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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)
 Secondorder 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 (EEﬂrlQ—CQ
[ ]=V“=Gt0r [win], [22]
{ } = tensor {g . g} The multiplication signs can be interpreted as follows: Multiplication sign Order of Result
None 2
x 21
2—2
24 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+11=1 which isa vector g; order is 2+24=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: llW 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 righthanded 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[cSic3‘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 nondiagonal 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 coordinates. Cylindrical: The following relations can be obtained by differentiating the relations between the unit vectors in the cylindrical coordinates 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 coordinates 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 35 =5 cosgﬁ 3—5 = 5 sinQé‘ cost?
6gb r (if. a¢ 1'9 415 a¢ z r 6 The deﬁnition of V in Cartesian coordinates is: 6 6 6
V: axazgygy—+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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 Spring '11
 Wereley

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