HOMOGENIZATION OF A CAPILLARY PHENOMENA

We study the height of a liquid in a tube when it contains a great number of thin vertical bars and when its border is finely strained. For this, one uses an epi-convergence method.


Introduction
Let T be a cylindrical tube of height infinity, containing a compressible liquid and a great number of thin cyclindrical bars of radius r which are distributed periodically with a period such that r < (see Figure 1).One supposes the bars are of the same material which is specified by the coefficient λ (see (1) below).In section 2 we show that (see Proposition 2 and Theorem 1) when ( , rλ) tends to (0, 0, 0) the height of the liquid grows indefinitely, does not grow or take a certain height according to the rate rλ 2 being +∞, 0 or finite (see Proposition 2 and Theorem 1).Suppose now that the border of the tube is made of two alternate periodic bands with a period .The band 1 of length r is encrusted in the other of length − r (r < ) (see Figure 3).The different materials that constitute the bands are specified by the coefficients k 1 and k 2 .
Border of the tube The tube with the bars.

Problem.
Let T be a cylindrical tube of base Ω with radius R and of height infinity, containing a compressible liquid of a finite volume.Let us divide Ω by a net of side and let us set in the center x ,j of the net Y ,j (j ∈ I the integer part of the real |Ω| 2 ) an homogeneous vertical cylindrical bar T h,j of radius r (r < ) where h = ( , r, λ) (see Figures 1 and 2).

Let us denote the union of the bars T h,j by T h and let Ω
The height of the liquid is stabilizing when it satisfies the following minimization problem (see [3], [4]): where ) the space of functions of bounded variation defined in Ω (see [5] for the definition and the properties of BV (Ω)): where C 1 0 (Ω) is the space of functions defined in Ω, of class C 1 with a compact support included in Ω, The capillary constant c is related to the pressure term of the liquid, λ and k are respectively the cosine of the contact angles (liquid, bar), (liquid, border) and which specify respectively the bars material and the tube border.The constants c, λ and k are supposed real positive numbers.
Problem.We look for the limit behavior of the height of the liquid when h = ( , r, λ) tends to (0, 0, 0).

Existence and extension of the solution.
Proposition 1. (See [3], [4]).The problem (1) has a unique positive solution in BV (Ω h ).Furthermore u h ∈ C 2 (Ω h ) and is constant on the boundary ∂T h,j of the bar T h,j for all j ∈ I.
Let us cite a trace lemma in the space BV (Ω) which will be useful: Lemma 1. (See [4]).Suppose that Ω is of class C 2 and satisfying an internal sphere condition of radius r (i.e. for any boundary point x ∈ ∂Ω, there is a ball B of radius r such that B ⊂ Ω and x ∈ ∂B).Then there exists a positive constant ct such that for any v ∈ BV (Ω), Let us now state some estimates of u h : ii) If a is finite, then Ω h u h and Ω h |Du h | are uniformly bounded for λ bounded.
Proof: i) The problem ( 1) is equivalent to the Euler equation (see [4]) where 1+|Dv| 2 and n the outer normal.
ii) According to Lemma 1 ( 6) Let ũh be the extension of u h in Ω defined by where u h,j is the value of u h on ∂T h,j .
Using again Lemma 1, one has (9) where the last inequality of (9) is due to (6).The inequality and ( 9) yield Hence the assertion ii) holds.
The following proposition yields an extension of u h and of J h in Ω which will be useful afterwards.

Proposition 3. Suppose that lim h→0
λ is finite.Let ũh be the extension of u h in Ω which is defined by ( 7), ( 8) and Jh the extension of J h in BV (Ω) defined by where v i is the trace of the restriction of v in Ω h .Then i) Jh attains its minimum at the point ũh .
Proof: i) For the existence and the regularity of the minimum of Jh in BV (Ω) see [3], [4].The minimum value of Jh is attained at ũh since Jh (ũ h ) = J h (u h ).

Epi-limit of the functional Jh .
The following proposition characterizes the epi-limit of the sequence Jh (see [1], [2] for the definition and the properties of the epi-limit) when h → (0, 0, 0): Proposition 4. Suppose that a = lim h→0 λr 2 is finite.Then the sequence of the functionals Jh epi-converges to the functional J in BV (Ω) endowed with the topology of L 1 (Ω) where 2) Let v and v h ∈ BV (Ω) such that v h converges to v in L 1 (Ω) when h tends to (0, 0, 0).Let J h the functional defined in BV (Ω) by where v i is the trace, on ∂T h , of the restriction of v to Ω h .
Using the inequality ( 6), Now, passing to the limitinf in (12), using the lower semi continuity of the functionsl v → Ω 1 + |Dv| 2 − k ∂Ω v in BV (Ω) according to the topology of L 1 (Ω) (see [6]), Fatou Lemma, the fact that v h converges to v in L 1 (Ω) and that lim h→0 λ is finite, one deduces that

Limit behavior.
According to the properties of the epi-convergence (see [1], [2]), one deduces the limit behavior of the problem (1) from Propositions 2 and 3.

Theorem 1. If lim h→0 λr
2 is finite, then the minimum Jh (ũ h ) of J converges to the minimum J(u) and ũh converges to u in L 1 (Ω) when h tends to (0, 0, 0).

Physic interpretation.
When the section, of radius r, of the bars T h,j satisfies r 2 λ , the liquid rises indefinitely; when r 2 λ , it does not rise and when r a 2 λ , a ∈ (0, +∞), the capillary problem in the tube with the bars is approximated by a capillary problem in the tube without bars with the same capillary constant 2c but with −2πa as Lagrange parameter.

Tube with a strained border
Let n ∈ N * and = |Ω| n .Suppose now that the border of the tube is formed by two bands which are distributed alternately and periodically with the period (see Figures 3 and 4): the boundary ∂Ω is formed by n arcs Y ,j , j ∈ {1, . . ., n} of length .In the center of any arc Y ,j is encrusted an arc T h,j (band 1), h = ( , r), of length r (r < ).Suppose that the arcs T h,j are of the same material specified by the coefficient k 1 and the union of Y ,j \T h,j (band 2) is of another material specified by the coefficient k 2 (see ( 13)).Let us denote T h = ∪T h,j and (∂Ω) h = ∪(Y ,j \T h,j ).
The height of the liquid is stabilizing when it satisfies the minimization problem (see [3], [4]): where Proof: The proof of the existence is similar than the one of Proposition 1.
The problem (13) is equivalent to the equation where 1+|Dv| 2 and n the outer normal.

Band 1
Band 2 ∂Ω r Ω Figure 4 3.2.Epi-limit of the functional F h .The following proposition characterizes the epi-limit of the sequence F h when h → (0, 0): Proposition 6.The sequence of the functionals F h epi-converges to the functional F in BV (Ω) endowed within the topology of L 1 (Ω) where 2) Let v and v h ∈ BV (Ω) such that v h converges to v in L 1 (Ω) when h tends to (0, 0).
Using Lemma 1 Now, passing to the limitinf in (19), using the lower semi continuity of the functional v → Ω 1 + |Dv| 2 in BV (Ω) according to the topology of L 1 (Ω) (see [5]), Fatou Lemma and the fact that v h converges to v in L 1 (Ω), one deduces that

Limit behavior.
From Proposition 6, one deduces the limit behavior of the problem (13).
Theorem 2. i) The minimum of F h converges, when h tends to (0, 0), to the minimum of F in BV (Ω) where ii) The functional F attains its minimum at the point u and u h converges to u in L 1 (Ω) when h tends to (0, 0).