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transformation of the coordinate
system, the fluid is isotropic.
V I . CONSTITUTIVE EQUATIONS FOR FLUIDS
In fluid dynamics, a constitutive equation expresses the relation between the
stress tensor and the deformation-rate tensor of a fluid. We will now explain the ba-
sic logics involved.
1. The stress tensors in incompressible and compressible fluids are somewhat dif-
ferent, expecially the two pressures are radically different.
For an incompressible fluid, the stress tensor can be decomposed into two parts
u =- P I f z ( I . 2. 28)
where p=scalar , I=identity matrix, and t=order-2 bias (or viscosity) stress ten-
sor. The first term on the right-hand side is a diagonal matrix expressing the isotropic
part of the stress tensor. The diagonal element p is determined by
p = - tru/3 = - (uli + uZ2 + u3,>/3 (1. 2. 29)
where tru=sum of diagonal elements of the matrix u , called trace. When the fluid is
motionless, t = 0 and p denotes the static pressure. For a moving fluid, p is called the
dynamic pressure, which is intrinsically a mechanical quantity which does not depend
on other thermodynamic state variables. Hence, after the movement has stopped,
stress tensor will approach that in static state.
If the flow is compressible, p is generally not equal to the value given by the
above equation. It is necessary to consider the influence of the additional viscosity
stemming from dilatation of the fluid. Difference between the two values of p often is
a linear function of the inflation rate of the volume. However, we usually adopt
Stokes' hypothesis, which ignores this factor, leading to a Stokes' fluid.
Another feature of the compressible flow is that p not only satisfies balance rela-
tions among stresses, but also depends on other thermodynamic parameters, thus
gaining its name of thermodynamic pressure. Since we neglect the variation of tem-
perature, the dependency can be expressed by an equation of state f (p , p ) = 0. A flu-
id for which p is only a function of p is called a barotropic fluid. Specifically, for an
isentropic flow (adiabatic and reversible) of a perfect polytropic gas, the equation of
state can be formulated as
- = const (1. 2. 30)
PY
where y = C,/C,, is the ratio of specific heats under constant pressure C, and under
constant volume C,,.
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Under isentropy conditions, another thermodynamic parameter, c , related to p ,
is defined as
(1. 2. 31)
For perfect gases, if Eq. (1. 2. 30) is introduced into the above equation, we
obtain c = &&I. As the right-hand side is the coefficient of order-1 term in a
Taylor's series expansion of p , c denotes the speed of propagation of a small distur-
bance, the speed of sound in gas dynamics. However, the application of Eq. ( I. 2.
31) is conditional. The condition of isentropy also requires that there is no strong
shock wave (discontinuity) occurring in the flow.
The second term on the right-hand side of Eq. (1. 2. 28) represents the non-
isotropic part of the stress tensor. Its diagonal elements are called (viscosity) normal
stress, while nondiagonal ones are (viscosity) shear stress. The total normal stress in
any coordinate direction equals the sum of negative pressure - p and viscosity normal
stress, while the total shear stress is just viscosity shear stress. The above-mentioned
Stokes' hypothesis is equivalent to saying that the mean value of the viscosity normal
stress vanishes.
2. In a Newtonian fluid, at any point, the components of bias stress tensor T
are homogeneous linear functions of the components of a second-order tensor L , the
gradient of the velocity, &/ax,, called deformation rate tensor.
A velocity field determines not only the translation and rotation of fluid ele-
ments viewed as rigid bodies, but also the relative motions between fluid particles. In
other words, the velocity gradient determines the deformation rate, including dilata-
tion and shearing. Obviously, the assumption of linearity between bias stress tensor
and deformation rate tensor is a reasonable 3-D generalization of Newton's law in one
space dimension, thus called generalized Newton law. Now shear stress is proportion-
al to velocity gradient in a shear layer, whilst in the elastic-solid case stress is propor-
tional to deformation, but not its rate.
The deformation rate tensor L can be expressed as a sum of its symmetric part E
and its asymmetric part D (an order-2 tensor called the rotation tensor, vorticity ten-
sor or spin tensor)
L = E + Q
E = - ( L + LT) , D = - ( L - LT) 1 1 2 2
(1. 2. 32)
(1. 2. 33)
(1. 2. 34)
(1. 2. 35)
The relation between the symmetric term E and the velocity gradient tensor V Ti
is written as
1
E = -[Vv 4- (VTi)T] 2
t.
(1. 2. 36)
12
The diagonal elements of E denote the change rates of the relative extension or con-
traction in the three coordinate directions respectively, and their sum (trace of matrix
E) is just the divergence, V V , showing the inflation rate of the fluid element. A
nondiagonal element e,, denotes the shear deformation rate of the angle made by a pair
of coordinate axes (relative to the indices) divided by -2, or equivalently, a shear
velocity in the x,-direction of two neighboring particles located at the x,-axis divided
by -2. Among the components of E, a consistency condition must be satisfied, i. e. ,
the integrability condition that velocity can be obtained by integrating Eq. (1. 2. 36).
In the 2-D case, it can be expressed as
(1. 2. 37)
As for the rotation tensor, Q, because of asymmetry there are only three inde-
pendent components, which are equal to the components of vorticity V X V divided
by 2, and can be used as a measure of the angular velocity of the fluid element rotat-
ing as a rigid body. It should be noted that vorticity is a local property of the flow
field, independent of the curvature of the streamlines.
Indeed, the bias stress tensor T depends on the symmetric tensor E only, and not
on the rotation tensor Q. For a Newtonian fluid a general relation holds
ztj = t g j k l e k l (1. 2. 38)
where T is an order-4 viscosity tensor whose elements are related to temperature on-
ly, and not to u and E. As temperature is assumed here to be fixed, T is a constant
tensor. In the 3-D case, T has 3'=81 elements. For an isotropic fluid, according to
Eq. (1. 2. 1 6 ) , there are only two independent constants, denoted by A and ,u. By
mathematical derivation, Eq. (1. 2. 38) can be expressed as
z = 2 p ~ + a ( ~ - v ) z (1. 2. 39)
so we have
a,, = - pa,, t- 2w,, + k k k f f , ] (1. 2. 40)
For incompressible fluids, as div V = V V = 0 (or e,, = 0 ) , and under the as-
sumption that 3h+2p= 0 ,
(1. 2. 4 1 ) z = 2,uE - -(V V)Z = , u [VV + ( V V ) T ] - \$(V V ) I
c =- PI + 2hE (1. 2. 42)
V . a = - V p + div(2,uE) (1. 2. 42a)
,u is called the dynamic viscosity. In the 1-D case, it is the proportional constant be-
tween shear stress and shear deformation rate, and it can be interpreted as the mo-
mentum of the viscous force exerted on a unit area. It is a macroscopic characteristic
2,u
3
13
of momentum exchanges stemming from molecular motions. Due to their common
mechanism, there is a relation between viscosity ,u, heat conductivity k (featuring
energy exchange) and mass diffusivity coefficient D (featuring material transporta-
tion
(1. 2. 43)
The dimension of is MILT, while its unit is P = lg/(cm s ) = 1 dyne s/cm2 in
the CGS system, or Pa s (or denoted by Pi) = 1 newton s/mz = 10 P in the SI u-
nit system. The value ,u of water at 20 'C and 1 atmospheric pressure equals 0. 001
Pa s . When temperature varies only slightly, ,u can be viewed as a constant, and is
correlated slightly with pressure. For isotropic fluids, a constant ,u can be applied to
all coordinate directions in the 3-D case; otherwise, an order-2 viscosity tensor
should be introduced.
Moreover, we define kinetic viscosity by v=,u/p,