VectorsAndTensors

# VectorsAndTensors - SUMMARY OF VECTOR AND TEN SOR...

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Unformatted text preview: SUMMARY OF VECTOR AND TEN SOR 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 it The following types of parenthesis will also be used to denote the results of various operations. ()mscalaf (E-EﬂrlQ—CQ [ ]=V“=Gt0r [win], [2-2] { } = tensor {g . g} The multiplication signs can be interpreted as follows: Multiplication sign Order of Result None 2 x 2-1 2—2 2-4 Scalars can be interpreted as 0th order tensors, and vectors as ﬁrst order tensors. Examples: s3; order is O+2£2 which is a 2nd order tensor gxw order is l+1-1=1 which isa vector g; order is 2+2-4=0 which is a scalar Deﬁnition of a Vector: A vector is deﬁned as a quantity of a given magnitude and direction. lul is the magnitude of the vector 11 Two vectors are equal when their magnitudes are equal and when they point in the same direction. Addition and Subtraction of Vectors: ll-W Dot Product of two Vectors: (u - W) = IUI IWl COSGP) Area=(u.w) commutative (u . v) = (v . u) .um::.'::!121. not associative (u . v)w :3 u(v . w) distributive (u . [v + w]) = (u . v) + (u . w) Cross Product of two Vectors: [uxw] = |u| |w| sin(q)) n where n is a vector (unit magnitude) normal to the plane containing 11 and w and pointing in the direction that a right-handed screw will move if we turn 11 toward w by the shortest route. W Area of this equals the length offuxw] ll not commutative [uxw] = -[wxu] not associative [u x [v x w]] at [[u x v] x w] distributive [[u + v] x w] 2 [u x w] + [v x w] Addition of vectors: E+ﬂ = Ziﬁiui+ZiJiWi = Zi6i(ui+Wi) Multiglication of a Vector by a Scalar: SE = S[Zi5aui] = Zj5;(sui) Dot Product: (Edit) : [ZiSiui1-[Zj5jo] = ZiZj(§i-§j)Uin = = Zi2j5ijmwj' = Ziuiwi Cross Product: [2X32] = [(chS‘juJ')X(Zk5ka)] = 2j2k[6jx5k]ujwk = Zizjzk£ijk5iujwk 51 52 53 U1 “2 113 VECTOR DIFFERENTIAL OPERATIONS Deﬁne ﬁrst the E operator, which is a vector 6 6 6 6 v=51—u+5—+5———=25 5x1 Zaxz 3ax3 L laXi The Gradient of a Scalar Field: 6 s 6 s 6 s 65 VS = 5I_+ 2_‘+53_"— = Z: 1— 6 X1 6 x2 5 x3 5 x1 not commutative Vs i sVL not associative (Vr)s ¢V(rs) distributive V(r+s) = Vr + Vs The Divergence of a Vector Field: 6 (V-“)=|:Zj85ia-:l[2j5ju_i]= ZZJ-[cSi-c3‘j16—u aix Xi _ 6 _ 6”- — Zijcha—muj " Zia—3: not commutative (V.u);t(u . V) not associative (V. s )u¢ (Vs.u) distributive V.(u+w)m(V.u)+(V.w) The Curl of a Vector Field: a a [VXH] : Zj5j_ X[Zk5kuk} = ZjZk[5jX5k]—Uk an Ex]- 51 52 53 = i _a__ 1 6);} 6X2 6x3 u; uz L13 au3 6112] [all] (3113] [6112 6111] =5 —-— +5 ———— +5 -——— {5x2 5X3 2 6x3 5x1 3 5x1 axz [ V x u ] = curl (u) = rot (n) It is distributive but not commutative or associative. The Laplacian Operator: The Laplacian of a scalar is: The Laplacian of a vector is: V2u= V(V.u)—[Vx[qu]] The Substantial Derivative of a Scalar Field: If u is assumed to be the local ﬂuid velocity then: D a —=—+ .v Dr 5: (u ) The substantial derivative for a scalar is: DS 63 as Dl' 61 2;“: axj The substantial derivative for a vector is: Du _ 3U Dt at +(u.V)u=Zi§i [%i+(u.V)ui] This expression is only to be used for rectangular co—ordinates. For all co—ordinates: (u.V)u : £01.14) — [u x [V x u]] 10 11 SECOND - ORDER TENSORS A vector u is speciﬁed by giving its three components, namely 11], 1.12, and u;. Similarly, a second- order tensor 1 is speciﬁed by giving its nine components. In 1'12 r13 7: 721 1'22 1'23 731 2'32 2'33 The elements I] 1, 1'22, and 1333 are called diagonal while all the others are the non-diagonal elements of the tensor. If T12=T21, 1:31:1713, and 132=1723 then the tensor is symmetric. The transpose of 1: is deﬁned as: 1'11 1‘2: I31 r = Tl2 722 T32 713 2'23 7733 If T is symmetric then T=T*. DIFFERENTIAL OPERATIONS IN CURVILIN EAR COORDINATES The operator V will now be derived in cylindrical and spherical co-ordinates. Cylindrical: The following relations can be obtained by differentiating the relations between the unit vectors in the cylindrical co-ordinates with those in the Cartesian ones. a a 6 6r 5r Br 6" ‘32 a a _ = —.._ =.. :0 an 5’ 5" aa ‘59 5’ aa 52 a a The deﬁnition of V in Cartesian co-ordinates is: a a a V = 6XE+6Y5§+6ZE Substituting 5x, 5,,, and 62 in terms of 5,, 69, and 62 and simplifying we obtain V for cylindrical co- ordinates, that is: 23 a 1 a a V: _+ __+ _ ‘5’ar 59:69 azaz Spherical Coordinates The following relations can be obtained by differentiating the relations between the unit vectors in the Spherical coordinates with those in the Cartesian ones. 6 a 8 ——-6 :0 "—5 =0 —5 =0 6r r 6r 6 ar 45 B 6 6 n...— = — :— —. =0 35 =5 sing 3-5 =5 cosgﬁ 3—5 = 5 sinQ-é‘ cost? 6gb r (if. a¢ 1'9 415 a¢ z r 6 The deﬁnition of V in Cartesian co-ordinates is: 6 6 6 V: axa-z-gygy-—+52E Substituting 6x, 63,, and 62 in terms of 5,, 69, and 6,9, and simplifying we obtain V for spherical co- ordinates, that is: Vz5r2+5ﬁli+5 1 6 ar r649 wrsinﬁgqt—S For more details see: 1. RB. Bird, W.E. Stewart and EN. Lightfoot, "Transport Phenomena," Wiley, New York, 1960. 2. RB. Bird, R.C. Armstrong and O. Hassager, ”Dynamics of Polymeric Liquids," Vol.1, "Fluid Mechanics," Wiley, New York, 1977. ...
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