary Conditions

6.2 Typical Simplifications for Some Problems

6.3 Conditions on the Discontinuity Surfaces

6.4 Discontinuity Surfaces in Ideal Compressible Media

6.5 Dimensions of Physical Quantities

6.6 Parameters that Determine the Class of the Phenomenon

6.7 Similarity and Modeling of Phenomena

Chapter 7: Hydrostatics

7.1 Equilibrium Equations

7.2 Equilibrium in the Gravitational Field

7.3 Force and Moment that Act on a Body from the Surrounding Fluid

7.4 Equilibrium of a Fluid Relative to a Moving System of Coordinates

Chapter 8: Stationary Continuum Movement of an Ideal Fluid

8.1 Bernoulli’s Integral

8.2 Examples of the Application of Bernoulli’s Integral

8.3 Dynamic and Hydrostatic Pressure

8.4 Flow of an Incompressible Fluid in a Tube of Varying Cross Section

8.5 The Phenomenon of Cavitation

8.6 Bernoulli’s Integral for Adiabatic Flows of an Ideal Gas

8.7 Bernoulli’s Integral for the Flow of a Compressible Gas

Chapter 9: Application of the Integral Relations on Finite Volumes within the Continuum for a Stationary Movement

9.1 Integral Relations

9.2 Interaction of Fluids and Gases with Bodies Immersed in the Flow

Chapter 10: Potential Flows for Incompressible Fluids

10.1 The Cauchy–Lagrange Integral

10.2 Some Applications for the General Theory of Potential Flows

10.3 Potential Movements for an Incompressible Fluid

10.4 Movement of a Sphere in the Unlimited Volume of an Ideal, Incompressible Fluid

10.5 Kinematic Problem of the Movement of a Solid Body in the Unlimited Volume of an Incompressible Fluid

10.6 Energy, Movement Parameters and Moments of Movement Parameters for a Fluid during the Movement of a Solid Body in the Fluid

Chapter 11: Stationary Potential Flows of an Incompressible Fluid in the Plane

11.1 Method of Complex Variables

11.2 Examples of Potential Flows in the Plane

11.3 Application of the Method of Conformal Mapping to the Solution of Potential Flows around a Body

11.4 Examples of the Application of the Method of Conformal Mapping

11.5 Main Moment and Main Vector of the Pressure Force Exerted on a Hydrofoil Profile

Chapter 12: Movement of an Ideal Compressible Gas

12.1 Movement of an Ideal Gas Under Small Perturbations

12.2 Propagation of Waves with Finite Amplitude

12.3 Plane Vortex-Free Flow of an Ideal Compressible Gas

12.4 Subsonic Flow around a Thin Profile

12.5 Supersonic Flow around a Thin Profile

Chapter 13: Dynamics of the Viscous Incompressible Fluid

13.1 Rheological Laws of the Viscous Incompressible Fluid

13.2 Equations of the Newtonian Viscous Fluid and Similarity Numbers

13.3 Integral Formulation for the Effect of Viscous Fluids on a Moving Body

13.4 Stationary Flow of a Viscous Incompressible Fluid in a Tube

13.5 Oscillating Laminar Flow of a Viscous Fluid through a Tube

13.6 Simplification of the Navier–Stokes Equations

Chapter 14: Flow of a Viscous Incompressible Fluid for Small Reynolds Numbers

14.1 General Properties of Stokes Flows

14.2 Flow of a Viscous Fluid around a Sphere

14.3 Creeping Spatial Flow of a Viscous Incompressible Fluid

Chapter 15: The Laminar Boundary Layer

15.1 Equation of Motion for the Fluid in the Boundary Layer

15.2 Asymptotic Boundary Layer on a Plate

15.3 Problem of the Injected Beam

Chapter 16: Turbulent Flow of Fluid

16.1 General Information on Laminar and Turbulent Flows

16.2 Momentum Equation of a Viscous Incompressible Fluid

16.3 Equations of Heat Inflow, Heat Conduction and Diffusion

16.4 The Condition for the Beginning of Turbulence

16.5 Hydrodynamic Instability

16.6 The Reynolds Equations

16.7 The Equation of Turbulent Energy Balance

16.8 Isotropic Turbulence

16.9 The Local Structure of Fully Developed Turbulence

16.10 Models of Turbulent Flow

References

Appendix A: Foundations of Vectorial and Tensorial Analysis

A.1 Vectors

A.2 Tensors

A.3 Curvilinear Systems of Coordinates and Physical Components

A.4 Calculation of Lengths, Surface Areas and Volumes

A.5 Differential Operators and Integral Theorems

Appendix B: Some Differential Geometry

B.1 Curves on a Plane

B.2 Vectorial Definition of Curves

B.3 Curvature of a Curve in the Plane

B.4 Curves in Space

B.5 Curvature of Spatial Curves

B.6 Surfaces in Space

B.7 Fundamental Forms of the Surface

B.8 Curvature of a Curve on the Surface

B.9 Internal Geometry of a Surface

B.10 Surface Vectors

B.11 Geodetic Lines on a Surface

B.12 Vector Fields on the Surface

B.13 Hybrid Tensors

Appendix C: Foundations of Probability Theory

C.1 Events and Set of Events

C.2 Probability

C.3 Common and Conditional Probability, Independent Events

C.4 Random Variables

C.5 Distribution of Probability Density and Mean Values

C.6 Generalized Functions

C.7 Methods of Averaging

C.8 Characteristic Function

C.9 Moments and Cumulants of Random Quantities

C.10 Correlation Functions

C.11 Poisson, Bernoulli and Gaussian Distributions

C.12 Stationary Random Functions and Homogeneous Random Fields

C.13 Isotropic Random Fields

C.14 Stochastic Processes, Markovian Processes and the Integral Equation of Chapman–Kolmogorov

C.15 Differential Equations of Chapman–Kolmogorov, Kolmogorov–Feller, Fokker–Planck and Liouville

C.16 Stochastic Differential Equations and the Langevin Equation

Appendix D: Basics of Complex Analysis

D.1 Complex Numbers

D.2 Complex Variables

D.3 Elementary Functions

D.4 Integration of Complex Variable Functions

D.5 Representation of a Function as a Series

D.6 Singular Points

D.7 Conformal Transformations

D.8 Application of the Theory of Complex Variables to Boundary-Value Problems

D.9 Physical Representations and Formulation of Problems

References to Appendix

Index

Related Titles

Leonov, E.G., Isaev, V.I.

Applied Hydroaeromechanics in Oil and Gas Drilling

2009

ISBN: 978-0-470-48756-3

Sinaiski, E.G., Zaichik, L.I.

Statistical Microhydrodynamics

2008

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Zaichik, L.I., Alipchenkov, V.M., Sinaiski, E.G.

Particles in Turbulent Flow

2008

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Sinaiski, E.G., Lapiga, E.J.

Separation of Multiphase, Multicomponent Systems

2007

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Kee, R.J., Coltrin, M.E., Glarborg, P.

Chemically Reacting Flow

Theory and Practice

2003

ISBN: 978-0-471-26179-7

Gregory, G.A., Radus, C. (Eds.)

Multiphase Technology (British Hydromechanics Research

(BHR) Group), Publication 40

2000

ISBN: 978-1-86058-252-3

The Author

Prof. Dr. Emmanuil Sinaiski

Leipzig, Germany

The Translator

Prof. Moritz Braun

University of South Africa, Dept. of Physics

Unisa, Republic of South Africa

Cover picture

by Volker Weitbrecht, Zürich

Influence of Dead-Water Zones on the

Dispersive Mass Transport in Rivers

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Dedication

Devoted to the memory of my teachers, academicians of the USSR Academy of Science, Leonid Ivanivitch Sedov and Georgiy Ivanivitch Petrov

Preface

Continuum mechanics is a part of mechanics devoted to the motion of gaseous, liquid and solid deformable matters. In theoretical mechanics, one studies the motion of mass points, of a discrete system of mass points and of absolute solid bodies. Continuum mechanics addresses the motion of the mass point filling the space continuously with the change of distances between mass points during the motion with the help of, and on the basis of methods developed in theoretical mechanics,

In addition to ordinary material media (gas, fluid, solid body), continuum mechanics is concerned with special (peculiar) media, for example, electromagnetic field, radiation field and so on.

Let us indicate some developed problems of continuum mechanics which gave rise to a self-reliant line of investigation.

The forces acting on a body as viewed from the surrounding fluid or gas are determined by the motion of fluid or gas. Thus, the study of motion of the body in fluid or gas is connected with the motion of a fluid. The last problem is the essence of one of the areas in continuum mechanics, that is, hydrodynamics. The problems of motion of gas form the field of gas dynamics. A special impetus in the rapid progression of both of these parts of a continuum medium provide problems regarding the motion of airplanes, helicopters, rockets, ships, submarines and so on.

The problems regarding the motion of fluid and gas in pipes and inside different machines are directly relevant to the designing and calculation of the gas- and oil-pipelines, the turbines, the compressors, and other hydraulic engines. The applied problems in this region form the basis of hydraulics.

The motion of fluid and gas through soil and other porous media (filtration) plays a great role in oil- and gas-field problems. The field devoted to investigations of fluid and gas flows in porous media is called underground hydromechanics.

Great theoretical and applied importance has the problem of wave propagation in various media: in solid bodies, in fluid and in gas. This field of continuum mechanics is called wave mechanics.

One task of continuum mechanics includes facing problems of flows of multi-component, multi-phase mixtures with regard to interaction between phases, with mass- and heat-exchanges, phase transitions, chemical reactions. These queries form the basis of physicochemical hydrodynamics.

The problems of motion of conducting and charged mixtures (fluid, gas, plasma) in electric and magnetic fields enter into the direction of magneto- hydrodynamics and electro-hydrodynamics.

The study of motion of bodies in rarefied gas, in space, in the atmosphere of stars and planets is also possible with methods of continuum mechanics. The problems associated with such motions are confronted by the mechanics of rarefied gas and space hydrodynamics.

Recently, a new direction within mechanic has been formed, that is, biomechanics, in which one investigates the mechanics of biological objects, in particular, the motion of blood in human organs, the contraction of muscles and, the task of constructing mechanical models of internal human organs.

Other areas that employ the methods of continuum mechanics are the weather forecast, the theory of turbulence, the motion of sand in a desert, the motion of snow-slip, the motion of mixtures with chemical reactions, the theory of detonation and many others.

A major part of continuum mechanics is devoted to the investigation of motion and the equilibrium of deformable bodies. The theory of elasticity is a basis for calculations regarding construction and machinery design. Ever-growing importance is focused on the realms of continuum mechanics dedicated to studying the elastic properties of the body of complex composition, accounting for inelastic properties in solid bodies. The theory of plasticity examines residual deformations in a body. The hereditary property of a substance is an internal property that in some types of matter is retained throughout its motion. In this sense, it is analogous to the hereditary properties in living organisms that are retained throughout their growth and development. With the advent of new materials with complex internal structure, for example, polymers, composites and others, came into demand to develop models of these materials with regard to their internal structure.

It is impossible to list all of the problems and applications of continuum mechanics. However, even the above-mentioned examples are enough to make a conclusion that continuum mechanics embraces a great range of theoretical and applied problems of science and engineering.

In a theoretical course in continuum mechanics, the motions of deformable media are considered. A set of notions which characterizes and uniquely determines the motion of a continuum medium are introduced. As examples serve concepts of velocity field, pressure, temperature and so on. In continuum mechanics, there are derived methods to reduce mechanical problems to the mathematical ones. The solution of the latter allows one to obtain the general properties and the laws of motion of deformable bodies.

Continuum mechanics has had a profound impact on the development of a variety of lines in mathematics. For example, the theory of airfoil has a pronounced effect on the progress of some divisions of the functions of complex variables, the problem of viscous fluid flow has stimulated investigations in the field of boundary-value problems of the equations with partial derivatives, some problems of the elasticity theory – in the line of the theory of the integral equations and many others. A great influence on the advance of the numerical methods has served the solution of boundary-value problems of Navier-Stokes equations describing the flow of viscous fluids. The solution of the majority of applied problems in continuum mechanics are at present impossible without use of numerical methods and high-performance computers.

The book, bringing to the reader’s notice, contains theoretical fundamentals of a part of continuum mechanics, hydromechanics. It is based on lecture course, delivered for students of Gubkin State University of Oil and Gas, Russia, Moscow.

Leipzig, October 2010

Emmanuil G. Sinaiski

List of Symbols

aVelocity of sounda0Velocity of sound in the medium at resta*Stagnation speed of sound[a]Dimension of quantity aaAcceleration vectora∞Velocity of sound at infinityComponents of acceleration in cylindrical coordinatesComponents of acceleration in spherical coordinatesAWork done by the system on the external bodyAVector field, driving forceWork done by external forcesWork done by internal forcesJacobianAijklComponents of a tensor of fourth orderbWidth of a beambijStructure functionbLL(r)Longitudinal structure functionbNN(r)Transverse structure functionBPointBMatrix, solenoidal vector field (Δ×B=0)Bij(r, t)Components of the correlation tensor of second orderBij, k(r, t)Correlation tensor of third orderBijklComponents of a tensor of fourth orderBLL(r, t)Longitudinal correlation functionBNN(r, t)Transverse structure functionMoment of N-th degreeBer(z), Bei(z)Kelvin functioncVelocity, profile chordcfFriction coefficientcpSpecific heat at constant pressurecυSpecific heat at constant volumecyLift coefficientcy0Lift coefficient in an incompressible fluidCPoint, contour, concentration of a passive additiveCMatrixC1, C2, C3Constants of integration, parametersElements of an orthogonal matrixCkContourCpPressure coefficientDCurrent volume, velocity of the shock waveDMolecular diffusion coefficientD0Initial volumed/dt, D/DtSubstantial/total derivativedA(e)Infinitesimal work done by external macroscopic mass and surface forces, Infinitesimal work done by external macroscopic forcesInfinitesimal work done by external surface forcesdA(i)Infinitesimal work done by internal mass and surface forcesInfinitesimal work done by internal surface forcesdEChange of kinetic energydPForce acting on the surface element dΣdq(e)Infinitesimal inflow of external energydq(i)Density of work done by internal surface forcesdq**Energy inflow per mass unitExternal energy inflow per mass unitdQ′Non-compensated heat, dissipation of mechanical energydQ(e)Infinitesimal heat flow from/to the outsidedQ*Infinitesimal heat inflow from the outsidedQ**External energy inflowdsLinear vector element along a curvedeSIncrement of entropy due to external processesdiSIncrement of entropy due to internal processesdT/dsDerivative in the direction sdUDensity of internal energydUmChange of internal energydσkInfinitesimal surface elementsdΣSurface elementdτInfinitesimal volumeDDisplacement velocity of the discontinuity surfaceDtCoefficient of turbulent diffusionEYoung’s modulus of elasticity, kinetic energy of the medium volume V, explosive energyEV2/2Density of kinetic energyE(k, t)Spectrum of mean energyEtTurbulent energyeDensity of internal energyeiMain components of the strain velocity tensoreiBasis vectors of the observer’s system of coordinates, basis vectors of the Cartesian system of coordinateseijComponents of the strain velocity tensorekiComponents of a symmetric tensor, components of velocity of relative extension of a distance along the Xi axis, velocity of angular reduction for a right angleeρVelocity of relative extension of a distance within a deforming bodyEStrain tensorStrain velocity tensorE°Initial strain tensorÊCurrent strain tensorEkDensity of kinetic energyEsDensity of the kinetic energy of averaged turbulent movementEtAverage density of the kinetic energy of the fluctuating movementEuEuler’s numberFFree energyF(x)Harmonic functionF*Mean value of the function FFForce, main vector of mass forcesFiEffective force on i-th material pointi-th component of external forcei-th component of internal forceFαGeneralized thermodynamic forceFij(k, t)Components of the spectral tensorFLL(k, t)Longitudinal spectrumFNN(k, t)Transverse spectrumFij, k(k)Spectrum of the correlation tensor of third order of an isotropic velocity fieldFsurfForce due to surface tensionfinDensity of inertial forcefmForce per unit massfVForce per unit volumefX, fY, fZCartesian components of the force acting on the unit volumeAmplitudeFrFroude numbergAcceleration due to gravityDeterminants of the fundamental matricesgij, gksComponents of the metric tensorComponents of the metric tensor in the initial stategωAmplitude at frequency ωGMass flow of fluid in beam, shear modulusGMetric tensor, shear force, Archimedes buoyancyG′Centripetal attractive forceGBGravity of a bodyGDGreen’s function for the Dirichlet problemGNGreen’s function for the Neumann problemhMass force pairs per unit massiEnthalpyi*Stagnation enthalpyi, j, kBasis unit vectors of the Cartesian system of coordinatesIMoment of intertia of a sphereI1, I2, I3Invariants of the stress tensorInvariants of the strain tensor E°Invariants of the strain tensor ÊI1(e)First invariant of the strain velocity tensorI1(ε)First invariant of the strain tensorIαGeneralized thermodynamic currentsIUnit tensorIiDiffusive current of the i-th component of a mixtureIm(z)Imaginary part of a complex numberIm(ω)Imaginary part of angular frequencyJ0Beam momentumkScalar, Boltzmann constant, wave numberk, nConstants in the Ostwald–Reiner rheological equationkPermeability of a layerkDensity of proper or internal angular momentak1, k2, …, knPhysical constantsKAngular momentum of a point, angular momentum of a volumeK*Inertial system of coordinatesK*Angular momentum of all points in the volume V relative to the center of mass O*lCoefficient of relative stretching, inertial radius of the sphere relative to center of rotation, mixing length in the Prandtl model, microscale of fluctuationsl, m, nDirection cosines of the normal vectorl0Internal microscale, Kolmogorov microscalel1Characteristic length in the Taylor modelliCoefficients of relative stretching in the direction of the axes ξiLCurrent line, characteristic linear scale, linear body measureL0Initial lineL1Longitudinal integral scaleL2Transverse integral scaleLαβQuadratic symmetric matrixmMass, porositymBMass of a bodymiMass of the i-th componentmnCoefficients of the Laurent seriesΔmMass of an infinitesimal volume ΔVMMach numberMPointMTorque of a force, resulting torque, main torqueM′PointM∞Mach number at infinitymVMomentummiViMomentum of the i-th material pointnNumber of components, polytropic exponentnUnit normal vector(n, t1, t2Orthogonal basis vectors of the local system of coordinatesniComponents of the unit normal vector, covariant components of the external unit normal vectornkNormal vector to elementary surface dσkO*Center of mass of a bodypDensity of surface forcespPressureMean value of the pressurep1, p2, p3Main components of the stress tensorpDensity of surface forcesp1, p2, p3Stresses at the point M on surface elements parallel to coordinate planespiStress vector on the surface element with unit normal vector eipdCavitation pressurepdynDynamic pressurep′Small pressure perturbationpcapCapillary pressurepstHydrostatic pressurep*Dynamic pressurepiMomentumpkiComponents of the expansion of the vector pi in basis vectors of the Cartesian system of coordinates, contravariant components of the stress tensorpminMinimum pressurepnDensity of surface forces at the point M, stress on the surface element at the point MpnnNormal component of the force of internal stressespntTangential component of the force of internal stressespx, py, pzComponents of the stress vectorPMatrix of pki, stress tensor, resulting force (main vector)Δ·PDivergence of a vectorP:ETotal scalar product of the stress tensor and the strain velocity tensorPMeasure of the probability for a state under consideration, pressure functionΔpPressure gradientPeDDiffusion Peclet numberPeTThermal Peclet numberPrPrandtl numberqVector of heat flow through an elementary surfaceqεDensity of internal heat sourcesQVolume flow per second, strength of a stream tube, flow through a stream tube, volume flow, flow through a contourQMomentum of a system of material pointsQmV*Momentum of the center of massqSurface mass densityqnSurface force pair per unit surfacerRadial coordinate, polar coordinaterRadius vectorRMain vectorr*Radius vector of the center of massRRadius of a continuum sphere, gas constantricRadius vector of the i-th point relative to the center of massr, , ZCylindrical coordinatesr, , Spherical coordinatesdrInfinitesimal element of the radius vector, differentialΔrIncrement of the radius vectorR1, R2Main radii of curvatureRyBuoyancyReReynolds numberRcrCritical Reynolds numberRe(z)Real part of a complex numberRelReynolds number of the microscale lsDensity of entropysVectors°Unit vectorSEntropy, current surfaceSVector of an entropy currentS0Initial surfaceSminMinimal cross section of a tubeScSchmidt numberShStrouhal numbertTimetTangential vectort1, t2Unit tangential vectorsTTemperatureTSpherical tensorT*Stagnation temperatureΔTGradient of a scalar function, vector gradientu, v, wVelocity components in the space point (X, Y, Z)Mean value of the velocityMean value of the velocity vectoru(X, t)Velocity vectoruiComponents of the velocity vector in Cartesian coordinatesu′Small perturbation of the velocity vectorulVelocity on the micro scale lSmall perturbations of the velocity componentsUDensity of internal energyU∞Flow velocity at infinityU, V, WVelocity components in Cartesian coordinatesU∞, V∞Velocity components at infinityUiPhysical components of the velocity vectorUr, U, UzVelocity components in cylindrical coordinatesVVelocity vector, velocity vector of the whole mixtureVVolume, volume in the current stateV*Moving volumeV′VolumeV*Velocity of the center of mass, velocity relative to the moving system of coordinates K*V0Volume in the initial stateV0Translatory velocityViComponents of the velocity vectorViVelocity of the i-th component of the mixture, (i=1, 2, …, n)V*Velocity of pure deformation, velocity of the center of massVcVelocity of the center of mass of a system of material points, velocity of the point M relative to O*VcrCritical velocityVvoidPore volumeWTotal energy flowVdefVelocity of the pure deformationVicVelocity of the i-th point relative to the center of mass system of coordinatesVmaxMaximum flow velocityVnNormal component of the velocityVrotRotating velocityVSComponent of the vector V in the direction swComplex potentialW(Z)Characteristic functionWDisplacement vector, resistance force of the bodyWavAverage velocityWkContravariant component of the displacement vectorXSpace pointX, Y, ZCoordinates of a point in Cartesian coordinatesXiCoordinates of a pointCartesian system of coordinatesYh(X)Contour of a bodyzComplex variableαAngleβAngle of inclination of the shock waveΓCirculation of the vector along the contour, velocity circulation along a closed contour, circulation of the vector along the contourΓCStrength of a vortex tubeChristoffel symbol of the second kindγAdiabatic exponent, density of vortices per unit lengthΔDeterminantΔAngular aperture of a wedge, thickness of a boundary layerΔ*Thickness of a laminar layer/tLocal derivativeKronecker symbolΔijComponents of the unit tensorεInfinitesimal quantity, dissipative function, small parameter, angle of attack of a profile, shear strainMean specific energy dissipationε0Initial deformationε1ε2Basis of the accompanying system of coordinatesεiBasis vectors of the accompanying (deformed, frozen, curvilinear) coordinate systemsBasis vectorεiMain components (main values) of the strain tensorεiVectors of the covariant basisεijComponents of the strain tensorShear velocityεkjiComponents of the permutation symbolMain values in an initial stateMain values in a current stateεsSpecific energy dissipation of the average turbulent movement due to viscous forcesMean specific energy dissipation of the fluctuating movementςCoefficient of volume viscosityηSimilar variable, dimensionless coordinateηiCoordinates in an orthogonal system of coordinatesCoefficients of volume expansion, angle, inclination angle to the X-axis of the velocity behind the shock wave, inclination angle of the tangent to the profileκHeat conductivityκtCoefficient of turbulent heat conductivityΛFunction of the state parameters of a system, total energy of the system, Loizianski integral, Loizianski invariantλEigenvalue, scalar, Lamé elasticity constant, dimensionless parameter, Taylor microscale, microscale of energy dissipationλ1Parameter of the Navier–Stokes law, longitudinal differential scaleλ2Transverse differential scaleλikVirtual mass coefficientsμCoefficient of dynamic viscosity, Lamé elasticity constant, virtual mass of a sphereμ(M)Density of a double layerμ′Coefficient of structural viscosityμ1Parameter of the Navier–Stokes lawμ1, μ2, …, μnState parameters, variable parametersνKinematic viscosity coefficient, complex velocityConjugate complex variablev*Velocity at the boundary between a lower layer and a turbulent coreνtTurbulent viscosity coefficientξ, η, ςNew coordinatesξ1, ξ2, ξ3Coordinates of a point in a curvilinear system of coordinatesξiLagrange coordinatesCurvilinear (accompanying) system of coordinatesΠDimensionless quantityρTrue (local) mass density, density of a whole mixtureAverage mass densityρiMass density of the i-th component (i=1, 2, …, n) of the mixtureMass change of the i-th component of the mixture per unit time and unit volumeρiInfinitesimal distanceρBMass density of the bodyρ*Stagnation pressureρRadius vector OO1ρ′1Radius vector O′1O′1ρ∞Mass density at infinityΔρChange of a radius vectorΣSurface, cross-sectional surface of the stream tubeΣ′SurfaceσTangential surface, Poisson number, coefficient of surface tensionσijComponents of the viscous stress tensorτCharacteristic period of fluctuations, tangential stress on the surface of the body, tangential stress, shear stress, timeτ0Friction force on the wall per surface unit, limiting tensionτijReynolds stressesComponents of the turbulent additional stressesτfFriction force on the wall per surface unitτ′Reynolds stress (shear stress)τwStress on the wallAverage stress due to frictionτiParameters of the Navier–Stokes Law, Main components of the viscous stress tensorτijComponents of the viscous stress tensorτhCharacteristic hydrodynamic relaxation timeυSpecific volumeυiComponents of the displacement velocityυ*Small volume elementΦQuadratic form, potential, physical characteristic, dissipation function, quadratic form of the generalized thermodynamic forces, potential of the external mass forces, Airy function, Newton’s potentialΦinPotential of the inertial forceΦDensity of the physical characteristicΦFlow potential, polar coordinate, scalar potentialBPotential value at point BAPotential value at point APotential of the internal Dirichlet problemPotential of the internal Neumann problemPotential of the external Dirichlet problemPotential of the external Neumann problemnPotential of a multipole flowtrPotential for a translatory flow/nNormal derivativeχCavitation numberχCoefficient of heat conductivityχiPhysiochemical parametersχijDifference between ψij and the right angleψQuadratic formψGibbs thermodynamic potential, angleψStream functionψijAngle between basis vectors εi and εjΔ4ψ=0Biharmonic equationΩAsymmetric tensor, curl of vector AΩ1, Ω2FunctionsωVector of the instantaneous angular velocity of the body, axial vector, vortex vector, angular velocity of the bodyωAngular velocity, complex frequencyωiComponents of the vector ωωijComponents of the antisymmetric tensorωnNormal component of the vector ωΔiCovariant derivativeΔ·VDivergence of the velocity vector V, velocity of a relative volume change, first invariant of the tensor of strain velocitiesΔ×ACurl of vector AΔLaplace operatorLaplace operator in Cartesian coordinates

Chapter 1

Introduction

1.1Goals and Methods of Continuum Mechanics

Continuum mechanics is that part of mechanics that deals with the movement of deformed gaseous, liquid and solid bodies. In theoretical mechanics, one studies the movement of a mass point, of a discrete system of a such points and of rigid bodies. On the other hand, in continuum mechanics, using the methods and results obtained in theoretical mechanics, one deals with the movement of such material bodies that fill the space continuously, and where the distance between points changes during the movement.

In addition to the usual media, that is, gas, fluid and solid, we also consider unusual media such as the electromagnetic field, the gravitational field and the radiation field.

In the following, we will discuss the most important problems that have given rise to the independent branches of continuum mechanics.

Since the forces that are exerted on a solid body by the gas or fluid that surrounds it depend on the movement of the gas or fluid, the study of the movement of solid bodies in a gas or a fluid is therefore closely linked with the investigation of the movement of a gas or a fluid. Hydromechanics as one of several branches of continuum mechanics deals with the movement of a fluid. Finally, the movement of a gas is discussed in gas dynamics. The problems of the movement of planes, helicopters, rockets, ships, submarines and so on gave the impetus for the massive boost in this research area.

The problems of movement of gas and fluid in tubes and different machines are of great relevance for the planning and calculation of natural gas pipelines, oil pipelines, pumps, compressors, turbines and other hydraulic machines. Applied problems in this field are the foundation for hydraulics.

The movement of gas and fluids through the ground and other porous media (filtration) is a very hot topic in the production of oil and natural gas. Thus, this field that deals with movement of gas and fluids through porous bodies is called subterranean hydromechanics.

Large theoretical and applied importance has the problem of wave propagation in the media mentioned above, that is, gases, fluids and solid bodies. This field is referred to as wave mechanics.

Finally, another part of continuum mechanics deals with the movement of multicomponent and multiphase mixtures caused by different interactions between phases, exchange of heat and matter, phase transitions and chemical reactions. These problems are the foundation of physicochemical hydrodynamics.

The problems connected to the movement of conducting and charged media in magnetic and electric fields are dealt with by the field of magneto and electrohydrodynamics.

The methods of continuum mechanics can also be applied to the movement of bodies in a gas of low density, in outer space, and in stellar and planet atmospheres. The fields of continuum mechanics dealing with these problems are known as the mechanics of diluted gases and cosmic hydrodynamics.

Recently, a new field of biomechanics has come into the forefront which investigates the mechanics of biological objects, the flow of blood in living organisms, the contraction of muscles and which attempts to construct mechanicals models of the internal organs.

Among other important problems, the following needs to be mentioned: weather forecasting, theory of turbulence, movement of sand dunes in the desert, avalanches, burning and detonation as well as the theory of explosions.

Large parts of continuum mechanics are devoted to the investigation of the dynamics and the equilibrium of deformed solid bodies. The theory of elasticity provides the computational foundation for the planning of machines and buildings. Those parts of continuum mechanics that deal with the elastic properties of bodies having a complicated composition and with making provision for inelastic effects in solid bodies become more and more important. The theory of plasticity investigates the behavior of a solid body beyond the elastic limit. Of high importance is also the investigation of various types of material fatigue and taking the the memory effect for the dynamics and equilibrium of a solid body into account. With the invention of new compounds and polymers with composite internal structures, the need to develop new models for these compounds based on their internal structures has become evident.

It is impossible to enumerate all problems and applications of continuum mechanics. However, the above-mentioned examples are sufficient to make the conclusion that continuum mechanics is involved when dealing with a large set of theoretical and applied problems in science and engineering.

Continuum mechanics has also had a large influence on the development of a number of areas of in mathematics. For example, the theory of the wing of a plane influenced the theory of functions of a complex variable, the movement of a viscous fluid gave impetus to research about boundary conditions of partial differential equations and some problems of elasticity theory influenced the research of the theory of integral equations.

The solution of many important questions for applied problems has always necessitated the use of numerical algorithms. Therefore, continuum mechanics has boosted the development of numerical methods.

The following series of lectures about hydromechanics as part of continuum mechanics covers the research methods applicable to the dynamics of deformed bodies. We start by introducing a number of concepts, that is, the fields of velocity, pressure and temperature. We further show how mechanical problems can be reduced to mathematical ones whose solutions determine the properties and dynamics of the deformed bodies.

1.2The Main Hypotheses of Continuum Mechanics

When investigating the dynamics of bodies, one has to exploit their real properties that depend on their internal structure. Every material body consists of different molecules and atoms which are in constant irregular movement. Certain interactions exist between the particles. For a gas, those are mostly determined by collisions. For a fluid, the particles are closer together than in a gas and thus the molecular forces, that is, London–Van der Waals attractive forces and repulsive electrostatic forces are important in this case. The strength and elasticity of solids are due to forces that are electrostatic in nature. If all forces are known, then it is, in principle, possible to create a theory of the dynamics of the material body. However, the complicated internal structure of the body and the huge number of molecules in the volume of interest are substantial hurdles for creating models of the medium. Therefore, it is practically impossible to consider the movement of one particle while taking into account the interactions with all others. Fortunately, however, when dealing with technical applications, there is no need to know the movement of all particles. One only needs to know mean values. In this connection, there are two methods to investigate the dynamics of the medium. The one is the statistical method that has been developed in physics and where the concept of a probability distribution is applied to the system under investigation, and one considers mean values of system data taken over all possible realizations of the system. The second method consists of arriving at a phenomenological macroscopic theory based on experimental observations and laws. In continuum mechanics, the second method is mostly used. However, in some areas, that is, when investigating the dynamics of heterogeneous multiphase media (emulsion, suspensions etc.), both methods are combined.

The methods of continuum mechanics are founded on two hypotheses. The first hypothesis is the continuum hypothesis which assumes that since the number of particles in any volume of practical interest of each material body is extremely large, the body can be considered as a medium that fills the volume in a continuous manner. Such a medium is referred to as a continuous medium. This idealization of the real medium makes it possible to employ the mathematical devices of continuous functions, and differential and integral equations for modeling the processes of interest in the medium.

The second hypothesis is the space-time hypothesis. In continuum mechanics, one normally considers the Euclidean space concept to be applicable so that a homogeneous global system of coordinates can be introduced. The version of mechanics that has been developed on the basis of this hypothesis is called Newtonian mechanics. In general, however, the time does depend on the coordinate system if relativistic effects are taken into account. By neglecting relativity, the time passes in the same manner for all observers. Such a time is referred to as absolute time. Normally, continuum mechanics investigates the dynamics of the continuum in Euclidean space, that is, continuum mechanics is based on Newtonian mechanics. However, in some cases, it is necessary to use the methods on non-Euclidean geometry and the theory of relativity, for example, when investigating the movement and deformation of boundary between different phases of a medium and also when considering the movements of objects in outer space at very high velocity.