A course in fluid mechanics with vector field theory d prieve

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Đây là bộ sách tiếng anh về chuyên ngành vật lý gồm các lý thuyết căn bản và lý liên quan đến công nghệ nano ,công nghệ vật liệu ,công nghệ vi điện tử,vật lý bán dẫn. Bộ sách này thích hợp cho những ai đam mê theo đuổi ngành vật lý và muốn tìm hiểu thế giới vũ trụ và hoạt độn ra sao. A Course in Fluid Mechanicswith Vector Field TheorybyDennis C. PrieveDepartment of Chemical EngineeringCarnegie Mellon UniversityPittsburgh, PA 15213An electronic version of this book in Adobe PDF® format was made available tostudents of 06-703, Department of Chemical Engineering,Carnegie Mellon University, Fall, 2000.Copyright © 2000 by Dennis C. Prieve06-703 1 Fall, 2000Copyright © 2000 by Dennis C. PrieveTable of ContentsALGEBRA OF VECTORS AND TENSORS 1VECTOR MULTIPLICATION 1Definition of Dyadic Product 2DECOMPOSITION INTO SCALAR COMPONENTS 3SCALAR FIELDS 3GRADIENT OF A SCALAR 4Geometric Meaning of the Gradient 6Applications of Gradient 7CURVILINEAR COORDINATES 7Cylindrical Coordinates 7Spherical Coordinates 8DIFFERENTIATION OF VECTORS W.R.T. SCALARS 9VECTOR FIELDS 11Fluid Velocity as a Vector Field 11PARTIAL & MATERIAL DERIVATIVES 12CALCULUS OF VECTOR FIELDS 14GRADIENT OF A SCALAR (EXPLICIT) 14DIVERGENCE, CURL, AND GRADIENT 16Physical Interpretation of Divergence 16Calculation of ∇.v in R.C.C.S 16Evaluation of ∇×v and ∇v in R.C.C.S 18Evaluation of ∇.v, ∇×v and ∇v in Curvilinear Coordinates 19Physical Interpretation of Curl 20VECTOR FIELD THEORY 22DIVERGENCE THEOREM 23Corollaries of the Divergence Theorem 24The Continuity Equation 24Reynolds Transport Theorem 26STOKES THEOREM 27Velocity Circulation: Physical Meaning 28DERIVABLE FROM A SCALAR POTENTIAL 29THEOREM III 31TRANSPOSE OF A TENSOR, IDENTITY TENSOR 31DIVERGENCE OF A TENSOR 32INTRODUCTION TO CONTINUUM MECHANICS* 34CONTINUUM HYPOTHESIS 34CLASSIFICATION OF FORCES 36HYDROSTATIC EQUILIBRIUM 37FLOW OF IDEAL FLUIDS 37EULER'S EQUATION 38KELVIN'S THEOREM 41IRROTATIONAL FLOW OF AN INCOMPRESSIBLE FLUID 42Potential Flow Around a Sphere 45d'Alembert's Paradox 5006-703 2 Fall, 2000Copyright © 2000 by Dennis C. PrieveSTREAM FUNCTION 53TWO-D FLOWS 54AXISYMMETRIC FLOW (CYLINDRICAL) 55AXISYMMETRIC FLOW (SPHERICAL) 56ORTHOGONALITY OF ψ=CONST AND φ=CONST 57STREAMLINES, PATHLINES AND STREAKLINES 57PHYSICAL MEANING OF STREAMFUNCTION 58INCOMPRESSIBLE FLUIDS 60VISCOUS FLUIDS 62TENSORIAL NATURE OF SURFACE FORCES 62GENERALIZATION OF EULER'S EQUATION 66MOMENTUM FLUX 68RESPONSE OF ELASTIC SOLIDS TO UNIAXIAL STRESS 70RESPONSE OF ELASTIC SOLIDS TO PURE SHEAR 72GENERALIZED HOOKE'S LAW 73RESPONSE OF A VISCOUS FLUID TO PURE SHEAR 75GENERALIZED NEWTON'S LAW OF VISCOSITY 76NAVIER-STOKES EQUATION 77BOUNDARY CONDITIONS 78EXACT SOLUTIONS OF N-S EQUATIONS 80PROBLEMS WITH ZERO INERTIA 80Flow in Long Straight Conduit of Uniform Cross Section 81Flow of Thin Film Down Inclined Plane 84PROBLEMS WITH NON-ZERO INERTIA 89Rotating Disk* 89CREEPING FLOW APPROXIMATION 91CONE-AND-PLATE VISCOMETER 91CREEPING FLOW AROUND A SPHERE (Re→0) 96Scaling 97Velocity Profile 99Displacement of Distant Streamlines 101Pressure Profile 103CORRECTING FOR INERTIAL TERMS 106FLOW AROUND CYLINDER AS RE→0 109BOUNDARY-LAYER APPROXIMATION 110FLOW AROUND CYLINDER AS Re→ ∞ 110MATHEMATICAL NATURE OF BOUNDARY LAYERS 111MATCHED-ASYMPTOTIC EXPANSIONS 115MAE’S APPLIED TO 2-D FLOW AROUND CYLINDER 120Outer Expansion 120Inner Expansion 120Boundary Layer Thickness 120PRANDTL’S B.L. EQUATIONS FOR 2-D FLOWS 120ALTERNATE METHOD: PRANDTL’S SCALING THEORY 120SOLUTION FOR A FLAT PLATE 120Time Out: Flow Next to Suddenly Accelerated Plate 120Time In: Boundary Layer on Flat Plate 120Boundary-Layer Thickness 120Drag on Plate 12006-703 3 Fall, 2000Copyright © 2000 by Dennis C. PrieveSOLUTION FOR A SYMMETRIC CYLINDER 120Boundary-Layer Separation 120Drag Coefficient and Behavior in the Wake of the Cylinder 120THE LUBRICATION APPROXIMATION 157TRANSLATION OF A CYLINDER ALONG A PLATE 163CAVITATION 166SQUEEZING FLOW 167REYNOLDS EQUATION 171TURBULENCE 176GENERAL NATURE OF TURBULENCE 176TURBULENT FLOW IN PIPES 177TIME-SMOOTHING 179TIME-SMOOTHING OF CONTINUITY EQUATION 180TIME-SMOOTHING OF THE NAVIER-STOKES EQUATION 180ANALYSIS OF TURBULENT FLOW IN PIPES 182PRANDTL’S MIXING LENGTH THEORY 184PRANDTL’S “UNIVERSAL” VELOCITY PROFILE 187PRANDTL’S UNIVERSAL LAW OF FRICTION 189ELECTROHYDRODYNAMICS 120ORIGIN OF CHARGE 120GOUY-CHAPMAN MODEL OF DOUBLE LAYER 120ELECTROSTATIC BODY FORCES 120ELECTROKINETIC PHENOMENA 120SMOLUCHOWSKI'S ANALYSIS (CA. 1918) 120ELECTRO-OSMOSIS IN CYLINDRICAL PORES 120ELECTROPHORESIS 120STREAMING POTENTIAL 120SURFACE TENSION 120MOLECULAR ORIGIN 120BOUNDARY CONDITIONS FOR FLUID FLOW 120INDEX 21106-703 1 Fall, 2000Copyright © 2000 by Dennis C. PrieveAlgebra of Vectors and TensorsWhereas heat and mass are scalars, fluid mechanics concerns transport of momentum, which is avector. Heat and mass fluxes are vectors, momentum flux is a tensor. Consequently, the mathematicaldescription of fluid flow tends to be more abstract and subtle than for heat and mass transfer. In aneffort to make the student more comfortable with the mathematics, we will start with a review of thealgebra of vectors and an introduction to tensors and dyads. A brief review of vector addition andmultiplication can be found in Greenberg,♣ pages 132-139.Scalar - a quantity having magnitude but no direction (e.g. temperature, density)Vector - (a.k.a. 1st rank tensor) a quantity having magnitude and direction (e.g. velocity, force,momentum)(2nd rank) Tensor - a quantity having magnitude and two directions (e.g. momentum flux,stress)VECTOR MULTIPLICATIONGiven two arbitrary vectors a and b, there are three types of vector productsare defined:Notation Result DefinitionDot Product a.b scalar ab cosθCross Product a×b vector absinθn where θ is an interior angle (0 ≤ θ ≤ π) and n is a unit vector which is normal to both a and b. Thesense of n is determined from the "right-hand-rule"♦Dyadic Product ab tensor ♣ Greenberg, M.D., Foundations Of Applied Mathematics, Prentice-Hall, 1978.♦ The “right-hand rule”: with the fingers of the right hand initially pointing in the direction of the firstvector, rotate the fingers to point in the direction of the second vector; the thumb then points in thedirection with the correct sense. Of course, the thumb should have been normal to the plane containingboth vectors during the rotation. In the figure above showing a and b, a×b is a vector pointing into thepage, while b×a points out of the page.06-703 2 Fall, 2000Copyright © 2000 by Dennis C. PrieveIn the above definitions, we denote the magnitude (or length) of vector a by the scalar a. Boldface willbe used to denote vectors and italics will be used to denote scalars. Second-rank tensors will bedenoted with double-underlined boldface; e.g. tensor T.Definition of Dyadic ProductReference: Appendix B from Happel & Brenner.♥ The word “dyad” comes from Greek: “dy”means two while “ad” means adjacent. Thus the name dyad refers to the way in which this product isdenoted: the two vectors are written adjacent to one another with no space or other operator inbetween.There is no geometrical picture that I can draw which will explain what a dyadic product is. It's bestto think of the dyadic product as a purely mathematical abstraction having some very useful properties:Dyadic Product ab - that mathematical entity which satisfies the following properties (where a,b, v, and w are any four vectors):1. ab.v = a(b.v) [which has the direction of a; note that ba.v = b(a.v) which has the direction ofb. Thus ab ≠ ba since they don’t produce the same result on post-dotting with v.]2. v.ab = (v.a)b [thus v.ab ≠ ab.v]3. ab×v = a(b×v) which is another dyad4. v×ab = (v×a)b5. ab:vw = (a.w)(b.v) which is sometimes known as the inner-outer product or the double-dotproduct.*6. a(v+w) = av+aw (distributive for addition)7. (v+w)a = va+wa8. (s+t)ab = sab+tab (distributive for scalar multiplication also distributive for dot and crossproduct)9. sab = (sa)b = a(sb) ♥ Happel, J., & H. Brenner, Low Reynolds Number Hydrodynamics, Noordhoff, 1973.* Brenner defines this as (a.v)(b.w). Although the two definitions are not equivalent, either can beused as long as you are consistent. In these notes, we will adopt the definition above and ignoreBrenner's definition.06-703 3 Fall, 2000Copyright © 2000 by Dennis C. PrieveDECOMPOSITION INTO SCALAR COMPONENTSThree vectors (say e1, e2, and e3) are said to be linearly independent if none can be expressedas a linear combination of the other two (e.g. i, j, and k). Given such a set of three LI vectors, anyvector (belonging to E3) can be expressed as a linear combination of this basis:v = v1e1 + v2e2 + v3e3where the vi are called the scalar components of v. Usually, for convenience, we chooseorthonormal vectors as the basis:ei.ej = δij = 10 if if i ji j=≠RSTalthough this is not necessary. δij is called the Kronecker delta. Just as the familiar dot and crossproducts can written in terms of the scalar components, so can the dyadic product:vw = (v1e1+v2e2+v3e3)(w1e1+w2e2+w3e3)= (v1e1)(w1e1)+(v1e1)(w2e2)+ = v1w1e1e1+v1w2e1e2+ (nine terms)where the eiej are nine distinct unit dyads. We have applied the definition of dyadic product toperform these two steps: in particular items 6, 7 and 9 in the list above.More generally any nth rank tensor (in E3) can be expressed as a linear combination of the 3n unit n-ads. For example, if n=2, 3n=9 and an n-ad is a dyad. Thus a general second-rank tensor can bedecomposed as a linear combination of the 9 unit dyads:T = T11e1e1+T12e1e2+ = Σi=1,3Σj=1,3TijeiejAlthough a dyad (e.g. vw) is an example of a second-rank tensor, not all2nd rank tensors T can be expressed as a dyadic product of two vectors.To see why, note that a general second-rank tensor has nine scalarcomponents which need not be related to one another in any way. Bycontrast, the 9 scalar components of dyadic product above involve only sixdistinct scalars (the 3 components of v plus the 3 components of w).After a while you get tired of writing the summation signs and limits. So anabbreviation was adopted whereby repeated appearance of an index implies summation over the threeallowable values of that index:T = Tijeiej06-703 4 Fall, 2000Copyright © 2000 by Dennis C. PrieveThis is sometimes called the Cartesian (implied) summation convention.SCALAR FIELDSSuppose I have some scalar function of position (x,y,z) which is continuously differentiable, thatisf = f(x,y,z)and ∂f/∂x, ∂f/∂y, and ∂f/∂z exist and are continuous throughout some 3-D region in space. Thisfunction is called a scalar field. Now consider f at a second point which is differentially close to thefirst. The difference in f between these two points iscalled the total differential of f:f(x+dx,y+dy,z+dz) - f(x,y,z) ≡ dfFor any continuous function f(x,y,z), df is linearly relatedto the position displacements, dx, dy and dz. Thatlinear relation is given by the Chain Rule ofdifferentiation:dffxdxfydyfzdz= + +∂∂∂∂∂∂Instead of defining position using a particular coordinatesystem, we could also define position using a position vector r:r ijk=++xyzThe scalar field can be expressed as a function of a vector argument, representing position, instead of aset of three scalars:f = f(r)Consider an arbitrary displacement away from the point r, which we denote as dr to emphasize that themagnitude dr of this displacement is sufficiently small that f(r) can be linearized as a function ofposition around r. Then the total differential can be written as06-703 5 Fall, 2000Copyright © 2000 by Dennis C. Prievedffdf=+−()()rrrGRADIENT OF A SCALARWe are now is a position to define an important vector associatedwith this scalar field. The gradient (denoted as ∇f) is definedsuch that the dot product of it and a differential displacementvector gives the total differential:dfdf≡∇r.EXAMPLE: Obtain an explicit formula for calculating the gradient in Cartesian* coordinates.Solution: r = xi + yj + zkr+dr = (x+dx)i + (y+dy)j + (z+dz)ksubtracting: dr = (dx)i + (dy)j + (dz)k∇f = (∇f)xi + (∇f)yj + (∇f)zkdr.∇f = [(dx)i + ].[(∇f)xi + ]df = (∇f)xdx + (∇f)ydy + (∇f)zdz (1)Using the Chain rule: df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz (2)According to the definition of the gradient, (1) and (2) are identical. Equating them and collecting terms:[(∇f)x-(∂f/∂x)]dx + [(∇f)y-(∂f/∂y)]dy + [(∇f)z-(∂f/∂z)]dz = 0Think of dx, dy, and dz as three independent variables which can assume an infinite number of values,even though they must remain small. The equality above must hold for all values of dx, dy, and dz. Theonly way this can be true is if each individual term separately vanishes:** *Named after French philosopher and mathematician René Descartes (1596-1650), pronounced "day-cart", who first suggested plotting f(x) on rectangular coordinates** For any particular choice of dx, dy, and dz, we might obtain zero by cancellation of positive andnegative terms. However a small change in one of the three without changing the other two would causethe sum to be nonzero. To ensure a zero-sum for all choices, we must make each term vanishindependently.06-703 6 Fall, 2000Copyright © 2000 by Dennis C. PrieveSo (∇f)x = ∂f/∂x, (∇f)y = ∂f/∂y, and (∇f)z = ∂f/∂z,leaving ∇ = + +ffxfyfz∂∂∂∂∂∂i j kOther ways to denote the gradient include:∇f = gradf = ∂f/∂rGeometric Meaning of the Gradient1) direction: ∇f(r) is normal to the f=const surface passing through the point r in the direction ofincreasing f. ∇f also points in the direction of steepest ascent of f.2) magnitude: |∇f| is the rate of change of f withdistance along this directionWhat do we mean by an "f=const surface"? Consider anexample.Example: Suppose the steady state temperature profilein some heat conduction problem is given by:T(x,y,z) = x2 + y2 + z2Perhaps we are interested in ∇T at the point (3,3,3)where T=27. ∇T is normal to the T=const surface:x2 + y2 + z2 = 27which is a sphere of radius 27 .♣Proof of 1). Let's use the definition to show that these geometric meanings are correct.df = dr.∇f ♣ A vertical bar in the left margin denotes material which (in the interest of time) will be omitted from thelecture.[...]... there is no displacement in position during the time interval dt As time proceeds, different material points occupy the spatial point r material derivative (a. k .a substantial derivative) - rate of change within a particular material point (whose spatial coordinates vary with time): Df  df  = Dt  dt d r = vdt   where the subscript dr = v dt denotes that a displacement in position (corresponding to... circular arc labelled θ has radius r and is subtended by the angle θ) The angle φ (measured in the xy-plane) is the angle the second blue plane (actually it’s one quadrant of a disk) makes with the xyplane (red) This plane which is a quadrant of a disk is a φ =const surface: all points on this plane have the same φ coordinate The second red (circular) arc labelled φ is also subtended by the angle φ... Solid-body rotation is simply the velocity field a solid would experience if it was rotating about some axis This is also the velocity field eventually found in viscous fluids undergoing steady rotation Ω Imagine that we take a container of fluid (like a can of soda pop) and we rotate the can about its axis After a transient period whose duration depends on the dimensions of the container, the steady-state... integration This last equality is only valid if the boundaries are independent of t Now mass enters through the surface A Subdividing A into small area elements: n = outward unit normal n.v da = vol flowrate out through da (cm3/s) ρ(n.v)da = mass flowrate out through da (g/s)  rate of    = ρ ( n.v ) da = ∫ n.( ρv ) da = ∫ ∇.(ρ v ) dV mass leaving  ∫  A A V The third equality was obtained by applying... a boundary which is impermeable) • determine the rate of change along some arbitrary direction: if n is a unit vector pointing along some path, then ∂f n.∇f = ∂s is the rate of change of f with distance (s) along this path given by n ∂f ∂s is called the directed derivative of f CURVILINEAR COORDINATES In principle, all problems in fluid mechanics and transport could be solved using Cartesian coordinates... system, denoted by V can be macroscopic (it doesn’t have to be differential) The boundaries of the system are the set of fixed spatial points denoted as A Of course, fluid may readily cross these mathematical boundaries Subdividing V into many small volume elements: dm = ρdV z z M = dm = ρ dV V  dM d  ∂ρ =  ∫ ρ dV  = ∫ dV  dt dt  ∂t V  V where we have switched the order of differentiation and integration... here v denotes the local fluid velocity As time proceeds, the moving material occupies different spatial points, so r is not fixed In other words, we are following along with the fluid as we measure the rate of change of f A relation between these two derivatives can be derived using a generalized vectorial form of the Chain Rule First recall that for steady (independent of t) scalar fields, the Chain... and spherical coordinates The results are tabulated in Appendix A of BSL (see pages 738741) These pages are also available online: Copyright © 2000 by Dennis C Prieve 06-703 22 Fall, 2000 rectangular coords cylindrical coords: spherical coords: Physical Interpretation of Curl To obtain a physical interpretation of ∇×v, let’s consider a particularly simple flow field which is called solid-body rotation... n.v da A PARTIAL & MATERIAL DERIVATIVES Let f = f(r,t) represent some unsteady scalar field (e.g the unsteady temperature profile inside a moving fluid) There are two types of time derivatives of unsteady scalar fields which we will find convenient to define In the example in which f represents temperature, these two time derivatives correspond to the rate of change (denoted generically as df /dt) measured... measured with a thermometer which either is held stationary in the moving fluid or drifts along with the local fluid partial derivative - rate of change at a fixed spatial point: Copyright © 2000 by Dennis C Prieve 06-703 15 ∂f  df = ∂t  dt  Fall, 2000   dr =0 where the subscript dr=0 denotes that we are evaluating the derivative along a path* on which the spatial point r is held fixed In other words, . 106FLOW AROUND CYLINDER AS RE→0 109BOUNDARY-LAYER APPROXIMATION 110FLOW AROUND CYLINDER AS Re→ ∞ 110MATHEMATICAL NATURE OF BOUNDARY LAYERS 111MATCHED-ASYMPTOTIC. example, if n=2, 3n=9 and an n-ad is a dyad. Thus a general second-rank tensor can bedecomposed as a linear combination of the 9 unit dyads:T = T11e1e1+T12e1e2+
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