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Chapter 2: Vectors and Tensors 2/5/2018 Chapter 2: Vectors and Tensors 2/5/2018 Review: We: Now… • Defined rheology • Contrasted with Newtonian and non‐Newtonian behavior • Saw demonstrations (film) © Faith A. Morrison, Michigan Tech U. Key to deformation and flow is the momentum balance: ⋅ ⋅ • Linear Newtonian fluids: • Instantaneous • Non-Newtonian fluids: • Non-linear • Non-instantaneous • ? (missing piece) 2 © Faith A. Morrison, Michigan Tech U. 1 Chapter 2: Vectors and Tensors 2/5/2018 Key to deformation and flow is the momentum balance: ⋅ ⋅ • Linear Newtonian fluids: • Instantaneous We’re going to be trying• to identify the constitutive Non-Newtonian fluids: equation for non- Newtonian fluids. • Non-linear • Non-instantaneous • ? (missing piece) 3 © Faith A. Morrison, Michigan Tech U. Key to deformation and flow is the momentum balance: ⋅ ⋅ • Linear Newtonian fluids: • Instantaneous We’re going to be trying• to identify the constitutive Non-Newtonian fluids: equation for non- Newtonian fluids. • Non-linear We’re going to need to • Non-instantaneous calculate how different • ? (missing piece) guesses affect the predicted behavior. 4 © Faith A. Morrison, Michigan Tech U. 2 Chapter 2: Vectors and Tensors 2/5/2018 Key to deformation and flow is the momentum balance: ⋅ ⋅ • Linear Newtonian fluids: • Instantaneous We’re going to be trying• to identify the constitutive Non-Newtonian fluids: equation for non- Newtonian fluids. • Non-linear We’re going to need to • Non-instantaneous calculate how different • ? (missing piece) guesses affect the We need to understand predicted behavior. and be able to manipulate this 5 mathematical© Faith A. Morrison, notation. Michigan Tech U. Chapter 2: Mathematics Review CM4650 Polymer Rheology 1. Vector review Michigan Tech 2. Einstein notation 3. Tensors Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 6 © Faith A. Morrison, Michigan Tech U. 3 Chapter 2: Vectors and Tensors 2/5/2018 Chapter 2: Mathematics Review 1. Scalar – a mathematical entity that has magnitude only e.g.: temperature T speed v time t density r – scalars may be constant or may be variable Laws of Algebra for yes commutative ab = ba Scalars: yes associative a(bc) = (ab)c yes distributive a(b+c) = ab+ac 7 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology 2. Vector – a mathematical entity that has magnitude and direction e.g.: force on a surface f velocity v – vectors may be constant or may be variable Definitions magnitude of a vector – a scalar associated with a vector v v f f This notation unit vector – a vector of unit length ,, is called v Gibbs notation. vˆ v a unit vector in the direction of v 8 © Faith A. Morrison, Michigan Tech U. 4 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Laws of Algebra for Vectors: 1. Addition a+b a b 2. Subtraction a+(-b) a b 9 © Faith A. Morrison, Michigan Tech U. Laws of Algebra for Vectors (continued): 3. Multiplication by scalar v yes commutative v v yes associative v v v yes distributive v w v w 4. Multiplication of vector by vector 4a. scalar (dot) (inner) product v w vwcos v Note: we can find w magnitude with dot product 2 vv vv cos0 v v v vv 10 © Faith A. Morrison, Michigan Tech U. 5 Chapter 2: Vectors and Tensors 2/5/2018 Laws of Algebra for Vectors (continued): 4a. scalar (dot) (inner) product (con’t) yes commutative v w wv NO associative v w z no such operation yes distributive z v w z v z w 4b. vector (cross) (outer) product v w vwsin eˆ v eˆ is a unit vector w perpendicular to both v and w following the right-hand rule 11 © Faith A. Morrison, Michigan Tech U. Laws of Algebra for Vectors (continued): 4b. vector (cross) (outer) product (con’t) NO commutative v w w v NO associative vwz vw z v wz yes distributive z v w z v z w 12 © Faith A. Morrison, Michigan Tech U. 6 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Coordinate Systems •Allow us to make actual calculations with vectors Rule: any three vectors that are non-zero and linearly independent (non-coplanar) may form a coordinate basis Three vectors are linearly dependent if a, b, and g can be found such that: a b c 0 for , , 0 If , , and are found to be zero, the vectors are linearly independent. 13 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology How can we do actual calculations with vectors? Rule: any vector may be expressed as the linear combination of three, non-zero, non-coplanar basis vectors coefficient of a in the eˆy direction a any vector x a axeˆx ayeˆy azeˆz ay a This notation z xyz is called matrix a1eˆ1 a2eˆ2 a3eˆ3 notation. 3 ˆ a je j j1 14 © Faith A. Morrison, Michigan Tech U. 7 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Trial calculation: dot product of two vectors ab a1eˆ1 a2eˆ2 a3eˆ3 b1eˆ1 b2eˆ2 b3eˆ3 a1eˆ1 b1eˆ1 b2eˆ2 b3eˆ3 a2eˆ2 b1eˆ1 b2eˆ2 b3eˆ3 a3eˆ3 b1eˆ1 b2eˆ2 b3eˆ3 a1eˆ1 b1eˆ1 a1eˆ1 b2eˆ2 a1eˆ1 b3eˆ3 a2eˆ2 b1eˆ1 a2eˆ2 b2eˆ2 a2eˆ2 b3eˆ3 a3eˆ3 b1eˆ1 a3eˆ3 b2eˆ2 a3eˆ3 b3eˆ3 If we choose the basis to be orthonormal - mutually perpendicular and of unit length - then we can simplify. 15 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology If we choose the basis to be orthonormal - mutually perpendicular and of unit length, then we can simplify. eˆ1 eˆ1 1 eˆ1 eˆ2 0 eˆ1 eˆ3 0 a b a1eˆ1 b1eˆ1 a1eˆ1 b2eˆ2 a1eˆ1 b3eˆ3 a2eˆ2 b1eˆ1 a2eˆ2 b2eˆ2 a2eˆ2 b3eˆ3 a3eˆ3 b1eˆ1 a3eˆ3 b2eˆ2 a3eˆ3 b3eˆ3 a1b1 a2b2 a3b3 We can generalize this operation with a technique called Einstein notation. 16 © Faith A. Morrison, Michigan Tech U. 8 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Einstein Notation a system of notation for vectors and tensors that allows for the calculation of results in Cartesian coordinate systems. a a1eˆ1 a2eˆ2 a3eˆ3 3 ajeˆj This notation j1 called Einstein ajeˆj ameˆm notation. •the initial choice of subscript letter is arbitrary •the presence of a pair of like subscripts implies a missing summation sign 17 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology Einstein Notation (con’t) The result of the dot products of basis vectors can be summarized by the Kronecker delta function eˆ eˆ 1 1 1 1 i p eˆ eˆ 0 eˆ eˆ 1 2 i p ip 0 i p eˆ1 eˆ3 0 Kronecker delta 18 © Faith A. Morrison, Michigan Tech U. 9 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Einstein Notation (con’t) To carry out a dot product of two arbitrary vectors . Detailed Notation Einstein Notation a b a1eˆ1 a2eˆ2 a3eˆ3 b1eˆ1 b2eˆ2 b3eˆ3 a b ajeˆj bmeˆm a1eˆ1 b1eˆ1 a1eˆ1 b2eˆ2 a1eˆ1 b3eˆ3 a e be a e b e a e b e 2ˆ2 1ˆ1 2ˆ2 2ˆ2 2ˆ2 3ˆ3 aj jmbm a3eˆ3 b1eˆ1 a3eˆ3 b2eˆ2 a3eˆ3 b3eˆ3 ab a b a b 1 1 2 2 3 3 ajbj 19 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology 3. Tensor – the indeterminate vector product of two (or more) vectors e.g.: stress velocity gradient – tensors may be constant or may be variable Definitions dyad or dyadic product – a tensor written explicitly as the indeterminate vector product of two vectors a d dyad A general representation This notation of a tensor , is also part of Gibbs notation. 20 © Faith A. Morrison, Michigan Tech U. 10 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Laws of Algebra for Indeterminate Product of Vectors: NO commutative a v v a yes associative b a v b a v b a v yes distributive a v w a v a w 21 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology How can we represent tensors with respect to a chosen coordinate system? Just follow the rules of tensor algebra a m a1eˆ1 a2eˆ2 a3eˆ3 m1eˆ1 m2eˆ2 m3eˆ3 a1eˆ1m1eˆ1 a1eˆ1m2eˆ2 a1eˆ1m3eˆ3 a2eˆ2m1eˆ1 a2eˆ2m2eˆ2 a2eˆ2m3eˆ3 a3eˆ3m1eˆ1 a3eˆ3m2eˆ2 a3eˆ3m3eˆ3 3 3 ˆ ˆ ak ek mwew kw1 1 3 3 Any tensor may be written as the ˆ ˆ sum of 9 dyadic products of basis ak mw ek ew kw1 1 vectors 22 © Faith A. Morrison, Michigan Tech U. 11 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology What about ? Same. 3 3 A Aij eˆieˆj ij1 1 Einstein notation for tensors: drop the summation sign; every double index implies a summation sign has been dropped. A Aij eˆieˆj Apk eˆpeˆk Reminder: the initial choice of subscript letters is arbitrary 23 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology How can we use Einstein Notation to calculate dot products between vectors and tensors? It’s the same as between vectors. a b a u v b A 24 © Faith A. Morrison, Michigan Tech U. 12 Chapter 2: Vectors and Tensors 2/5/2018 Mathematics Review Polymer Rheology Summary of Einstein Notation 1. Express vectors, tensors, (later, vector operators) in a Cartesian coordinate system as the sums of coefficients multiplying basis vectors - each separate summation has a different index 2. Drop the summation signs 3. Dot products between basis vectors result in the Kronecker delta function because the Cartesian system is orthonormal. Note: •In Einstein notation, the presence of repeated indices implies a missing summation sign •The choice of initial index (, , , etc.) is arbitrary - it merely indicates which indices change together 25 © Faith A. Morrison, Michigan Tech U. Mathematics Review Polymer Rheology 3. Tensor – (continued) Definitions Scalar product of two tensors A : M Aipeˆieˆp : M kmeˆk eˆm carry out the dot Aip M kmeˆieˆp : eˆk eˆm products indicated Aip M km eˆp eˆk eˆi eˆm Aip M km pkim “p” becomes “k” “i” becomes “m” Amk M km 26 © Faith A.
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