Abstract
We consider systems of elliptic partial differential equations in divergence form with Dirichlet’s boundary conditions in doubly-connected domain of the plane with modulus \({\mu }\). We prove an invariance property of the corresponding global flows in the class of domains with the same modulus. Applications are given to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour’s effects.
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1 Introduction
The solution of the problem
where \({{\mathcal O}}=\{({\rho },{\theta }); 0<R_1<{\rho }<R_2, 0\le {\theta }<2\pi \}\), is given by
Moreover,
If \(V\) represents the difference of potential applied to a metallic specimen \({{\mathcal O}}\), Eq. (1.1) tells us that the total current crossing \(\{{\rho }=R_2\}\) depends, in addition to \(V\), only on the ratio \(R_2/R_1\). In this paper we prove that this property of invariance remains true for much more complex domains and systems of PDE.
Let us consider the plane bounded doubly-connected domain \(\Omega \) with boundary formed by the two simple closed curves \({\Gamma }_1\) and \({\Gamma }_2\). We assume that physical state of \(\Omega \) is determined by \(n\) parameters \(u_i(x,y)\), \(i=1\ldots n\) via the \(n\) fluxes densities
The functions \(a_{ij}(u_1\ldots u_n)\) are given and regular. The conservation laws
are assumed to hold. Supposing Dirichlet’s boundary conditions for \(u_i\) on \({\Gamma }_1\) and \({\Gamma }_2\) we arrive at the following nonlinear boundary value problem
where \(u_i^{(1)}\) and \(u_i^{(2)}\) are \(2n\) functions given respectively on \({\Gamma }_1\) and \({\Gamma }_2\). In many cases what is of interest in the solutions of (1.4) and (1.5) are the \(n\) global quantities
where \({\mathbf{n}}\) denotes the unit vector normal to the boundary of \(\Omega \) pointing outward with respect to \(\Omega \). In view of (1.3) we have
Example 1
Consider a very long cylinder whose cross-section is the doubly-connected domain \(\Omega \). The cylinder is made of a material capable of conducting heat and electricity. Between the two lateral surfaces a difference of potential \(V\) is applied and they are kept at a constant temperature. The thermal conductivity \({\kappa }\) and the electric conductivity \({\sigma }\) are given functions of the temperature \(u\). In this problem are relevant the current density \({\mathbf{J}}\) given by the Ohm’s law
and the density of the flow of energy [12]
By the conservation of charge and energy we obtain the so-called thermistor problem, which has been thoroughly studied by many authors, [3, 4, 10] and recently by [6, 7] among others. In particular in [9] the invariance by conformal mappings of the problem is noted. The thermistor problem reads
In this case
gives the total electric current crossing \({\Gamma }_1\) in the unit time.
Example 2
We consider, as a second example, the heat and mass transfer occurring in a Darcy’s flow in a porous medium occupying the cylinder of Example 1. The first flow is the velocity given by the Darcy’s law [1]
where \(p(x,y)\) is the pressure. We take into account the Soret and Dufour’s effects [1], which in certain cases are not negligible [5]. Thus we have for the densities of the mass and heat flow
where \(c(x,y)\) is the concentration, \(u(x,y)\) the temperature, \({\beta }\) the Fick’s coefficient, \(S\) and \(D\) the Soret and Dufour’s coefficients. They all are supposed, as \(K\), to be given positive functions of \(p,\ c\) and \(u\). If we take into account (1.7) we have
By (1.3) we obtain from (1.7) and (1.8) the boundary value problem
In this problem we are interested in the global quantities
In this paper we prove a property of invariance of the quantities \(I_i\) defined in (1.6) which, together with the notion of modulus of a doubly-connected domain (see[2, 8, 14]), permits to compute \(I_i\) reformulating the problem (1.4) and (1.5) in a simpler doubly-connected domain of the same modulus. In Sect. 2 this method is applied to the problem of Example 1 and in Sect. 3 to Example 2.
2 Invariance properties
Let \(w=f(z)\) be the conformal mapping \(f(z)={\Phi }(x,y)+\Psi (x,y), z=x+iy, w=X+iY\) such that \(|f'(z)|>0\) in \(\bar{\Omega }\). Let \({\omega }=f(\Omega )\), \({\gamma }_1=f({\Gamma }_1)\), \({\gamma }_2=f({\Gamma }_2)\). Assume \(u_i(x,y), i=1\ldots n\) to be a solution of problem (1.4), (1.5). We set \({{\mathcal U}}_i(X,Y)=u_i(x,y)\) and \(X={\Phi }(x,y)\), \(Y=\Psi (x,y)\). Using the Cauchy–Riemann equations and their consequences we find
Thus we have, using (2.1) and (2.2),
Therefore the Eq. (1.4) is invariant under conformal mappings. Let \(z=\tilde{z}(s)\) be the parametric representation of the curve \({\Gamma }_1\) in term of the arc length \(s\) and \(w=f(\tilde{z}(s))\) the corresponding parametric representation of the curve \({\gamma }_1\) on the \(w\) plane. If \(S\) denotes the arc length on \({\gamma }_1\) we have
After a simple calculation, we have, if \({\mathbf{N}}\) denotes the unit vector normal to \({\omega }\),
This gives
Therefore we have
Summing up we have the following
Theorem 2.1
The boundary value problem
and the quantities
are invariant under conformal mappings.
To apply this result we recall the definition and properties of the modulus of a doubly-connected domain [8]. If \(\Omega \) is a doubly-connected domain bounded by two non-degenerate curves it is always possible to map \(\Omega \) conformally in a one-to-one manner on the annulus \(1<|w|<{\mu }\). The number \({\mu }\), the modulus of \(\Omega \), is a characteristic constant of \(\Omega \), i.e. to every \(\Omega \) there corresponds one and only one number \({\mu }>1\). This determines a partition into equivalence classes of all doubly-connected domains, in particular all the annuli of radii \(R_2>R_1>0\), such that \(\frac{R_2}{R_1}={\mu }\) belongs to the same class of modulus \({\mu }\). Hence, by Theorem 2.1 the global fluxes \(I_i\) related to problem (1.4), (1.5) can be computed solving the same problem in an annulus of radii \(1\) and \({\mu }\).
3 Total flows in the problem of electric heating of a conductor
In this section we consider the problem of Example 1 i.e.
If \({\sigma }(u)\in C^1({\mathbf{R}^1})\), \({\kappa }(u)\in C^1({\mathbf{R}^1})\), \({\sigma }(u)>0\), \({\kappa }(u)>0\) and
problem (3.1–3.4) has one and only one solution [3]. We wish to compute the total current
crossing the device for a generic doubly-connected domain of modulus \({\mu }\). Define
By (3.5) \(F\) maps one-to-one \([0,\infty )\) onto \([0,\infty )\). In terms of \({\theta }\), \({\varphi }\) and \(u\) we can restate problem (3.1–3.4) as follows
The simple functional relation
exists between \({\theta }\) and \({\varphi }\) (see [3]). Hence, we have
thus (3.6) can be rewritten
with the boundary conditions
To this problem we can apply the Kirchhoff’s reduction. More precisely, define
We have
and, in view of (3.9) and (3.10),
Moreover, recalling (3.8) we obtain
If \(v\) is the solution of the problem
we have
Hence, by (3.11)
and
On the other hand, \(I\) is invariant in the class of the doubly-connected domains of modulus \({\mu }\). It is therefore enough to compute \(\int _{{\Gamma }_1}\frac{dv}{dn}ds\) in the annulus of radii \(1\) and \({\mu }\). We easily find
Hence, by (3.12)
This gives the total current crossing any doubly-connected domain of modulus \({\mu }\) if all the others data in problem (3.1–3.4) remain unchanged.
Remark 3.1
In problem (3.1–3.4) it is interesting to consider also the density of the heat flow as given by the Fourier’s law
and the corresponding global quantities, a priori not necessarily equal,
We have
and, by (3.7),
On the other hand,
Hence, in view of the condition \({\varphi }=0\) on \({\Gamma }_1\) and of (3.12)
Moreover, since \({\varphi }=V\) on \({\Gamma }_2\) and \(\int _{{\Gamma }_1}{\mathbf{J}}\cdot {\mathbf{n}}\ ds=-\int _{{\Gamma }_2}{\mathbf{J}}\cdot {\mathbf{n}}\ ds\) we have
4 Invariance properties in the Soret-Dufour’s problem
Theorem 2.1 can be applied to the problem of Example 2 i.e.
Therefore the total flows of mass and heat depend on \(\Omega \) only via its modulus. In this section we consider a special case of problem (4.1–4.4). We suppose \({\beta }\), \({\kappa }\), \(D\), \(S\) and \(K\) to be positive constants. We have
Theorem 4.1
Let
Suppose
Then the problem
has one and only one solution. Moreover, if all the data are of class \(C^\infty \) then also the corresponding solution is of class \(C^\infty \).
Proof
We compute \(p(x,y)\) from the problem
By (4.5) we have \(p\in H^2(\Omega )\). Denote \(c_b\) and \(u_b\) the solutions of the problems
Setting \(h=c-c_b\) and \(z=u-u_b\) we restate problem (4.7–4.10) with homogeneous boundary conditions
Define the bilinear form
where \((h,z)\in H^1_0(\Omega )\times H^1_0(\Omega )\) and \((v,w)\in H^1_0(\Omega )\times H^1_0(\Omega )\). \(a((h,z),(v,w))\) is coercive and bounded. In fact, we have by (4.11)
Thus
Similarly
Hence
On the other hand, the matrix
of the quadratic form
is definite positive since the determinants of the principal minors i.e.
are all positive by (4.6). Hence there exists a positive constant \(L\) such that
It is also easy to prove that \(a((h,z),(v,w))\) is bounded. Let us define the linear functional of \((H^1_0(\Omega )\times H^1_0(\Omega ))'\)
We can rewrite (4.14), (4.15) as follows
The Lax-Milgram lemma [13] can be applied and we conclude that (4.7–4.10) has one and only one solution. Using standard regularity results for elliptic system [11] this weak solution can be regularized.
Remark 4.1
If the condition \(\Bigl (\frac{S+D}{2}\Bigl )^2<{\beta }{\kappa }\) is not satisfied, problem (4.7–4.10) may not have solutions at all. This fact is already apparent in the one-dimensional counterpart of problem (4.7–4.10) i.e.
where \(a\) is a given constant. In fact, suppose \({\beta }={\kappa }=D=S=a=1\). We have by difference \((c-u)'=0\) which is not always compatible with the boundary conditions \( c(0)=c^{(1)},c(1)=c^{(2)}\), \( u(0)=u^{(1)},u(1)=u^{(2)}\).
Let (4.6) be satisfied and let us take in problem (4.7–4.10) \( u^{(1)}, u^{(2)},\ c^{(1)},c^{(2)} \) as constants and \(p^{(1)}=0, p^{(2)}=\bar{P}\). We compute the total flows of mass and heat in a generic doubly-connected domain of modulus \({\mu }\). For this goal we solve the problem in the annulus of radii \(1\) and \({\mu }\). Denote \(\rho \) and \(\theta \) the polar coordinates. Since the solution is unique by Theorem 4.1 a solution which depends only on \({\rho }\) is the only possible one. Therefore, we have
In polar coordinates the Eqs. (4.8) and (4.9) become
where
From (4.6) and the inequality \(\sqrt{DS}<\frac{S+D}{2}\) we obtain \(DS-{\beta }{\kappa }\ne 0\). This permits to solve the system of ordinary differential Eqs. (4.18), (4.19). We obtain
where
The four constants of integration \(A_1\), \(A_2\), \(A_3\) and \(A_4\) are in a one-to-one correspondence with the boundary values \( u^{(1)},\ u^{(2)},\ c^{(1)},c^{(2)} \). They can easily be computed explicitly. Thus we obtain for the total flows of mass \(Q_{m}\) and heat \(Q_{h}\) in an arbitrary domain of modulus \({\mu }\)
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Cimatti, G. Invariance of flows in doubly-connected domains with the same modulus. Boll Unione Mat Ital 7, 217–226 (2014). https://doi.org/10.1007/s40574-014-0012-y
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DOI: https://doi.org/10.1007/s40574-014-0012-y
Keywords
- Systems of PDE in divergence form
- Doubly-connected plane domain
- Thermistor problem
- Porous media
- Darcy law
- Soret effect
- Dufour effect