FLUID_DYNAMICS_CRETE

FLUID_DYNAMICS_CRETE - SUMMARY OF VECTOR AND TENSOR...

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SUMMARY OF VECTOR AND TENSOR NOTATION -Bird, Stewart and Lightfoot "Transport Phenomena" -Bird, Armstrong and Hassager "Dynamics of Polymeric Liquids" The Physical quantities encountered in the theory of transport phenomena can be categorised into: - Scalars (temperature, energy, volume, and time) - Vectors (velocity, momentum, acceleration, force) - Second-order tensors (shear stress or momentum flux tensor) While for scalars only one type of multiplication is possible, for vectors and tensors several kinds are possible which are: - single dot . - double dot : - cross x The following types of parenthesis will also be used to denote the results of various operations. ( ) = scalar (u . w ), ( σ : τ ) [ ] = vector [ u x w ], [ τ . u ] { } = tensor { σ . τ } The multiplication signs can be interpreted as follows: Multiplication sign Order of Result None Σ x Σ -1 . Σ -2 : Σ -4 ________________________________________ Scalars can be interpreted as 0th order tensors, and vectors as first order tensors. Examples: s τ order is 0+2=2 which is a 2nd order tensor u xw order is 1+1-1=1 which is a vector σ : τ order is 2+2-4=0 which is a scalar
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2 Definition of a Vector: A vector is defined as a quantity of a given magnitude and direction. |u| is the magnitude of the vector u Two vectors are equal when their magnitudes are equal and when they point in the same direction. Addition and Subtraction of Vectors: w w u + w u - w u u Dot Product of two Vectors: (u . w) = |u| |w| cos( φ ) w ϕ u commutative (u . v) = (v . u) Area=( u.w ) not associative (u . v)w u(v . w) distributive (u . [v + w]) = (u . v) + (u . w) Cross Product of two Vectors: [uxw] = |u| |w| sin( φ ) n where n is a vector (unit magnitude) normal to the plane containing u and w and pointing in the direction that a right-handed screw will move if we turn u toward w by the shortest route. ϕ w u Area of this equals the length of [uxw]
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3 not commutative [uxw] = -[wxu] not associative [u x [v x w]] [[u x v] x w] distributive [[u + v] x w] = [u x w] + [v x w]
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4 VECTOR OPERATIONS FROM AN ANALYTICAL VIEWPOINT Define rectangular co-ordinates: 1, 2, 3 x, y, z respectively Many formulae can be expressed more compactly in terms of the kronecker delta δ ij and the alternating unit tensor ε ijk, which are defined as: δ ij = 1 if i=j δ ij =0 if i j and ε ijk =1 if ijk=123, 231, 312 ε ijk = -1 if ijk=321, 132, 213 ε ijk =0 if any two indices are alike We will use the following definitions, which can be easily proved: δ ε ih hjk ijk k j 2 = and The determinant of a three-by-three matrix may be written as: jm in jn im mnk ijk k - = α 3k 2j 1i ijk k j i 33 32 31 23 22 21 13 12 11 =
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5 DEFINITION OF A VECTOR AND ITS MAGNITUDE: THE UNIT VECTORS A vector u can be defined completely by giving the magnitudes of its projections u 1 , u 2 , and u 3 on the co-ordinate axis 1, 2, and 3 respectively. Thus one may write u = u + u + u i i 3 1 = i 3 3 2 2 1 1 δ = u where δ 1 , δ 2 , and δ 3 are the unit vectors in the direction of the 1, 2 and 3 axes respectively. The following identities between the vectors can be proven readily: All these relations can be summarized as: 0 = . = . = .
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FLUID_DYNAMICS_CRETE - SUMMARY OF VECTOR AND TENSOR...

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