Intro to Fluid Mechanics
Intro to Fluid Mechanics ME 6601
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Introduction INTRODUCTION This course is intended to introduce to you the fundamentals of uid mechanics M graduate level As such it will provide a broad physically and mathematically based overview of the subject the foundation that you will need to take more advanced and specialized uids courses such as I Viscous Flows ME 6602 I Viscous Fluid Flows and Turbulent Flows AE 6009 and 6012 I Hydrodynamic Stability and Turbulence CEE 6293 I Computational Fluid Dynamics MECEE 7751 I Transport Phenomena in Multiphase Flow ME 7301 I Hydrodynamic Stability ME 7602 We do not require in ME 6601 that you have any prior knowledge of uid mechanics You are however expected to have a working knowledge of engineering mechanics eg Newton s Laws and engineering mathematics vector calculus multivariable calculus differential equations and complex analysis If you re rusty in or unfamiliar with some of these subjects be prepared to invest significant additional time to catch up the Course References and Communications handout on the class Tsquare site suggests some supplemental references Homework 1 gives a sample of the math that you will be using in this course What makes this a graduate level course I The course emphasizes the basis of the equations and models that many of you have already seen in undergraduate uid mechanics courses What is the basic concept eg Newton s 2n Law conservation of mass energy balance expressed by these equations How are they derived What are the assumptions required to reduce a basic concept to common equations such as the Bernoulli or NavierStokes Equations I You will be expected in your HWs to go beyond the material covered in lectureiin other words you will have to apply the basic principles covered in lecture to new situations I In a few cases the lectures will cover material that is not in the text One final comment some of you may be taking this course to prepare for the qualifying exam in uid mechanics If so please be aware that the FL qualifying exams are restricted to material in the undergraduate course ME 3340 and having taught both courses and chaired the FL qualifying exam I believe that sitting in on ME 3340 is a better use of your time if you want to prepare for the FL quals Review Syllabus I Everything is on TSquare A oncampus and Q Distance Learning sections have different TSquare pages because they have different schedules I Both A and Q sections have access to videotaped lectures through the DL media server Lectures should be posted within 48 h I 1 Copyright 2012 by Minami Yoda Introduction I Please use Piazza available as a Course Tool on TSquare as the primary venue for HW questions and discussions I will be regularly monitoring these discussions You are expected to refer rst to Piazza if you have a HW question Please post your question there and let other students in the class benefit from your questions What is a uid A uid E a substance that deforms continuously when sheared 139 e when a tangential force is applied no matter how small the shear stress tangential force per unit area Examples of uids gases and liquids such as air water oil and bloodiand mixtures of such uids including multiphase uids such as soda water Plasmas ionizeddissociated molecules and powders granular materials can also be modeled as uids In this class we will only talk about a singlephase chemically homogeneous gas or liquid Fluid mechanics E The study of uid motion translation rotation and deformation and how this motion is related to the force and moment distributions exerted by and on th e uid Continuum Hypothesis In most uid mechanics problems and in this course we assume that the uid is a continuumior that all the properties of the uid vary continuously throughout the uid Physically this means that the smallest uid particle that we will consider must still consist of many many thousands of uid molecules In a gas the mean free path is a measure of the distance over which a single gas molecule moves For nitrogen N2 modeled an ideal gas the mean free path is about 10 7 m or 100 nm Assuming that one needs to average over about 10 gas molecules to obtain statistically converged properties the minimum spatial scale that is resolvable in air as a continuum is about 10 times this value or 1 pm For liquid water the intermolecular spacing based on number density is about 7 nm so the minimum spatial scale that can be resolved in water M continuum is about 70 nm Mathematically this means that all the changes in uid properties can be expressed in terms of spatially continuous derivatives with respect to spatial variables Consider a steady 139 e timeindependent uid density eld at a position vector ixyzx1x2x3i p poi pxy 2 If density varies continuously with position the change in density a small distance E dx dv dz away from x is dppltiE pi 12 Copyright 2012 by Minami Yoda Introduction dp the total di erential of the density can then be expressed in terms of its continuous spatial derivatives dpa p clx dya p dz 6x y 6y 2 w Coordinate Systems Fluid mechanics considers M quantities that vary over both space and time such as the velocity Vxt The spatial dependence of this eld quantity is expressed in terms of a position vector usually denoted by x which must be de ned in terms of a coordinate system By de nition a threedimensional coordinate system requires three independent coordinate axes and an origin This course will for the most part consider two coordinate systems 1 Caltesian coordinates x y z where i x0 yo zo x 2 Cylindrical polar coordinates r 0 z where i 709020 Occasionally two other coordinate systems will also be brie y considered 3 Spherical coordinates p 6 p Where i po90po 4 The 2D tangential normal coordinate system s n All these coordinate systems are orthogonal curvilinear coordinate systems where coordinate isosurfaces or coordinate surfaces corresponding to a constant value of a given coordinate are orthogonal ie at right angles to isosurfaces for the other coordinates Copyright 2012 by Minami Yoda Introduction CARTESIAN INDEX NOTATION The velocity eld in a uid ow is a vector Vectors are examples of quantities known as tensors which can be transformed in a speci c manner from one coordinate system to another This course will in almost all cases consider only the three lowestrank tensors namely 1 scalars rank 0 tensors 2 vectors rank 1 tensors 3 matrices rank 2 tensors The ra of a tensor describes how many indices are required to specify that tensor Considering only tensors expressed in Cartesian coordinates there is a compact way to represent these tensors using Cartesian index notation Up to now most of you have learned how to expresswrite vectors and matrices in another type of notation known as Gibbs notation for vectors and matrices Our text and most advanced uid mechanics courses instead use Cartesian index notation instead to write these and other types of tensorsiCartesian index notation is a compact and elegant way to write and analyze the basic principles of uid mechanics and more generally continuum mechanics Gibbs notation Index notation Cartesian Scalar a a Cartesian Vector a a Cartesian Matrix 2 or A ail We will use both Gibbs and index notation by the end of the course we will be primarily using index notation to express tensors in Cartesian coordinates We will only use Gibbs notation for tensors in cylindrical polar or spherical coordinates You are not expected to be familiar with tensors in general but by the end of this course you should be familiar with Cartesian index notation and using this notation to carry out various operations involving scalars vectors and matrices First consider a vector i that is the product of a matrix E and another vector 6 In Gibbs notation this is written as 526 a1 b1 1 b1 2 b13 C 1 a2 bZI 22 b23 02 a3 b3 1 bill b33 C 3 I 4 Copyright 2012 by Minami Yoda Introduction This can be written as a system of three equations a1 bll Cl b12 CZ bl3 03 a2 b21 01 bZZ 02 b23 03 a3 b31 C1 b32 02 baa 03 We can collapse these three equations into the following single equation 3 6121 1123 1 where we m as denoted by the summation symbol 2 over the summation index j as j takes on the values of l 2 and 3 This single equation represents thri independent equations because the range index 1 can take on three valuesiagain l 2 and 3iand each distinct value of 1 represents one of the three independent equations written above So there are two different types of indices in this equation the range also known as free index 1 and the summation also known as dummy index j Moreover inspecting this equation we see that the summation index appears in two different variables within a single term in the equation namely by w c If we then distinguish between the two different types of indices by assuming that any index that is repeated within a single term in the equation is a summation index and any index that is not repeated within a single term in the equation is a range index we can further collapse this equation to at by c Formally speaking the following rules and conventions are used to reduce 3 qzhjcj 1123 F1 to the shorter Cartesian index notation form of albljcj V Range Convention a rangefree index here 1 is understood to take on the values 1 2 and 3 Range Rule all terms in an equation must have the same rangefree indices Einstein Summation Convention all summation dummy indices here j are understood to denote summation over the range from 1 to 3 4 Summation Rule summationdummy indices may be any letter as long as they only appear twice in a single term in the equation and the letter is different from any free indices LAN VV Copyright 2012 by Minami Yoda Introduction By these rules and conventions the product of a vector and matrix can be written in Cartesian index notation as a bYZ CZ where x is a rangefree index and z is a summationdummy index This expression represents three separate equations where x takes on the values of l 2 and 3 and the term on the RHS of this expression represents the w of three terms corresponding to the summation over the range 2 l to 3 Note that indices can only occur once for a range index or twice for a summation index within a single term in an equation written in index notation If any index appears more than twice in a single term you have made a mistake somewhere Next consider a matrix 2 that is the dyadic or outer product of two vectors e and f In Gibbs notation we would write this as dn diz d13 61 elfl eifz elfS d21 dzz dza 82 f2 ezfi ezfz ezfa d3 1 d3 2 d3 3 63 53 f1 53 f2 63 f3 In index notation this would be written as dab eafb Note here that we have two range indices a and b and m summation indices This expression therefore represents 9 independent equations Vectors and their Components A vector i is speci ed by its magnitude El and a unit vector describing its direction a 55 lil The vector can be expressed in terms of its components in various coordinate systems for example in Cartesian coordinates a 01 a2 3 03 12 or in Gibbs notation In index notation 51 would be written as a where the index 139 can have the values of l 2 or 3 139 can either be a range or summation index depending on the context a represents all three vector components The choice of 139 as the index variable is not sacred any other letter is ne ab for example Copyright 2012 by Minami Yoda Introduction Matrices and their Components A matrix E is specified in terms of its components note that the variables denoting matrices are usually capitalized while the variables denoting its components are lower case For a 3 X 3 matrix bu bll b13 E 2 bn bzz bzs b31 bsz b33 In index notation B would be written as b 0p where both 0 and p are indices that can have the values of l 2 0r 3 The transpose of a matrix bu bu b31 T bu bzz bsz bp0 b13 bzs b33 Symmetric and Antisymmetric Tensors A matrix Q is by de nition symmetric if 00d Alternatively a symmetric matrix is identical to its transpose 0m A matrix gis by de nition antisymmetric if pfg pgf Alternatively an antisymmetric matrix is the negative of its transpose In general any matrix I can be written as the sum of or decomposed into a symmetric matrix and an antisymmetric matrix A where s and a Although we mainly consider the symmetry of matrices rank 2 tensors any tensor of rank greater than unity 1 can be described as symmetric or antisymmetric as follows I If switching two indices does not affect the value of that tensor element it is a symmetric tensor I If switching two indices changes the sign but not magnitude of that tensor element it is an antisymmetric tensor Note that most tensors are neither symmetric nor antisymmetric 17 Copyright 2012 by Minami Yoda Introduction Identity Tensor The identity substitution tensor 1 can be written in Gibbs notation as l 0 0 l 0 l 0 0 0 1 In index notation g is usually written instead as the Kronecker delta 8 0 if 139 j Ul1 ifij39 The identity tensor is of rank 2 isotropic invariant under rotation symmetric and has the sifting property where 5m b a Alternating Unit Tensor The alternating unit tensor written as 8 or g 8123 8312 8231 1 cycllc mdlces 8321 8132 8213 l antlcycllc 1nd1ces 8 0 in all other cases The alternating unit tensor is of rank 3 and antisymmetric Out of the 27 possible independent components of this rank 3 tensor all but 6 are zero As you will see shortly this tensor is used in index notation to calculate the vector cross product and the curl of a vector Identity The product of two alternating unit tensors can be written in terms of the identity tensor SykSzm S sm 8m8kl Note that j k l m are all range indices and 139 is a sumation indexiand that 139 must be the rst index of both alternating unit tensors Copyright 2012 by Minami Yoda Introduction VECTOR PRODUCTS There are three ways to multiply two vectors 1 The dotscalar product which gives a scalar rank 0 tensor 2 The crossvector product which gives a vector rank 1 3 The outer dyadic product which gives a matrix rank 2 Scalar Product In index notation the dot scalar product of two vectors a and q can be expressed in Gibbs and Cartesian index notations as follows 5 a39q alql a2q2 a3q3 S at qn 39 n is a summation index and that s is a scalar The scalar product is a contracted product or a product that reduces the rank of the tensor or the number of indices by 1 Vector Product The cross vector product of two vectors a and j is also a vector g In Gibbs notation g a X j In index notation gr th ab qc Here b and c are both summation indices and h is a range index h must be the rst index of the alternating unit tensor just as the second index of the alternating unit tensor must be the same index as that of the rst vector 5 and the third index of the alternating unit tensor must be the same index as that of the second vector 1 Dyadic Product The outer dyadic product of two vectors a and 1 can be written in Gibbs and index notation 01 611 01 2ifl 61261 CIqu 612 a3q1 Clqu 613 z39 and j are both range indices and 2 is a matrix Copyright 2012 by Minami Yoda Introduction MATRIX PRODUCTS Inner Product The inner product of two matrices tensors I and is a scalar de ned as l g Tm Sm Here m and n are both summation indices The inner product is an example of a doubly contracted product or a product that reduces the rank of the tensor by 2 Note that the indices of the two matrices in the inner product are not in the same order andthat T S T S mnm mnm Instead T S ElST mm Theorem The inner product of a symmetric matrix and an antisymmetric matrix A is always the scalar zero 0 This theorem can be extended to the doubly contracted product of any tensor of rank 2 or greater I 10 Copyright 2012 by Minami Yoda Introduction TENSOR VECTOR CALCULUS OPERATIONS Spatial and Temporal Derivatives In proper Cartesian index notation the partial derivative of a component of a matrix S sab with respect to a spatial variable xm is written 6 sab Sab m 39 696m 39 Note that Panton s text uses instead 6 sab amsab 696m Panton also uses a subscript of 0 zero to indicate partial derivatives with respect to time 1 63gb 60Sab Gradient In general the gradient g gunk of atensor l tymk increases the rank of a tensor by 1 The gradient can be written in Cartesian index notation as atlk a t 6x 2 zjnJc If 2 is ofrank N its gradient Q will be ofrank Nl t Um a11indices free indices gymZ For example the gradient of a scalar 3 rank 0 is a vector 7 rank 1 In Gibbs notation the gradient is written as V Vs V is known as the del 0r nabla operator In Cartesian index notation Va 6assa 696a 39 The gradient 6f 6x fa of a scalar function f i has two interesting properties I The direction of if xa points toward the maximum value of f I The magnitude of if Bxa is the maximum spatial rate of change in f l l 1 Copyright 2012 by Minami Yoda Introduction To illustrate these properties consider the total differential 5le of the position vector x x1 which can be written as the product of its magnitude dr and a unit vector i dxz 0tz The total d erential df of a scalar function at position x1 f x1 is the difference in the value of f between two points very close to each other x1 and x1 dxl Mathematically 6 defOc dxgt fxgta dx f dx 1W f W 699 39 For the direction of 6f Bxa let x and x1 de both be points on a curve defined by the equation f constant By definition df 0 on such a curve and 5le will be tangent to this curve since x1 and x1 de are very close to each other So t1 will be the unit vector tangent to the curve Then from the above equation 6 i t 0 Bxi If the dot product of Bf 6x1 and a vector tangent to the curve of constant f is zero the gradient of f must be normal to all the contours curves of constant f So the direction of the gradient points towards an extremum minimum or maximum of f Since the gradient is positive if f is increasing we can conclude that the gradient of f is a vector that points toward the maximum value off Second let the points x1 and x1 de define a vector along the gradient off Based on the first part of this proof 1 C 711 Bxi where C is the magnitude of if 6x1 and n1 is the unit vector normal to all the contours of constant f And the direction of 5le must for this case also be along n1 dxz dr z Then 6f 6V 6 61x Cndrn Cdr xi Since the total differential of f for this case points towards the maximum value of f again based on the first part of this proof it represents the maximum change in fat x1 dfm C dr C dme dr So C the magnitude of if 6x1 is the maximum spatial rate of change inf I 12 Copyright 2012 by Minami Yoda Introduction Divergence The divergence Q of atensor I contracts the tensor or reduces its rank by l at Umklm dzkm aztzmktm tzfuklmJ 096 If 2 is ofrankN 2 is ofrank N l one summation index I The divergence of a vector V is a scalar s In Gibbs notation the divergence is written as s V V In index notation 6Vb 6 v v axb b b bb Curl The curl of a vector 7 with three components is a vector 5 In Gibbs notation i j k E V X v 3 3 3 6x By 62 u v w In index notation avk C zjk a 8kaka Alternatively 6v 0 8ka g Szjkvjk Note that the negative sign is used here because of the order of the indices 0 Sz c vk ng ijc Szk vjvk We will discuss the physical meaningsproperties of the divergence and curl of vectors such as the velocity 7 later in this course Laplacian The divergence of the gradient of a scalar s is known as the Laplacian operator 62 st VVs 6 akaks SJ xk Vector I 13 Copyright 2012 by Minami Yoda Introduction The Laplacian is an operator that preserves the rank of the tensor and can be applied to higherrank tensors The secondorder linear homogeneous partial differential equation PDE Z V23 6 j 0 696k is called Laplace s Equation The inhomogeneous form of this equation is known as Poisson s Equation Index Notation Example One of the bene ts of Cartesian index notation is that it greatly simplifies proofs of vector identities We illustrate this with two examples involving the cross product of two vectors a and q gtltq One of the properties of the cross product is that it is anticommutative i x q q X E This can be shown 39 39 39 11g qk aqxa the alternating unit tensor is antisymmetric so 8 Slk Another property of the cross product is that g aim is normal to both i E q We can show this using Cartesian index notation In index notation the cross product is 811 8111 a q h range index 139 summation indices To show that g is normal to i we will show that g5 0 g393 g11 ah 8hlj a qa11 811U qa1 ah h 1 j all summation indices Now 8M q is an antisymmetric tensor of rank 2 since the alternating unit tensor is antisymmetric 8M q by h 1 range indices j summation index and alah is a symmetric tensor of rank 2 h 1 range indices I 14 Copyright 2012 by Minami Yoda Introduction Since 721 can be rewritten as the inner product of an antisymmetric tensor and a symmetric tensor both of rank two g5th cm 0 by the theorem that the inner product of an antisymmetric tensor and a symmetric tensor 1s zero In these proofs we have used only two concepts I the antisymmetry and symmetry of various tensors I the theorem about the inner product of a symmetric and antisymmetric tensor We can use similar arguments to show that gr 0 and thereby prove that g is normal to IMPORTANT VECTOR CALCULUS THEOREMS Gauss Divergence Theorem This theorem relates the volume integral of any vector and tensor to a surface integral Consider an arbitrary volume VL bounded by a closed surface S Next consider a small piece of the surface of area dA with a unit normal vector which by convention points out or away from 5 n For any vector field b h Gauss Theorem states that ISB d4jSB LVBW Since VT is the divergence of 5 this theorem is also often called the Divergence Theorem Physically the volume integral of the divergence of b is equal to a surface integral that is in the form of an ef J ie out ow for example if BpV this surface integral would be the mass ef ux So the divergence of a vector is related to the ef ux per unit volume of the vector Gauss Theorem applies not just to vectors but to any tensor field For a vector 1 j bl n 544 j 451v S V ext 139 summation index For a higherrank tensor 57 J 0N V 6x1 139 summation index all other indices range indices IT ndAj s 1 l I 15 Copyright 2012 by Minami Yoda