Flow structure beneath periodic waves with constant vorticity under normal electric ﬁelds M. V. Flamarion1 T. Gao2 R. Ribeiro-Jr3 A. Doak4

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Flow structure beneath periodic waves with constant vorticity

under normal electric ﬁelds

M. V. Flamarion1, T. Gao2, R. Ribeiro-Jr3& A. Doak4

1Unidade Acadêmica do Cabo de Santo Agostinho, UFRPE/Rural Federal University of Pernambuco, BR 101 Sul, Cabo de Santo

Agostinho-PE, Brazil, 54503-900

marcelo.ﬂamarion@ufrpe.br

2School of Computing and Mathematical Sciences, University of Greenwich, London SE10 9LS, UK.

t.gao@gre.ac.uk

3UFPR/Federal University of Paraná, Departamento de Matemática, Centro Politécnico, Jardim das Américas, Caixa Postal

19081, Curitiba, PR, 81531-980, Brazil

robertoribeiro@ufpr.br

4Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.

add49@bath.ac.uk

Abstract

Waves with constant vorticity and electrohydrodynamics ﬂows are two topics in ﬂuid dynamics that have

attracted much attention from scientists for both the mathematical challenge and their industrial applica-

tions. The coupling of electric ﬁelds and vorticity is of signiﬁcant research interest. In this paper, we study

the ﬂow structure of steady periodic travelling waves with constant vorticity on a dielectric ﬂuid under the

eﬀect of normal electric ﬁelds. Through the conformal mapping technique combined with pseudo-spectral

numerical methods, we develop an approach that allows us to conclude that the ﬂow can have zero, two or

three stagnation points according to variations in the voltage potential. We describe in detail the recircu-

lation zones that emerge together with the stagnation points. Besides, we show that the number of local

maxima of the pressure on the bottom boundary is intrinsically connected to the saddle points.

1 Introduction

The problem of surface water waves dates back to Stokes [40], who ﬁrst considered surface gravity waves in

deep water with a very large lengthscale in kilometres. The research on this subject has been extended in many

directions. In particular, capillary waves, in which the surface tension is dominant with a typical wavelength

in millimetres, were studied ﬁrst by Crapper [7] where an exact solution was derived, and then by Kinnersley

[28] who generalised the results to the ﬂuid sheet problem. When gravity and surface tension are equally

important, it leads to a much more mathematical challenging problem named capillary-gravity wave, with a

small lengthscale about a centimetre, which plays a vital role in transferring energy momentum and material

ﬂuxes across the ocean surface [4, 29]. A good monograph can be found in [44] for a review.

The aforementioned works assumed that the ﬂuid is inviscid and the ﬂow is incompressible and irrotational.

In practice, a theory with non-zero vorticity is often more physically realistic. The most common choice on the

vorticity is a constant value. A number of works have been achieved under such assumption for gravity waves.

For example, Constantin and Strauss[5] conducted a rigorous analytical study. Thomas et al. [43] investigated

the modulational instability of such waves in the framework of a nonlinear Schrödinger equation, and recently a

more comprehensive work of stability analysis was achieved by Francius and Kharif [13]. Numerically speaking,

a boundary integral equation method was employed to compute fully nonlinear travelling-wave solutions with

constant vorticity in this context by many authors [39, 42, 45, 46, 47]. In [2], time-dependent simulations

were conducted by a hodograph transformation method, ﬁrst pioneered by Dyachenko et al. [11], to examine

the modulational instabilities numerically. In the presence of surface tension, i.e. capillary-gravity waves with

constant vorticity, there have been fruitful achievements by diﬀerent authors. For example, steady-state solutions

were computed by Kang and Vanden-Broeck [26] via a boundary integral method. A Nonlinear Schrödinger

Equation was derived in [25] to study the modulational instabilities. Time-dependent wave interactions in this

context were investigated numerically in [15] in which various travelling-wave solutions were also computed.

All the studies in the last paragraph were interested solely in the shape of the free-surface, and integral

properties such as wave energy and momentum. More recent studies have considered the ﬂow structure beneath

the waves [37, 36]. These structures are particularly interesting in the case of waves with constant vorticity. Of

arXiv:2210.10825v1 [physics.flu-dyn] 19 Oct 2022

note, stagnation points beneath the surface – points within the bulk of the ﬂuid that travels with the same speed

as the wave. Such conﬁgurations can be also observed in waves with a point vortex [22, 38, 8, 48]. Another

peculiarity of ﬂows with constant vorticity is the emergence of pressure anomalies, such as the maxima and

minima of pressure on the bottom boundary being attained at locations other than beneath the crests and

troughs of the free surface respectively.

In some practical situations is of interest to couple the ﬂuid motion with electric ﬁelds. This subject of study

is known as Electrohydrodynamics (EHD) and it can be applied widely in chemical engineering such as cooling

systems in conducting pumps [16], coating process [21] and etc. A good review paper was recently published

by Papageorgiou [31]. Beginning with the early work and experiments of Taylor and McEwan [41], researchers

have been interested in the eﬀect of electric ﬁelds on ﬂuids with varying electrical properties. In that particular

paper, normal electric ﬁelds were shown to be capable to destabilise the interface between two ﬂuids. It was

quickly followed by work on tangential electric ﬁelds by Melcher and Schwarz [30], where, on the contrary, the

interfacial waves can be stabilised due to such eﬀects. Kelvin-Helmholtz and Rayleigh-Taylor instabilities were

shown to be controlled and suppressed by tangential electric ﬁelds in [51] and [1, 3] respectively. Besides many

works were achieved theoretically by multi-scale technique, e.g. [12, 23, 20, 24, 35, 32, 27] in which weakly

nonlinear model equations were derived for the two-dimensional problem. The readers are referred to the work

by Wang [50] for a comprehensive review. Travelling waves were computed by a boundary integral method, e.g.

[33, 34, 10, 9], for diﬀerent conﬁgurations. Of note, in [27, 9], the electric ﬁelds were shown to be capable of

changing the type of the associated nonlinear Schrödinger equation from focusing to defocusing such that dark

solitary waves bifurcate at the phase speed minimum. Nonlinear wave interactions in this context were studied

theoretically and numerically by Gao et al. [19]. Unsteady simulations were performed based on the time-

dependent conformal mapping technique in [14, 18] for a special case of capillary-gravity waves on a dielectric

ﬂuid of ﬁnite and inﬁnite depth respectively covered above by a conducting gas layer. More recently, Gao et al.

[17] followed to investigate the singularity formation of the capillary-gravity wave due to the eﬀect of normal

electric ﬁelds.

In this work, we consider a dielectric ﬂuid in a two-dimensional space bounded below by an electrode, and

above by a conducting passive gas, which in turn is bounded above by another electrode. A potential diﬀerence

is applied between the electrode such that the electric ﬁeld is imposed in the direction perpendicular to the

undisturbed ﬂuid surface. This conﬁguration is well known for destabilising the interface between the ﬂuids. The

ﬂuid is assumed to be under the eﬀect of constant vorticity. Our aim is to study numerically the ﬂow structure

beneath waves with constant vorticity under the inﬂuence of normal electric ﬁelds. More speciﬁcally, we make

use of the conformal mapping technique and pseudo-spectral numerical methods to compute the locations of

stagnation points and the pressure. The rest of this paper is structured as follows. The mathematical formulation

is given in section 2. The linear theory is studied in section 3. The numerical scheme is presented in section 4.

Section 5 is devoted to the results on the full nonlinear governing equations. Finally, concluding remarks are

given in section 6.

2 Mathematical Formulation

An inviscid and incompressible dielectric ﬂuid with density ρof mean depth hand electric permittivity 1is

considered in a two-dimensional space. It is bounded below by a wall electrode and surrounded by a region

occupied by perfectly conducting gas and enclosed by another wall electrode atop. In the electrostatic limit

of Maxwell’s equations, the induced magnetic ﬁelds are negligible so the electric ﬁeld is irrotational due to

Faraday’s law. A potential function Vof the electric ﬁelds ~

Ecan be then introduced such that ~

E=∇V. It

follows that Vsatisﬁes the Laplace Equation in the dielectric ﬂuid layer. The potential diﬀerence between the

two electrodes is denoted by V0. The voltage potential is invariant in the conducting gas layer. Without losing

generality, we set V=−V0on the bottom boundary and V= 0 in the gas layer. A schematic of the problem is

presented in Figure 1 . The gravitational acceleration and the surface tension coeﬃcient are denoted by gand

T, respectively. A Cartesian x−ycoordinate system is introduced with the gravity pointing in the negative

y-direction and y= 0 being the undisturbed free-surface. It follows that the bottom boundary of the ﬂuid is

at y=−h, and the upper boundary is free to move and denoted by η(x, t). We consider a periodic wavetrain

propagating at a constant speed in the positive x-direction. Also, we let x= 0 be at a wave crest such that the

wave proﬁle is symmetric by the y-axis.

The velocity ﬁeld is assumed to be an irrotational perturbation of a shear ﬂow, namely

U= ˆu+~u, (1)

where the vorticity is constant that equals γ, and ˆu= (−γ(y+b),0) with bbeing another constant. The velocity

ﬁeld ~u is irrotational so that there exists a potential function φsatisfying ~u =∇φ. We seek steady waves of

wavelength λin a frame of reference moving with the wave speed c. This is achieved by a change of variables:

wall electrode

y=−h

wall electrode

y= 0

V= 0

V=−V0

V= 0

∇2φ= 0,∇2V= 0.

Dielectric ﬂuid

Permittivity 1

Conducting gas

y=η(x, t)

Figure 1: Schematic of the problem.

X=x−ct,Y=y. It immediately follows that η(x, t) = η(X). The ﬂuid motion can be characterised by the

potential function φand the voltage potential Vas follows

∇2φ= 0 in −1< Y < η(X),(2)

∇2V= 0 in −1< Y < η(X),(3)

−cηX+ (φX−γ(η+b)) ηX=φYfor Y=η(X),(4)

φY= 0 for Y=−1,(5)

V= 0 for Y=η(X),(6)

V=−1for Y=−1,(7)

where we have chosen h,ph/g and V0as the reference length, time and voltage potential, respectively. Fur-

thermore, the continuity of pressure on the free-surface yields the dynamic boundary condition, which can be

written as

−cφX+1

2(φ2

X+φ2

Y) + η−γ(η+b)φX+γψ −τηXX

(1 + η2

X)3/2+Eb

2|∇V|2=B. (8)

Here, ψis the harmonic conjugate of φ. Parameter τand Ebare the non-dimensional Bond and Electric Bond

numbers respectively, given by

τ=T

ρgh2, Eb=1V2

ρgh3.(9)

The hydrodynamic pressure in ﬂuid body is computed by the Bernoulli equation

p=−−cφX+1

2(φ2

X+φ2

Y) + Y−γ(Y+b)φX+γψ −B,(10)

and the stream function can be written as

ψs(X, Y ) := ψ(X, Y )−γY 1

2Y+b−cY. (11)

3 Linear theory

In this section, a linear theory is developed from the governing equations. A trivial solution takes the form of

φ0(X, Y )=0, V0(X, Y ) = Y, η0(X)=0,(12)

which is perturbed by a small disturbance, namely











η(X) = ˆη,

φ(X, Y ) = ˆ

φ,

V(X, Y ) = Y+ˆ

V ,

(13)

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FlowstructurebeneathperiodicwaveswithconstantvorticityundernormalelectriceldsM.V.Flamarion1,T.Gao2,R.Ribeiro-Jr3&A.Doak41UnidadeAcadêmicadoCabodeSantoAgostinho,UFRPE/RuralFederalUniversityofPernambuco,BR101Sul,CabodeSantoAgostinho-PE,Brazil,54503-900marcelo.amarion@ufrpe.br2SchoolofComputingandMat...

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Flow structure beneath periodic waves with constant vorticity under normal electric ﬁelds M. V. Flamarion1 T. Gao2 R. Ribeiro-Jr3 A. Doak4

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