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directions. V may be viewed as a vector entering into operations formally. V
V is a dot (scalar) product of Vand V , called divergence and also denoted by div
V . V X V is a cross (vector) product of V and V , called vorticity and also denoted
by rot V. If V = (uI , u2 ,u3 IT is in terms of the Cartesian coordinates ( x l , z2, z 3 ) ,
then
7
(1. 2. 10)
In this case the above theorems can be reduced to relations between integrals
over a plane region and its boundary curve, and still retain its name, Gauss' theo-
rem.
All the integral laws stated above can be considered as specific forms of the gen-
eral transport equation
(1.2.11)
where J, is the density of a transported quantity, f is the flux density of that quanti-
t y , and cp is its source density. By using Green's formula, the second term can be re-
duced to a volume integral
so that under some differentiability condition we have the differential form
3 + , . f = y
at
(1. 2. 12)
N . BRIEF INTROWCTION TO CARTESIAN TENSOR
A tensor defined in a Cartesian coordinate system is called a Cartesian tensor. A
scalar is zeroth order (order-0) tensor. A vector is an order-1 tensor. In a three-di-
mensional space, an order-2 tensor T is composed of 32 = 9 components { t,, } ( z , j =
1 ,2,3) . The definitions of scalar, vector and tensor are based on how their compo-
nents change under a rotation of the coordinate system. A scalar does not change at
all. Any set of three scalars forms a vector only if they are subject to the well-known
coordinate rotation formula. An order-2 tensor can be regarded as a matrix such
that, when coordinates ( r l , zz, r 3 ) are changed into ( G I , i z , &) under a rotational
transformation, the relation between its old and new components { t,, } and { t,, } (z, j
= 1 , 2 , 3 ) is given by
t,, = ~mnB,mRB,8 (1. 2. 13)
where B,, is the direction cosine of the angle made by the $,-axis with the r,-axis. For brevity of notation, the Einstein summation convention is adopted in the above equa- tion and also hereafter. If a pair of common indices appears in a certain term, then it - " a u z a3 a, axl az ax3 denotes a summation over the index, e. g. , 2 = - + - + - . The notation of tensor is similar to that of vector. A tensor is denoted either by an uppercase only or by a lowercase with subscripts. To the former, called a symbol notation, no coordinate system is assigned. It has the merit of conciseness, but we should follow some special rules in writing a tensor equation. The latter, named an index notation, can show clearly the order of tensor, the arrangement of its compo- nents, and the role of the coordinate system. When writing component equations for the convenience of numerical solution, there is no special requirement as each compo- 8 nent is a scalar. One reason for using the tensor as our tool is that if some physical law has been formulated in the form of a Cartesian tensor equation, then it must hold in any or- thogonal coordinate system. Partial derivatives of a Cartesian tensor constitute a new higher-order tensor. For example, those of a first-order tensor {ti} satisfy the definition of second-order tensor (1. 2. 1 4 ) An order-2 tensor whose components can be arranged to form a symmetric ma- trix is called a symmetric tensor. A coordinate system can be found to annul its non- diagonal elements. Such coordinates are called major axes, while its diagonal ele- ments are called major values. Suppose that under an arbitrary orthogonal transformation of rectangular coordi- nates, the values of all components of a given tensor do not change; then it is an isotropic tensor. There exists no order-1 isotropic tensor. Any order-2 isotropic ten- sor can be written in the form @$, , where p is a scalar, while 4, denotes the Kroneck-
er delta
s,, = O(2 # .7), 6, = l ( 2 = 3 ) (1. 2. 15)
Moreover, a symmetric and isotropic order-4 tensor can be expressed in a form con-
taining only two scalars :
t r j k l = nsb]sU + p(ackdjL + 6'16jk) (1. 2. 16)
Chief operations defined for order-2 tensors include :
a = t,,s,, (denoted as LI = T: S )
T,k = S,,t,k(denOted aS R = S T)
(1 ) scalar product of two tensors
(2) inner product of two tensors
( 3 ) vector product of a tensor and a vector
( 4 ) tensor product of two vectors (diad)
(1. 2. 17)
(1. 2. 18)
(1. 2. 19)
(1.2. 20)
u, = v,t,,(denoted as u = V T)
t,, = u,v,(denoted as T = uv)
V. DIFFERENTIAL FORMS OF GOVERNING EQUATIONS IN FLUID DYNAMICS
Assume that: (i) the integral conservation laws hold for any bounded control
volume selected within some region; (ii) there is no singularity in the solution for
that region. Applying the integral forms of governing equations to an infinitesimal
fluid element, and using the Gauss theorem for changing area integrals into volume
integrals, we arrive at the desired differential forms, in which the left-hand sides are
just the integrands under volume integral symbol (the right-hand sides are zero).
Such an example has been given in Eq. (1. 2. 12) . The derivation is based on the
theorem that if an integral of a given function over any control volume vanishes,
then that function is identically equal to zero at all of its points of continuity. There-
fore, when some derivative or nonhomogeneous term in a differential equation suffer
a discontinuity or approaches infinity, we must still use the integral form.
9
1. Equation of continuity
(1.2.21)
Defining the material derivative (also called the substantial derivative) as D / D t
= a/at + V V (note that in Eqs. (1. 2. 1 ) - (1. 2. 4 ) the derivative d / d t should be
understood as D / L & ) , then the above equation can be written as
b + p v * v = o Dt (1.2. 22)
It should be noted that, when we use the integral conservation laws, the order
of the two operations, material derivative and integral, generally cannot be inversed.
For an incompressible fluid, Eq. (1. 2. 21) reduces to
v.v = 0 (1. 2. 23)
A velocity vector field satisfying this equation is called a solenoidal vector field. In-
deed, we can use the equation as a definition of generalized incompressible flow, in-
stead of constancy of density.
2. Equation of motion
P E = Dv v u + pFf l (1. 2. 24)
where u denotes a symmetric order-2 Cartesian tensor, called the stress tensor, acting
on an infinitesimal fluid element. Its nine components {a,, } fully describe the stress
behavior at a point. Its diagonal elements correspond to normal stresses, while the
nondiagonal ones correspond to shear stresses.
The right-hand side of the above equation represents the net external force exert-
ed on the fluid element per unit volume. According to the d'Alember Principle, the
left-hand side can be moved to the right-hand side, and considered as a part of the ex-
ternal force, called the inertial force. As for the integral conservation laws, the re-
sultant of the inertial forces should be exerted on the centre of the mass. Then the
moving element can be viewed as if it were in equilibrium.
The left-hand side of the above equation can be expanded to yield some different
forms as follows:
Eulerian form (also called convective form or nonconservative form)
] ~7 + P F B av P[X + (v v>v = v
which can be applied to Cartesian coordinate systems only.
Conservation form
(1. 2. 25)
(1. 2. 26)
where VV denotes a diad having the associative property that A (BC) = ( A B ) C .
Lamb-Gromeko or Bernoulli form
10
(1. 2. 27)
The above forms are suitable for any fluid. For an incompressible fluid, the constant
p can be moved outside the derivation symbol.
As a stress tensor appears in the equation of motion, the system, Eqs. (1. 2.
21 ) and (1. 2. 24) , is open, and cannot be solved. It is thus necessary to introduce
a constitutive equation, a relation between stress tensor and strain tensor. If that e-
quation does not itself change under any orthogonal