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arXiv:1608 [physics-dyn] 17 Dec 2016
A regularity criterion for solutions of the three-dimensional
Cahn-Hilliard-Navier-Stokes equations and associated computations
John D. Gibbon 1 , Nairita Pal 2 , Anupam Gupta 3 , and Rahul Pandit 2 1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK. 2 Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, 560 012, India. 3 Department of Physics, University of Rome ‘Tor Vergata’, 00133 Roma, Italy. (Dated: August 27, 2021)
We consider the 3D Cahn-Hilliard equations coupled to, and driven by, the forced, incompressible 3D Navier-Stokes equations. The combination, known as the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical mechanics to model the motion of a binary fluid. The potential development of singularities (blow-up) in the contours of the order parameter φ is an open problem. To address this we have proved a theorem that closely mimics the Beale-Kato-Majda theorem for the 3D incompressible Euler equations [Beale et al. Commun. Math. Phys., 94 6166 (1984)]. By taking an∫ L∞ norm of the energy of the full binary system, designated as E∞, we have shown that t 0 E∞(τ ) dτ governs the regularity of solutions of the full 3D system. Our direct numerical simula- tions (DNSs), of the 3D CHNS equations, for (a) a gravity-driven Rayleigh Taylor instability and (b) a constant-energy-injection forcing, with 128 3 to 512 3 collocation points and over the duration of our DNSs, confirm that E∞ remains bounded as far as our computations allow.
PACS numbers: 47.10-,47.27-,47.27,Navier-Stokes equations, Integro-partial differential equations, Existence, uniqueness, and regularity theory, Two-phase and multiphase flows
I. INTRODUCTION
The Navier-Stokes (NS) equations [1–6], the funda- mental partial differential equations (PDEs) that govern viscous fluid dynamics, date back to 1822. Since its in- troduction in 1958 the Cahn-Hilliard (CH) PDE [7], the fundamental equation for the statistical mechanics of bi- nary mixtures, has been used extensively in studies of critical phenomena, phase transitions [8–12, 15], nucle- ation [16], spinodal decomposition [17–21], and the late stages of phase separation [21, 22]. If the two compo- nents of the binary mixture are fluids, the CH and NS equations must be coupled, where the resulting system of PDEs is usually referred to as Model H [9] or the Cahn- Hilliard-Navier-Stokes (CHNS) equations.
The increasing growth of interest in the CHNS equa- tions arises from the elegant way in which they allow us to follow the spatio-temporal evolution of the two fluids in the mixture and the interfaces between them. These interfaces are diffuse, so we do not have to im- pose boundary conditions on the moving boundaries be- tween two different fluids, as in other methods for the simulation of multi-phase flows [12–14]. However, in addition to a velocity field u, we must also follow the scalar, order-parameter field φ, which distinguishes the two phases in a binary-fluid mixture. Here, interfacial regions are characterized by large gradients in φ. The CHNS equations have been used to model many binary- fluid systems that are of great current interest : exam- ples include studies of (a) the Rayleigh-Taylor instabil- ity [23, 24] ; (b) turbulence-induced suppression of the phase separation of the two components of the binary fluid [19] ; (c) multifractal droplet dynamics in a tur- bulent, binary-fluid mixture [25] ; (d) the coalescence of
droplets [26] ; and (e) lattice-Boltzmann treatments of multi-phase flows [19, 27]. The system of Cahn-Hilliard-Navier-Stokes (CHNS) equations are written as follows [24, 28–30] :
(∂t + u · ∇) u = −∇p/ρ + ν∇ 2 u − αu − (φ∇μ) −Ag + f , (1) (∂t + u · ∇) φ = γ∇ 2 μ , (2)
where p is the pressure, and ρ(= 1) is the constant density, together with the incompressibility condition ∇·u = 0. In Eq. (1), u ≡ (ux, uy , uz ) is the fluid velocity and ν is the kinematic viscosity. In the 2D case uz = 0 and α, the air-drag-induced friction, should be included, but in 3D we set α = 0. φ(x, t) is the order-parameter field at the point x and time t [with φ(x, t) > 0 in the lighter phase and φ(x, t) < 0 in the heavier phase]. The third term on the right-hand side of Eq. (1) couples u to φ via the chemical potential μ(x, t), which is related to the the free energy F of the Cahn-Hilliard system as follows :
μ = δF [φ]/δφ(x, t) , (3)
F [φ] = Λ
∫
V
[ 1
2 |∇φ|
2 + (φ 2 − 1) 2 /(4ξ 2 )] dV , (4)
where Λ is the energy density with which the two phases mix in the interfacial regime [24], ξ sets the scale of the interface width, σ = 2(
1 2 )Λ/ 3 ξ is the surface tension, γ is the mobility [29] of the binary-fluid mixture, A = (ρ 2 − ρ 1 )/(ρ 2 + ρ 1 ) is the Atwood number, and g is the acceleration due to gravity. While solutions of the CHNS equations have been shown to be regular in the 2D-case [31, 32], with an equiv- alent body of literature associated with the CH equations
alone (mainly 2D) (see, e., Ref. [34]) a critical issue for the 3D-CHNS system (1) - (4) revolves around the smoothness of the contours of φ packed together within the fluid interfaces. The regularity of the solutions of the 3D Navier-Stokes (NS) equations alone is in itself a major open problem [6]; a coupling of the CH and the NS equa- tions poses additional severe difficulties. For instance, how do we know whether a slope discontinuity, such as a cusp, might develop in a finite time in arbitrarily large spatial derivatives of φ, thereby affecting the smooth- ness of these contours? Moreover, if such singularities do develop, how closely are they associated with the break- down of regularity of the solutions of the 3D NS equa- tions themselves? To answer such questions, we follow a strategy that is closely connected to an issue that once arose in studies of the incompressible 3D Euler equations (for a survey of the Euler literature see Refs. [35–37]). Since the time of Leray [2, 5, 6] it has been known that the finiteness of
∫
V |ω|
2 dV pointwise in time controls the
regularity of solutions of the 3D incompressible NS equa- tions, where ω = ∇ × u is the vorticity. There are also a variety of alternative time integral criteria, such as the
finiteness of
∫ T
0 ‖u‖
2 ∞ dτ or
∫ T
0
(
‖ω‖ 24 /‖ω‖ 2
)
dτ. In ad- dition, other conditions exist involving the pressure [38]. In contrast, prior to 1984, it was not known what vari- ables control the regularity of solutions of the 3D Euler equations. Beale, Kato, and Majda [39] then proved that
the time integral
∫ T ∗
0 ‖ω‖∞ dτ is the key object : if this integral becomes infinite at a finite time T ∗, then solu- tions have lost regularity at T ∗ (i., blow-up occurs), but there exists a global solution if, for every ∫ T > 0, T 0 ‖ω‖∞ dτ < ∞. This result is now generally referred to as the BKM theorem. Its practical value is that only one simple integral needs to be monitored numerically. It also discounts the possibility that very large spatial derivatives of u could develop a discontinuity if the inte- gral is finite.
The main result of this paper is that we have shown that there exists a similar result for the 3D-CHNS sys- tem. It can be expressed very simply and takes its mo- tivation from the energy E(t) of the full system, which can be written as
E(t) =
∫
V
[
12 Λ|∇φ| 2 + Λ 4 ξ 2
(
φ 2 − 1
) 2
- 12 |u| 2
]
dV. (5)
Given that this can be viewed as a combination of squares of L 2 -norms, it suggests a corresponding L∞ version, which we call the maximal energy[55] :
E∞(t) = 12 Λ‖∇φ‖ 2 ∞ +
Λ
4 ξ 2
(
‖φ‖ 2 ∞ − 1
) 2
- 12 ‖u‖ 2 ∞. (6)
In Sec. III we prove a theorem which says that ∫ T ∗ 0 E∞(τ ) dτ is the key object that controls regularity of solutions of the 3D CHNS equations exactly in the same
fashion as
∫ T ∗
0 ‖ω‖∞ dτ does for the 3D Euler equations [39]. The proof of the theorem is technically complicated,
so this is given in Appendix A. Our numerical calcula- tions in that Section (Fig. 1 (left)) suggest that E∞ is indeed finite. In order to make a comparison with 3D Navier-Stokes results, we also calculate the time dependence of scaled L 2 m-norms of other fields, such as the fluid vorticity ω. The study of similar scaled norms has led to fruitful in- sights into the solutions of the 3D NS [40–42] and the 3D MHD equations [43]. We find that plots of all these norms, versus time t, are ordered as a function of m (curves with different values of m do not cross) ; and, as m → ∞, these curves approach a limit curve that can be identified as the scaled L∞ norm. The remainder of this paper is organized as follows: In Sec. II we discuss the numerical methods that we use to study its solutions. Section III is devoted to the statement of our E∞ theorem and associated numerical results together with plots of the L 2 m norms mentioned in the last paragraph. Section IV contains concluding remarks. In the Appendix we describe the details of the proof of the theorem
II. NUMERICAL METHODS
We carry out direct numerical simulations (DNSs) of the 3D CHNS equations. For this we use a simulation domain that is a cubical box with sides of length 2π and periodic boundary conditions in all three directions. We use N 3 collocation points, a pseudo-spectral method with a 1/2- dealiasing rule, and a second-order Adams- Bashforth method for time marching. In our DNSs we use the following two types of forcing: (a) In the first type, we use the gravity-driven Rayleigh Taylor instabil- ity (RTI) of the interface of a heavy fluid that is placed initially on top of a light fluid ; this instability is of great importance in inertial-confinement fusion [44, 45], astro- physical phenomena [23], and in turbulent mixing, es- pecially in oceanography [46]. (b) In the second type, we have a forcing that yields a constant energy-injection rate [47]. In our RTI studies, there is a constant gravita- tional field in the ˆz direction ; here we stop our DNS just before plumes of the heavy or light fluid wrap around the simulation domain in the ˆz direction because of the periodic boundary conditions. Most of the DNSs of such CHNS problems, e., CHNS studies of the RTI, have been motivated by experiments [48–50]. To the best of our knowledge, no studies have investigated the growth of L 2 m norms of the quantities we have mentioned above. (For the RTI problem, some of these norms have been studied [51] by using the DNS results of Ref. [52] for the miscible, two-fluid, incompressible 3D NS equations.) It behooves us, therefore, to initiate such DNS investiga- tions of L 2 m norms of fields in the 3D CHNS equations. The last-but-one term in Eq. (1) is used in our DNSs of the RTI ; in these studies we set the external force f = 0. We also carry out DNSs, with no gravity, but
(a) (b)
(c) (d)
FIG. 1: (Color online) Isosurface plots of the φ-field in the 3D CHNS equations illustrating the development of the RTI with large-wavelength perturbations in 3D (DNS run T1 in Table I), with 256 3 collocation points, at times (a) t = 1, (b) t = 10, (c) t = 25, and (d) t = 36. The spatiotemporal development of this field is given in the Video RTI Atwood=5e-1 in You Tube [53].
with the additional relation that includes the time- dependent exponents λm
Am(t) =
λm(t)(m − 1) + 1 4 m − 3
. (16)
It was observed numerically [41] that the maxima of the λm lay in the range 1. 15 − 1 .45. For purposes of compar- ison between those calculations and our RTI simulation,
we plot Am(t) versus t, in Fig. 3(b), where
Am(t) = ln Dm/ ln D 1. (17)
We observe that the Am do not change significantly with t but that they depend on m. We also give the plot of Am(t) versus t for the case of constant-energy-injection in Fig. 3(e). As in DNSs of the 3D Navier-Stokes equation [41], we find, for the 3D CHNS system, that D 1 lies well above the other Dm (see Fig. 3(a) and Fig. 3(c)). We give the plots for λm(t) in Fig. 3(c) (for the RTI case)
10
0 10
1 10
102
−
10
−
10 −
100
t
E
m
m= m= m= m= m= m=
(a)
10
0 10
1
10
0
t
E
m
m= m= m= m= m= m=
(b)
0 10 20 30 40 50
2
4
6
8
10
12
14
t
−1lm
m= m= m= m= m= m=
(c)
10 20 30 40 50
2
4
6
8
10
12
14
t
−1lm
m= m= m= m= m= m=
(d)
FIG. 2: (Color online) (a) Plots against time t of Em according to Eq. (11) for m = 1 (blue curve with squares), m = 2 (green curve with inverted triangles), m = 3 (red curve with diamonds), m = 4 (light blue curve with pentagrams), m = 5 (magenta curve with crosses), and m = 6 (yellow curve with open circles). These plots are for an RTI flow. (b) Plots against time t of Em according to Eq. (11) for m = 1 (blue curve with squares), m = 2 (green curve with inverted triangles), m = 3 (red curve with diamonds), m = 4 (light blue curve with pentagrams), m = 5 (magenta curve with crosses), and m = 6 (yellow curve with open circles). These plots are for a flow with a constant-energy-injection forcing scheme (see Eq. (7)), with no gravity. (c) Plots against time t of ℓ− m 1 for m = 1 (blue curve with squares), m = 2 (green curve with inverted triangles), m = 3 (red curve with diamonds), m = 4 (light blue curve with pentagrams), m = 5 (magenta curve with crosses), and m = 6 (yellow curve with open circles). These plots are for an RTI flow. (d) Plots against time t of ℓ− m 1 for m = 1 (blue curve with squares), m = 2 (green curve with inverted triangles), m = 3 (red curve with diamonds), m = 4 (light blue curve with pentagrams), m = 5 (magenta curve with crosses), and m = 6 (yellow curve with open circles). These plots are for a flow with a constant-energy-injection forcing scheme.
and in Fig. 3(f) (for the constant-energy-injection forcing scheme). In the 3D NS case, the λm are related to the spectral exponents for the inertial-range, power-law form of the energy spectra [41] ; the analogous relation for the 3D CHNS case is not straightforward because the power- law ranges in such spectra depend on several parameters in the CHNS equations (see, e., Ref. [25]).
We also compute the temporal evolution of the L 2 m- norms of the gradients of φ by using the definition of the
inverse length scale ℓ− m 1
ℓ−m 2 m=
∫
V |∇φ|
2 mdV ∫ V |φ|
2 mdV. (18)
Figures 2(a)-(d) show plots of Em and ℓ− m 1 versus time t for different values of m. These are qualitatively similar to those for Dm in so far as curves for different values of m do not cross ; they are ordered in m such that ℓ− m 1 < ℓ− m 1 +1. Furthermore, both ℓ− m 1 and Em approach limiting curves as m → ∞. (We mention in passing that errors increase, as m increases. We present data for values of m
ents of φ, the order parameter. The E∞ theorem, stated in Sec. III and proved in Appendix A, is a conditional- regularity criterion on periodic boundary conditions that is realistically computable. The motivation for this re- sult lies in the BKM theorem for the 3D Euler equations. Constantin, Fefferman and Majda [33] have reduced the ‖ω‖∞ within the BKM criterion to ‖ω‖p for finite p ≥ 2, but at the heavy price of introducing technically com- plicated, local constraints on the direction of vorticity, which are difficult to compute. Thus, the original form of the BKM theorem, with its single requirement of ‖ω‖∞ being finite, remains the simplest regularity criterion to this day. Our E∞ theorem is the equivalent result for the 3D CHNS system. Our curves for Em versus time in Fig. 2 (left) suggest convergence to E∞, with increasing values of m, thereby indicating that solutions remain regular for as long as our DNSs remain valid, even though more resolution would
be desirable in the future to investigate the delicate issue of possible finite-time singularities in solutions of the 3D CHNS equations.
Acknowledgments
We thank Prasad Perlekar, Samriddhi Sankar Ray, Ak- shay Bhatnagar, and Akhilesh Kumar Verma for discus- sions. J.D. thanks F ́ed ́eration Doeblin for support and the International Centre for Theoretical Sciences, Banga- lore for hospitality during a visit in which this study was initiated. N. and R. thank University Grants Com- mission (India), Department of Science and Technology (India), and Council of Scientific and Industrial Research (India), for support and SERC (IISc) for computational resources.
Appendix A: Proof of Theorem 1
In the following proof the coefficients in Eqs. (1)-(4) in the main paper are set to unity to avoid needless complication. First, we recall the definitions of Hn and Pn in Eq. (8). In addition to these we define
Xn = Hn + Pn+1. (A1)
The proof uses the method of BKM [39], which is by contradiction. The strategy is the following : suppose there exists an interval [0, T ∗) on which solutions are globally regular with the earliest loss of regularity at T ∗. Assume that ∫ T ∗ 0 E∞(τ ) dτ < ∞, and then show that a consequence of this is that Xn(T
∗) < ∞, which contradicts the statement
that solutions first lose regularity at T ∗. This falsifies the assumption of the finiteness of the integral. We proceed in 3 steps.
Step 1 : We begin with the time evolution of Pn (the dot above Pn denotes a time derivative):
12 P ̇n = −Pn+2 + Pn+1 +
∫
V
(∇nφ)∇n∆(φ 3 ) dV −
∫
V
(∇nφ)∇n(u · ∇φ) dV ; (A2)
and then we estimate the third term on the right as
∣ ∣ ∣ ∣
∫
V
(∇nφ)∇n∆(φ 3 ) dV
∣
∣
∣
∣ ≤ ‖∇
nφ‖ 2
n∑+
i,j=
C i,jn+2 ‖∇iφ‖p|∇j φ‖q‖∇n+2−i−j φ‖r , (A3)
where 1/p + 1/q + 1/r = 1/2. Now we use a sequence of Gagliardo-Nirenberg inequalities
‖∇iφ‖p ≤ cn,i‖∇n+2φ‖a 21 ‖φ‖ 1 ∞− a 1 , ‖∇j φ‖q ≤ cn,j ‖∇n+2φ‖a 22 ‖φ‖ 1 ∞− a 2 , (A4) ‖∇nφ‖r ≤ cn,i,j ‖∇n+2−i−j φ‖a 23 ‖φ‖ 1 ∞− a 3 ,
where, in d dimensions,
1 p
=
i d
- a 1
(
1
2
−
n + 2 d
)
,
1
q
=
j d
- a 2
(
1
2
−
n + 2 d
)
, (A5)
1
r
=
n + 2 − i − j d
- a 3
(
1
2
−
n + 2 d
)
.
By summing these and using 1/p + 1/q + 1/r = 1/2, it is seen that a 1 + a 2 + a 3 = 1. Thus, we have ∣ ∣ ∣ ∣
∫
V
(∇nφ)∇n+2(φ 3 ) dV
∣
∣
∣
∣ ≤ cn‖∇
nφ‖ 2 ‖∇
n+2φ‖ 2 ‖φ‖
2 ∞ ≤ 12 Pn+2 + cnPn‖φ‖
4 ∞ , (A6)
and so Eq. (A2) becomes (here and henceforth coefficients such as cn are multiplicative constants),
12 P ̇n = − 12 Pn+2 + Pn+1 + cn‖φ‖ 4 ∞Pn +
∣
∣
∣
∣
∫
V
(∇nφ)∇n(u · ∇φ) dV
∣
∣
∣
∣. (A7)
Estimating the last term in Eq. (A7) we have ∣ ∣ ∣ ∣
∫
V
(∇nφ)∇n(u · ∇φ) dV
∣
∣
∣
∣ =
∣
∣
∣
∣−
∫
V
(∇n+1φ)∇n− 1 (u · ∇φ) dV
∣
∣
∣
∣
≤ ‖∇n+1φ‖ 2
n∑− 1
i=
Cin ‖∇iu‖p‖∇n− 1 −i(∇φ)‖q , (A8)
where 1/p + 1/q = 1/2. Now we use two Gagliardo-Nirenberg inequalities in d dimensions to obtain
‖∇iu‖p ≤ c ‖∇n− 1 u‖a 2 ‖u‖ 1 ∞− a, (A9) ‖∇n− 1 −i(∇φ)‖q ≤ c ‖∇n− 1 (∇φ)‖b 2 ‖∇φ‖ 1 ∞− b. (A10)
Equations (A9) and (A10) follow from
1 p
=
i d
- a
(
1
2
−
n − 1 d
)
, (A11)
1
q
=
n − 1 − i d
- b
(
1
2
−
n − 1 d
)
. (A12)
Because 1/p + 1/q = 1/2 then a + b = 1. Thus Eq. (A3) turns into
∣ ∣ ∣ ∣
∫
V
(∇nφ)∇n(u · ∇φ) dV
∣
∣
∣
∣ ≤ cnP
1 / 2 n+1H
a/ 2 n− 1 P
b/ 2 n ‖u‖
1 −a ∞ ‖∇φ‖
1 −b ∞
≤ P n 1 /+1 2
[
cnHn− 1 ‖∇φ‖ 2 ∞
]a/ 2 [ Pn‖u‖ 2 ∞
]b/ 2
≤ 12 Pn+1 + 12 acnHn− 1 ‖∇φ‖ 2 ∞ + 12 bPn‖u‖ 2 ∞ , (A13)
and Eq. (A7) becomes
12 P ̇n = − 12 Pn+2 + 32 Pn+1 + cn, 1 ( 12 ‖φ‖ 4 ∞ + ‖u‖ 2 ∞ ) Pn + cn, 2 Hn− 1 ‖∇φ‖ 2 ∞. (A14)
Step 2 : Now we look at Hn defined in Eq. (8) using Eq. (A15) with f = −zφˆ. The easiest way is to use the 3D NS equation in the vorticity form as in Doering and Gibbon [6] to obtain the ‖u‖ 2 ∞-term in Eq. (A16), where gradient terms have been absorbed into the pressure term, which disappears under the curl-operation :
(∂t + u · ∇) ω = ∆ω + ω · ∇u + ∇φ × ∇∆φ − ∇⊥φ. (A15)
Therefore,
1 2 H ̇n ≤ −
1 2 Hn+1 + cn‖u‖
2 ∞Hn +
∣
∣
∣
∣
∫
V
(∇n− 1 ω)
[
∇n− 1 (∇φ × ∆∇φ)
]
dV
∣
∣
∣
∣
+
∣∣
∣
∣
∫
V
(∇n− 1 ω)
[
∇n− 1 ∇⊥φ
]
dV
∣∣
∣
∣. (A16)
Beginning with the third term on the right-hand side of Eq. (A16), we obtain
∣ ∣ ∣ ∣
∫
V
(∇n− 1 ω)∇n− 1 (∇φ × ∆∇φ) dV
∣
∣
∣
∣ ≤ ‖∇
n− 1 ω‖ 2
n∑− 1
i=
Cni ‖∇i(∇φ)‖r ‖∇n+1−i(∇φ)‖s. (A17)
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[55] In [42] a set of multiplicative positive constants Cm were included.
1608 - Cours
Course: comptabilté approfondie (comp12)
University: Université Ibn Zohr
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