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```An Introduction to Mathematical Modeling: A Course in Mechanics
by J. Tinsley Oden
нвннм
CHAPTER 8
^
Μ
Ι
EXAMPLES AND
APPLICATIONS
At this point, we have developed all of the components of the general
continuum theory of thermomechanical behavior of material bodies. The
classical mathematical models offluidmechanics, heat transfer, and solid
mechanics can now be constructed as special cases. In this chapter, we
develop several of the more classical mathematical models as special
applications of the general theory. Our goal is merely to develop the
system of equations that govern classical models; we do not dwell on
the construction of solutions or on other details.
8.1 The Navier-Stokes Equations for Incompressible Flow
We begin with one of the most important example in continuum mechanics: The equations governing the flow of a viscous incompressible fluid.
These are the celebrated Navier-Stokes equations for incompressible
flow. They are used in countless applications to study the flow of fluids,
primarily but not exclusively water or even air under certain conditions.
We write down the balance laws and the constitutive equation:
An Introduction to Mathematical Modeling: A Course in Mechanics, First Edition. By J. Tinsley Oden
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.
63
64
CHAPTER 8. EXAMPLES AND APPLICATIONS
Conservation of Mass
(Recall (2.4))
| | + d i v ( f w ) = 0.
(8.1)
For an incompressible fluid, we have
0 = — / dx =
dctFdX
dtJnt
Jn0
= / detFdivvdX= /
Jn0
Jnt
divvdx,
which implies that
div v = 0.
(8.2)
Thus
°ρ
dp
,
dg
0 = — + v ■ gradp + pdivv = — +v· gradρ= —,
σί
σί
dt
which means that ρ is constant.
Conservation of Momentum
dv
g— + pvg™dv-divT
(8.3)
(Recall (4.8))
= f,
V(a;,t)eii t x(0,T),
T = TT.
Constitutive Equation
(Recall (7.11))
Τ = - ρ Ι + 2μϋ,
(8.5)
Zi
The Navier-Stokes equations Finally, introducing (8.5) into (8.4)i
and adding (8.2) gives the celebrated Navier-Stokes equations,
dv
at
div v = 0,
where Δ = vector Laplacian = div grad.
(8.6)
8.1 THE NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLOW
65
Figure 8.1: Geometry of the backstep channel flow.
Application: An initial-boundary-value problem As a standard
model problem, consider the problem of viscous flow through a channel
with a "backstep" as shown in Fig. 8.1.
Initial conditions:
v(x,0) = v0(x),
(8.7)
where the initial field must satisfy div v0 = 0, which implies that
/
v0-ndA = 0.
(8.8)
Boundary conditions:
1. On segment ВС and DE U EF U FA, we have the no-slip boundary
condition v = 0.
2. On the "in-flow boundary" AB, we prescribe the Poiseuille flow
velocity profile (see Fig. 8.1):
Mo^t)=\i-[-^)
ν 2 (0,χ 2 ,ί) = 0.
2z 2 x
2>
)u0,
(89)
66
CHAPTER 8. EXAMPLES AND APPLICATIONS
3. On the "out-flow boundary" CD, there are several possibilities. A
commonly used one is
-ρ + μ
v2{L,x2,t)
x\ = L
= T e i | r i = L = o,
(8.10)
= 0,
where L is the length of the channel, i.e., L = |BC
8.2
Flow of Gases and Compressible Fluids:
The Compressible Navier-Stokes Equations
In the case of compressible fluids and gases, the so-called compressible
Navier-Stokes equations can be derived from results we laid down earlier. These are the equations governing gas dynamics, aerodynamics,
and the mechanics of high-speed flows around objects such as aircraft,
missiles, and projectiles.
Typical constitutive equations for stress, heat flux, and entropy are
T = - p i + A tr(D)I + 2μΌ - αθΐ,
9η
_2
я " = Q P,
where p is now the thermodynamic pressure, usually given as a function
of Θ and ρ by an "equation of state," A and μ are bulk and shear viscosities, a a thermal parameter, and к the thermal conductivity. Here Θ
is usually interpreted as a change in temperature from some reference
state. Because (8.1 l)i is linear in the deformation rate D, (8.1 l)i is said
to describe a Newtonian fluid. Relation (8.11)2 is the classical Fourier's
Law.
The equations of balance of mass, linear momentum, and energy now
8.3 HEAT CONDUCTION
67
assume the forms
— + div(er) =0,
dv
+ μΑυ — a grad Θ,
*И*+|р) =кАв + r + X (trD)
2
+ 2^tr(D 2 ).
Here ρ = dg/dt, θ = άθ/dt, and t r D = div«. Typical boundary and
initial conditions are of the form
ρ(χ,0) = ρο(χ),
v{x,0) = v0(x),
θ{χ,0) = θ0(χ),
(8.13)
for given data, ρο,ν0,θ0, with ж e Ω С I 3 , and
n ■ ( - pi + λ div « I + 2μΌ\ = g(x, t)
v(x,t)
-n-kWe
= v(x,t)
=q
on Γι x (0, r),
о п Г 2 х ( 0 , г ) , (8·14)
опГ3х(0,г),
where g, v, and q are given boundary data and n is a unit extension
normal to domain Ω, and Γι, Γ 2 , and Г 3 are subsets of the boundary <9Ω
οίΩ.
8.3
Heat Conduction
Ignoring motion and deformation for the moment, consider a rigid body
being heated by some outside source. We return to the Helmholtz free
energy of (7.13) and the constitutive equations (7.19). The constitutive
equations are assumed to be
Φο = ~(θ-
θ0)2 - β(θ - θ0) + 7(0ο),
(8.15)
68
CHAPTER 8. EXAMPLES AND APPLICATIONS
where a and ß are positive constants, θ0 is a given uniform reference
temperature, and -у(90) is a constant depending only on θ0. The heat flux
is given by
(Fourier's Law).
q0 = -fcV(0 - Θ0)
(8.16)
Then,
дфо
= a(9 - θ0) + β,
3Θ
m=
«a
n
д
^°
п
(8.17)
(irrelevant).
The specific heat с of a material is the change in internal energy of
a unit mass of material due to a unit change in temperature at constant
deformation gradient F and is defined by
(8.18)
с = θθ'
where e0 is the internal energy per unit mass in the reference configuration. Using the definition of ψ0, we easily show that
θ-
дв
= αθ.
(8.19)
Finally, turning to the equation for the conservation of energy
(e.g., (5.5)), we have
de0οθ
= QocO
= - Div q0 + r 0
= -Div(-fcVÖ)+r 0
or
3Θ_
Qocdt
V · к^в = r0.
(8.20)
This is the classical heat equation. In general, one can consider using
the approximate specific heat с = αθ « αθ0.
8.4 THEORY OF ELASTICITY
69
8.4 Theory of Elasticity
We consider a deformable body В under the action of forces (body
forces f and prescribed contact forces σ(η, χ) = g(x) on dilt). The
body is constructed of a material which is homogeneous and isotropie
and is subjected to only isothermal (Θ = constant) and adiabatic (q = 0)
processes. The sole constitutive equation is
φ = free energy = Φ(Χ, t, Θ, E) = Φ(Ε).
Let Φ denote the free energy per unit volume: Φ = ρ0^.
constitutive equation for stress is
S =
дФ(Е)
дЕ
(8.21)
So the
(8.22)
sym
In this case, the free energy is^called the stored energy function, or the
strain energy function. Since Φ(·) (and дф/дЕ) must be form-invariant
under changes of the observer and since В is isotropie, Φ (E) must depend
on invariants of E:
4!{K) =
W{!E,IE,ME).
(8.23)
Or, since E = (С—/)/2, we could also write Ф as a function of invariants
ofC:
_
_
Ъ = W{Ic,ic,Mc).
(8.24)
The constitutive equation for stress is then
dW_ dlE_ dW_ дПЕ
~Ъ1^'Ш
+
+
dW_ дШЕ
дТЕ'~дЁ дЙЕ'''дЁ
_ dW dlc^ Ж_ Шс_ dW_ дШ^
~ д1с"дС+Ъ1^"дС+
дШс ' дС '
(8.25)
70
CHAPTER 8. EXAMPLES AND APPLICATIONS
and we note that
dIE
dE
t
T
'
= (trE-')I-- E " T C o f E ,
£-«*
(8.26)
Materials for which the stress is derivable from a stored energy potential
are called hyperelastic materials.
The governing equations are [recall (4.14)]
dW
Div (I + Vw)
dE
with
sym/
+ f0 —
d2u
QQ-^T,
Otl
F = (I + V«),
dW
S =
= T(/E,#E,#E,E),
dE sym
1
E = ^(Vu + VuT + VurVu).
(8.27)
(8.28)
Linear Elasticity One of the most important and widely applied theories of continuum mechanics concerns the linearized version of the
equations of elasticity arising from the assumptions of "small" strains.
In this case,
Ezie
= - ( V u + Vu T ),
W = -Етеекее^,
Su
(Hooke's Law),
IJ = -— = Eijkeeke = EijkeTr[r
de
дХ,
ч
and we assume
Ецы — Eiike — Еце
■'kCij ■
ijlk к = Е\
(8.29)
J
(8.30)
8.4 THEORY OF ELASTICITY
71
Figure 8.2: Elastic body in equilibrium.
Then, for small strains, we have
(8.31)
For isotropie materials, we have
Eijke = ^.SijSke + ß{öik5ji + StfSjk),
(8.32)
where A, μ are the Lamé constants:
A=
uE
{l + v){l-2vY
rμ
E
2(1 + 1/)'
(8.33)
E is Young's Modulus and v Poisson's Ratio. Then
S = A (trE)I + 2μΕ = A divw I +
2ß{Vu)sym.
(8.34)
Suppose (8.34) holds and the body is in static equilibrium
(d2u/dt2 = 0). Then (8.31) reduces to the Lamé equations of elastostatics. In component form, these can be written
(8.35)
72
CHAPTER 8. EXAMPLES AND APPLICATIONS
A typical model boundary-value problem in linear elastostatics is
suggested by the body in Fig, 8.2. The body is fixed against motions
on a portion TD of the boundary, is subjected to surface tractions g on
the remainder of the boundary surface, Г \ = dil \ To, and is subjected
to body forces / 0 . Then equations (8.35) are augmented with boundary
conditions of the form
njEijki
щ =0
duk
= 9i
on FD,
on Г Ν·
(8.36)
```
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