2.1 Notation and hypothesis
Let us now introduce some notation following paper [19], where the existence and uniqueness for the solution of similar problem are treated. We denote by the domain occupied by the solid in a reference configuration. We assume that is an open connected set with boundary, and we choose a system of coordinates with the origin at the mass center of . For the deformable solid, we suppose that the motion is given by a smooth mapping
which satisfies
(1)
where for every , is the trajectory of the mass center, and represents the angle giving the orientation of the solid. denotes the matrix associated to the rotation of angle θ. In (1), denotes an appropriate smooth mapping, representing the undulatory deformation of the creature. The rigid solid is obtained by considering the map
where I denotes the identity map.
Throughout this paper, the deformable body will be called the creature or sometimes just the body in particular, when considering the rigid body case.
In the remaining part of this work, the functions ξ, θ are unknowns to be determined from the governing equations below, whereas the undulatory motion will be supposed to be known and to satisfy several assumptions as in [19], which will be recalled in the sequel.
(H1) For every , the mapping is a diffeomorphism from onto , where . Moreover, for every , the mapping is of class and .
For every we denote by the inverse of , i.e., the diffeomorphism satisfying
for every , and .
(H2) The total volume of the creature is preserved, i.e.,
(2)
Denote by the undulatory velocity of creature, written as a vector field on , i.e.,
(3)
Let be the density field of the solid in the reference configuration , and let be the density field of . The mass conservation principle applied to the whole body gives
(4)
whereas the local form of the conservation of mass yields
(5)
where stands for the Jacobian matrix of .
(H3) for all .
(H4) for all , where we denote by the vector for .
Conditions (H3), (H4) correspond to the so-called self-propelling conditions which are natural requirements for understanding swimming viewed as a self-propelled phenomena.
In particular, hypotheses (H1) and (H3) imply that the position of the center of mass of the creature is not affected by the undulatory motion, that is,
(6)
From (1), it follows that the region occupied by the creature at time t is given by
(7)
Moreover, by differentiating equation (1) with respect to t, it follows that the Eulerian velocity field of the solid is given for every by
(8)
where
(9)
The Eulerian density field of the body is given by
(10)
with given by (5). The mass M of the body and its moment of inertia with respect to an axis orthogonal to the plane of the motion and passing by the mass center of , are as usually given by
(11)
(12)
Let us notice that from (4), (10) and (11), we have that
Remark 2.1 In the case (rigid solid), all hypotheses (H1)-(H4) are satisfied, and the undulatory velocity field is equal to zero.
2.2 Equations
Let Ω be an open bounded set in representing the domain occupied by the solid-fluid system. Recalling that is the domain occupied by the solid at instant t, we have that the fluid fills, at instant t, the domain .
With the notation above, the full system describing the self-propelled motion of the creature can be written as
(13)
(14)
(16)
(17)
(18)
In the system above, and stand for the density and the viscosity of the fluid, which are supposed to be constant, u is the Eulerian velocity field of the fluid, and p denotes the pressure field of the fluid. A prime stands for the derivation operator with respect to time. By using the classical notation
(19)
the stress tensor field σ is defined by
where Id is the identity matrix in . Moreover, for and we denote by the unit normal to oriented towards the solid. Recall that the mass M and the moment of inertia of the solid at instant t are defined by (11) and (12).
System (13)-(20) is completed by the initial conditions
(21)
(22)
Remark 2.2 In the case of rigid solid, equation (16) becomes
because the undulatory velocity field w is equal to zero.
2.3 Weak formulation
Let , be two functions such that for all . In the sequel, we define and . Moreover, if no confusion is possible, we define
Let be a mapping such that for every , the function is a -diffeomorphism from ℱ onto and such that the derivatives
exist and are continuous. The existence of such a function is due, in particular, to the fact that for all t (see [19]). We can now define the following functions spaces:
where denotes the function defined by for .
In order to introduce the weak formulation, we first define some additional functions spaces. For every , let be an arbitrary position of the creature at time t, such that . We denote
(23)
(24)
where is the strain rate tensor defined by (19).
Let be a solution of (13)-(22). The vector velocity field u and the pressure p can be extended to Ω by setting
(25)
(26)
The extended vector belongs to . In the remaining part of this paper, the solution u and p of (13)-(22) will be extended as above.
We also need to extend the density field of the creature (defined in (10)) to the whole domain Ω by setting
(27)
By a slight variation of the argument in Ladyzhenskaya [[20], p.27], it can be shown that for every , there exists a continuous function such that, for every , the map is on and such that the function is of class for every and
(28)
For every , let be an arbitrary position of the creature at time t, such that . We then define by
(29)
Then the function Λ satisfies
(30)
An important ingredient of the numerical method we use is given by the characteristic functions whose level lines are the integral curves of the velocity field. More precisely (see, for instance, [10, 11]) the characteristic function is defined as the solution of the initial value problem
(31)
It is well known that the material derivative of the velocity field u at instant satisfies:
(32)
Remark 2.3 By using a classical result of Liouville (see, for instance, [[21], p.251]), if
are such that for any , we have and
then we get
(33)
where we have denoted by
the Jacobian matrix of the transformation .
In order to give the global weak formulation of our problem, we need to introduce the bilinear forms
defined by
Proposition 2.4
Assume that
and that u and p are extended to Ω as above.
Then is the solution of (13)-(22) if and only if for all , , , and satisfies
(34)
(35)
for a.e. .
More details on the existence and uniqueness of the solution and the complete proof of this result could be found in [19].
In the remainder of the paper, we suppose that f and satisfy
(36)
where and , are given as initial data in (22). Let us also assume that the corresponding solution of problem (13)-(22) satisfies the following regularity properties:
(37)
Moreover, we suppose that there exists a nonempty open connected subset of Ω such that for any , we have
(38)
Using this notation, we assume that
(39)
Remark 2.5 The hypotheses (37) and (39) imply the existence of such that
(40)