The roots of Eq. (4) are chosen so that

1.2 The Green’s functions and its multipoles

We define now a Cartesian (,,)x y z coordinate system fixed on the centre of the solid. The x-axis is the longitudinal axis of the ellipsoid while the y-axis is pointing toward and normally to the wall. The Green’s function that satisfies the Laplace equation and the zero-velocity condition on the wall is

where d is the distance between the centre of the ellipsoid and the rigid wall. Accordingly, the multipoles of the Green’s function are obtained by using Miloh’s theorem on even exterior ellipsoidal harmonics[12]. It should be noted that only the even harmonics are retained as the problem is symmetric in z and hence only the Lamé functions which are even in z must be considered. The Lamé functions which are even in z are those of classes Kand L. Thus we have

where 0S is the area of the focal ellipse (4). Using the Miloh’s theorem[12], Eq. (7) yields

where denotes the multipoles of the Green’s function, , are the Lamé functions of the first and the second kind, of order m and degree n,while denotes the source distribution over the fundamental ellipse (4), given by[12]

where

The integral in Eq. (8) will be denoted by(λ, μ ,ν) . Taking the Fourier Transform of the last component we obtain

Given that the extremes of y in the fundamental ellipse are ±a2, the above formula is valid for y＜2d -a2

1.3 Expansion of regular terms in ellipsoidal harmonics

Let us further assume that the exponential term exp[τy+iτ( x c osφ + z sin φ )] is expanded using interior ellipsoidal harmonics according to

where (,)τ φ are coefficients to be determined. To obtain those coefficients, Eq. (11) is evaluated on the surface of the ellipsoid, defined at 1=aλ which in turn allows us to exploit the fundamental relation of orthogonality[3]. Subsequently, the coefficients sought will be given by

where are the normalization (orthogonality)constants and the integration is taken over the surface of the ellipsoid Sa, while ( x0 , y0, z0 ) denote the Cartesian representation of that surface evaluated by Eq. (2) using λ=a1.

Introducing Eqs. (12), (13) into the regular terms of Eq. (10) the latter can be written as

and the coefficients are obtained by

where we let d S 0 =dξ d η and ( x0 , y0, z0 ) are understood as functions of (μ ,ν).

The Green’s function multipoles are now expressed by

The coefficients can be relatively simplified by calculating two of the involved integrals. One may use the following formulae

to yield

1.4 The velocity potential

The ellipsoid was assumed to translate steadily(along x) close to a wall. Hence, the velocity potential will be composed by the equivalent uniform stream and the disturbance potential that is constructed in ellipsoidal harmonics using the multipole potentials of Eq. (16). In particular, it holds that

The unknown expansion coefficients α nm are to be determined. We note that these are dimensional and are expressed in length units. Again, the velocity potential must satisfy the Neumann condition on the surface of the ellipsoid, and in other words

W 1e 1n oμtνe =t hEa t( μ1 )1E (ν ) Thuafter introducing Eq. (20) into the boundary condition of Eq. (21)and making use of the orthogonality relation[3], the following linear system is derived

where δns is the Kronecker’s delta function. Eq. (22)is to be solved in terms of the unknown expansion coefficients . Having calculated the latter, the total velocity potential is immediately derived through Eq. (20). In fact, the derivation of completes the solution of the problem. Clearly, the linear system (22)must be truncated to a sufficient maximum degree to be solved using standard matrix techniques.

1.5 The attraction force

“Attraction force” is designated the force that is applied normal to the direction of motion, which thrusts the body toward the rigid wall. The disturbance here, has a more complicated structure than the usual flow lines that occur when the body moves steadily(or is subjected to a uniform stream) in an unbounded mass of liquid. The attraction force, coined yF, is convenient to be calculated by the steady Lagally theorem[8-10], namely

where σ(x, y) is the source distribution on the fundamental ellipse, while the integration is performed over the area S0 of that ellipse. In addition, φ~(x, y,0) denotes the regular term of the disturbance potential evaluated explicitly on the fundamental ellipse letting z = 0 The source distribution is obtained by

while the regular term of the disturbance potential is obtained using Eq. (6) as

where denotes the second term of the right-hand side of Eq. (6). Assuming finally unit velocity U = -1, the normalized attraction force will be given by

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