Port-Hamiltonian formulation of two-phase flow models

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Systems & Control Letters

journal homepage:www.elsevier.com/locate/sysconle

Port-Hamiltonian formulation of two-phase flow models

✩

H. Bansal

a,∗

, P. Schulze

b

_{, M.H. Abbasi}

a

_{, H. Zwart}

c,d

_{, L. Iapichino}

a

_{, W.H.A. Schilders}

a

_,

N. van de Wouw

d,e

a_{Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands} b_{Institut für Mathematik, Technische Universität Berlin, Germany}

c_{Department of Applied Mathematics, University of Twente, The Netherlands}

d_{Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands} e_{Department of Civil, Environmental and Geo-Engineering, University of Minnesota, USA}

a r t i c l e i n f o Article history:

Received 14 April 2020

Received in revised form 30 September 2020 Accepted 18 January 2021

Available online xxxx

Keywords:

Two-Fluid Model Drift Flux Model

Non-quadratic Hamiltonian Skew-adjoint

Stokes–Dirac structures Port-Hamiltonian

a b s t r a c t

Two-phase flows are frequently modelled and simulated using the Two-Fluid Model (TFM) and the Drift Flux Model (DFM). This paper proposes Stokes–Dirac structures with respect to which port-Hamiltonian representations for such two-phase flow models can be obtained. We introduce a non-quadratic candidate Hamiltonian function and present dissipative Hamiltonian representations for both models. We then use the structure of the corresponding formally skew-adjoint operator to derive a Stokes–Dirac structure for the two variants of multi-phase flow models. Moreover, we discuss the difficulties in deriving a port-Hamiltonian formulation of the DFM with general slip conditions, and argue why this model may not be energy-consistent.

© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

In this paper, we develop a port-Hamiltonian (pH) formulation for modelling multi-phase flow dynamics in pipes. Multi-phase flows are important in a large range of industrial applications, such as within the oil and gas industry, chemical and process industry (including heat-pumping systems) as well as the safety analysis of nuclear power plants [1–3]. Within the oil and gas industry, such models are used for virtual drilling scenario test-ing [1,2]. The multi-phase aspect is particularly relevant in these applications in case of gas influx occurring from a reservoir.

A pH model formulation is known to provide a modular framework for multi-physics and interconnected systems [4]. The pH structure allows for non-zero energy flow through the boundary and guarantees power preservation [5]. Furthermore, structure-preserving methods for discretization and the model ✩ _{The first author has been funded by the Shell NWO/ FOM Ph.D. Programme}

in Computational Sciences for Energy Research. The second author is supported by the DFG, Germany Collaborative Research Center 1029, project A02. The third author has carried out this research in the HYDRA project, which has received funding from European Union’s Horizon 2020 research and innovation program under grant agreement No 675731.

∗ _{Corresponding author.}

E-mail addresses: h.bansal@tue.nl(H. Bansal),pschulze@math.tu-berlin.de

(P. Schulze),m.h.abbasi@tue.nl(M.H. Abbasi),h.j.zwart@utwente.nl(H. Zwart),

l.iapichino@tue.nl(L. Iapichino),w.h.a.schilders@tue.nl(W.H.A. Schilders),

n.v.d.wouw@tue.nl(N.v.d. Wouw).

order reduction of infinite-dimensional pH systems can preserve original system-theoretic properties such as stability and passiv-ity [6,7]. Moreover, the pH framework supports the development of control strategies [8].

There exists a pH modelling framework both for finite- and infinite-dimensional systems. In the scope of this paper, we will only focus on the state-of-the-art for infinite-dimensional sys-tems, particularly in the context of two/multi-phase flows.

In the past, both linear and non-linear partial differential equations, arising in several domains of science and engineer-ing, have been formulated as an infinite-dimensional pH model. For instance, some well-known fluid dynamical systems such as the shallow water equations [7], reactive Navier Stokes equa-tions [9], reaction diffusion processes [10], and single-phase flow models [11] have already been formulated in the pH formal-ism. A pH model representation is also prevalent in the fields of structural dynamics [8] and fluid–structure interaction [12]. Furthermore, several conservation laws, including the systems subject to heat and mass transfer, have previously been converted to pH representations [13–15]. Moreover, some work on Hamil-tonian modelling for multi-phase hydrodynamics has been done in [16]. However, (dissipative) Hamiltonian and pH representa-tions do not yet exist for two frequently used two-phase flow models, namely the Two-Fluid Model (TFM) and the Drift Flux Model (DFM) [17].

By now there are different approaches of formulating dis-tributed parameter pH systems, for an overview we refer to [18]. https://doi.org/10.1016/j.sysconle.2021.104881

In this paper, we adopt the Stokes–Dirac structure approach for developing pH formulations for the two-phase flow models of interest. From a theoretical point of view, we seek to generalize the definition of Stokes–Dirac structures introduced for constant coefficient Hamiltonian matrix operators in [19].

The main contributions of this paper are as follows: (i) (dis-sipative) Hamiltonian representations of the TFM and the DFM, and (ii) proposition of state-dependent Stokes–Dirac structures for both the TFM and the DFM.

The paper is organized as follows. In Section2, we introduce the two mathematical models governing 1-D multi-phase flow dynamics and mention under which conditions these are equiva-lent. The (dissipative) Hamiltonian representations of these mod-els are presented in Section3. Then, the corresponding geometric properties are discussed and proved in Section 4. Afterwards, Section 5 deals with the reasons behind formulating the DFM without slip between the two phases in a pH representation instead of a general DFM with the Zuber–Findlay slip conditions. Finally, Section6closes with conclusions.

Notations:L2(Ω) is the space of square-integrable functions over the spatial domainΩ, and

L2(Ω)p

=

L2(Ω)

×

L2(Ω)

× · · · ×

L2(Ω) (p-times)

.

(1)

H1_{(Ω) denotes the Sobolev space of functions that also possess a}

weak derivative, H1

0(Ω) denotes the functions in H1(Ω) that have

zero boundary values, H1_(Ω)p _{is defined analogous to L}2_(Ω)p_,

and R denotes the space of real numbers.

2. Multi-phase flow models

Different variants of two-phase flow models exist in the lit-erature; see [1] and the references therein. Here, we focus on the following two variants of two-phase flow models: the TFM [17] and the DFM [17]. These models are popularly employed in the scope of drilling applications, see [20]. The reader is referred to [1] for insights into the physical assumptions leading to the aforementioned mathematical models.

2.1. Two-fluid model

The TFM is a set of Partial Differential Equations (PDEs) and algebraic closure relations. The PDEs expressing mass and mo-mentum balance for each phase are as follows:

∂

t

(

α

g

ρ

g

) +

∂

x

(

α

g

ρ

g

v

g

) =

0

,

(2a)

∂

t

(α

ℓ

ρ

ℓ

) + ∂

x

(α

ℓ

ρ

ℓ

v

ℓ

) =

0

,

(2b)

∂

t

(

α

g

ρ

g

v

g

) +

∂

x

(

α

g

ρ

g

v

2g

) = −

∂

x

(

α

gp

) +

Mg

,

(2c)

∂

t

(αℓρℓvℓ) + ∂

x

(

αℓρℓv

2 ℓ

) = −

∂

x

(αℓ

p

) +

Mℓ, (2d)

where t

∈

_R≥0and x

∈

Ω

= [

a

,

b

]

are, respectively, the temporal

and spatial variables (a and b refer to the location of the left and the right boundary of the one-dimensional spatial domain). The model contains seven unknown variables, namely, liquid and gas void fraction,

αℓ

and

α

g, liquid and gas phase velocity,

vℓ

and

v

g, liquid and gas phase density,

ρℓ

and

ρ

g, and the common

pressure, p.

To complete the model, we use one set of the most widely applied closure laws as in [17]:

α

g

+

αℓ

=

1

,

(3a) Mg

+

Mℓ

=

0

,

(3b) Mg

=

p

∂

x

α

g

+

Mig

,

(3c) Mig

=

bMg(

vℓ

−

v

g)

,

with bMg

≥

0

,

(3d)

ρ

g

=

pc −2 g

,

(3e)

ρℓ

=

ρℓ

₀

+

(p

−

p_ℓ0)c −2 ℓ

,

(3f)

where(3a)expresses that any pipe segment is occupied by the combination of gas and liquid. The terms Mgand Mℓwith the

con-stant bM

g in(3b)–(3d)account for the force interaction between

the phases. Finally,(3e)–(3f)define the equation of state of each phase with the reference density and pressure as

ρℓ

0and pℓ0, and

cg and cℓare the constant speeds of sound in the gas and liquid

phases, respectively.

Remark 2.1. We do not consider gravitational and frictional

effects in the above TFM description for the sake of simplic-ity. However, in principle, the TFM can be formulated with the additional terms accounting for these effects [17].

The TFM, governed by the set of equations(2)and(3), can be written in terms of only four physical variables. We introduce the following shorthand notations: mg

:=

α

g

ρ

g, mℓ

:=

αℓρℓ

,

Ig

:=

mg

v

g, and Iℓ

:=

mℓvℓ.

Assumption 1. The gas void fraction, the liquid void fraction, the

liquid and the gaseous phase densities along with

β = ρℓ

0c_ℓ2

−

pℓ0

are positive.

Lemma 2.2. By considering mg, mℓ, Ig and Iℓas state variables,

the system of equations(2)and(3)can be rewritten in the following form:

∂

tmg

+

∂

xIg

=

0

,

(4a)

∂

tmℓ

+

∂

xIℓ

=

0

,

(4b)

∂

tIg

+

∂

x

(

I_g2 mg

)

+

mg

∂

x

(

c2_gln

(

p c2 g

))

=

bM_g

v

r

,

(4c)

∂

tIℓ

+

∂

x

(

I2 ℓ m_ℓ

)

+

mℓ∂x

(

c_ℓ2ln

(

p

+

β

c2 ℓ

))

= −

bM_g

v

r

,

(4d) where

v

r

=

(

v

ℓ

−

v

g

)

, and p

(

mg

,

mℓ, αg

) =

mgc2g

+

mℓcℓ2

−

β (

1

−

α

g

(

mg

,

mℓ

))

,

(5)

α

g

(

mg

,

mℓ

) = −

mg c2 g 2

β

−

mℓ c_ℓ2 2

β

+

1 2

+







√

(

mg c2 g 2

β

+

mℓ c_ℓ2 2

β

−

1 2

)

2

+

mg c2 g

β

.

(6)

We refer to [21] for the proof of the lemma, and to [2] for the derivation of the expression for

α

g

(

mg

,

mℓ

)

. In summary, the set of equations(4)is equivalent to(2)and(3).

2.2. Drift flux model

The DFM can be obtained from the TFM via a slip relation of the form

v

g

−

vℓ

=

Φ

(

mg

,

mℓ, vg

)

,

(7)

where mgand mℓare as introduced above. Since the slip relation

(7) determines the coupling between the velocities of the two phases, only one momentum equation is required contrary to the two momentum equations in the TFM(2). Several models of the form(7) exist depending on the choice of the functionΦ [17]. In the simplest case, without slip,Φ

:=

0. Another case is the Zuber–Findlay relation [17]:

Φ

:=

(K

−

1)

v

g

+

S

K

αℓ

→

v

g

=

K (

α

g

v

g

+

α

ℓ

v

ℓ)

+

S

,

(8)

where K and S are flow-regime dependent parameters. The governing equations for the DFM are:

∂

tmg

+

∂

xIg

=

0

,

(9a)

∂

tmℓ

+

∂

xIℓ

=

0

,

(9b)

∂

t

(

Ig

+

Iℓ

) +

∂

x

(

Ig

v

g

+

Iℓvℓ

) = −

∂

xp

+

Qg

+

Qv (9c)

completed with closure laws(3a),(3e),(3f),(7), and gravitational effects Qg and frictional effects Qvdefined by:

Qg

= −

g

(

mg

+

mℓ

)

sin

θ,

(10a) Q_v

= −

32

µ

m(

α

g

v

g

+

αℓvℓ

)

/

d2

,

(10b)

with gravitational constant g, space-dependent pipe inclination

θ

(x), mixture viscosity

µ

m

>

0, and pipe diameter d.

Remark 2.3. Similar toLemma 2.2, the governing equations(9)

associated with

v := v

g

=

vℓ

(DFM without slip), the closure

equations (3a), (3e), (3f) and (10), upon elimination of auxil-iary variables, can be rewritten as an explicit system of balance equations.

The DFM can be interpreted as a TFM with a specific choice of the momentum source term Mig, as is detailed in the following

theorem.

Theorem 2.4 ([17]). Under zero gravitational and frictional effects, the DFM(9)together with (3a),(3e),(3f), and(7)is equivalent to the TFM(2)with(3a)–(3c),(3e),(3f), and

Mig

= −

α

g

αℓ

ρ

g

−

ζρℓ

mg

+

ζ

mℓ

∂

xp

−

mgmℓ mg

+

ζ

mℓ

(vℓ∂

x

vℓ

−

ζv

g

∂

x

v

g

+

µ

g

∂

x(mg

v

g)

+

µℓ∂

x(mℓvℓ)

),

(11) with

µ

g

:=

_∂∂Φmg,

µℓ

:=

∂Φ ∂mℓ,

ζ :=

1

−

∂v∂Φg.

Remark 2.5. The equivalence of the DFM and the TFM can also

be shown in the presence of gravitational and frictional effects; see [17] for further details.

The model equivalence, stated above, will play a crucial role in drawing a conclusion about the behaviour of the Hamiltonian along the solutions of the DFM by using the theoretical analysis conducted for the TFM (see Section5).

3. Dissipative Hamiltonian formulations

We consider dissipative Hamiltonian systems (with state vari-able z) of the form

∂

tz

=

(

J(z)

−

R(z)

)δ

zH(z)

,

(12)

whereHis the Hamiltonian functional,

δ

zHits variational

deriva-tive, andJ(z) andR(z) are matrix differential operators. Further-more, for every z,J(z) is formally skew-adjoint with respect to the L2(Ω) inner product, i.e., for e1

,

e2 sufficiently smooth and

zero at the boundary there holds

∫

Ω eT₁(J(z)e2)dx

+

∫

Ω eT₂(J(z)e1)dx

=

0

,

(13)

where Ω refers to the spatial domain. Moreover, forJ(z) to be classified as a Hamiltonian operator it should satisfy the Jacobi identity [22,23]

{{

F

,

G

}

,

H

} + {{

G

,

H

}

,

F

} + {{

H

,

F

}

,

G

} =

0

,

(14)

for all F

,

G

,

H in a certain function space (e.g. infinitely

differen-tiable functionals). Here,

{

F

,

G

}

, the Poisson bracket evaluated at

F and G, is defined as follows:

{

F

,

G

} =

∫

Ω

(

δ

F

δ

z

)

T J(z)

(

δ

G

δ

z

)

dx

.

(15)

Moreover, R(z) is formally self-adjoint with respect to theL2(Ω) inner product and positive semi-definite.

The dissipation inequality, which expresses that energy cannot be generated within the system, is a property which directly follows from the definition of a pH system. In particular, ignoring the boundary conditions,

dH dt

=

∫

Ω (

δ

zH(z))T

∂

tz dx (12)

=

∫

Ω (

δ

zH(z))T

(

(J(z)

−

R(z))

δ

zH(z)

)

dx

=

∫

Ω (

δ

zH(z))T(

−

R(z))

δ

zH(z) dx

≤

0

.

(16)

Thus,Ris the dissipative component of the system.

Derivation of the Hamiltonian functional

The kinetic, gravitational potential and internal energy con-tribute to the construction of the energy/Hamiltonian functional for the TFM and the DFM. To derive the internal energy of the system, consider the following remark.

Remark 3.1. The internal energy ui

,

i

∈ {

ℓ,

g

}

, can be interpreted

as the energy causing the expansion of the ith compressed phase or compression of the ith expanded phase. In order to derive this energy component, the Gibbs relation [24] under barotropic and isentropic flow considerations for an infinitesimal part of the phase is used, i.e.,

ρ

2

idui

=

pd

ρ

i

,

i

∈ {

ℓ,

g

}

.

Using(3e)–(3f)and integrating the above equation leads to

u_ℓ

= −

pℓ0

ρℓ

+

cℓ2ln

ρℓ

+

ρℓ

0 c2 ℓ

ρℓ

+

K1

,

(17a) ug

=

cg2ln

ρ

g

+

K2

,

(17b)

where K1and K2are the integration constants.

Considering the total energy of the system (neglecting the gravitational potential energy), we define a candidate for the Hamiltonian as follows: H

:=

∫

Ω

(α

g

ρ

g

v

2 g 2

+

αℓρℓ

v

2 ℓ 2

+

α

g

ρ

gug

+

αℓρℓ

uℓ

)

dx

.

(18) Inserting(17)into (18), the Hamiltonian for a flow across a (unit) constant cross-section takes the following form:

H

:=

∫

Ω

(α

g

ρ

g

v

2 g 2

+

α

ℓ

ρ

ℓ

v

2 ℓ 2

+

α

g

ρ

g

(

c_g2ln

ρ

g

+

K2

) +

αℓρℓ

(

c_ℓ2ln

ρℓ

+

K1

) +

αℓ

(c_ℓ2

ρℓ

0

−

pℓ0)

)

dx

.

(19)

It should be noted that when

ρ

i

→

0,

ρ

iln

ρ

i

→

0. The

term

ρ

iln

ρ

i is bounded from below, i.e.,

ρ

iln

ρ

i

≥ −

1

/

e. So,

the Hamiltonian (19)is bounded from below. Due to the high bulk modulus of the liquid phase, we usually have

ρℓ

0c_ℓ2

≫

p_ℓ0 [25]; therefore, the positivity of the Hamiltonian(19)can be

ensured by appropriately choosing K1and K2or even adding some

constants to the Hamiltonian. For simplicity, we set K1

=

0 and

K2

=

0 henceforth. 3

3.1. Dissipative Hamiltonian formulation for the two-fluid model

The Hamiltonian (19) can be re-written in terms of q

=

[

q1

,

q2

,

q3

,

q4

]

T

:= [

mg

,

mℓ, Ig

,

Iℓ

]

T as follows: HT(q1,q2,q3,q4):= ∫ Ω q2 3 2q1 + q 2 4 2q2 +q1cg2ln ( p(q1,q2, αg(q1,q2)) c2 g ) +q2c2ℓln ( p(q1,q2, αg(q1,q2))+β c2 ℓ ) +( 1−αg(q1,q2) ) βdx, (20)

where p(q1

,

q2

, α

g(q1

,

q2)) and

α

g(q1

,

q2) can be replaced by the

relations(5)and(6), respectively.

We now present the dissipative Hamiltonian framework for the TFM.

Theorem 3.2. The governing equations(4)can be written in the following dissipative Hamiltonian form:

∂

tq

=

(

JT(q)

−

RT

) δ

qH(q) (21)

with q

= [

q1

,

q2

,

q3

,

q4

]

T

= [

mg

,

mℓ

,

Ig

,

Iℓ

]

T, the Hamiltonian

functional(20), and where

JT(q)

= −

⎡

⎢

⎣

0 0

∂

x(q1

·

) 0 0 0 0

∂

x(q2

·

) q1

∂

x(

·

) 0

∂

x(q3

·

)

+

q3

∂

x(

·

) 0 0 q2

∂

x(

·

) 0

∂

x(q4

·

)

+

q4

∂

x(

·

)

⎤

⎥

⎦

is a formally skew-adjoint Hamiltonian operator with respect to the

L2_{inner product, and}

RT

=

⎡

⎢

⎢

⎢

⎢

⎣

0 0 0 0 0 0 0 0 0 0 bM g

−

bMg 0 0

−

bM_g bM_g

⎤

⎥

⎥

⎥

⎥

⎦

is a symmetric and positive semi-definite matrix.

Proof. The variational derivatives ofHT are given by:

δ

HT

δ

q1

= −

1 2 q2₃ q2 1

+

c_g2ln

(

p c2 g

)

+

c_g2

,

δ

HT

δ

q3

=

q3 q1

,

δ

H_T

δ

q2

= −

1 2 q2₄ q2 2

+

c_ℓ2ln

(

p

+

β

c2 ℓ

)

+

c2_ℓ

,

δ

HT

δ

q4

=

q4 q2

.

Next, we prove the claim. The first lines of(21)read

∂

tmg

= −

∂

x

(

mg

(

m_g

v

_g mg

))

= −

∂

_x(mg

v

g)

,

(22)

∂

tmℓ

= −

∂

x

(

m_ℓ

(

mℓvℓ m_ℓ

))

= −

∂

_x(mℓvℓ)

.

(23)

The third line yields

∂

t(mg

v

g)

= −

mg

∂

x

(

δ

H_T

δ

q1

)

−

∂

x

(

mg

v

g

(

δ

H_T

δ

q3

))

−

mg

v

g

∂

x

(

δ

H_T

δ

q3

)

−

bM_g

δ

HT

δ

q3

+

bM_g

δ

HT

δ

q4

.

By substituting the variational derivatives into the above equa-tion, we have

∂

t(mg

v

g)

= −

mg

∂

x

(

−

1 2

v

2 g

+

cg2ln

(

p c2 g

)

+

c2 g

)

−

∂

x(mg

v

2g)

−

mg

v

g

∂

x

v

g

−

bMg(

v

g

−

vℓ

)

.

Simplifying the above equation leads to:

∂

tIg

= −

∂

x(mg

v

g2)

−

mg

∂

x

(

c_g2ln

(

p c2 g

))

−

bM_g(

v

g

−

vℓ

)

.

(24)

Similarly, the fourth line of(21)reads

∂

tIℓ

= −

∂

x(mℓv2_ℓ)

−

mℓ∂x

(

c_ℓ2ln

(

p

+

β

c2 ℓ

))

+

bM_g(

v

g

−

vℓ

)

.

(25)

The claim follows by observing that(22),(23),(24)and(25)are identical to(4a),(4b),(4c)and(4d), respectively.

To prove formal skew-adjointness of J_T, we check whether

⟨

e1

_,

_J

Te2

⟩

L2_(Ω)

+ ⟨

JTe1

,

e2

⟩

L2_(Ω)

=

0 for smooth e1

,

e2which are

zero at the boundary, where we define ei

=

(ei₁

,

ei₂

,

ei₃

,

ei₄)T. Here, the variable ei_jrefers to the jth element of ei. By straightforward computations, we obtain:

⟨

e1

,

J_Te2

⟩

_L2_(Ω)

+ ⟨

J_Te1

,

e2

⟩

_L2_(Ω)

=

(26)

−

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

[

e1₁ e1₂ e1₃ e1₄

]

⎡

⎢

⎢

⎢

⎢

⎣

0 0 q1 0 0 0 0 q2 q1 0 2q3 0 0 q2 0 2q4

⎤

⎥

⎥

⎥

⎥

⎦







Q

⎡

⎢

⎢

⎢

⎢

⎣

e2₁ e2 2 e2 3 e2₄

⎤

⎥

⎥

⎥

⎥

⎦

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

|

b_a

,

which vanishes under our assumptions on the boundary condi-tions.

The proof of the symmetric and positive semi-definite nature ofR_T is straightforward.

In order to check whether J_T satisfies the Jacobi identity,

(14)should hold. To this end, under consideration of vanishing variational derivatives at the boundary, we first write the Poisson bracket

{

F

,

G

}

, which is of the following form:

{

F

,

G

} = −

∫

Ω

[

q1

(

δ

F

δ

q3

∂

x

δ

G

δ

q1

−

δ

G

δ

q3

∂

x

δ

F

δ

q1

)

+

q3

(

δ

F

δ

q3

∂

x

δ

G

δ

q3

−

δ

G

δ

q3

∂

x

δ

F

δ

q3

)

+

q2

(

δ

F

δ

q4

∂

x

δ

G

δ

q2

−

δ

G

δ

q4

∂

x

δ

F

δ

q2

)

+

q4

(

δ

F

δ

q4

∂

x

δ

G

δ

q4

−

δ

G

δ

q4

∂

x

δ

F

δ

q4

)]

dx

.

(27) Next, we compute the brackets

{{

F

,

G

}

,

H

}

,

{{

G

,

H

}

,

F

}

and

{{

H

,

F

}

,

G

}

. For this the variational derivative of e.g.

{

F

,

G

}

with respect to the state variables must be evaluated. Even these ex-pressions become very long, but after extensively lengthy compu-tations, which include, a.o. integration by parts, it can be observed that (14)holds and, hence, the operatorJT satisfies the Jacobi

identity, and is a Hamiltonian operator. For the detailed proof, see [26]. □

3.2. Dissipative hamiltonian formulation for the drift flux model

We will now deal with the DFM under the gravitational and frictional effects, and present a corresponding dissipative Hamil-tonian formulation. For the DFM, we focus only on case in which there is no slip between the phases, i.e.,

v := v

g

=

vℓ

(the reason

for adopting this no-slip assumption is provided in Section 5). Since gravitation is considered, the gravitational potential energy needs to be added to the Hamiltonian, which now takes the following form: H_D(mg

,

mℓ,Itot)

=

∫

Ω

[

(

mg

+

mℓ

)

(∫

x a g sin(

θ

(

ξ

))d

ξ

)

+

I2 tot 2(mg

+

mℓ)

+

m_ℓc_ℓ2ln

(

p(mg

,

mℓ, αg(mg

,

mℓ))

+

β

c2 ℓ

)

+

mgcg2ln

(

p(mg

,

mℓ, αg(mg

,

mℓ)) c2 g

)

+

(1

−

α

_g(mg

,

mℓ))

β

]

dx

,

(28) where Itot

:=

Ig

+

Iℓ

=

(mg

+

mℓ)

v

. 4

Using the above candidate Hamiltonian functionHD, a

dissi-pative Hamiltonian representation of a special case of the DFM is shown below.

Theorem 3.3. The governing equations(9) with

v := v

g

=

vℓ

,

the closure equations (3a), (3e), (3f)and (10)can be written in dissipative Hamiltonian form as follows:

∂

tzD

=

(

JD(zD)

−

RD(zD)

) δ

zDHD(zD) (29)

with zD

:= [

mg

,

mℓ, Itot

]

T, the Hamiltonian functional(28), where

J_D(zD)

= −

⎡

⎢

⎣

0 0

∂

x

(

mg

·

)

0 0

∂

x

(

mℓ

·

)

mg

∂

x(

·

) mℓ

∂

x(

·

)

∂

x(Itot

·

)

+

Itot

∂

x(

·

)

⎤

⎥

⎦

is a formally skew-adjoint Hamiltonian operator with respect to the

L2inner product, and

RD(zD)

=

⎡

⎣

0 0 0 0 0 0 0 0 32µm d2

⎤

⎦

is a symmetric and positive semi-definite matrix.

Proof. The variational derivatives ofHDare given by:

δ

HD

δ

mg

= −

I 2 tot 2(mg

+

mℓ)2

+

c_g2ln

(

p c2 g

)

+

c2_g

+

Gr

,

δ

HD

δ

m_ℓ

= −

I2 tot 2(mg

+

mℓ)2

+

c_ℓ2ln

(

p

+

β

c_ℓ2

)

+

c_ℓ2

+

Gr

,

δ

HD

δ

Itot

=

v,

with the short-hand notation Gr

=

∫

x

ag sin(

θ

(

ξ

))d

ξ

.

Next, we prove the claim. The first lines of(29)read

∂

tmg

=−

∂

x

(

mg

δ

H_D

δ

Itot

)

= −

∂

_x

(

mg

v),

(30)

∂

tmℓ

=−

∂

x

(

m_ℓ

δ

HD

δ

Itot

)

= −

∂

_x

(

mℓv) . (31)

The third line of(29)yields

∂

tItot

= −

mg

∂

x(

δ

H_D

δ

mg )

−

mℓ∂x(

δ

H_D

δ

m_ℓ)

−

∂

x(Itot

δ

H_D

δ

Itot )

−

Itot

∂

x(

δ

H_D

δ

Itot )

−

32

µ

m d2

δ

H_D

δ

Itot

.

(32) On substituting the variational derivatives in(32), we have

∂

tItot

= −

mg

∂

x

(

−

v

2 2

+

c 2 gln

(

p c2 g

)

+

c2 g

)

−

mℓ∂x

(

−

v

2 2

+

c 2 ℓln

(

p

₊

β

c2 ℓ

)

+

c2_ℓ

)

−

∂

_x((mg

+

mℓ)

v

2)

−

(mg

+

mℓ)

v∂

x

v −

(mg

+

mℓ)g sin

θ

(x)

−

32

µ

m

v

d −2

_.

The above equation can be simplified to:

∂

t((mg

+

mℓ)

v

)

= −

∂

x((mg

+

mℓ)

v

2)

−

∂

xp

−

(mg

+

mℓ)g sin

θ

(x)

−

32

µ

m

v

d−2

,

(33)

where we have used the identity

−

mgcg2

∂

x(ln p c2 g )

−

m_ℓc_ℓ2

∂

x(ln p

+

β

c_ℓ2 )

= −

∂

xp

.

The claim follows by observing that (30), (31), and (33) are identical to(9a),(9b), and(9c), respectively.

The symmetric and positive semi-definite nature of RD

fol-lows immediately from the positivity of

µ

m. The formal

skew-adjointness of JD essentially follows from integration by parts

and neglecting the boundary conditions. The operatorJDcontains

terms similar to the skew-adjoint operatorJT, the formal

skew-adjointness of which was discussed extensively in the proof of

Theorem 3.2. For the sake of brevity, we refer the reader to follow similar lines of reasoning to show the formal skew-adjointness of

J_D, and just present the final result.J_Dis formally skew-adjoint with respect to theL2inner product because

⟨

e1

,

JDe2

⟩

L2_(Ω)

+ ⟨

J_De1

,

e2

⟩

_L2_(Ω)

=

−

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

[

e1 1 e12 e13

]

⎡

⎢

⎢

⎣

0 0 mg 0 0 m_ℓ mg mℓ 2Itot

⎤

⎥

⎥

⎦







QD

⎡

⎢

⎣

e2 1 e2 2 e2₃

⎤

⎥

⎦

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

|

b_a

,

(34)

which vanishes under smooth e1

_{= [}

_e1

1

,

e12

,

e13

]

T

,

e2

= [

e21

,

e22

,

e2

3

]

T that are zero at the boundary.

In order to check whetherJDsatisfies the Jacobi identity,(14)

should hold. To this end, we first define the Poisson bracket

{

F

,

G

}

as follows:

{

F

,

G

} = −

∫

Ω

[

mg

(

δ

F

δ

Itot

∂

x

(

δ

G

δ

mg

)

−

δ

G

δ

Itot

∂

x

(

δ

F

δ

mg

))

+

m_ℓ

(

δ

F

δ

Itot

∂

x

(

δ

G

δ

m_ℓ

)

−

δ

G

δ

Itot

∂

x

(

δ

F

δ

m_ℓ

))

+

Itot

(

δ

F

δ

Itot

∂

x

(

δ

G

δ

Itot

)

−

δ

G

δ

Itot

∂

x

(

δ

F

δ

Itot

))]

dx

.

(35) By following the procedure similar to the one pursued for check-ing whetherJT is a Hamiltonian operator, it is found that JD

satisfies the Jacobi identity and, hence, is also a Hamiltonian operator. □

4. Geometrical properties of the system

We now define a generalization of symplectic and Poisson structures, namely, Dirac and Stokes–Dirac structures.

Definition 4.1 ([19,27]). ConsiderFandEas real Hilbert spaces which are dual of each other. The subspace D

⊂

F

×

E is a Dirac structure if D

=

D⊥

, where D⊥

denotes the orthogonal complement which is defined as follows:

D⊥

:= {

(

˜

f

, ˜

e)

∈

F

×

E

|

⟨⟨

(

˜

f

, ˜

e)

,

(f

,

e)

⟩⟩ =

0

∀

(f

,

e)

∈

D

}

,

(36) where

⟨⟨

(

˜

f

, ˜

e)

,

(f

,

e)

⟩⟩

is defined as follows:

⟨⟨

(

˜

f

, ˜

e)

,

(f

,

e)

⟩⟩ := ⟨˜

f

|

e

⟩ + ⟨

f

| ˜

e

⟩

,

(37) with

⟨

f

|

e

⟩

indicating a non-degenerate bilinear form defined on the bond spaceB

=

F

×

E.

Definition 4.2 ([19,27,28]). Let F_Ω and E_Ω denote the real Hilbert spaces of flow variables and effort variables defined on the domainΩ, respectively. Also, letF_∂ and E_∂ denote the real Hilbert spaces of flow variables and effort variables defined on the boundary

∂

Ω, respectively. Furthermore, consider the spaces of flow variablesF_S and of effort variablesE_S, whereF_S

=

F_Ω

×

F_∂, andE_S

=

E_Ω

×

E_∂, respectively. Moreover, letE_S be the dual of

FS [27,28]. Finally, we endow the bond spaceBS

=

FS

×

ES with

the pairing:

⟨⟨

(f

,

f∂,e

,

e_∂)

,

(f

˜

, ˜

f∂, ˜e

, ˜

e_∂)

⟩⟩ = ⟨

f

| ˜

e

⟩ + ⟨

e

| ˜

f

⟩ +

⟨

f_∂

| ˜

e_∂

⟩ + ⟨

e_∂

| ˜

f_∂

⟩

.

(38)

Then, the subsetDSofBSis a Stokes–Dirac structure with respect

to the non-degenerate bilinear form (38)if DS

=

D⊥S, where

D_S⊥denotes the orthogonal complement ofDSand is defined like

(36).

Dirac structures relate the composing elements of a system in a power-conserving manner [7]. Such geometric structures often have a compositionality property [7,29,30].

For (f

,

e) element of a Dirac structure, it is easy to see that

⟨

f

|

e

⟩ =

0, and thus there is a close relation to (formally) skew-adjoint operators, see also(13). However, if f

=

Je for all

(f

,

e)

∈

D, andJ is formally skew-adjoint, then D

⊂

D⊥. To make such aDinto a Dirac structure, it is required thatD

=

D⊥

holds. The formally skew-adjoint part of a pH system will form the foundation of the associated Dirac structure, as we will show as well.

In the existing results [19,27,28], the effort variables e belong to the function class H1_{(Ω) since they dealt with constant matrix}

differential operators of first order. Given the state-dependent nature of the (first-order) skew-adjoint operators in(21)and(29), unlike in [19,27,28], a combination of the state and the effort variables has to belong to the function class H1_{(Ω) or suitable}

conditions have to be imposed on the state variables in order to have effort variables belonging to the function class H1_(Ω)

(see Theorems 4.5and 4.6). Boundary port-variables have been parametrized in [19] using the trace operators. However, such an elegant parametrization is limited to the case of a non-singular matrix Q (synonymous to (26)) arising in linear problems with state-independent operators. To the best of our knowledge, [28] is the only work in the scope of parametrization of boundary port-variables for a singular matrix Q , thereby enlarging the class of systems that can be dealt. Villegas in [28] demonstrated the approach to define the non-degenerate bilinear form under sin-gular Q and consequently modified the definition of the boundary port-variables. However, [28] was limited to the setting of

state-independent Stokes–Dirac structures. In this work, we extend

the definition of boundary port-variables to eventually obtain

state-dependent Stokes–Dirac structures with boundary ports for

non-linear problems with non-quadratic Hamiltonian functional. It should be mentioned that the authors in [5] have also consid-ered state-dependent Stokes–Dirac structures for problems (for instance, ideal isentropic fluid) with non-quadratic Hamiltonian functional by using a differential geometric viewpoint. We, how-ever, use the matrix/operator-theoretic viewpoint [19] in the consideration of such geometric structures in the scope of the compressible two-phase flow models.

Remark 4.3. Boundary port-variables, in our setting, will re-main unchanged in the presence of dissipation. This is only true since our resistive operator (R) does not include any differential operator. In general, the boundary ports could also include con-tributions from the resistive part. In this work, we only consider Stokes–Dirac structures without accounting for resistive ports (for the above mentioned reason) and finally arrive at a defi-nition of the boundary port-variables, which is practical for pH representations.

We recall the following fundamental lemma of calculus of variations, which is extensively used in order to prove that our structure is a Stokes–Dirac structure.

Lemma 4.4. If the pair (h

,

m)

∈

L2(Ω)2satisfies

∫

b a

[

h(x)

∂

xf (x)

+

m(x)f (x)

]

dx

=

0

,

(39) for all f

∈

H1 0(Ω), then h

∈

H1(Ω)

,

and

∂

xh

=

m(x)

.

(40)

The proof toLemma 4.4 follows the derivation of Lemma 4 in [31].

Using the above mathematical preliminaries, we first propose a Stokes–Dirac structure for the TFM and present its proof, and then we propose it for the DFM without slip.

4.1. Stokes-Dirac structure representation for the two-fluid model

We, first, introduce the notations

ft

=

[

fmg f_mℓ fIg f_Iℓ f B a,t fbB,t

]

T

,

(41a) et

=

[

emg emℓ eIg eIℓ e B a,t eBb,t

]

T (41b) with ft

∈

Ft, et

∈

Et whereFt

=

Et

=

L2(Ω)4

×

R2

×

R2. On

Ft

×

Etthe following non-degenerate bilinear form is defined:

⟨

ft

|

et

⟩ =

∫

Ω

(fmgemg

+

fmℓemℓ

+

fIgeIg

+

f_Iℓe_Iℓ)dx

+

(f_bB_,t)TeB_b,t

+

(f_aB_,t)TeB_a,t

.

(42) Using these notations, the Stokes–Dirac structure corresponding to the dissipative Hamiltonian representation of the TFM can be expressed as follows.

Theorem 4.5. Let Zt

=

L2(Ω)4 be the state space. Consider

the bond space, a trivial bundle over Z_t:B_t

=

Z_t

×

(F_t

×

E_t),

whereFt and Et are as introduced above. Moreover, assume that

mg

,

mℓ,Ig

,

Iℓ

=:

q1

,

q2

,

q3

,

q4

∈

H1(Ω). We also assume that

q1

,

q2

>

0 onΩ. Then, for any q

∈

Zt,

D_t,q

=

{

(ft

,

et)

∈

Ft

×

Et

|

⎛

⎜

⎝

emg e_mℓ eIg e_Iℓ

⎞

⎟

⎠

∈

H 1_(Ω)4

_,

⎛

⎜

⎝

fmg f_mℓ fIg f_Iℓ

⎞

⎟

⎠

= −

⎡

⎢

⎣

0 0

∂

x(mg

·

) 0 0 0 0

∂

x(mℓ

·

) mg

∂

x(

·

) 0

∂

x(Ig

·

)

+

Ig

∂

x(

·

) 0 0 mℓ∂x(

·

) 0

∂

x(Iℓ

·

)

+

Iℓ∂x(

·

)

⎤

⎥

⎦

⎛

⎜

⎝

emg e_mℓ eIg e_Iℓ

⎞

⎟

⎠ ,

(

fB b,t eB_b,t

)

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

f_b1B_,t fB b2,t eB_b1,t eB b2,t

⎞

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

⎛

⎜

⎜

⎜

⎜

⎜

⎝

mg 0 Ig 0 0 m_ℓ 0 I_ℓ 0 0 1 0 0 0 0 1

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎜

⎜

⎝

emg e_mℓ eIg e_Iℓ

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎠

(b)

,

(

fB a,t eB a,t

)

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

fB a1,t f_a2B_,t eB a1,t eB_a2,t

⎞

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

⎛

⎜

⎜

⎜

⎜

⎜

⎝

mg 0 Ig 0 0 m_ℓ 0 I_ℓ 0 0

−

1 0 0 0 0

−

1

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎜

⎜

⎝

emg e_mℓ eIg e_Iℓ

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎟

⎟

⎟

⎟

⎟

⎠

(a)

}

,

(43)

is a pointwise Stokes–Dirac structure with respect to the symmetric pairing given by

⟨⟨

(ft

,

et)

,

(

˜

ft

, ˜

et)

⟩⟩ = ⟨

ft

| ˜

et

⟩ + ⟨˜

ft

|

et

⟩

,

(ft

,

et)

,

(f

˜

t

, ˜

et)

∈

Ft

×

Et

,

where the pairing

⟨· | ·⟩

is given in(42).

Proof. The proof is divided into two parts. We first prove that D_t,q

⊂

D⊥_t,q.

We consider two pairs of flow and effort variables belonging to the Stokes–Dirac structure, i.e., (ft

,

et)

∈

Dt,qand (

˜

ft

, ˜

et)

∈

Dt,q.

Using the introduced notations, we obtain:

⟨⟨

(ft

,

et)

,

(

˜

ft

, ˜

et)

⟩⟩ =

(44)

∫

Ω (fmge

˜

mg

+

f_mℓe

˜

_mℓ

+

fIge

˜

Ig

+

f_Iℓe

˜

_Iℓ)dx

+

∫

Ω (

˜

fmgemg

+ ˜

fmℓemℓ

+ ˜

fIgeIg

+ ˜

fIℓeIℓ)dx

+

(f_aB_,t)T

˜

eB_a,t

+

(f_bB_,t)Te

˜

B_b,t

+

(f

˜

_aB_,t)TeB_a,t

+

(f

˜

_bB_,t)TeB_b,t

.

Substituting the mappings between the flow and the effort vari-ables, as in(43), it is straightforward to see that Eq.(44)equals zero and soDt,q

⊂

Dt⊥,q. This concludes the first part of the proof.

We now prove the converse part, i.e., D⊥

t,q

⊂

Dt,q. For this,

we follow the steps similar to Proposition 4.1 in [27]. The proof consists of several repeated steps, which are summarized be-low. We take (

˜

ft

, ˜

et)

∈

D⊥t,q i.e., (

˜

ft

, ˜

et)

∈

Ft

×

Et such that

⟨⟨

(ft

,

et)

,

(f

˜

t

, ˜

et)

⟩⟩

=

0

∀

(ft

,

et)

∈

Dt,q. To this end, we use

the freedom in the choice of the effort variables and exploit

Lemma 4.4.

Step 1: Let (ft

,

et)

∈

Dt,q with e_mℓ

,

eIg

,

e_Iℓ

=

0 and emg(a)

=

emg(b)

=

0. Using(44), we find that

∫

Ω

−

(mg

∂

xemg)

˜

eIg

+ ˜

fmgemgdx

=

0

∀

emg

∈

H 1 0(Ω)

.

(45) Lemma 4.4gives mg

˜

eIg

∈

H 1_{(Ω) and f}

˜

mg

= −

∂

x(mg

˜

eIg)

.

(46)

Using mg

∈

H1(Ω) along with mg

>

0 on Ω, we obtain that

˜

eIg

∈

H1(Ω).

Step 2: Considering (ft

,

et)

∈

Dt,q with emg

,

eIg

,

e_Iℓ

=

0 and

e_mℓ

∈

H₀1(Ω), and following the procedure as in the previous step, we obtain:

m_ℓ

˜

e_Iℓ

∈

H1_{(Ω) and f}

˜

mℓ

= −

∂

x(mℓe

˜

Iℓ)

.

(47)

As before, using m_ℓ

∈

H1_{(Ω) along with m}

ℓ

>

0 onΩ, we have

that

˜

e_Iℓ

∈

H1_(Ω).

Step 3: For (ft

,

et)

∈

Dt,qwith emg

,

e_mℓ

,

e_Iℓ

=

0 and eIg

∈

H01(Ω),

we obtain:

∫

Ω

−

(

∂

xmg)(eIge

˜

mg)

−

(

∂

xIg)(eIg

˜

eIg)

−

(

∂

xeIg)

·

(

mge

˜

mg

+

2Ige

˜

Ig

)

+ ˜

fIgeIgdx

=

0

∀

eIg

∈

H 1 0(Ω)

.

As a result ofLemma 4.4, we have that mge

˜

mg

+

2Ige

˜

Ig

∈

H

1_(Ω).

Moreover, we obtain the following identity:

˜

fIg

= −

∂

x(mge

˜

mg

+

2Ige

˜

Ig)

+ ˜

emg

∂

xmg

+ ˜

eIg

∂

xIg

.

(48) Using mg

,

Ig

, ˜

eIg

∈

H 1_{(Ω) and that m} g

>

0, it can easily be deduced that

˜

emg

∈

H

1_{(Ω), and so(48)can be written as}

˜

fIg

= −

mg

∂

x

˜

emg

−

∂

x(Ige

˜

Ig)

−

Ig

∂

x

˜

eIg

.

(49)

Step 4: Considering (ft

,

et)

∈

Dt,qwith emg

,

emℓ

,

eIg

=

0 and eIℓ

∈

H1

0(Ω), and usingLemma 4.4, we have that q2e

˜

_mℓ

+

2q4e

˜

_Iℓ

∈

H1(Ω)

and also obtain:

˜

f_Iℓ

= −

∂

_x(m_ℓe

˜

_mℓ

+

2I_ℓe

˜

_Iℓ)

+ ˜

e_mℓ

∂

xmℓ

+ ˜

eIℓ

∂

xIℓ

.

(50)

Using mℓ,Iℓ, ˜e_Iℓ

∈

H1_{(Ω) and that m}

ℓ

>

0, it can easily be deduced that

˜

e_mℓ

∈

H1_{(Ω) and so}

˜

f_Iℓ

= −

mℓ∂xe

˜

_mℓ

−

∂

x(Iℓ

˜

eIℓ)

−

Iℓ∂xe

˜

_Iℓ

.

(51)

Step 5: Let (ft

,

et)

∈

Dt,q with e_mℓ

=

eIg

=

e_Iℓ

=

0 and

emg(a)

=

0

,

emg(b)

̸=

0. Using the procedure outlined above, we

obtain the following identity:e

˜

B

b1,t

= ˜

eIg

|

b.

Step 6: Let (ft

,

et)

∈

Dt,q with emg

=

eIg

=

e_Iℓ

=

0 and

e_mℓ(a)

=

0

,

e_mℓ(b)

̸=

0. We now observe that

˜

eB

b2,t

= ˜

eIℓ

|

bholds.

Step 7: Let (ft

,

et)

∈

Dt,q with emg

=

e_mℓ

=

e_Iℓ

=

0 and

eIg(a)

=

0

,

eIg(b)

̸=

0. Using the outlined procedure, we now

obtain:

−

(mg

˜

emgeIg)

|

b

−

(Ige

˜

IgeIg)

|

b

+˜

fb1B,teIg

|

b

=

0

.

(52)

Finally, we obtain the following identity:

˜

f_b1B_,t

=

(

mge

˜

mg

+

Ige

˜

Ig

)

|

b

.

(53)

Step 8: Let (ft

,

et)

∈

Dt,q with emg

=

e_mℓ

=

eIg

=

0 and

e_Iℓ(a)

=

0

,

e_Iℓ(b)

̸=

0. Using the outlined procedure, we now obtain the following identity:

˜

f_b2B_,t

=

(

m_ℓe

˜

_mℓ

+

I_ℓe

˜

_Iℓ

)

|

_b

.

(54)

The boundary port-variables fB

a1,t

,

fa2B,t

,

eBa1,t and eBa2,t can be

ob-tained in a manner similar to the one outlined for computing the boundary port-variables at the right boundary of the spatial domainΩ.

Thus, in summary we have shownD_t⊥_,q

⊂

Dt,qand, hence,Dt,q

is a Stokes–Dirac structure. □

The boundary flow and effort variables can be interpreted physically. Ignoring the sign associated to the boundary port variables; the effort variables eB_a1 and eB_a2 (eB_b1 and eB_b2) can be interpreted as gas and liquid volumetric flow rates, respectively, at the left (and right) end of the spatial domain. The flow variables at the left boundary: fB

a1and fa2B, and the corresponding variables

at the right boundary: fB

b1 and fb2B have physical dimensions of

energy per unit mass.

In the absence of dissipation, the behaviour of the Hamil-tonian, H_T, along the solutions of the mathematical model, is governed by the following power balance equation:

dHT dt

=

(

eIg(mgemg

+

IgeIg)

+

e_Iℓ(mℓe_mℓ

+

Iℓe_Iℓ)

)

|

_x=a

−

(

eIg(mgemg

+

IgeIg)

+

e_Iℓ(mℓe_mℓ

+

Iℓe_Iℓ)

)

|

_x=b

,

(55)

which can be obtained by using the definition of the port-Hamiltonian system with respect to the Stokes–Dirac structure and(42). Furthermore, if e1_{and e}2 _{in(26)are as follows: e}1

₌

e2

₌

_e

t

:= [

emg e_mℓ eIg e_Iℓ

]

T, then it is straightforward to see that

the right-hand side of(55)is equivalent to

−

1₂

(

eT tQet

)

, where Q is the matrix introduced in(26).

We now discuss the representation of the Stokes–Dirac struc-ture corresponding to the skew-adjoint operatorJ_Din the Drift Flux Model without slip.

4.2. Stokes-Dirac structure representation for the drift flux model

We introduce the notations

fd

=

[

fmg,d f_mℓ,d fItot,d faB,d f B b,d

]

T

,

(56a) ed

=

[

emg,d emℓ,d eItot,d e B a,d eBb,d

]

T

.

(56b)

A Stokes–Dirac structure for the dissipative Hamiltonian repre-sentation of the DFM can be expressed as follows.

Theorem 4.6. LetZd

=

L2(Ω)3be the state space. Consider the

bond space, a trivial bundle overZd:Bd

=

Zd

×

(Fd

×

Ed), where

Fd

=

Ed

=

L2(Ω)3

×

R2. We assume that mg, mℓ

∈

H1(Ω). The

non-degenerate bilinear form onFd

×

Edis defined in the following

way: