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Optimal shape of a variable condenser Anvar Kacimov Proc. R. Soc. Lond. A 2001 457, 485-494 doi: 10.1098/rspa.2000.0677
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10.1098/rspa.2000.0677
Optimal shape of a variable condenser By Anvar Kacimov Department of Soil and Water Sciences, Sultan Qaboos University, PO Box 34, Al-Khod 123, Sultanate of Oman (
[email protected]) Received 24 January 2000; revised 5 July 2000; accepted 3 August 2000
The shape of a condenser of maximal cross-sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary-value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova–Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor–Saffman bubble. In the limit of high capacitance, the Polubarinova–Kochina contour tends to the Saffman– Taylor finger, which in its own turn coincides with the Morse–Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non-isoperimetric optimum) and of maximal confined area (isoperimetric extremum). Keywords: optimization; global extremum; shaping; Laplace’s equation; Dirichlet’s boundary-value problem
1. Introduction Optimal shape design of surfaces exposed to physical fields improves the efficiency of many constructions (Haslinger & Neittaanmaki 1988; Pironneau 1984). An airfoil of minimal drag or maximal lift (Elizarov et al . 1997), a developed surface dissipating a maximal amount of heat (Aziz 1992), a soil channel of minimal seepage losses (Preissmann 1957), a bar of a maximum torsional rigidity (Polya & Szego 1951), an underground waste repository of a minimum plume size (Kacimov 1993a), among others, have been determined according to the so-called isoperimetric approach. Optimal forms are aimed at achieving an extremum of a chosen criterion (objective) under certain isoperimetric and non-isoperimetric restrictions. In mathematical physics, the governing equation should also hold in the domain bounded by optimal shapes. Hence the shapes should satisfy certain boundary conditions, which are imposed by the nature of the field. For example (Polya & Szego 1951), a sphere is a body of a minimal capacitance (criterion) at a given volume (restriction) if the potential (temperature, concentration, hydraulic head, etc.) is a harmonic function (governing equation), which becomes constant along the body surface (the Dirichlet boundary condition). A sphere in three-dimensional fields and a circle in two-dimensional fields appear as extrema in many optimal shape design problems (Polya & Szego 1951) which c 2001 The Royal Society
Proc. R. Soc. Lond. A (2001) 457, 485–494
485
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486
A. Kacimov y
Ψ = V0
B
A F1
Gz B
D1
Ψ= 0 D
Y = Q/2
F
a
H h
S
(z)
O
E1
E
A
x
Y = −Q/2
2l 2L B
Ψ = V0
A
Figure 1. A variable condenser of finite size, physical plane (z).
follow the Dido trick. Less trivial shapes, deduced as rigorous analytic solutions of optimal shape design problems, are surprisingly rare, i.e. in few cases, the extremal curve or surface can be derived explicitly with a proper proof of the necessary and sufficient condition of optimality in a reasonable admissible class. To the best of our knowledge, the first non-trivial optimal shape for the Laplace equation was found by Preissmann (1957), who considered a free-boundary (phreatic surface) problem and searched for an equipotential (channel bottom) confining maximal area (channel capacity). The technique of Preissmann was further developed and rigorous explicit non-trivial (non-circular) extrema were found for equipotentials, stream lines or isobars (Ilyinsky & Kacimov 1991, 1992a–c; Ill’inskii & Kasimov 1991; Kacimov 1991, 1993b; Kasimov 1991, 1992; Kacimov & Nicolaev 1992; Kasimov & Tartakovsky 1993) with applications in hydrology and geotechnical engineering. In the processes governed by the Laplace equation, the Polubariniova-Kochina contour (PKC) of the base of a weir (Polubarinova-Kochina 1977) appeared as a fascinating optimum in many flow problems. PKC delivers a maximum to the seepage rate, plume size, heat dissipation and other criteria in flows under dams, toward drains, through highly permeable lenses in aquifers, conduction from cooling surfaces with grooves, viscous flows near a surface with longitudinal ribs, and Taylor– Saffman flows of bubbles in a Hele-Shaw apparatus (Kacimov 1993a, 1994; Kacimov & Obnosov 1997). In this paper we show that PKC is also an exact extremum in a problem of minimal capacitance of a variable condenser (Morse & Feshbach 1953, pp. 1247– 1250). The possibility of breakdown of the dielectric should be minimized for such condensers and, as was shown by Morse & Feshbach (1953), the optimal cross-section of the central plate (extending infinitely in one direction) is of a half-thickness of the whole gap between two parallel plates. Amazingly, the optimal contour of Morse & Feshbach (1953) coincides with the Saffman–Taylor (Saffman & Taylor 1958) finger, which develops in a Hele-Shaw apparatus and models the phenomenon of a different physical nature! We shall call this plate (finger) the MFST contour. We prove that in a more general problem, when the central plate is of finite horizontal extension, the shape of minimal capacitance is just PKC. In the limit of a semi-infinite plate, our optimal curve degenerates into a ‘finger’, which coincides with the MFST contour. Proc. R. Soc. Lond. A (2001)
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It means that the MFST curve possesses both the property of minimal capacitance and optimal field intensity.
2. Optimization We consider a variable condenser consisting of a central plate DFED placed between two parallel plates of an aperture 2a (figure 1) and we will adhere to all denotations of Morse & Feshbach (1953). We assume that DFED is symmetric about both Cartesian axes originated at the centre of the plate O. The potential Ψ satisfies the equation ∂2Ψ ∂2Ψ + =0 ∂x2 ∂y 2
(2.1)
in the double connected domain in figure 1 and the boundary conditions Ψ =0 Ψ = V0
along DFED, along AB and BC,
(2.2)
where V0 is a constant. The complex potential F is defined as F = Ψ + iΥ where Υ is the flow function (Morse & Feshbach 1953, p. 1180). The capacitance is defined as dF 1 C= ds, (2.3) V0 dz Ψ =0 where integration is performed over the whole contour and s is the arc coordinate of DFED. Assume that the cross-sectional area 2S confined by DFED is fixed. Due to symmetry, we consider only the upper half (y > 0) of the condenser. The corresponding field domain is Gz . Along BD and EA the flow function Υ = ± 12 Q. Owing to (2.2) in the F -plane, we have a rectangle, GF , corresponding to Gz (figure 2). Clearly, the capacitance (2.3) of the whole system is C = 2Q/V0 (in heat conduction and subsurface flows, C is called an effective conductivity). Problem 2.1. Determine the shape of FED such that C is extremal at given values of S and V0 . The problem should be solved in a class of admissible curves, which is as broad as possible. We shall describe this class later. Morse & Feshbach (1953) optimized the shape of the inner plate in the class of equipotential curves which are generated by a line sink (a semi-infinite plate of zero thickness). Hence their class is quite narrow and there is no guarantee that the extremum they found is global. Note that the objective of Morse & Feshbach (1953) was the modulus of the field intensity |dF/dz| and their optimization was non-isoperimetric. First, we map conformally GF onto the upper half-plane v > 0, Gw , of an auxiliary plane w = u + iv (figure 2) with the following correspondence of points A, B → ±m, E, D → ±1. From the Schwartz–Christoffel formula, the mapping function is F (w) = − Proc. R. Soc. Lond. A (2001)
iQ F (arcsin w, λ), 2K
(2.4)
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488
A. Kacimov Y
v
(F)
Q/2
D
−Q/2
B GF
F
V0 A
E
Ψ
−m
−1
B
D
(w)
Gw F
1 E
m A
u
Figure 2. Complex potential plane (F ) and an auxiliary plane (w).
where F (arcsin w, λ) is an elliptic integral of the first type with modulus λ = 1/m. The modulus is determined from (2.4) at w = m, i.e. from the equation K(λ) Q √ . = 2 2V K( 1 − λ ) 0
(2.5)
We can not use the second conformal mapping as Morse & Feshbach (1953) because the shape of DFE in Gz is not fixed but has to be found as a part of the solution. Consider the function z1 = z − ia = x + i(y − a), which is an analytic function of w. The boundary conditions for z1 in Gw are at |u| > m, Im z1 = 0 at 1 < |u| < m, Im z1 = −a Im z1 = −a + f (u) at |u| < 1.
(2.6)
The first two conditions in (2.6) are clear, while the last condition reflects the variation of the shape of DFE. In other words, f (u) is a control function. We assume it to be from the Holder class (Polubarinova-Kochina 1977) that guarantees the existence of the Cauchy integrals below. The Holder class covers all reasonable plate shapes (including non-smooth boundaries, see Polubarinova-Kochina (1977)) occurring in practice. For points D and E in the z-plane, we have to impose the constrain f (±1) = 0. No other restrictions are necessary and the curve DFE defined by f can be extravagant (as shown in figure 1 by a dashed line). Thus we have a standard Dirichlet problem for z1 (w) in Gw , the integral solution of which is (Polubarinova-Kochina 1977) 1 ∞ Im z1 (τ ) dτ z1 (w) = + x(∞), (2.7) π −∞ τ −w where τ is a dummy variable. Obviously, x(∞) = 0 in (2.7). Substituting (2.6) into (2.7) and returning to z, we determine a m−w 1 1 f (τ ) dτ z(w) = − ln + . (2.8) π m + w π −1 τ − w Now we expand the control function, which is the Cauchy integral kernel in (2.8), by a series of Chebyshev polynomials of second type Un (u) = sin(n arccos u) as y = f (u) = a
∞
bn Un (u),
n=0
where bn are the coefficients to be found. In what follows we drop the limits of summation. Proc. R. Soc. Lond. A (2001)
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In the limit w → u, |u| < 1, we derive from (2.8) a parametric equation of DFE: a m−u x = − ln y=a bn Un (u). (2.9) −a bn Tn (u), π m+u Due to the symmetry about the Oy-axis, only odd coefficients should be retained in (2.9). The cross-sectional area confined by DFE and the Ox-axis is 1 S= y(x) dx = y(u)x (u) du. −1
DFE
From (2.9),
2am a nbn Un (u) √ x (u) = − π(m2 − u2 ) 1 − u2
and 2a2 m bn S= π
1
−1
Un (u) du − a2 m2 − u 2
1
−1
nb U (u) √ n n bk Uk (u) du. 1 − u2
In the last expression, due to the property of orthogonality of Chebyshev polynomials, the second integral is non-zero only at n = k and is calculated directly. The first integral is 1 π Un (u) sin nx sin x dx du = 2 2 2 2 −1 m − u 0 m − cos x and can be calculated directly according to eqn (67) of Prudnikov et al . (1986, p. 417). Thus we arrive at S = 2a2 b2n−1 (m − m2 − 1)2n−1 − 12 πa2 b22n−1 (2n − 1). (2.10) We assume that Q and V0 (and hence m) are given and consider S as a function of the shape, i.e. of b1 , b3 , b5 , . . . . The necessary condition of extremum is written for the first variation as DS = 0 or ∂S = 0 for all n. ∂bn Consequently, from (2.10), the coefficients of a potential extremum are √ 2(m − m2 − 1)2n−1 b2n−1 = . 2n − 1
(2.11)
From (2.10), ∂ 2 S/∂b22n−1 < 0 and ∂ 2 S/(∂bn ∂bj ) = 0 at n = j. Therefore, the second variation D2 S > 0 and, hence, the coefficients (2.11) do give a unique global maximum of S. It is easy to show (see Il’inskii & Kasimov (1984) for details) that the same b2n−1 provide a solution to the adjoint problem (minimum of Q and, hence, C at fixed S). Substitution of (2.11) back into (2.9) and convolution of series according to eqn (6) of Prudnikov et al . (1986, p. 738) yields a parametric equation of the extreme curve, xm 1 m + cos θ , = ln a 2π m − cos θ Proc. R. Soc. Lond. A (2001)
ym 1 sin θ = arctan √ , a π m2 − 1
(2.12)
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490
A. Kacimov y V0 upper pool h E
A a
tail water
2l dam F
x D
B foundation soil
A
B
Figure 3. Equivalent problem of seepage beneath a concrete dam.
where 0 θ π. If we shift the origin of coordinates to point D, then (2.12) becomes xm m − 1 m cos(πym /a) + 1 − m2 sin(πym /a) 1 ln . (2.13) = a 2π m + 1 m cos(πym /a) − 1 − m2 sin(πym /a) We recall now the known flow pattern beneath a concrete dam (figure 3). Water seeps in a homogeneous soil from the upper pool AE to the lower pool DB. The head drop is V0 , the depth of an impermeable horizontal bottom AB is a, the flow rate is Q. Polubarinova-Kochina (1962) used the hodograph method and determined the shape of an impermeable contour EFD from the condition of a constant hydraulic gradient along it (see also Gurevich (1965) for equivalent problems in the theory of jets). Note that the hydraulic gradient in subsurface mechanics is equivalent to the field intensity in electrostatics. Unfortunately, her solution contained a mistake, which she corrected in the second (Russian) edition (Polubarinova-Kochina 1977). Namely, instead of 2T in the equation of EFD in Polubarinova-Kochina (1962), one should set T . As a result, the contour of constant gradient is expressed by eqn (9.6) of Polubarinova-Kochina (1977, p. 194). As Q → ∞, the contour in figure 3 tends to a semicircle. If Q → 0, then, as we shall show later, EFD tends to two ‘counter-looking’ Saffman–Taylor (Saffman & Taylor 1958) half-fingers with their tips at points E and D (the fingers are ‘matched’ at point F). Comparing (2.12) with the constant-gradient contour of Polubarinova-Kochina (1977), we see that our extremum coincides with PKC. Contours (2.12) for different values of m are plotted by Polubarinova-Kochina (1977, fig. 133) and Kacimov (1994). Kacimov (1993a) showed that (2.12) coincides with the famous Taylor–Saffman (Taylor & Saffman 1959) bubble. Substitution of (2.11) into (2.10) and summation gives the area 1 m Sm = ln √ . 2 a π m2 − 1
(2.14)
Our figure 4 displays C/(2V0 ) as a function of S/a2 for the extreme condenser calculated according to (2.14) and (2.5). Because the extremum is unique and global, it can be used for isoperimetric estimations similar to Polya & Szego (1951). Indeed, if the plate area 2S of an arbitrary condenser is known, then its capacitance can be immediately estimated from below by our extremum. Besides, ‘chain inequalities’ (Carleman 1918; Polya & Szego 1951; Ilyinsky & Kacimov 1992a) can be applied to bound C of an arbitrary condenser from below and above by C values of inscribed and confining contours. Proc. R. Soc. Lond. A (2001)
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C/(2V0 )
1.5
1.0
0.5
0
0.01
0.02
0.03
0.04
S/a2 Figure 4. Non-dimensional capacity C/(2V0 ) as a function of a non-dimensional area S/a2 .
Alas, optimization in the class of arbitrary curves is not always achievable. For example, if AB in figure 1 is not horizontal, then it is hardly possible to find the global maximum in an explicit form. As a remedy, optimization can still be performed in particular classes of shapes generated by a linear sink as Morse & Feshbach (1953) did for their optimal plate, by a point sink (Fujii & Kacimov 1998), or even in classes where the shape is varied directly in the physical plane—for ellipses, trapezia, triangles, etc. (Aziz 1992; Kacimov 1992, 1997; Kacimov & Obnosov 1994, 1997). Since (2.1) is ‘stable’ to small variations of the domain boundaries, one can hope that these particular extrema are close to the unknown global one. A nice feature of the extremal curve (2.12) is that its horizontal and vertical sizes 2l and h are exactly twice as small as the horizontal and vertical cuts of the same Q. In other words, if we fix Q and solve the problem for a horizontal plate D1 E1 (figure 1), then its length 2L = 4l. Similarly, a vertical plate (one-half of it, OF1 , is shown in figure 1) of capacity C has the height H = 2h. The feature of ‘halved’ sizes (as compared with geometrically degenerating shapes) of optimal contours was found for optimal soil channels, drains and dams (Il’inskii & Kasimov 1984; Ilyinsky & Kacimov 1992b, c; Kacimov 1993a). Note that sometimes optimal shapes appear as equipotentials or streamlines generated by simple hydrodynamic singularities. For example, it was shown that it is a point source that generates the Preissmann extreme equipotential contour (Kacimov 1993b). Obviously, PKC becomes a semicircular streamline generated by a vortex if the depth of the impermeable bed tends to infinity in Polubarinova-Kochina (1977, fig. 133). In our figure 1, it means that at m → ∞ the curve (2.12) degenerates into an uninteresting circle. Now we consider the limit S → ∞ and, hence, m → 1. First note that the contour of half-thickness determined by Morse & Feshbach (1953) in their fig. 10.22 is exactly the Saffman–Taylor finger. Indeed, if we put ξ = 12 V0 in the parametric equations (10.2.54) of Morse & Feshbach (1953) and shift the origin of coordinates from the tip of the infinitesimally thin plate T to the tip F of the condenser in figure 5 (as we did transforming (2.12) into (2.13)), then (10.2.54) for the curvilinear plate is reduced to one explicit formula x/a = −0.5π −1 ln cos(πy/a). Proc. R. Soc. Lond. A (2001)
(2.15)
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492
A. Kacimov y
curvilinear plate xc
F a/2
D
T
A
x
generating plate of infinitesimal thickness Figure 5. The Morse–Feshbach–Taylor–Saffman contour and the generating line sink.
Obviously, equation (2.15) is just what Saffman & Taylor (1958) found in their HeleShaw model for a viscous finger of a half-size of the apparatus (see also Tanveer 2000). Hence the Saffman–Taylor curve is an equipotential of a line sink extending infinitely in one direction, as it follows from Morse & Feshbach (1953). The semiinfinite plate generating the finger is located distance xc = π −1 ln 2 from the tip, as figure 5 shows. Second, the MFST contour is the limiting case of PKC, a fact hardly discussed in the literature. Indeed, at m → 1, equation (2.13) reduces to (2.15). It implies, in particular, that the Saffman–Taylor finger is a limiting case of the Taylor–Saffman bubble and both are extremals of the above discussed problem of isoperimetric optimal shaping in the class of arbitrary curves. Note that both the Saffman–Taylor bubble and Taylor–Saffman finger solutions depend on one parameter (equivalent to our m), i.e. they are not unique. Considerable studies have been conducted to understand the ‘selection mechanism’ (Tanveer 2000), i.e. why in practical Hele-Shaw experiments a certain bubble size or finger width are observed. Our solution (2.12) is unique because the parameter (m) is immediately determined from the isoperimetric constraint (2.5). The extreme property of the Taylor–Saffman bubble and Saffman–Taylor finger was discussed by Aldushin & Matkowsky (1999). However, their selection of the halfaperture finger (observed in experiments of Saffman & Taylor (1958)) is based on optimization of a one-parameter function. In other words, Aldushin & Matkowsky (1999) have varied the shape in the class of curves specified by a phreatic-surface-type (Polubarinova-Kochina 1977) boundary condition. Our optimization is much broader because S in (2.10) depends on an infinite number of coefficients bn . Therefore, we have shown that the MFST contour can be viewed as a selection of the shape of a viscous finger from a general class defined by the control function f [u], which includes non-half-size Saffman–Taylor fingers.
3. Discussion and conclusion We have studied the problem of optimal shape design of an equipotential line, which corresponds to the inner plate of a condenser. We used a conformal mapping of the complex potential domain (rectangle) onto an auxiliary half-plane. Then, in this halfplane, we solved the Dirichlet boundary-value problem for the complex coordinate. The kernel of the Cauchy integral was taken as a control function, which governed the Proc. R. Soc. Lond. A (2001)
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shape of the plate. The condenser of maximal cross-sectional area at a given capacitance was found in an explicit form. The maximum discovered is global and unique because both the necessary and sufficient conditions for this maximum are satisfied in a strict way. The extreme curve is shown to coincide with the Polubarinova-Kochina contour of the dam of constant hydraulic gradient. In the limit of large cross-sectional area, the contour tends to the famous Morse–Feshbach–Taylor–Saffman curve, which in our electrostatic context is not only a plate of minimal capacity at a given crosssection but also a line of constant field intensity. This study was supported by Sultan Qaboos University, projects AGSWAT 9903, AGR/99/13. Helpful comments by Yu. V. Obnosov and P. Cookson are appreciated.
References Aldushin, A. P. & Matkowsky, B. J. 1999 Extremum principles for selection in the Saffman– Taylor finger and Taylor–Saffman bubble problems. Phys. Fluids 11, 1287–1296. Aziz, A. 1992 Optimum dimensions of extended surfaces operating in a convective environment. Appl. Mech. Rev. 42, 155–173. Carleman, T. 1918 Uber ein Minimalproblem der mathematischen Physik. Math. Z. 1, 208–212 (in German). Elizarov, A. M., Ilinskiy, N. B. & Potashev, A. V. 1997 Mathematical methods of airfoil design. In Inverse boundary-value problems of aerohydrodynamics. Academic. Fujii, N. & Kacimov, A. R. 1998 Analytic formulae for rate of seepage flow into drains and cavities. Int. J. Numer. Analytical Meth. Geomech. 22, 277–301. Gurevich, M. I. 1965 Theory of jets in ideal fluids. Academic. Haslinger, J. & Neittaanmaki, P. 1988 Finite element approximation for optimal shape design: theory and applications. Wiley. Il’inskii, N. B. & Kasimov, A. R. 1984 Seepage-limitation optimization of the shape of an irrigation channel by the inverse boundary-value problem method. Fluid Dynamics 19, 404–410. Il’inskii, N. B. & Kasimov, A. R. 1991 Estimates of backwater and drying levels in hydrodynamic model. Fluid Dynamics 1991, 26, 224–231. Ilyinsky, N. B. & Kacimov, A. R. 1991 Problems of seepage under dam. Proc. Indian Nat. Sci. Acad. A 57, 61–68. Ilyinsky, N. B. & Kacimov, A. R. 1992a The estimation of integral seepage characteristics of hydraulic structures in terms of the theory of inverse boundary-value problems. Z. Angew. Math. Mech. 72, 103–112. Ilyinsky, N. B. & Kacimov, A. R. 1992b Problems of seepage to empty ditch and drain. Water Resources Res. 28, 871–876. Ilyinsky, N. B. & Kacimov, A. R. 1992c Analytic estimation of ground water flow around cutoff wall and into interceptor ditches. J. Ground Water 30, 901–906. Kacimov, A. R. 1991 Steady, two-dimensional flow of ground water to a trench. J. Hydrol. 127, 71–83. Kacimov, A. R. 1992 Seepage optimization for a trapezoidal channel. J. Irrigation Drainage Am. Soc. Civ. Engng 118, 520–526. Kacimov, A. R. 1993a Inclusion shaping and extremal property of the Taylor–Saffman bubble. Fluid Dyn. 28, 741–743. Kacimov, A. R. 1993b Seepage optimization of the shape of an earth channel allowing for capillarity. J. Soviet Math. 63, 462–464. Kacimov, A. R. 1994 Optimization of the protrusion shape for a Couette type flow. Optimal Control Appl. Meth. 15, 193–203. Proc. R. Soc. Lond. A (2001)
Downloaded from rspa.royalsocietypublishing.org on July 17, 2011
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A. Kacimov
Kacimov, A. R. 1997 Optimization of seepage rate through a triangular core. Int. J. Numer. Analytical Meth. Geomech. 21, 443–451. Kacimov, A. R. & Nicolaev, A. N. 1992 Steady seepage near an impermeable obstacle. J. Hydrol. 138, 17–40. Kacimov, A. R. & Obnosov, Yu. V. 1994 Minimization of ground water contamination by lining of a porous waste repository. Proc. Indian Natn. Sci. Acad. A 60, 783–792. Kacimov, A. R. & Obnosov, Yu. V. 1997 Explicit, rigorous solutions to 2-D heat transfer: twocomponent media and optimization of cooling fins. Int. J. Heat Mass Transfer 40, 1191–1196. Kasimov, A. R. 1991 Optimization of irrigation in hydrodynamic filtering model. J. Appl. Math. Mech. 55, 273–275. Kasimov, A. R. 1992 Minimum dimension of total saturation bubble around an isolated source. Fluid Dyn. 27, 886–889. Kasimov, A. R. & Tartakovsky, D. M. 1993 Estimation of tracer migration time in ground water flow. Comput. Math. Math. Phys. 33, 1535–1541. Morse, P. M. & Feshbach, H. 1953 Methods of theoretical physics. Part II. New York: Pergamon. Pironneau, O. 1984 Optimal shape design for elliptic systems. Springer. Polubarinova-Kochina, P. Ya. 1962 Theory of ground-water movement. Princeton, NJ: Princeton University Press. Polubarinova-Kochina, P. Ya. 1977 Theory of ground-water movement. Moscow: Nauka (in Russian). Polya, G. & Szego, G. 1951 Isoperimetric inequalities in mathematical physics. Princeton, NJ: Princeton University Press. Preissmann, A. 1957 A propos de la filtration au-dessous des canaux. Houille Blanche 12, 181– 188 (in French). Prudnikov, A. P., Brychkov, Yu. V. & Marichev, O. I. 1986 Integrals and series, vol. 1. New York: Gordon and Breach. Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329. Tanveer, S. 2000 Surprise in viscous fingering. J. Fluid Mech. 409, 273–308. Taylor, G. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Math. 12, 265–279.
Proc. R. Soc. Lond. A (2001)
