The inner limit process of slightly compressible multiphase equations

Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 12, Number 3 (2016), pp. 2219–2241
© Research India Publications
http://www.ripublication.com/gjpam.htm
The inner limit process of slightly compressible
multiphase equations1
Hyeonseong Jin2
Department of Mathematics,
Jeju National University,
Jeju, 63243, South Korea.
Abstract
We discuss the motion of the slightly compressible multiphase flow near the initial
time. The formally uniformly valid inner limit expansions are derived for the solutions of compressible equations. Under the singular limit process the constitutive
asymptotic expansions and conditions are examined as requirements for the derivation of asymptotic expansions for compressible solutions.
AMS subject classification: 76Txx, 76F25, 76N99, 76Axx.
Keywords: Multiphase flow models, Closure, Constitutive laws, Averaged equations.
1.
Introduction
In this paper we consider a hyperbolic system of the compressible isentropic two-pressure
two-phase flow equations [2, 6, 7, 12, 13] in an appropriate nondimensional form
βkλ
βkλ ρkλ
1
∂ρkλ
∂t
∂vkλ
∂t
∂β λ
∂βkλ
+ v ∗λ k = 0,
∂z
∂t
λ
λ
λ
λ ∂ρk
λ λ ∂vk
λ λ
∗λ ∂βk
+ vk
+ βk ρk
+ ρk (vk − v )
= 0,
∂z
∂z
∂z
λ
∂β λ
∂v
∂pλ
+ vkλ k + λ2 βkλ k + λ2 (pkλ − p ∗λ ) k = βkλ ρkλ g(t),
∂z
∂z
∂z
(1)
(2)
(3)
This research was supported by Basic Science Research Program through the National Research
Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (No. NRF2009-0069783).
2
Corresponding author.
2220
Hyeonseong Jin
for the volume fraction βkλ , velocity vkλ , density ρkλ , and pressure pkλ of fluid k depending
on a large dimensionless parameter λ. Here pkλ = pk (ρkλ ) and an equation of state
γ
pk (ρk ) = Ak ρk k , γk > 1 is given with ∂pk /∂ρk (ρk ) > 0 for ρk > 0 and the entropy Ak
assumed to be constant within each fluid but A1 = A2 . The fluids are distinguished by
a subscript k, where k = 1 and k = 2 denote the light and heavy fluids, respectively.
Eqs. (1)–(3) have been nondimensionalized so that a typical mean fluid velocity, |vm |,
has been chosen as the ratio of time units to space units. Accordingly, the parameter λ is
essentially the reciprocal of the Mach number, M = |vm |(γp(ρm )/ρ)−1/2 , the ratio of
fluid speed to sound speed, where ρm is the mean density. In fact, λ = M −1 (γ A)−1/2 .
The interface quantities v ∗ and p∗ have been proposed [7,12,13] by closure relations
q ∗ = µ1 q2 + µ2 q1 ,
q
q
q
q
q
q = v, p,
(4)
q
where µ1 + µ2 = 1, µk ≥ 0 and that µk /βk is continuous on 0 ≤ βk ≤ 1 and for all t.
Here the primed index k denotes the fluid complementary to fluid k, i.e., k = 3 − k. The
q
q
q
µk thus depends on a single parameter dk . The closure for the constitutive law dk (t)
has been proposed [12, 13] and compared in a validation study based on simulation
data [1, 17]. For all the data sets of the 3D Rayleigh-Taylor and circular 2D RichtmyerMeshkov simulations, we have seen excellent validation agreement for the closures
proposed. We denote Zk = Zk (t) as the position of the mixing zone edge k, defined as
the location of vanishing βk and Vk = dZk /dt as the velocity of the edge k. At edge k,
the following boundary data holds vk = Vk (t) at z = Zk (t).
On the other hand, the incompressible flow equations are a distinctly different system
of the equations
∂β ∞
∂βk∞
+ v ∗∞ k = 0,
∂t
∂z
∞
∞
∂β
∂v
βk∞ k + (vk∞ − v ∗∞ ) k = 0,
∂z
∂z
∞
∞
∞
∞
∂β
∂vk
∂v
∂p
βk∞ ρk∞
+ vk∞ k + βk∞ k + (pk∞ − p ∗∞ ) k = βk∞ ρk∞ g(t)
∂t
∂z
∂z
∂z
(5)
(6)
(7)
for the volume fraction βk∞ , velocity vk∞ , and scalar pressure pk∞ , where ρk∞ is the
constant density of phase k. We assume that all state variables are piecewise C 1 functions
with discontinuous derivatives at the mixing zone edges z = Zk∞ (t) of incompressible
flow. Analytic solutions of the incompressible problem (5)–(7) have been obtained in
closed form [5, 6, 7].
One expects, under appropriate conditions, that the compressible multiphase flow
solutions βkλ , vkλ , pkλ converge to the incompressible solutions βk∞ , vk∞ , pk∞ as λ → ∞.
The zero Mach limit of the compressible multiphase flow equations is a time-singular
and layer-type problem which requires advanced techniques in asymptotics. Jin et al.
[4, 11] discussed the outer limit process of the weakly compressible multiphase flow
equations describing the fluid motions away from the initial time. The slow variables
in the outer limit asymptotic expansions have a slow scale of motion and they have
The inner limit process of slightly compressible multiphase equations
2221
been determined through second order in closed form. Information supplied from the
weakly compressible theory resolves underdetermination of incompressible pressures.
From a number of points of view [3, 8, 15, 16], the zero Mach limit of the single phase
compressible Euler equations has been studied in higher space dimensions. In contrast,
two-phase flow presents additional difficulties, so that even one dimensional two-phase
flow is nontrivial.
In this paper we discuss the limiting behavior of the solutions Ukλ ≡ (βkλ , vkλ , ρkλ , pkλ )tr
of the compressible equations (1)–(3) near the initial time as λ → ∞. For simplicity, we
suppress superscript λ’s of compressible variables from now on. We are concerned with
the derivation of inner limit asymptotic expansions for the solutions of the compressible
equations. In Sec. 4 the inner terms, uniformly valid in space, are evaluated in closed
form through first order in the expansion of volume fraction and pressure, and zero-th
order in the expansion of velocity. They are determined uniformly by matching the
inner limit in the exterior and the incompressible mixing zone and the fast transitionlayer expansions. The fast variables are defined as the inner expansion terms minus
the common terms resulting from the outer expansion evaluated in the inner limit to
each order. These variables oscillate on the fast time scale and are described by linear
wave equations. They are also defined uniformly in space by matching. Specifically, in
Sec. 4, the inner limit analysis is performed to show that there are no fast scale acoustical
oscillations in the asymptotic expansions of βk , ρk , pk through first order and in the
zero-th order of the expansion of vk . In Sec. 2 the constitutive asymptotic expansions
and conditions are examined as requirements of the derivation of formal asymptotic
expansions for compressible solutions.
The fluids are distinguished by a subscript k, where k = 1 and k = 2 denote the
light and heavy fluids, respectively. We assume existence of rigid wall at the top of a
finite but large domain D for compressible and incompressible two-phase flow. Then
velocity is zero and the pressure is unknown there. At the bottom of this domain, we
have conceptually an open container. This fixes the pressure at some ambient value, but
not the velocity at the bottom of D. This leads to the boundary conditions
v1 (z+∞ ) = 0 ,
p2 (z−∞ ) = const,
(8)
where z = z+∞ (z = z−∞ ) denotes the position of the upper (lower) wall of the domain
D.
2.
Constitutive Asymptotic Assumptions
We consider the limiting behavior of the solutions Uk ≡ (βk , vk , ρk , pk )tr of the compressible equations (1)–(3) as λ → ∞. The asymptotic expansions of compressible
2222
Hyeonseong Jin
solutions are introduced in the form
βk =
βk(0,s)
+ λ−1 βk(1,s)
+λ
−2
βk(2,s)
(2,f )
+ βk
+ O(λ−3 ),
(1,f )
vk = vk(0,s) + λ−1 vk(1,s) + vk
+ O(λ−2 ),
(2,f )
ρk = ρk(0,s) + λ−1 ρk(1,s) + λ−2 ρk(2,s) + ρk
+ O(λ−3 )
(2,f )
(2,s)
(0,s)
−1 (1,s)
−2
pk = pk + λ pk + λ
+ O(λ−3 ).
pk + pk
(9)
The equation of state gives the relations
pk(0,s) = pk (ρk(0,s) ),
pk(2,s)
pk(1,s) = ck2 (ρk(0,s) )ρk(1,s)
1 ∂ 2 pk (0,s) (1,s)2
(ρk )ρk
+ ck2 (ρk(0,s) )ρk(2,s) ,
=
2
2 ∂ρk
where ck2 (ρ) =
(2,f )
pk
(2,f )
= ck2 (ρk(0,s) )ρk
,
(10)
∂pk
(ρ). We assume the initial conditions
∂ρk
βk (z, 0) = βk∞ (z, 0) + λ−1 βk(1,s) (z, 0) + λ−2 βk(2,s) (z, 0) ,
vk (z, 0) = vk∞ (z, 0) + λ−1 vk(1,s) (z, 0) + vk(1) (z) ,
ρk (z, 0) = ρk∞ (z, 0) + λ−2 ck−2 (ρk∞ )ρk∞ (z, 0) + ρk(2) (z) ,
(2)
∞
−2
∞
pk (z, 0) = pk (ρk ) + λ
pk (z, 0) + pk (z)
(11)
for the compressible solutions, where vk(1) (z), ρk(2) (z), and pk(2) (z) belong to C 1 on
(−1)k z ≤ (−1)k Zk (0) and |vk(1) (z)| = O(1), |ρk(2) (z)| = O(1) and |pk(2) (z)| =
O(1).
tr
The variables Uk(m,s) ≡ βk(m,s) , vk(m,s) , ρk(m,s) , pk(m,s) , m = 0, 1, 2, have a slow
scale of motion and solve linearized incompressible problems. They have been deter(1,f )
(2,f )
(2,f )
mined in closed form through second order [11]. The variables vk , βk , ρk ,
(2,f )
contain fast scale acoustical oscillations on the fast time scale τ ≡ λt and they
pk
are solved in closed form in Sec. 4. We show by the inner limit analysis that there are
no fast scale acoustical oscillations in the asymptotic expansions of βk , ρk , pk through
first order and in the zero-th order of the expansion of vk .
The two phase flow model depends on the motions Zk of the mixing zone edges and
q
closure for the interfacial averages with the constitutive law dk , q = v, p. Since the
Zk are not well characterized for compressible flows, the velocities or trajectories of the
edges of the mixing zone must be provided as data. The compressible constitutive factor
q
dk is asymptotically assumed with a specific limit term.
The inner limit process of slightly compressible multiphase equations
2223
We assume a uniformly valid asymptotic expansion for the compressible mixing zone
edge,
(2,f )
Zk (t) = Zk(0,s) (t) + λ−1 Zk(1,s) (t) + λ−2 Zk(2,s) (t) + Zk (t, λ) + O(λ−3 ). (12)
Here
m
λ−j Zk
(j,s)
, m = 0, 1, 2, denotes the location of vanishing βk(m,s) . Thus Zk and
j =0
(2,f )
are input to the model equations.
each of the expansion coefficients Zk(m,s) and Zk
We assume that the compressible edge moves faster than the incompressible edge with
no initial perturbation. A similar assumption is applied to any finite number of terms in
the expansion (11). We assume that the zero-th order term in the expansion (11) equals
to the incompressible edge trajectory Zk∞ (t). Thus we require
Zk(0,s) (t) = Zk∞ (t), Zk (0) = Zk∞ (0),
(−1)k Zk(m,t) (t) ≥ 0, m = 1, 2, t = s, f.
(13)
The variables Zk(m,s) , m = 0, 1, 2, are the slow variables with a slow scale of motion while
(2,f )
(2,f )
is oscillatory on the fast time scale τ ≡ λt. We assume that the fast variable Zk
Zk
decays exponentially in τ away from the initial curve t = 0. We observe [9] that the fast
variables can appear in the second or higher order of the asymptotic expansion (12). The
formally uniformly valid expansion (12) leads to the outer and inner expansion under
the corresponding limit process. The reduced expansion is assumed in the derivation of
inner and outer expansions for compressible solutions in Sec. 4 and [11, 9], respectively.
Under the outer limit process, λ → ∞ with t fixed = 0, (12) leads to the outer expansion
(0,s)
Zk (t) = Zk
(t) + λ−1 Zk(1,s) (t) + λ−2 Zk(2,s) (t) + O(λ−3 )
(14)
valid away from the initial curve t = 0. Specifically, following the inner limit process
λ → ∞ with τ fixed = ∞, (12) leads to the inner limit expansion, valid near t = 0:
Zk (τ ) = Ẑk(0) (τ ) + λ−1 Ẑk(1) (τ ) + λ−2 Ẑk(2) (τ ) + O(λ−3 )
= Zk(0,s) (0) + λ−1 τ Żk(0,s) (0)
(15)
2
τ
(2,f )
+ λ−2
Z̈ (0,s) (0) + τ Żk(1,s) (0) + Zk(2,s) (0) + Zk (τ ) + O(λ−3 ).
2 k
Refer to [9]. The initial conditions associated with (13) are
Ẑk(0) (0) = Zk∞ (0), Ẑk(m) (0) = 0, m ≥ 1.
(2,f )
(16)
Notice that the fast variable Zk
consist of the inner term Ẑk(2) minus common terms to
order λ−2 . The assumptions (13) imply that (−1)k Ẑk(1) and (−1)k Ẑk(2) are nonnegative.
2224
Hyeonseong Jin
The edge velocity of the compressible flow satisfies Vk = Żk = vk (Zk , t) and
therefore, it must have an asymptotic expansion associated with the expansion (11) in
the form
(1,f )
(t, λ)
Vk (t) =Vk(0,s) (t) + λ−1 Vk(1,s) (t) + Vk
(17)
(2,f )
(2,s)
−2
+λ
(t, λ) + O(λ−3 ).
Vk (t) + Vk
From the expansion of vk in (17), we see that the leading order term of the asymptotic
expansion of Vk must be vk(0,s) (Zk(0,s) , t), where Zk(0,s) = Zk(0,s) (t) denotes the location
of vanishing βk(0,s) (z, t). Thus Vk(0,s) (t) = vk(0,s) (Zk(0,s) , t). We obtain that the formal
asymptotic expansion (17) leads to the outer limit expansion
Vk (t) = Vk(0,s) (t) + λ−1 Vk(1,s) (t) + λ−2 Vk(2,s) (t) + O(λ−3 )
(18)
which is valid away from the initial curve t = 0 under the limit process λ → ∞ with t
fixed = 0, and to the inner limit expansion
Vk (τ ) = V̂k(0) (τ ) + λ−1 V̂k(1) (τ ) + O(λ−2 )
(1,f )
(0,s)
(0,s)
(1,s)
−1
(τ ) + O(λ−2 ).
τ V̇k (0) + Vk (0) + Vk
= Vk (0) + λ
(19)
which is valid near the initial curve t = 0 under the limit process λ → ∞ with τ fixed
= ∞.
Comparing (14), (15) with (18) and (19), we note that for m ≥ 0,
dZk(m,s) (t)
= Vk(m,s) (t),
dt
(m+2,f )
dZk
(τ )
(m+1,f )
(τ ),
= Vk
dτ
d Ẑk(0) (τ ))
d Ẑk(m+1) (τ )
= 0,
= V̂k(m) (τ ),
dτ
dτ
(m+2,f )
dZk
(t, λ)
(m+1,f )
(t, λ).
= Vk
dτ
(20)
From (20) it is reasonable to assume that the fast variables can appear in the second or
higher order of the asymptotic expansion (12). If the expansion (12) has a fast variable
(0,f )
(1,f )
in the first order, it implies that a fast variable Vk
exists in the zero-th order
Zk
of the asymptotic expansion (17). This contradicts the fact that the zero-th order fast
(0,f )
is suppressed by the initialization (11), as discussed in Sec. 4.
variable vk
p
We assume that the constitutive laws dkv (t) and dk (t) have a formally uniformly valid
asymptotic expansion as follows
q(0,s)
q(1,s)
q(1,f )
q
(t) + λ−1 dk
(t) + dk
(t, λ) + O(λ−2 ), q = v, p, (21)
dk (t, λ) = dk
q(m,f )
q(m,s)
q(0,s)
q∞
(t), dk
(τ ) ∈ ([0, ∞)), and we assume dk
(t) = dk (t). Similarly,
where dk
we obtain [9] that the expansion (21) leads to the outer limit expansion
q
q(0,s)
dk (t) = dk
(t) + λ−1 dk
q(1,s)
(t) + λ−2 dk
q(2,s)
(t) + O(λ−3 ), q = v, p,
(22)
The inner limit process of slightly compressible multiphase equations
2225
and to the inner limit asymptotic expansion
q
(τ ) + λ−1 d̂k (τ ) + (λ−2 )
q
dd
(0,
s)
q(1,s)
q(1,f )
q(0,s)
(0) + dk
(τ ) + O(λ−2 ).
(0) + dk
(0) + λ−1 τ k
= dk
dt
(23)
q(0)
q(1)
dk (τ ) = d̂k
3.
Inner Expansions and Transitional Layers
To understand the motion of the fast variables, we make the change of variables to the
fast time scale τ ≡ λt. The The compressible equations (1)–(3) become
∂βk
∂βk
= 0,
+ v∗
∂z
∂τ
∂ρk
∂vk
∂βk
+ vk
+ βk ρk
+ ρk (vk − v ∗ )
= 0,
∂z
∂z
∂z
∂βk
∂vk
∂pk
+ vk
+ λ2 βk
+ λ2 (pk − p ∗ )
= βk ρk g(t),
∂z
∂z
∂z
λ
∂ρk
βk λ
∂τ
∂vk
βk ρk λ
∂τ
(24)
(25)
(26)
for Uk (z, τ ). We assume the initial data (11) for the compressible solutions and the
q
inner limit asymptotic expansions (15), (19) and (23) for Zk , Vk and dk , q = v, p
to derive the inner limit asymptotic expansions, valid near t = 0, of the solutions of
the compressible equations (24)-(26), uniformly valid in z. We introduce inner limit
asymptotic expansions associated with the inner limit λ → ∞ with τ fixed = ∞:
Uk (z, τ ) = Ûk(0) (z, τ ) + λ−1 Ûk(1) (z, τ ) + λ−2 Ûk(2) (z, τ ) + · · · ,
Ûk(m) (z, τ )
(m) (m)
(m)
(m) tr
β̂k , v̂k , ρ̂k , p̂k
,
where
≡
has the expansion relations
p̂k(0) = pk (ρ̂k(0) ),
p̂k(1) = ck2 (ρ̂k(0) )ρ̂k(1) ,
(27)
m = 0, 1, 2, . . .. The equation of state
p̂k(2) =
1 ∂ 2 pk (0) (1)2
(ρ̂k )ρ̂k + ck2 (ρ̂k(0) )ρ̂k(2) ,
2
2 ∂ρk
(28)
2226
Hyeonseong Jin
between the terms in the expansions of ρk , pk . From (27) and (28), the inner limit
q
q
expansion for the mixing coefficient µk (βk , dk ), q = v, p,
q
q
q(0,s)
µk (βk , dk ) =µ̂k
+ λ−1 µ̂k
q(1,s)
q
+ λ−2 µ̂k
q(2,s)
+ O(λ−3 )
q(0,s)
)
=µk (β̂k(0,s) , d̂k
q
∂µk (0,s) q(0,s) (1,s)
(β̂
, d̂k
)β̂k
+
+ λ−1
∂βk k
q
∂µk (0,s) q(0,s) (2,s)
+ λ−2
(β̂
, d̂k
)β̂k
+
∂βk k
q
∂µk
(0,s) q(0,s) q(1,s)
, d̂k
)d̂k
q (β̂k
∂dk
q
1 ∂ 2 µk (0,s) q(0,s) (1,s) 2
(β̂
, d̂k
)β̂k
2 ∂βk2 k
q
q
∂ 2 µk
1 ∂ 2 µk (0,s) q(0,s) q(1,s) 2
(0,s) q(0,s) (1,s) q(1,s)
, d̂k
)d̂k
)
β̂
+
(β̂
d̂
+
(
β̂
,
d̂
q
k
k
k
2 ∂d q 2 k
∂βk ∂dk k
k
q
∂µ
q(0,s) q(2,s)
+ qk (β̂k(0,s) , d̂k
)d̂k
+ O(λ−3 ).
∂dk
(29)
Ûk(m) (z, τ )
(m) (m)
(m)
(m) tr
β̂k , v̂k , ρ̂k , p̂k
,
The equations for the inner limit terms
≡
m = 0, 1, are derived by repeated application of the inner limit expansions (27) to the
compressible equations (24)–(26), and by equating terms of the same order of λ. Within
a single power of λ, the inner terms are defined as a solution of simple differential
equations. We first solve the inner term β̂k(0) , ρ̂k(0) , p̂k(0) equations in Sec. 4.1. The
inner terms v̂k(0) , β̂k(1) , ρ̂k(1) , p̂k(1) , uniformly valid in z are found by solving a subsystem
of hyperbolic differential equations and by matching at the outer and inner edges of
regions in Sec. 4.2. In [10], we will completely determine the higher order terms v̂k(1) ,
β̂k(2) , ρ̂k(2) , p̂k(2) , uniformly valid in z.
In higher order in λ−1 , there exist fast transitional layers in the inner expansion
similarily to the transition regions in the outer asymptotic expansion. The first order
inner expansion is defined by five regions,
(1)
(1)
(1)
Ê(1)
k ∪ T̂k ∪ M̂ ∪ T̂k ∪ Êk ,
including two transition-layers through z = Ẑi(1,s) , i = k, k . The regions are defined by
Ê(1)
k
Ẑk(1)
= (z, t) : (−1)
≤ (−1) z ,
k (0)
k
k (1)
T̂k(n)
Ẑ
Ẑ
=
(z,
t)
:
(−1)
≤
(−1)
z
<
(−1)
,
k
k
M̂ = (z, t) : Ẑ1(0) < z < Ẑ2(0) ,
k
k
(30)
(31)
(32)
The inner limit process of slightly compressible multiphase equations
where
Ẑk(1) (t)
≡
1
λ−j Ẑk
(j,s)
2227
(33)
.
j =0
denotes the position of boundaries of the fast transition-layers. With the new inner spatial
variables
ζ̂i(1) = λn z − Ẑi(1) , i = k, k ,
(34)
we make the change of variables from (z, τ ) to (ζ̂i(1) , τ ), reducing (24)-(26) to the
equations
∂β
∂βk
k
= 0,
(35)
− λ V̂k(1) − v ∗
λ
(1)
∂τ
∂ ζ̂i
∂ρ
∂ρk
∂vk
k
∗ ∂βk
ρ
+
λρ
(v
−
v
) (1) = 0, (36)
− λ V̂k(1) − vk
+
λβ
βk λ
k
k
k
k
∂τ
∂ ζ̂i(1)
∂ ζ̂i(1)
∂ ζ̂i
∂v
∂vk
∂pk
k
3
3
∗ ∂βk
β
+
λ
(p
−
p
) (1) = βk ρk g.
− λ V̂k(1) − vk
+
λ
βk ρk λ
k
k
∂τ
∂ ζ̂i(1)
∂ ζ̂i(1)
∂ ζ̂i
(37)
Here the edge velocity of the transition-layers is defined by
(1)
V̂k(1) (t)
d Ẑ k
≡
dt
=
1
λ−j V̂k .
(j )
(38)
j =0
We assume fast transitional layer expansions of the form
(ζ̂i(1) , τ ) + O(λ−3 ),
(39)
i+1 (1)
i (1)
in the region (−1) Ẑi ≤ (−1) ζ̂i ≤ 0. The fast transitional variables of each order
of λ−1 in the expansions (39) satisfy simple differential equations and they are solved
in closed form in Sec. 4.2.2. The fast variables are matched continuously order by order
at the boundaries of these layers in Sec. 4.2.4. This matching process of inner limit and
fast transitional expansions determines uniformly valid outer expansions in z.
(0,f t)
Uk (ζ̂i(1) , τ ) = Ûk
3.1.
(ζ̂i(1) , τ ) + λ−1 Ûk
(1,f t)
(ζ̂i(1) , τ ) + λ−2 Ûk
(2,f t)
Transitional Effective Variable
In this section we solve a simple system of ODEs to be used in the analysis of the fast
(0,f t)
(0,f t)
, pk
in Sec. 4.2.2. Let the effective transitional variable qk(t)
transitional terms vk
satisfy the system
βk(t)
∂β (t)
∂qk(t)
q(t)
k
+ µk
=0
qk(t) − qk(t)
∂ζi
∂ζi
(40)
2228
Hyeonseong Jin
n the region (−1)i+1 Zi(1) ≤ (−1)i ζi ≤ 0. Here βk(t) (ζi , t) is a given C 1 function in the
region and
q(t)
q(0,s)
)
µk (βk(t) , dk
q(t)
=
βk(t)
q(0,s) (t)
βk βk(t) + dk
(41)
q(t)
satisfying µk +µk = 1. This system can be decoupled and solved in closed form [11]
by the introduction of two linear combinations of the effective transitional variables as
follows
Proposition 3.1. Let qk(t) (ζi , t) satisfy the system (40) in (−1)i+1 Zi(1) ≤ (−1)i ζi ≤ 0.
Then the solution is
q(t)
qk(t) (ζi , t)
4.
=
µk
(1)
(t)
(t)
(1)
βk(t)
(−Z
,
t)
−
β
i
k qk (−Zi , t)
βk(t)
q(t)
(t)
(1)
q
(−Z
,
t)
.
+ βk(t) (−Zi(1) , t) + dk βk(t)
k
i
(42)
Expansion Procedure of the Inner Limit
We derive the inner terms in the inner limit asymptotic expansions (27) for the solutions
of the compressible equations. The inner terms, uniformly valid in space, are determined
by matching the inner limit in the exterior and the incompressible mixing zone and the
fast transition-layer expansions. The fast variables are defined as the inner expansion
terms minus the common terms resulting from the outer expansion evaluated in the
inner limit to each order [14]. These variables oscillate on the fast time scale τ = λt
and are described by linear wave equations. They are also defined uniformly in space by
matching. We substitute the inner limit asymptotic expansions (3.4) into the compressible
equations (24)–(26), and equate powers of λ. Since λ is arbitrary, the coefficient of λm
for each order m must vanish, defining equations for the inner terms to each order.
The zero-th order terms are determined by initial data of the incompressible flow. The
inner terms of higher order in λ−1 satisfy simple differential equations. Specifically the
(0,f )
(1,f )
fast variables satisfy linearized compressible equations. The variables vk
and vk
solve a hyperbolic IBVP. They are solved by the method of characteristics and a Picard
(1,f )
iteration. The remaining equations can thus be viewed as equations for βk
alone.
The solutions are expressed in terms of the initial and the boundary data. The boundary
data is provided by matching at the outer and inner edges of regions which exist in order
λ−m expansions. The first order inner terms in the expansion (3.4) of βk and pk and
the zero-th order inner term of vk have the fast transitional layers in the region T̂i(1) ,
i = k, k , introduced in Sec. 3. The fast transitional variables for each order in λ−1
satisfy simple differential equations. The inner terms are matched continuously order
by order at the boundaries of these layers. This matching process for the inner limit and
The inner limit process of slightly compressible multiphase equations
2229
the fast transitional expansions determines uniformly valid inner expansions in z. The
main result is summarized in the following theorem.
Theorem 4.1. Assume (8), (15), (19), (23), and the initial data (11). Then the compressible solutions have the uniformly valid inner expansions in z to O(λ−1 ) for βk , ρk , pk
and to O(1) for vk of the form
βk = βk∞ (z, 0) + λ−1 β̂k(1) + O(λ−2 ),
vk = vk∞ (z, 0) + O(λ−1 ),
(43)
ρk = ρk∞ + O(λ−2 ),
pk = p(0) + O(λ−2 ),
where p (0) = pk (ρk∞ ). The inner term β̂k(1) (z, τ ) satisfies

0





∂βk∞ ∞

∞

(0)
(Zk (0) + (−1)k 0, 0)
λ
z
−
Z

k


∂z

∞
(1)
β̂k (z, τ ) = +τ ∂βk (Z ∞ (0) + (−1)k 0, 0)

k

∂t



∂βk∞
(1,s)


(z,
0)
+
τ
(z, 0)
β

k

∂t


(z, t)
−βk(1,s)
4.1.
(1)
in Ê(1)
1 , Ê2
in T̂k(1)
,
(44)
in M̂
in T̂k(1)
The Zero-th Order Inner Terms
We show that the inner limit terms β̂k(0) , ρ̂k(0) , p̂k(0) in the expansions (27) describing the
incompressible limit process near the initial time are determined by the incompressible
background solution. To this order, there is no transition zone, and no contribution from
(m,f t)
. Substituting the inner limit asymptotic expansions (3.4)
fast transitional terms Uk
into the compressible equations (24)–(26), we equate powers of λ. The coefficients of
λ in the interface and mass equations and the order λ2 terms in the momentum equation
define the equations
∂ β̂k(0)
= 0,
O(1) :
∂τ
∂ ρ̂ (0)
O(1) : β̂k(0) k = 0,
∂τ
(0)
(0)
p(0)
(0) ∂ p̂k
(0)
(0) ∂ β̂k
2
+ µ̂k
=0
p̂k − p̂k
O(λ ) : β̂k
∂z
∂z
(45)
(46)
(47)
for β̂k(0) , ρ̂k(0) , p̂k(0) . We note that v̂k(0) does not contribute here and it is found in Sec. 4.2.
The initial conditions associated with (11) are given as
β̂k(0) (z, 0) = βk∞ (z, 0),
ρ̂k(0) (z, 0) = ρk∞ ,
p̂k(0) (z, 0) = p(0) .
(48)
2230
Hyeonseong Jin
Using the initial data (48), we solve Eqs. (45)–(47).
Proposition 4.2. Assume (8), (15), (19), (23), and the initial data (11). Then the zero-th
order terms in the expansions (27) satisfy
ρ̂k(0) (z, 0) = ρk∞ ,
β̂k(0) (z, 0) = βk∞ (z, 0),
p̂k(0) (z, 0) = p(0) ,
(49)
where the universal constant p (0) = pk (ρk∞ ).
4.2.
The First Order Inner Terms
Using the zero-th order inner terms (49), we find the inner terms v̂k(0) , β̂k(1) , ρ̂k(1) , p̂k(1)
in the expansions (27), uniformly valid in z. The initial conditions associated with (11)
satisfy
v̂k(0) (z, 0) = vk∞ (z, 0),
β̂k(1) (z, 0) = βk(1,s) (z, 0),
(0,f )
Let us define the variables vk
and p̂k(1) as the following
(1,f )
, βk
(1,f )
, pk
ρ̂k(1) (z, 0) = 0 = p̂k(1) (z, 0).
(50)
(0)
which are oscillatory part of v̂k , β̂k(1) ,
(z, τ ) ≡ v̂k(0) − vk∞ (z, 0),
∂βk∞
(1,f )
(1)
(1,s)
βk (z, τ ) ≡ β̂k − βk (z, 0) + τ
(z, 0) ,
∂t
(0,f )
vk
(1,f )
pk
(51)
(z, τ ) ≡ p̂k(1) .
(0,f )
(1,f )
and pk
satisfy a subsystem of hyperbolic PDFs with no
The fast variables vk
(1,f )
source terms. The remaining equations are solved for βk . Using the method of
(0,f )
and
characteristics and the process of Picard iteration, we show that the variables vk
(1,f )
are suppressed under the initial condition (50).
pk
We consider the transitional regions T̂i(1) , i = k, k , by discussion of the fast transitionlayers through z = Ẑi(1,s) in Sec. 4.2.2. The matching the inner edge of the inner limit
expansions in the exterior with the outer edge of the transitional expansions and the
inner edge of the transitional expansions with the edge of the incompressible mixing
zone determines the first order inner terms uniformly in space in Sec. 4.2.4.
Proposition 4.3. Assume (8), (15), (19), (23), and the initial data (11). Then β̂k(1) (z, τ )
satisfies (44) and the inner terms satisfy
∞
(1)
vk (z, 0) in Ê(1)
k , T̂k , M̂
(0)
,
(52)
v̂k (z, τ ) =
Vk∞ (0)
in T̂k(1)
p̂k(0) (z, τ ) = ρ̂k(0) (z, τ ) = 0
(1)
(1)
in Ê(1)
k , T̂k , M̂, T̂k .
(53)
The inner limit process of slightly compressible multiphase equations
2231
4.2.1 The Inner Terms in the Exterior Domain
In this section, we find the inner terms v̂k(0) and p̂k(1) in the single phase region Ê(1)
k . We
substitute the asymptotic expansions (27) into (24)–(26) and equate powers of λ. Using
(49), we isolate the order zero terms in the mass equations and the order λ terms in the
momentum equation. Since λ is arbitrary, the coefficient of λ must vanish, leading to
the equations
(1)
(0)
∂ ρ̂k
∞ ∂ v̂k
= 0,
+ ρk
O(1) :
∂z
∂τ
(0)
∂ p̂k(1)
∞ (0) ∂ ρ̂k
+
= 0,
O(λ) : ρk v̂k
∂τ
∂z
(54)
(55)
for v̂k(0) , ρ̂k(1) , p̂k(1) in Ê(1)
k . We note that
β̂k(1) = 0
in Ê(1)
k i = 1, 2.
(56)
Substituting (51) into (54), (55), and multiplying (54) by ak2 ≡ dpk /dρk (ρk∞ ) and (55)
by 1/ρk∞ , the equations reduce to the system of the wave equations,
(1,f )
∂uk
∂τ
for
(1,f )
uk
=
(1,f ) (0,f ) tr
pk , vk
,
(1,f )
0 ∂uk
+ Ak
∂z
=0
(57)
where A0k is the constant coefficient matrix
A0k
0
ρk∞ ak2
=
0
1/ρk∞
From (50) the initial conditions follow
tr
(1,f )
(0,f )
(1,f )
uk (z, 0) = pk (z, 0), vk (z, 0) = 0.
(58)
(59)
We discuss the IBVP (57)-(59) in Ê(1)
k using the method of characteristics. The matrix
¥Ák0 has two distinct eigenvalues 0k,i ≡ (−1)i ak , i = 1, 2. We define k0 to be the
nonsingular matrix whose column vectors consist of linearly independent eigenvectors
of A0k ,
1/2
1/2
0
(60)
k =
−1/(2ρk∞ ak ) 1/(2ρk∞ ak ).
The characteristic curves Ck,i , i = 1, 2, satisfy
Ck,i :
dX
= 0k,i .
ds
(61)
2232
Hyeonseong Jin
Therefore the backward characteristics Ck,i , i = 1, 2, through a point (z, τ ) have equations for 0 ≤ s ≤ τ ,
(62)
Ck,i : X(s) = αk,i (s, z, τ ) = (−1)i ak (s − τ ) + z.
t
(1,f )
(1,f )
(0,f )
(1,f )
= Uk,1 , Uk,2
≡ (k0 )−1 uk
Introducing a new unknown column vector Uk
and multiplying (57) by (k0 )−1 , we obtain
(1,f )
(1,f )
∂Uk
∂U
+ 0k k
=0
∂τ
∂z
(1,f )
(1,f )
Uk
(z, 0) = (k0 )−1 uk (z, 0) = 0.
(63)
(64)
By the method of characteristics we find the solution
(1,f )
(1,f )
Uk,k (z, τ ) = Uk,k (αk,k (0, z, τ ), 0)
(0,f )
(1,f )
k ∞
= pk
(z + (−1)k ak τ, 0) = 0,
+ (−1) ρk ak vk
(65)
 (1,f )
 Uk,k (αk,k (0, z, τ ), 0) = 0 if (−1)k αk,k (0, z, τ ) ≥ Ẑk(0)
(1,f )
,
Uk,k (z, τ ) =
(1,f )
(1)
(1)
 Uk,k
Ẑk(1)
if (−1)k αk,k (0, z, τ ) ≤ Ẑk(0)
(τk ), τk
(66)
(1)
in Ê(1)
k , where τk
(1)
= τk(1) (z, τ ) satisfies αk,k (τk(1) , z, τ ) = Ẑk(1)
(τk ). From (65), we see
(1,f )
that the solution Uk,k is completely determined by initial data along the incoming char
(1,f )
acteristic αk,k . Furthermore, this solution provides the boundary data Uk,k
Ẑk(1)
,
τ
which will be used to solve the related inner term system in M̂ in Sec. 4.2.3 once the vari(1,f )
are determined in the fast transitional regions T̂i(1) , i = k, k . The solution
ables uk
(1,f )
Uk,k depends upon boundary data at the edge z = Ẑk(1)
to be completely determined in
the single phase region Ê(1)
k along the outgoing characteristic αk,k . The boundary data
(1,f )
(0,f )
(1,f )
(1)
k ∞
Uk,k Ẑk(1)
Ẑ
,
τ
=
p
+
(−1)
ρ
a
v
,
τ
=0
(67)
k k k
k
k
will be shown later by matching in Sec. 4.2.4.
Lemma 4.4. Assume (8), (15), (19), (23), and the initial data (11). Assume the boundary
data (67). Then the fast variables satisfy
(0,f )
vk
(1,f )
= pk
=0
(68)
in Ê(1)
k . Thus, we obtain the inner terms
v̂k(0) = vk∞ (z, 0),
ρ̂k(1) = 0 = p̂k(1) .
(69)
The inner limit process of slightly compressible multiphase equations
2233
4.2.2 The Fast Transition-Layer Expansion
We discuss the fast transition-layer extending through z = Ẑi(1) , i = k, k , which defines v̂k(0) , β̂k(1) , p̂k(1) , continuously in T̂i(1) . We substitute the fast transition-layer expansions (39) into the compressible equations (eq:comp-1-tran)–(eq:comp-3-tran) and
equate powers of λ. Since λ is arbitrary, the coefficient of λn for each order n must
vanish, defining differential equations for the fast transitional terms. We first determine
(0,f t)
in the expansions (3.14). We isolate the order
the fast transitional limit solutions Uk
λ terms in the interface and mass equation and the order λ3 terms in the momentum
equation. Since the coefficients of λ and λ3 must vanish, the leading order terms satisfy
the coupled ODEs
(0,f t)
∂βk
O(λ) :
∂τ
O(λ) :
(0,f t)
βk
+
−V̂i(0)
(0,f t)
∂ρk
∂τ
+
(0,f t) v(0,f t)
+ρk
µk
O(λ ) :
3
(0,f t)
for Uk
v, p,
(0,f t)
(0,f t) ∂pk
βk
∂ ζ̂i(1)
+v
∗(0,f t)
(0,f t)
vk
= 0,
∂ ζ̂i(1)
∂ρ (0,f t)
(0,f t)
− vk p(0,f t)
+ µk
k
(0,f t)
+ vk
−V̂i(0)
∂β (0,f t)
k
∂ ζ̂i(1)
∂β (0,f t)
(0,f t)
pk
(0,f t)
− pk (0,f t)
(0,f t) (0,f t) ∂vk
ρk
+ βk
∂ ζ̂i(1)
= 0,
k
∂ ζ̂i(1)
(70)
∂β (0,f t)
k
∂ ζ̂i(1)
(71)
=0
(72)
(ζ̂i(1) , t) in the region (−1)i+1 Ẑi(1) ≤ (−1)i ζ̂i(1) ≤ 0. We note that for q =
(0,f t)
q(0,f t)
µk
=
βk
(0,f t)
βk
q∞ (0,f t)
+ d̂k βk
.
The matching conditions to O(1) are
(0,f t)
βk
ζ̂i(1) = 0, τ = β̂k(0) Ẑi(1) , τ = δik ,
(0,f t)
βk
ζ̂i(1) = −Ẑi(1) , τ = β̂k(0) Ẑi(0) , τ = δik ,
(0,f t)
(1)
(0)
(1)
(0)
ρk
ζ̂i = −Ẑi , τ = ρ̂k Ẑi , τ = ρk∞
(0,f t)
pk
ζ̂i(1) = −Ẑi(1) , τ = p̂k(0) Ẑi(0) , τ = p(0)
(0,f t)
(73)
(74)
by use of (49). We note that the zero-th order terms v̂k(0) and vk
are not discussed here.
(0)
We will see later that v̂k is coupled with the higher order term p̂k(1) and the boundary
(0,f t)
is found by matching in Sec. 4.2.4.
data of v̂k(0) and vk
2234
Hyeonseong Jin
Using (74), Eqs. (70)–(72) are solved as the following:
Lemma 4.5. Assume (8), (15), (19), (23), and the initial data (11). Assume the boundary
data (74). Then the fast transitional limit in the expansions (39) satisfies
(0,f t)
(1)
(75)
ζ̂i , τ = δik ,
βk
(0,f t)
ρk
(76)
ζ̂i(1) , τ = ρk∞ ,
(0,f t)
pk
(77)
ζ̂i(1) , τ = p(0) ,
(0,f t)
(0,f t)
(0,f t)
vk
(78)
ζ̂i(1) , τ = vk
ζ̂i(1) = −Ẑi(1) , τ = vk
ζ̂i(1) = 0, τ ,
where the universal constant p(0) = pk (ρk∞ ) and δik is the Kronecker symbol.
(0,f t)
(0,f t)
Proof. We first solve Eq. (72) for pk
by using Proposition 3.1 with qk(t) = pk
and the boundary data (74). Then we obtain the solution (77). The equation of state
(0,f t)
implies (76). Substituting (76) into (71) and multiplying the equation by 1/ρk
, it
reduces to
(0,f t)
(0,f t) ∂vk
∂ ζ̂i(1)
βk
(0,f t)
To decouple vk
(0,f t)
vk
(0,f t)
and βk
v(0,f t)
+ µk
(0,f t)
vk
(0,f t)
− vk ∂β (0,f t)
k
∂ ζ̂i(1)
= 0.
(0,f t)
, we use Proposition 3.1 with qk(t) = vk
(79)
. Thus,
µv(0,f t) (0,f t)
(0,f t)
(0,f t)
(1)
(1)
(1)
(−Ẑi , τ ) − βk (−Ẑi , τ )
ζ̂i , τ = k(0,f t)
βk vk βk
(0,f t)
(0,f t)
(0,f t)
(1)
(1)
(−Ẑi , τ ) + d̂kv∞ βk (−Ẑi , τ )
vk
+ βk

v(0,f t) 

(0,f t) (0,f t)
(1)
 µk
v∞ β (0,f t) v (0,f t) (−Ẑ (1) , τ )

v
(−
Ẑ
,
τ
)
+
1
+
d̂
−β

k
i
i
k
k
k
k
(0,f t)
βk
=




 µv(0,f t) v (0,f t) (−Ẑ (1) , τ ) + µv(0,f t) v (0,f t) (−Ẑ (1) , τ )
i
i
k
k
k
k
if i = k if i = k
(80)
by (74). Substituting (refeq:v(0,ft)-cal) into (70), we solve the quasilinear PDE for
(0,f t)
. The characteristic speed satisfies −V̂i(0) + v ∗(0,f t) . By the boundary condition
βk
(0,f t)
(0,f t)
is constant in space and time. Substitution of the βk
into
(74), the solution βk
(0,f t)
in (78 which is independent of space. (refeq:v(0,ft)-cal) implies the velocity vk
(1,f t)
Using (75)–(78), we now define the fast transitional variables Uk
in the expansions (39). We isolate the order zero terms in the interface equation, the order zero
terms and the order λ−1 terms in the continuity equation for i = k and i = k, and the
order λ2 terms and the order λ terms in the momentum equation for i = k and i = k,
The inner limit process of slightly compressible multiphase equations
2235
respectively. Since λ is arbitrary, the coefficients of λ2 , λ and λ−1 must vanish, leading
to the equations
(1,f t)
∂βk
O(1) :
∂τ
+
(1,f t)
∂ρk
O(1) :
∂τ
O(λ−1 ) :
O(λ2 ) :
O(λ) :
+
−V̂i(1)
−V̂i(1)
+v
∗(1,f t)
+v
∗(1,f t)
∂β (1,f t)
k
∂ ζ̂i(1)
= 0,
(81)
∂ρ (1,f t)
k
∂ ζ̂i(1)
(1,f t)
(1,f t)
(0,f t)
(0,f t) ∂βk
∞ ∂vk
∞
(82)
+ρk
+ ρk vk
− vk = 0, i = k ,
(1)
(1)
∂ ζ̂i
∂ ζ̂i
(1,f t)
(1,f t)
∂ρ (1,f t)
(1,f t) ∂ρk
(1,f t) ∞ ∂vk
(1)
k
∗(1,f t)
βk
ρk
+ −V̂i + v
+ βk
∂τ
∂ ζ̂i(1)
∂ ζ̂i(1)
(1,f t)
(0,f t)
(0,f t) ∂βk
∞ v(1,f t)
− vk = 0, i = k,
(83)
+ρk µk
vk
∂ ζ̂i(1)
(1,f t)
∂pk
= 0, i = k ,
(84)
(1)
∂ ζ̂i
(1,f t)
(1,f t) ∂pk
βk
= 0, i = k
(85)
∂ ζ̂i(1)
(1,f t)
(1,f t)
(ζ̂i(1) , t) in (−1)i+1 Ẑi(1) ≤ (−1)i ζ̂i(1) ≤ 0. Here we may assume that βk
for Uk
is nontrivial.
Solving (84) and (85), we obtain
Lemma 4.6. Assume (8), (15), (19), (23), and the initial data (11). Then
(1,f t)
pk
(1,f t)
(ζ̂i(1) , τ ) = pk
(1,f t)
(ζ̂i(1) = −Ẑi(1,s) , τ ) = pk
(0,f t)
(ζ̂i(1) = 0, τ ).
(86)
(1,f t)
From (78) and (86), the fast transitional terms vk
and pk
are constant in space
(1)
and depend on boundary data at the boundaries of the regions T̂i . By the matching
(0,f t)
and
process in Sec. 4.2.4, the boundary data will be provided and therefore, vk
(1,f t)
(0,f t)
(1,f t)
(1,f t)
will be determined completely. Using the vk
and pk
, the solution βk
pk
of (81)–(83) is found in Sec. 4.2.4.
4.2.3 The Inner Terms within the Mixing Zone
We find the inner terms v̂k(0) , β̂k(1) , ρ̂k(1) , p̂k(1) in the mixing zone M̂. Using (49), we isolate
the order zero terms in the interface and continuity equation and the order λ terms in the
2236
Hyeonseong Jin
momentum equation, defining the equations
(0)
∂ β̂k(1)
∗(0) ∂ β̂k
O(1) :
+ v̂
= 0,
(87)
∂τ
∂z
(0)
(1)
(0)
(0) v(0)
(0)
(0) ∂ β̂k
(0) ∂ ρ̂k
(0) (0) ∂ v̂k
= 0, (88)
+ β̂k ρ̂k
+ ρ̂k µ̂k
v̂k − v̂k
O(1) : β̂k
∂z
∂τ
∂z
∂ β̂ (0)
∂ v̂ (0)
∂ p̂(1)
p(0)
k
=0
(89)
p̂k(1) − p̂k(1)
O(λ) : β̂k(0) ρ̂k(0) k + β̂k(0) k + µ̂k
∂τ
∂z
∂z
for v̂k(0) , β̂k(1) , ρ̂k(1) , p̂k(1) in M̂.
Substitution of (51) into (87)-(89) reduces to the system
(1,f )
(1,f )
∂pk
∂τ
(0,f )
for vk
(0)
∂βk
∗(0,f ) ∂ β̂k
+v
= 0,
∂τ
∂z
(0,f )
(0)
∂v
(0,f )
(0,f ) ∂ β̂k
− vk + ρk∞ ak2 k
+ ρk∞ ak2 fkv (z) vk
= 0,
∂z
∂z
(0,f )
(0,f )
p
fk (z) (1,f )
∂vk
1 ∂pk
(1,f )
+ ∞
+ ∞ pk
− pk =0
∂τ
ρk
∂z
ρk
. Here ak2 = c2 (ρk∞ ) and for q = í, p,
q∞
q(0)
(0)
∞
µ̂
µ
∂
β̂
∂β
q∞
q
q
k
k
k
=
(z, 0) = −dk (0)fk (z).
fk (z) ≡ k(0)
∞ ∂z
∂z
β
β̂
k
(1,f )
, βk
(90)
(91)
(92)
(1,f )
and pk
(93)
k
(0,f )
vk
(1,f )
and pk
satisfy a subsystem of hyperbolic PDEs which are linObserve that
(1,f )
(0,f )
depends on the solution vk . The initial
earized compressible equations while βk
condition (11) implies the trivial initial data
(1,f )
βk
(0,f )
(z, 0) = vk
(1,f )
(z, 0) = pk
(z, 0) = 0
(94)
(1,f )
for the fast variables. We prove that this initialization suppresses the fast variables βk ,
(0,f )
(1,f )
and pk . In this section, we assume the boundary data
vk
= p̂k(1) + (−1)k ρk∞ ak v̂k(0) Ẑk(1)
(95)
p̂k(1) + (−1)k ρk∞ ak v̂k(0) Ẑk(0)
,τ
,τ ,
v̂k(0) Ẑk(0) , τ = v̂k(0) Ẑk(1)
= V̂k(0) .
(96)
,τ
These data will be shown later by matching in Sec. 4.2.4. We see from (19) and (51) that
the conditions (95), (96) are equivalent to the boundary data
(1,f )
(0,f )
(1,f )
(0,f )
(1)
k ∞
pk
Ẑ
+ (−1)k ρk∞ ak vk
,
τ
=
p
+
(−1)
ρ
a
v
,
τ
= 0,
Ẑk(0)
k k k
k
k
(0,f )
vk
(0,f )
Ẑk(0) , τ = vk
Ẑk(1) , τ = 0.
(97)
(98)
The inner limit process of slightly compressible multiphase equations
(1,f )
2237
(0,f )
Let us first solve the IBVP (91), (92), (94), (97), (98) for pk
and vk
in M̂ by using
the method of characteristics and by applying a Picard iteration. We can write (91), (92)
(1,f ) (0,f )
(1,f ) (0,f )
for unknown column vector u(1,f ) = (p1 , v1 , p2 , v2 )t in the form
∂u(1,f )
∂u(1,f )
+ A0
+ B 0 (z)u(1,f ) = 0
∂τ
∂z
(99)
where A0k was defined in (58),
0
0
A1 0
B1 −B10
0
ρk∞ ak2 fkv (z)
0
0
0
A =
, Bk (z) =
, B (z) =
.
p
−fk (z)/ρk∞
0
−B20 B20
0 A02
(100)
The constant coefficient matrix A0 has four distinct eigenvalues ±a1 , ±a2 . The nonsingular matrix 0 whose column vectors consist of linearly independent eigenvector of A0
has the form
0
0
1
0 =
,
(101)
0 20
where k0 was given in (60) and it satisfies
A0 0 = 0 0 ,
0 =
01
0
0
02
(102)
with the diagonal matrix 0k defined in (63), whose diagonal entries are eigenvalues of
A0 . The characteristics curves Ck,i , i = 1, 2, satisfy (61) and therefore, the backward
characteristics Ck,i , i = 1, 2 through a point (z, τ ) have the equations (62). Introducing
a new unknown vector U (1,f ) ≡ ( 0 )−1 u(1,f ) , we find that U (1,f ) satisfies semilinear
hyperbolic system
∂U (1,f )
∂U (1,f )
(0)
+ 0
+ B (z)U (1,f ) = 0
∂τ
∂z
(103)
with the initial condition
U (1,f ) (z, 0) = 0.
(104)
(Ẑ2(0) , τ ) = U1,1 (Ẑ2(1) , τ ) = 0,
(1,f )
(0,f )
(1,f )
(1,f )
U2 (Ẑ1(0) , τ ) = U1
+ 2ρ1∞ a1 v1
(Ẑ1(0) , τ ) = U1 (Ẑ1(0) , τ ),
(1,f )
(1,f )
(0,f )
(1,f )
(0)
∞
U3 (Ẑ2 , τ ) = U3
− 2ρ2 a2 v2
(Ẑ2(0) , τ ) = U4 (Ẑ2(0) , τ ),
(105)
and the boundary data
(1,f )
U1
(1,f )
U4
(1,f )
(1,f )
(Ẑ1(0) , τ ) = U2,2 (Ẑ1(1) , τ ) = 0
(106)
(107)
(108)
2238
Hyeonseong Jin
in M̂. Here B
(1,f )
(0)
= ( 0 )−1 B 0 0 , Uj
(1,f )
denotes the j −th component of the vector
(1,f )
Uk,k
and
was given in (65)
By the method of characteristics, the IBVP for U (1,f ) is solved implicitly. In this
case a Picard iteration converges to a solution in closed form.
U
Proposition 4.7. Assume (8), (15), (19), (23), and the initial data (11). Assume the
boundary data (95) and (96). Then the fast variables are
(0,f )
vk
(1,f )
= pk
= 0,
(109)
p̂k(1) = 0.
(110)
in M̂. Thus the inner terms satisfy
v̂k(0) = vk∞ (z, 0),
Proof. We solve the IBVP (103)-(108) for U (1,f ) in M̂. By the method of characteristics,
the solution is determined implicitly,
(1,f )
Uj
=−
τ
4
0 k=1
(1,f ) 0 αi,l (s, z, τ ), s ds, j = 1, 2, 3, 4.
B j,k αi,l (s, z, τ ) Uk
(111)
Here
denotes the (j, k) component of the matrix B , αi,l (s, z, τ ) was defined in
(62) and (i, l) = (1, 1), (1, 2), (2, 1), (2, 2) corresponding to j = 1, 2, 3, 4. Notice that
there is no contribution of the initial data (eq:sys-initial). The solution (111) can be
written symbolically as
(112)
U (1,f ) = SU (1,f )
0
B j,k
0
Here S is the integral operator taking a vector U (1,f ) into a vector SU (1,f ) and the j −th
component of SU (1,f ) . Using the process of a Picard iteration
U (1,f )[m+1] = SU (1,f )[m] , m ≥ 0,
U (1,f )[0] = U (1,f ) (z, 0) = 0,
(113)
(114)
we obtain the explicit solution
U
(1,f )
=
∞
U (1,f )[m] = 0.
(115)
m=0
Therefore the fast variables satisfy u(1,f ) = 0 U (1,f ) = 0.
(1,f )
under the initial data (94). Thus the inner
Using (109), Eq. (90) is solved for βk
(1)
term β̂k is determined in closed form. We note that v̂ ∗(0) = v ∗∞ (z, 0) from (49), (110).
The inner limit process of slightly compressible multiphase equations
2239
Proposition 4.8. Assume (8), (15), (19), (23), and the initial data (11). Assume the
boundary data (95) and (96). We obtain the fast variable
(1,f )
βk
= 0,
(116)
and therefore, the inner term satisfies
β̂k(1) = βk∞ (z, 0) + τ
4.2.4
∂βk∞
(z, 0)
∂t
in M̂.
(117)
Matching Process
We match the inner asymptotic in the exterior domain with the outer edge of the fast
transition layer and we match the inner edge of the fast transition layer with the edge of
the incompressible mixing zone M̂. The matching conditions determine the unknown
boundary data of the inner terms in the exterior and the incompressible mixing zone and
the fast transition-layer variables.
The matching conditions to O(λ−1 ) for βk and pk and to O(1) for vk are
(1,f t)
(118)
ζ̂i(1) = 0, τ = β̂k(1) Ẑi(1) , τ = 0,
βk
(0,f t)
vk
(119)
ζ̂i(1) = 0, τ = v̂k(0) Ẑi(1) , τ = δik V̂k(0) ,
(1,f t)
pk
ζ̂i(1) = 0, τ = p̂k(1) Ẑi(1) , τ = 0,
(120)
(1,f t)
βk
(121)
ζ̂i(1) = −Ẑi(1) , τ = β̂k(1) Ẑi(0) , τ ,
(0,f t)
vk
(122)
ζ̂i(1) = −Ẑi(1) , τ = v̂k(0) Ẑi(0) , τ ,
(1,f t)
pk
(123)
ζ̂i(1) = −Ẑi(1) , τ = p̂k(1) Ẑi(0) , τ .
The conditions (118)–(120) match the inner limit expansions in the exterior domain and
fast transition-layer expansions while (121)–(123) match the inner limit expansions in
the mixing zone M̂ and the fast transition-layer expansions. These matching conditions
give boundary data for the inner terms β̂k(1) , v̂k(0) , p̂k(1) and fast transition-layer variables
(1,f t) (0,f t)
(1,f t)
, vk
, pk
. Using (78), (119) and (122), we obtain the boundary condition
βk
(0,f t)
ζ̂i(1) = −Ẑi(1) , τ
v̂k(0) Ẑi(0) , τ = vk
(0,f t)
(1)
(0)
(1)
= vk
(124)
ζ̂i = 0, τ = v̂k Ẑi , τ = δik V̂k(0) .
Also,
(1,f t)
(1,f t)
ζ̂i(1) = −Ẑi(1) , τ = pk
ζ̂i(1) = 0, τ = p̂k(1) Ẑi(1) , τ
p̂k(1) Ẑi(0) , τ = pk
(125)
2240
Hyeonseong Jin
by use of (86), (120) and (123). Therefore the boundary data (95) and (96) are proved.
So we can use Proposition 4.7 to evaluate v̂k(0) , p̂k(1) in M̂. Then the boundary data satisfy
(0)
(0)
v̂k(0) Ẑk(1)
,
τ
= vk∞ Zk∞ (0), 0 = 0,
(126)
,
τ
=
v̂
Ẑ
k
k
p̂k(1) Ẑi(1) , τ = p̂k(1) Ẑi(0) , τ = 0.
(127)
(0,f t)
(1,f t)
and pk
These boundary data determine the fast transitional variables vk
(1,f t) (1)
(1)
(ζ̂i , τ ).
pletely in T̂i . Using (121) and (116), we solve Eq. (83) for βk
com-
Proposition 4.9. Assume (8), (15), (19), (23), and the initial data (11). Assume (118)–
(123). Then the fast transitional variables are
(0,f t)
(ζ̂i(1) , τ ) = δik V̂k(0) ,
(128)
(1,f t)
(ζ̂i(1) , τ ) = 0,
(129)
vk
pk
(1,f t)
βk
(ζ̂i(1) , τ ) =
∞
(1) ∂βk
ζ̂i
(Zi∞ (0) + (−1)i+1 0, 0).
∂z
(130)
Proof. Since v ∗(0,f t) = V̂i(0) , Eq. (83) reduces to
(1,f t)
∂βk
∂τ
=0
(131)
in (−1)i+1 Ẑi(1) ≤ (−1)i ζ̂i(1) ≤ 0. Using (121) and (116), evaluation of the boundary
data yields
∂β ∞
(1,f t)
ζ̂i(1) = −Ẑi(1) , τ = β̂k(1) Ẑi(0) , τ = −τ V̂i(0) k (Zi∞ (0) + (−1)i+1 0, 0)
βk
∂z
∞
∂β
= −Ẑi(1) k (Zi∞ (0) + (−1)i+1 0, 0).
∂z
(132)
(1,f t)
Therefore the solution βk
satisfies (130).
Proposition 4.3 is proved by summarizing Propositions 4.7-4.9 that determine the
inner terms β̂k(1) , v̂k(0) , p̂k(1) uniformly in space. The proof of Theorem 4.1 is complete
from Propositions 4.2 and 4.3. Thus the inner limit expansions of βk and pk up to O(λ−1 )
and of vk up to O(1) are nonoscillatory and depend on the initial data only.
References
[1] W. Bo, H. Jin, D. Kim, X. Liu, H. Lee, N. Pestieau, Y. Yan, J. Glimm, and J. Grove.
Comparison and validation of multi phase closure models Computers & Mathematics with Applications, 56:1291–1302, 2008.
The inner limit process of slightly compressible multiphase equations
2241
[2] Y. Chen, J. Glimm, D. H. Sharp, and Q. Zhang. A two-phase flow model of the
Rayleigh-Taylor mixing zone. Phys. Fluids, 8(3):816–825, 1996.
[3] D. Ebin. Motion of slightly compressible fluids in a bounded domain I. Comm. on
Pure Appl. Math., 35:451–485, 1982.
[4] J. Glimm and H. Jin. An asymptotic analysis of two-phase fluid mixing. Bol. Soc.
Bras. Mat., 32:213–236, 2001.
[5] J. Glimm, H. Jin, M. Laforest, F. Tangerman, and Y. Zhang. A two pressure numerical model of two fluid mixing. Multiscale Model. Simul., 1:458–484, 2003.
[6] J. Glimm, D. Saltz, and D. H. Sharp. Two-pressure two-phase flow. In G.-Q. Chen,
Y. Li, and X. Zhu, editors, Nonlinear Partial Differential Equations, pages 124–148.
World Scientific, Singapore, 1998.
[7] J. Glimm, D. Saltz, and D. H. Sharp. Two-phase modeling of a fluid mixing layer.
J. Fluid Mech., 378:119–143, 1999.
[8] D. Hoff. The zero-mach limit of compressible flows. Commu. Math. Phys., 192:543–
554, 1998.
[9] H. Jin. The singular limit of weakly compressible multiphase flow. Global J. Pure
Appl. Math., In Press, 2016.
[10] H. Jin. The singular perturbations of compressible multiphase flow equations. In
preparation, 2016.
[11] H. Jin, and J. Glimm Weakly compressible two-pressure two-phase flow. Acta. Math.
Sci, 29B(6):1497–1540, 2009.
[12] H. Jin, J. Glimm, and D. H. Sharp. Compressible two-pressure two-phase flow
models. Phys. Lett. A, 353:469–474, 2006.
[13] H. Jin, J. Glimm, and D. H. Sharp. Entropy of averaging for compressible twopressure two-phase flow models. Phys. Lett. A, 360:114–121, 2006.
[14] J. Kevorkian and J.D. Cole. Perturbation methods in applied mathematics. SpringerVerlag, New York, 1985.
[15] S. Klainerman and A. Majda. Singular limits of quasilinear hyperbolic systems with
large parameters and the incompressible limit of compressible fluids. Comm. Pure
Appl., 34:481–524, 1981.
[16] S. Klainerman and A. Majda. Compresible and incompressible fluids. Comm. Pure
Appl., 35:629–651, 1982.
[17] H. Lee, H. Jin,Y.Yu and J. Glimm. On the validation of turbulent mixing simulations
of Rayleigh-Taylor mixing. Phys. Fluids, 20:012102, 2008.

Download Report