Dilip Das

Department of Mathematics, Shibpur Dinobundhoo Institution (College), Shibpur, Howrah-711102, West Bengal, India

1 Introduction1

The motion of a body of any geometrical configuration,floating on the surface of water, is investigated in the literature assuming linearized theory of water waves. The problem of heaving motion of a long, horizontal circular cylinder on the surface of water was investigated by Ursell(1949) using the method of multipole expansion of the time-harmonic stream function. The corresponding velocity potential also has a similar expansion. In fact, for an infinitely long horizontal cylinder of arbitrary cross section floating on the surface of water, the potential function in general can be expressed in terms of a regular wave, a wave source, a dipole and wave-free potentials (Ursell, 1968;Athanassonlis, 1984). The wave-free potentials are singular at some point and tend to zero rapidly at infinity. Obviously these satisfy the free-surface condition. Two and three dimensional problem of multipole expansions in the theory of surface waves in infinite deep water and also in water of uniform finite depth water has been given by Thorne (1953).Expansions in terms of the wave source and an infinite set of wave-free potentials were introduced for the threedimensional problem involving a floating sphere halfimmersed and making periodic heaving oscillations by Havelock (1955). Ursell (1961a; 1961b), Bolton and Ursell(1973), Mandal and Goswami (1984) considered problems where the potential functions is expansion in terms of wave sources and wave-free potentials. Taylor and Hu (1991)described expansion of the velocity potential for two and three dimensional wave diffraction and radiation problems.Linton and McIver (2001) briefly described the construction of wave free potentials in the case of water of infinite and finite depth water with a free surface.

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems (BVP) associated with the Laplace equation (or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the method multipole expansion. In most of the wave-structure interaction problems, the governing equation is either the Laplace or the Helmholtz equation and, thus,the features of the orthogonal relation mainly depend upon the nature of the bottom and upper surface boundary conditions. Higher-order boundary conditions occur frequently in fluid-structure interaction problems when we deal with very large floating structures (VLFS). Evans and Porter (2003) analysed the oblique wave scattering caused by a narrow crack in ice sheets floating on water of finite depth with the eigenfunction expansion method. Chakrabarti(2000) analysed the problem of scattering of surface water waves by the edge of an ice cover and obtained the explicit solution with a singular, Carleman-type integral equation.Das and Mandal (2006; 2009) analysed the problem of water wave scattring by a circular cylinder with an ice-cover. Das and Mandal (2008) also studied the water wave radiation by a sphere submerged in water with an ice-cover. Das and Mandal (2010a), Mandal and Das (2010) presented in a systematic manner the construction of wave free potentials for two- dimensional deep water and finite depth water with free surface, or corresponding the effect of the surface tension at the free surface and also in water with an ice-cover. Dhillon and Mandal (2013) also presented the construction of a wave free potential for three dimensional deep water as well as finite depth water.

The multipoles and wave-free potentials have been described for single layer fluids, both in two and three dimensions and for infinite as well as finite depth water.More recently, however, interest has been extended to bodies which are floating, submerged or partially immersed in two-layer fluids, each fluid having a different density.The wave motion in a two-layer fluid has gained importance due to plans to construct under water pipe bridge across the Norwegian fjords. A fjord consists of a layer of fresh water on the top of a deep layer of salt water. For tow-dimension motions, Kassem (1982) presented the multipole expansions for two superposed fluids, each of finite depth. Linton and McIver (1995), Linton and Cadby (2002) have constructed a set of multipoles in the theory of surface waves in two-layer fluid with free surface for normal and oblique incident wave trains. Three-dimensional multipoles in two-layer fluid with free surface have been given by Cadby and Linton (2000).Das and Mandal (2007; 2010b) constructed the multipole potentials in their problems for two-dimension and three-dimension in two-layer fluid with an ice-cover.

In the literature, two-layer fluid water wave problems have not been studied for wave-free potential function construction. However, for the various classes of water wave problems in two-layer fluid many researchers may use the wave-free potentials in the mathematical analysis. For circular cylinder of arbitrary cross section floating on the surface of a two-layer fluid or half-immersed circular cylinder in a two-layer fluid, the potential function in general can be expressed in terms of a regular wave,wave-free potentialetc. Also in winter, a fjord is covered by a layer of ice, so that we have a cylindrical pipe bridge submerged below an ice-cover. Now if we consider the problem of partially or half-immersed circular cylinder in a two-layer fluid with an ice-cover, then potential functions may be expressed in terms of the regular potential as well as wave-free potential function. Thus it will study the problem of construction of wave-free potential in a two-layer fluid.Also in these problems (both single layer and two-layer) the higher-order boundary condition involves third order partial derivative (surface tension) and fifth order partial derivation(ice-cover). However, the boundary value problem involving higher-order boundary conditions more than fifth order partial derivative (Manamet al., 2006; Daset al., 2008)have not been extensively studied with a view to establish the multipole potentials and also wave-free potentials.

In this paper construction of wave-free potentials and multipoles are presented in a systematic manner. The cases of two-dimensional non-oblique and oblique waves in two-layer fluid with free surface condition with higher order partial derivative are considered. Also the cases of three-dimensional waves in two-layer fluid with free surface with higher order partial derivative are considered. When the higher order partial derivative reduces to first order (free surface) or fifth order partial derivative (ice-cover),multipoles exactly coincide with the multipoles for two-layer fluid with free surface (Linton and McIver, 1995;Linton and Cadby, 2002; Cadby and Linton, 2000), or for two-layer fluid with ice-cover (Das and Mandal, 2007;2010b).

2 Multipoles and wave-free potentials for non-oblique waves

In a two-layer fluid, both the upper and lower fluids are assumed to be homogeneous, incompressible and inviscid.LetρIbe the density of the upper fluid andρII(>gt;ρI) be the same for the lower fluid. Let the lower fluid extend infinitely downwards while the upper one has a finite heighthabove the mean interface. Lety-axis points vertically upwards from the undisturbed interfacey=0. Thus the upper layer occupies the region 0>lt;y>lt;hwhile the lower layer occupies the regiony>lt;0. Under the usual assumption of linear theory and irrotational two-dimensional motion,velocity potentialsbeing angular velocity, describing the fluid motion in the upper and lower layers exist. For a general BVP,I,IIjsatisfy

On the upper surface having the mean positiony=h,Ijsatisfies the free-surface condition with higher-order derivatives of the form (Landau and Lifshitz, 1959):

If the free-surface has an ice-cover modelled as a thin elastic plate, where,ρ0is the density of ice,ρis density of water,h0is the small thickness of ice-cover,E,vare the Young’s modulus and Poission’s ratio of the ice andbeing the acceleration due to gravity. A generalization of (2) for more higher-order derivatives has been introduced by Manamet al. (2006) and has the form:

wherecm(m=0, 1,…,m0) are known constants. Keeping in mind various physical problems involving fluid structure interaction, only the even order partial derivatives inxare considered in the differential operator.

The linearised boundary conditions at the interfacey=0 are

2.1 Singularities in the lower layer

We first consider solutions of Laplace equation in two dimensions (x,y) which are singular at (0,f>lt;0). Polar co-ordinates (r,θ) are defined in the (x,y)-plane by

Now for the case of normal incidence, the solutions of Laplace’s equation singular aty=f>lt;0 areand, and these have the integral representations(Thorne, 1953)

Also they represent outgoing waves as

The mutipoles are constructed as Linton and McIver(1995)

whereA(k),B(k),C(k) are functions ofkto be found such that the integrals exist in some sense and satisfy the generalized boundary condition (4) and the interface conditions (5) and (6) and are of outgoing nature at infinity.All the conditions are satisfied if we chooseA(k),B(k) andC(k) as

whereH(k) is given by

The path of the integration in the integrals in (14) to (17)is indented below the poles atk=k1andk=k2on the realk-axis to take care of their outgoing behaviour asandk1,k2are only two real positive roots of the equationH(k)=0 (Daset al., 2008).

The far-field forms of the multipoles, in the lower fluid, is given by

where

Using (22) and (23), we find

and

Lettingf→0 in (27) and (28) we obtain the symmetric and antisymmetric wave-free potentials with singularity near the interface between two-layer and are given by

and

whereC*(k) is the limiting value ofC(k) whenf→0.

2.2 Singularities in the upper layer

To develop multipoles singular aty=f>gt;0 and polar co-ordinates are again defined via (9). The solutions of Laplace’s equation singular aty=f>gt;0 areandand letanddenote the symmetric and antisymmetric multipoles satisfying (1), (2)except at (0,f) with boundary conditions (4) to (7) and

Also they represent outgoing waves as

The mutipoles are constructed as (Linton and McIver,1995)

where

The path of the integration in the integrals in (33) to (36)is indented below the poles atk=k1andk=k2on the realk-axis to take care of their outgoing behaviour as.

The far-field forms of the multipoles, in the upper fluid, is given by

where

where

Using (40) and (41), we find

Now using the representations (44) and (45) it can be shown that

and

where

The last contour integrals in (46) and (47) can be written in the form:

and

the integrals being in the sense of Cauchy principal value.

Thus (46) and (47) reduce to

and

the last integrals of (50) and (51) being in the sense of Cauchy principal value. Lettingf→hin (50) and (51) we obtain the symmetric and antisymmetric wave-free potentials with singularity in the upper surface of the two-layer and are given by

and

In particular, choosec0=1,ci=0,i=1, 2,…,m0, the BVP becomes the BVP for two-layer fluid with free surface(Linton and McIver, 1995) and the multipoles exactly coincide with those for the case of two-layer fluid with free surface (Linton and McIver, 1995) and the wave-free potentials become the wave-free potentials for two-layer fluid with free surface. Similarly, if choosec0=1?εK,c1=0,c2=D,ci=0,i=3, 4,…,m0, then the BVP becomes the BVP for two layer fluid with ice-cover boundary condition (3)(Das and Mandal, 2007) and obtain the corresponding multipoles (Das and Mandal, 2007) and wave-free potentials.If we letc0=1,ci=0,i=1, 2,…,m0andρ→0 in this problem then it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles evaluated by Thorne (1953) and wave-free potential evaluated by Das and Mandal (2010a). Thus by lettingρ→0 in the above analysis we recover the results for the single layer fluid.

3 Multipoles and wave-free potentials for oblique waves

Under the usual assumption of linear theory and irrotational two-dimensional motion, velocity potentialsis the wave number component along thez-direction, describing the fluid motion in the upper and lower layers exist. For a general BVP,satisfy

Eqs. (6) and (7) represent the linearized boundary conditions at the interfacey=0, while the free-surface condition with higher-order derivatives aty=his

where the differential operator

In (57),cm(m=0, 1,…,m0) are known constants. Keeping in mind various physical problems involving fluid structure interaction, only the even order partial derivatives inxare considered in the differential operator M and (8) is the bottom condition of the lower layer.

3.1 Singularities in the lower layer

whereKn(z) denotes the modified Bessel function of second kind.

The multipoles are constructed as (Linton and Cadby,2002)

The path of the integration in the integrals in (60) to (63)is indented below the poles at, where

The far-field forms of the multipoles, in the lower fluid, is given by

where

asx, whereCm1andCm2are given by

Using (70) and (71), we find

Thus

and

C(k)evysinh nksin(γ xsinhk)dk

Lettingf→0 in (75) and (76) we obtain the symmetric and antisymmetric wave-free potentials with singularity near the interface between two layers and are given by

C *( k)evycosh mkcos(γ xsinhk)dkand

C *(k )evysinhmksin(γ xsinhk)dk

3.2 Singularities in the upper layer

To develop multipoles singular aty=f>gt;0 and polar co-ordinates are again defined via (9). The solutions of Helmholtz equation singular aty=f>gt;0 areandLetdenote the symmetric and antisymmetric multipoles satisfying (54) and(55) for upper and lower fluid except at (0,f) with boundary conditions (56) and (6) to (8) and represent an outgoing waves at infinity. Near the point (0,f), the behaviours ofare given by

Also they represent outgoing waves as

The mutipoles are constructed as (Linton and Cadby,2002)

where

where the contour is indented below the polesk=μ1andk=μ2in the complexk-plane.

The far-field forms of the multipoles, in the upper fluid, is given by

where

asx? ±￥. Hereare the residues ofandrespectively atk=m1andk=m2, given in (42) and (43).

Using (88) and (89), we find

and

Now using the representations (90) and (91) it can be shown that

and

The last contour integrals in (92) and (93) can be written in the form

the integrals being in the sense of Cauchy principal value.

After substituting (94), (95) in (92), (93) respectively and lettingf→hin (92) and (93) we obtain the symmetric and antisymmetric wave-free potentials with singularity in the upper surface of the two-layer and are given by

and

the last integrals in (96) and (97) being in the sense of Cauchy principal value.

In particular, choosec0=1,ci=0,i=1, 2,…,m0, the BVP becomes the BVP for two-layer fluid with free surface(Linton and Cadby, 2002) and the multipoles exactly coincide with those for the case of two-layer fluid with free surface (Linton and Cadby, 2002) and the wave-free potentials become the wave-free potentials for two-layer fluid with free surface. Similarly, if choosec0=1?εK,c1=0,c2=D,ci=0,i=3, 4,…,m0, then the BVP becomes the BVP for two layer fluid with ice-cover boundary condition (3)(Das and Mandal, 2007) and obtain the corresponding multipoles (Das and Mandal, 2007) and wave-free potentials.If we letc0=1,ci=0,i=1, 2,…,m0andρ→0 in this problem then it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles evaluated by Thorne (1953) and wave-free potential evaluated by Das and Mandal (2010a). Thus by lettingρ→0 in the above analysis we recover the results for the single layer fluid.

4 Three dimensional multipoles and wave-free potentials

Here the velocity potential

describing the fluid motion exists whereφ(x,y,z) is a complex valued function andwis the angular frequency.Let the potential in the upper layer be φIm and that in the lower layer beIImj(m=0, 1, the potential functions for the heave and sway problems being denoted by φ0 and φ1 ,respectively). The potential functions satisfy the Laplace’s Eqs. (1), (2) with

and equations (6) and (7) represent the linearized boundary conditions at the interfacey=0, while the free-surface condition with higher-order derivatives aty=his

where the differential operator

wherecm(m=0, 1,…,m0) are known constants. Keeping in mind various physical problems involving fluid structure interaction, only the even order partial derivatives inxare considered in the differential operator N and (8) is the bottom condition of the lower layer.

4.1 Singularities in the lower layer

The velocity potential singular at (0,f, 0) describing the motion in the lower fluid, then multipole singular potential functions are solutions of the Laplace’s equation which are singular atr=0, satisfy the boundary conditions (6) to (8)and (98). These can be constructed using the method given by Thorne (1953). A solution of Laplace’s equation in the spherical polar co-ordinate system (r,θ,α) and singular atr=0 iswhereare associated Legendre functions. This has the integral representation, valid fory>gt;f(Thorne, 1953)

whereJmare Bessel functions and. Let the multipole potentials(in the notation of Cadby and Linton (2000),m=0, 1) be the singular solutions of the Laplace’s equation and satisfy the free surface boundary condition with higher order derivatives (98), the interface conditions (6) and (7) and behave as outgoing waves asR→∞. Thenare obtained as

where

The path of integration in the integrals in (101), (102) is chosen to be indented below the poles atk=k1andk=k2. It can be shown that asR→∞, only the contributions to the integrals from these indentations, to the potential functionsj, prevail and behave as waves having outgoing nature.

The far-field forms of the multipoles, in the lower layer, is given by

asR→∞, where

Ck1andCk2being the residues ofC(k) atk=k1andk=k2respectively, which are given by

Using (106), we find

Now using the representations (108), it can be shown that

This is the wave-free potential having singularity at (0,f,0). Lettingf→0 in (109) it is obtained the wave-free potentials having singularity near the interface between two-layer and is given by

4.2 Singularities in the upper layer

To develop multipoles singular aty=f>gt;0. The solution of Laplace’s equation singular at y=f>gt;0 is. Letis denote the multipoles satisfying (1), (2) except at (0,f, 0) with boundary conditions (6), (7) and (98). The mutipoles are constructed as (Cadby and Linton, 2000)

where

The path of the integration in the integrals in (111) and(112) is indented below the poles atk=k1andk=k2on the realk-axis.

The far-field forms of the multipoles, in the upper fluid, is given by

where

AsR→∞. HereA1(k1),A1(k2) andB1(k1),B1(k2) are the residues ofA1(k) andB1(k) respectively atk=k1andk=k2,given by

Using (116), it is found that

Now using the representation (119) it can be shown that

The last contour integrals in (120) can be written in the form:

the integral being in the sense of Cauchy principal value.

Thus (120) reduces to

This is the wave-free potential having singularity at (0,f,0). Lettingf→hin (122) it is obtained the wave-free potentials with singularity in the upper surface of the two-layer and are given by

In particular, choosec0=1,ci=0,i=1, 2,…,m0, the BVP becomes the BVP for two-layer fluid with free surface(Cadby and Linton, 2000) and the multipoles exactly coincide with those for the case of two-layer fluid with free surface (Cadby and Linton, 2000) and the wave-free potentials become the wave-free potentials for two-layer fluid with free surface. Similarly, if choosec0=1?εK,c1=0,c2=D,ci=0,i=3, 4,…,m0, then the BVP becomes the BVP for two layer fluid with ice-cover boundary condition (3)(Das and Mandal, 2010b) and obtain the corresponding multipoles (Das and Mandal, 2010b) and wave-free potentials. If we letc0=1,ci=0,i=1, 2,…,m0andρ→0 in this problem then it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles evaluated by Thorne (1953) and wave-free potential evaluated by Dhillon and Mandal (2013). Thus by lettingρ→0 in the above analysis we recover the results for the single layer fluid.

5 Conclusions

Wave-free potentials and multipoles in two-layer fluid with a free surface condition with higher order derivatives for non-oblique and oblique waves (two dimensions) and also three dimension are constructed in a symmetric manner.Appropriate modifications of the wave-free potentials can be made in the circumstances when the two-layer fluid are of uniformly finite depth for both the layers having a free surface conditions with higher order derivatives. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.

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