From the summary: "In the situation dealt with in Part I [European J. Appl. Math. 12 (2001), no. 6, 665--676; MR1877346 (2002m:76034); see the preceding review] we suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment that does not adjoin the vertex, so we have air at the vertex. We show that, if we assume the pressure to be constant along the free boundaries, the region occupied by the fluid is the image of a rectangle under a conformal map that can be expressed in terms of elliptic functions if $\alpha=1$ or $\alpha=2$, and in terms of theta functions if $0<\alpha<1$ or $1<\alpha<2$. The form of the function giving the map can be written down, and the parameters appearing in it then determined as the solution to a set of transcendental equations. The procedure is illustrated by a number of examples."

1. Bowman, F. (1961) Introduction to Elliptic Functions with Applications. Dover. MR0132214 (24 #A2060)
2. Hermite, C. (1912) \OE uvres de Charles Hermite, Tome III. Gauthier-Villars.
3. Rainville, E. D. (1971) Special Functions (2nd edition). Chelsea. MR0393590 (52 #14399)
4. Richardson, S. (1994) Hele-Shaw flows with time-dependent free boundaries in which the fluid occupies a multiply-connected region. Euro. J. Appl. Math. 5, 97--122. MR1285033 (95d:76043)
5. Richardson, S. (1996) Hele-Shaw flows with free boundaries driven along infinite strips by a pressure difference. Euro. J. Appl. Math. 7, 345--366. MR1413369 (97e:76088)
6. Richardson, S. (2000) Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region. Euro. J. Appl. Math. 11, 249--269. MR1844587 (2002d:76038)
7. Richardson, S. (2001a) Hele-Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region. Euro. J. Appl. Math. 12, 571--600. MR1870859 (2002m:76033)
8. Richardson, S. (2001b) Hele-Shaw flows with free boundaries in a corner or around a wedge. Part I: Liquid at the vertex. Euro. J. Appl. Math. 12, 665--676. MR1877346 (2002m:76034)

Summary: "Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle $\alpha\pi$; if $0<\alpha<1$ we have flow in a corner, while if $1<\alpha\leq 2$ we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment emanating from the vertex, so we have liquid at the vertex, and contemplate such a situation produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundary, the region occupied by the fluid is the image of a semidisc (a domain bounded by a semicircle and its diameter) in the $\zeta$-plane under a conformal map given by a function of the form $\zeta^\alpha$ times a rational function of $\zeta$. The form of this rational function can be written down, and the parameters appearing in it then determined as the solution to a set of algebraic equations. Examples of such flows are given (including one which shows that, in a certain sense, injection can produce a cusp), and the limiting situation in the wedge configuration as one injection point is moved to infinity is also considered."

\{For Part II see the following review [MR1877347 (2002m:76035)].\}

1. Cummings, L. J. (1999) Flow around a wedge of arbitrary angle in a Hele-Shaw cell. Euro. J. Appl. Math. 10, 547--560. MR1757941 (2001c:76041)
2. Howison, S. D. (1992) Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3, 209--224. MR1182213 (94f:76025)
3. Richardson, S. (1981) Some Hele Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263--278. MR0612095 (82d:76023)
4. Richardson, S. (1997) On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows. Quart. J. Appl. Math. 55, 313--319. MR1447579 (98f:76029)
5. Richardson, S. (2001a) Hele-Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region. Euro. J. Appl. Math. 12, 571--600. MR1870859 (2002m:76033)
6. Richardson, S. (2001b) Hele-Shaw flows with free boundaries in a corner or around a wedge. Part II: Air at the vertex. Euro. J. Appl. Math. 12, 677--688. MR1877347 (2002m:76035)

This paper considers the physical problem of singularity-driven Hele-Shaw flows of a multiply-connected fluid region. The physical free boundary problem is such that, under certain circumstances, the Cauchy transform of the fluid domain preserves its rational character under evolution. In this way, it is shown that reconstruction of the fluid boundary under evolution can be effected by constructing a conformal mapping from a canonical $N$-connected region to the fluid domain. The author elects to employ the Schottky model and constructs the mappings in various cases explicitly as an automorphic function invariant with respect to the relevant Schottky groups. This is done using a standard ratio of Poincaré (theta) series. Different examples are given and some physical issues discussed.

By far the most interesting aspect of this work is mathematical; although it discusses the problem of Hele-Shaw flows, this paper really provides an example of a mathematical method to construct multiply-connected quadrature domains---an issue of much broader significance beyond the realm of Hele-Shaw flows. The paper is clearly written and concisely documents the dextrous construction of automorphic functions on Riemann surfaces. The Poincaré series representations used are non-unique and the author gives careful details of his particular choices in the construction of his maps. Ultimately, a system of nonlinear equations (some of which are non-local) must be solved numerically for the parameters appearing in the mapping functions. One shortcoming of the paper is that the author provides very few details of the numerical method employed, or any numerical difficulties encountered. Also, it is not always clear precisely how large a system is solved in each particular example. In particular, some of the nonlinear equations to be solved are derived at certain zeros of a particular function and these zeros are not known a priori. This would appear to increase the number of equations to be solved and a discussion of such issues would have been helpful for any reader wishing to independently implement the method.

For general mathematical interest, it seems appropriate to document here the simultaneous appearance of closely related work [e.g., D. G. Crowdy, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2014, 2337--2359; MR1862658 (2002h:76029)] which, although presented in the context of a very different physical problem, also involves the mathematical construction of multiply-connected quadrature domains. Crowdy adopts a rather different algebro-geometric approach to the construction, exploiting the fact that the boundaries of quadrature domains are algebraic curves, and makes strategic use of the notion of "special points" of the domains. Interestingly, rather than being a repeat of the method used in Richardson's paper, Crowdy's paper nicely complements that of Richardson and provides a distinct alternative mathematical perspective. For a general reader, the two papers together provide a particularly instructive showcase of the possibilities for constructing multiply-connected quadrature domains---an endeavour which has direct application to a range of different physical problems.

1. Aharonov, D. & Shapiro, H. S. (1976) Domains on which analytic functions satisfy quadrature identities. J. Analyse Math. 30, 39--73. MR0447589 (56 #5899)
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3. Ford, L. R. (1951) Automorphic Functions (2nd edition). Chelsea.
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6. Poincaré, H. (1916) \OE uvres de Henri Poincar\'e, Tome II. Gauthier-Villars.
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8. Richardson, S. (1996) Hele-Shaw flows with time-dependent free boundaries involving a concentric annulus. Phil. Trans. R. Soc. Lond. A 354, 2513--2553. MR1419926 (98a:76028)
9. Richardson, S. (1997) Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension. Euro. J. Appl. Math. 8, 311--329. MR1471595 (98g:76023)
10. Richardson, S. (2000) Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region. Euro. J. Appl. Math. 11, 249--269. MR1844587 (2002d:76038)
11. Sakái, M. (1978) A moment problem on Jordan domains. Proc. Amer. Math. Soc. 70, 35--38. MR0470216 (57 #9974)
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Summary: "Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at $N$ points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are $N$ invariants; while the $N$ points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem."

1. Bowman, F. (1961) Introduction to Elliptic Functions with Applications. Dover. MR0132214 (24 #A2060)
2. Byrd, P. F. & Friedman, M. D. (1971) Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. Springer-Verlag. MR0277773 (43 #3506)
3. Carlson, B. C. & Todd, J. (1983) The degenerating behavior of elliptic functions. SIAM J. Numer. Anal. 20, 1120--1129. MR0723828 (85m:33002)
4. Crowdy, D. & Tanveer, S. (1998) A theory of exact solutions for annular viscous blobs. J. Nonlinear Sci. 8, 375--400. MR1629812 (2000c:76020b)
5. Gustafsson, B. (1988) Singular and special points on quadrature domains from an algebraic geometric point of view. J. Analyse Math. 51, 91--117. MR0963151 (90i:30066)
6. Lu, J. (1995) Complex Variable Methods in Plane Elasticity. World Scientific. MR1370444 (96k:73094)
7. Richardson, S. (1968) Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33, 475--493. MR0233549 (38 #1870)
8. Richardson, S. (1992) Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Euro. J. Appl. Math. 3, 193--207. MR1182212 (94f:76016)
9. Richardson, S. (1994) Hele-Shaw flows with time-dependent free boundaries in which the fluid occupies a multiply-connected region. Euro. J. Appl. Math. 5, 97--122. MR1285033 (95d:76043)
10. Richardson, S. (1996) Hele-Shaw flows with time-dependent free boundaries involving a concentric annulus. Phil. Trans. R. Soc. Lond. A, 354, 2513--2553. MR1419926 (98a:76028)
11. Richardson, S. (1997) Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension. Euro. J. Appl. Math. 8, 311--329. MR1471595 (98g:76023)
12. van de Vorst, G. A. L. (1993) Integral method for a two-dimensional Stokes flow with shrinking holes applied to viscous sintering. J. Fluid Mech. 257, 667--689.

Summary: "We present a new family of analytic solutions for Hele-Shaw flows with time-dependent free boundaries, a constant-pressure condition being supposed relevant on each free boundary. The basic starting configuration has the plan-view of the region occupied by fluid within the Hele-Shaw cell as an infinite strip bounded by parallel lines, these lines being the free boundaries. The subsequent motion is produced by the injection (or suction) of fluid at points within the strip and, perhaps, the imposition of a pressure difference between the two boundaries."

Summary: "We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface, the motion being driven solely by a constant surface tension acting at the free boundary. Of particular concern here are those flows that start from an initial configuration with the fluid occupying an array of touching circular disks. We show that, when there are $N$ such disks in a general position, the evolution of the fluid region is described by a conformal map involving $2N-1$ time-dependent parameters whose variation is governed by $N$ invariants and $N-1$ first-order differential equations. When $N=2$, or when the problem enjoys some special features of symmetry, the moving boundary of the fluid domain can be determined by solving purely algebraic equations, the solution of a single differential equation being needed only to link a particular boundary shape to a particular time. The analysis is aided by exploiting a connection with Hele-Shaw free boundary flows when the zero-surface-tension model is employed. If the initial configuration for the Stokes flow problem can be produced by injection (or suction) at $N$ points into an initially empty Hele-Shaw cell, as can the $N$-disk configuration referred to above, then so can all later configurations; the points where the fluid must be injected move, but the amount to be injected at each of the $N$ points remains invariant. The efficacy of our solution procedure is illustrated by a number of examples, and we exploit the method to show that the free boundary in such a Stokes flow driven by surface tension alone may pass through a cusped state."

1. Cummings, L. J., Howison, S. D. & King, J. R. (1997) Conserved quantities in Stokes flow with free surfaces. Phys. Fluids 9, 477--480. MR1437543 (97k:76033)
2. Hille, E. (1969) Lectures on Ordinary Differential Equations. Addison-Wesley. MR0249698 (40 #2939)
3. Hopper, R. W. (1984) Coalescence of two equal cylinders: Exact results for creeping viscous plane flow driven by capillarity. J. Am. Ceram. Soc. (Comm.) 67, C262--264.
4. Hopper, R. W. (1990) Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349--375. MR1051541 (91c:76033)
5. Hopper, R. W. (1991) Plane Stokes flow driven by capillarity on a free surface. Part 2. Further developments. J. Fluid Mech. 230, 355--364. MR1125711 (92h:76033)
6. Hopper, R. W. (1992) Stokes flow of a cylinder and half-space driven by capillarity. J. Fluid Mech. 243, 171--181.
7. Hopper, R. W. (1993) Capillarity-driven plane Stokes flow exterior to a parabola. Quart. J. Mech. Appl. Math. 46, 193--210. MR1125711 (92h:76033)
8. Howison, S. D. & Richardson, S. (1995) Cusp development in free boundaries, and two-dimensional slow viscous flows. Euro. J. Appl. Math. 6, 441--454. MR1363757 (96g:76018)
9. Klein Obbink, B. (1995) Moving boundary problems in relation with equations of Löwner-Kufareev type. PhD thesis, Eindhoven University of Technology.
10. Richardson, S. (1968) Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33, 475--493. MR0233549 (38 #1870)
11. Richardson, S. (1973) Two-dimensional bubbles in slow viscous flows. Part 2. J. Fluid Mech. 58, 115--127. MR0233549 (38 #1870)
12. Richardson, S. (1992) Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Euro. J. Appl. Math. 3, 193--207. MR1182212 (94f:76016)
13. Richardson, S. (1996a) Hele-Shaw flows with free boundaries driven along infinite strips by a pressure difference. Euro. J. Appl. Math. 7, 345--366. MR1413369 (97e:76088)
14. Richardson, S. (1996b) Hele-Shaw flows with time-dependent free boundaries involving a concentric annulus. Phil. Trans. R. Soc. Lond. A 354, 2513--2553. MR1419926 (98a:76028)
15. Richardson, S. (1997a) On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows. Quart. Appl. Math. 55, 313--319. MR1447579 (98f:76029)
16. Richardson, S. (1997b) Hele-Shaw flows with time-dependent free boundaries involving an infinite strip of fluid. Quart. Appl. Math. (to appear). cf. MR1686185 (2000a:76062)
17. Tanveer, S. & Vasconcelos, G. L. (1994) Bubble breakup in two-dimensional Stokes flow. Phys. Rev. Lett. 73, 2845--2848.
18. Tanveer, S. & Vasconcelos, G. L. (1995) Time-evolving bubbles in two-dimensional Stokes flow. J. Fluid Mech. 301, 325--344. MR1361287 (96i:76036)
19. van de Vorst, G. A. L. (1994) Modelling and numerical simulation of viscous sintering. PhD thesis, Eindhoven University of Technology. MR1254883 (95e:76024)

The author considers Hele-Shaw flow describing the evolution of a free boundary with zero surface tension. Initially, the fluid occupies a region that is bounded, connected and simply connected. The motion results from injection or suction at a single interior point. Interestingly, in the case of suction, the connectivity of the region occupied by the flow may be destroyed, resulting in a possibly infinite number of fluid islands isolated from the suction point. Such a solution is a consequence of the time-reversible nature of the flow. For example, suppose initially that the plane contains a number of disjoint islands of fluid. By beginning the injection of fluid from any point in the plane, the expanding boundary eventually captures and absorbs the islands. By reversing the process with suction, the boundary retreats and the original distribution is restored. The Cauchy transform is exploited to construct explicit analytic solutions that display these processes.

The paper addresses the Hele-Shaw flow in a doubly-connected region, driven by injection or suction at specific points, as well as by different pressures at the free boundaries, with a possibly variable area of the blob enclosed by the inner boundary (gas addition or removal). It is shown that, if the initial region is a concentric circular annulus, then there exists an analytic solution expressed in terms of time-dependent conformal mappings with parameters satisfying a set of transcendental equations. Some illustrative examples are presented.

Summary: "Consider the classical Hele-Shaw situation with two parallel planes separated by a narrow gap, and suppose the plan-view of the region occupied by fluid to be confined to an infinite strip by barriers in the form of two infinite parallel lines. With the fluid initially occupying a bounded, simply-connected region that touches both barriers along a single line segment, we seek to predict the evolution of the plan-view as the blob of fluid is driven along the strip by a pressure difference between its two free boundaries. Supposing the relevant free boundary condition to be one of constant pressure (but a different constant pressure on each free boundary), we show that the motion is characterized by (a) the existence of two functions, analytic in disjoint halfplanes, that are invariants of the motion and (b) the centre of area of the plan-view of the blob having a component of velocity down the infinite strip that is simply related to the imposed pressure difference. These features allow explicit analytic solutions to be found; generically, the mathematical solution breaks down when cusps appear in the retreating free boundary. A rectangular blob, of course, moves down the strip unchanged, with no breakdown, but if it encounters stationary blobs of fluid placed within the strip then, modulo multiply-connected complications, these are first absorbed into the advancing front of the rectangular blobs and then disgorged from its retreating rear, leaving behind stationary blobs of exactly the same form in exactly the same place as those originally present, but consisting of different fluid particles. This soliton-like interaction involves no phase change: with a given pressure difference driving the motion, the rectangular blob is in the same position at a given time after the interaction as it would have been had no intervening blobs been present."

Summary: "We consider a family of problems involving two-dimensional Stokes flows with a time-dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there is no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed."

{For the entire collection see MR1363753 (96g:76002).}

Summary: "We consider the classical Hele-Shaw situation with two parallel planes separated by a narrow gap. A blob of Newtonian fluid is sandwiched between the planes, and we suppose its plan-view to occupy a bounded, multiply connected domain; physically, we have a viscous fluid with the holes giving rise to the multiple connectivity occupied by relatively inviscid air. The relevant free boundary condition is taken to be one of constant pressure, but we allow different pressures to act within the different holes, and at the outer boundary. The motion is driven either by injection of further fluid into the blob at certain points, or by injection of air into the holes to change their area, or by a combination of these; suction, instead of injection, is also contemplated. A general mathematical theory of the above class of problems is developed, and applied to the particular situation that arises when fluid is injected into an initially empty gap bounded by two straight, semi-infinite barriers meeting at right-angles: injection into a quarter-plane. For a range of positions of the injection point, air is trapped in the corner and, invoking images, the problem is equivalent to one involving a doubly-connected blob. When there is an air vent in the corner, so that the pressure is the same on the two free boundaries in these circumstances, the air hole rapidly disappears, as might be expected. If, however, there is no air vent and we suppose the air to be incompressible, so that the area of the region occupied by the air in the plan-view remains constant, we find there to be no solution within the framework of our model. Other scenarios within this same geometry, involving both suction and injection of fluid at the injection point, and air at the corner, are also examined."

Summary: "Let $D$ be a bounded, simply-connected domain in the plane whose boundary $\partial D$ is an analytic curve. We show that, if the unit normal to $\partial D$ is the boundary value of a function which is analytic in $D$, then $D$ must be a circular disc."

This paper concerns analytic solution of free boundary value problems for two-dimensional quasi-steady Stokes flow of an incompressible surface-tension-driven Newtonian fluid. The boundary of the fluid is a free surface under constant surface tension so that in the absence of gravity one expects the evolution of a simply connected region to become circular. A differential equation describing the evolution of the fluid from its initial configuration is derived. One consequence is that if the fluid initially occupies a region that is the image of the unit disk under a rational map, the region has this property so long as the evolution remains simply connected. Since conformal maps can be approximated arbitrarily closely by rational functions, a procedure is implied for approximating well-posed problems. Two examples are provided to illustrate the procedure.

This paper continues the author's imaginative and elegant application of complex analysis to problems of Hele-Shaw flow. In two earlier papers [J. Fluid Mech. 56 (1972), no. 4, 609--618; J. Fluid Mech. 102 (1981), 263--278; MR0612095 (82d:76023)], the author considered injection of fluid into a Hele-Shaw cell at one or more isolated points, the aim being to determine the moving free boundary of the injected fluid domain $D$ for all time. Here, methods of these papers are extended to deal with injection (or suction) from a slit or slits, the injection being uniform along the slit. Surface tension effects are neglected and $D$ is assumed to be simply connected. The Schwarz function $S$ of the (analytic) boundary curve $\partial D$ [P. J. Davis, The Schwarz function and its applications, Math. Assoc. Amer., Buffalo, NY, 1974; MR0407252 (53 \#11031)] and the Cauchy transform of $D$ (that is, the double integral over $D$ of the Cauchy kernel) play a fundamental role. One interesting deduction is that the Cauchy transform of a domain $D$ provides a generalization of the Schwarz function when $\partial D$ is not analytic. If $S$ is decomposed as $S_{\rm i}+S_{\rm e}$, with $S_{\rm i}$ analytic inside $D$, $S_{\rm e}$ analytic outside $D$ and zero at infinity, then, in the present circumstances, as in the previous papers, it is shown that $S_{\rm e}$ is known, and the function mapping the interior of the unit circle one-to-one conformally onto $D$ is thereby determined. Thus the problem is solved. In the earlier work, $S_{\rm e}$ was a rational function; but here the slits produce logarithmic singularities. Analytical and graphical results are given in three cases, the first source being a single slit, the second consisting of two slits of equal length at right angles, emanating from a point. The third example involves injection both through a slit and from a neighboring point; it is shown that injection through the slit followed by suction at the point may lead to development of a cusp in the free boundary.

Singular perturbation has been highly successful in solving many difficult problems in science and engineering. However, the conditions for its validity have never been established, and it is well known that sometimes it does generate erroneous answers. This brief note provides two extremely simple and instructive examples of its failures. Both examples are ODE problems (for which the exact solutions are known) with a single small parameter $R$ over an infinite domain. The author shows that (1) a straightforward small $R$ perturbation analysis fails to obtain a solution at all for the infinite domain problem, and that (2) a similar analysis for a finite domain problem does uneventfully generate the first two terms of the expansion without protest but this two-term expansion yields an incorrect approximate solution in the limit when the domain is later taken to infinity; the first sign of trouble for this second procedure appears only when the third term is being evaluated.

This paper extends the results in a paper by \n R. F. Millar\en [same journal 15 (1990), no. 1, 1--10; see the preceding review]. For $N\in\bold Z$, denote by $\scr S_N$ the class of all simple closed analytic curves $C$ in the plane whose Schwarz function $S_C$ is holomorphic outside $C$ and is $O(|z|^N)$ as $|z|\to\infty$.

As mentioned in the above-cited paper, $\scr S_N$ is empty when $N\leq-2$ while, for $-1\leq N\leq 1$, each member of $\scr S_N$ is an ellipse. Let $N\geq1$. In this paper it is shown that $C\in\scr S_N$ if and only if the conformal mapping $\varphi$ of the exterior of the unit disk $\Delta$ onto the exterior of $C$, with $f(\infty)=\infty$, has a holomorphic extension inside $\Delta$ whose only finite singularity is a pole of order $n$ at 0. Moreover, denoting by $\scr S$ the class of all simple closed analytic curves $C$ in the plane whose Schwarz function $S_C$ is holomorphic outside $C$ with an essential singularity at $\infty$, the author also shows that $C\in\scr S$ if and only if the above conformal mapping has a holomorphic extension inside $\Delta$ whose only finite singularity is an essential singularity at 0. The author characterizes also those $C$ for which $S'_C$ has no zero outside $C$.

Consider the domain $D$ consisting of the upper half plane from which a triangle $BCD$ is removed, where $B$ and $D$ lie on the real axis, $C$ is in the upper half plane and the interior angles at $B$ and $D$ are $\alpha \pi$ and $\beta \pi$, respectively. The conformal mapping of $D$ onto the upper half $z$-plane, for which we have $\infty\to\infty$, $B\to -1$ and $C\to 0$, is effected by the Schwarz-Christoffel transformation $${ {dw}\over{ dz}}=K{{z^{\alpha+\beta}}\over{(z+1)^\alpha(z-r)^\beta}},$$ where $K$ is a scaling and orientation factor, but $r$ is determined by the conditions imposed. \n O. F. Hughes\en [SIAM J. Math. Anal. 6 (1975), 258--261; MR0364616 (51 \#870)] presented two proofs that $r=\alpha/\beta$, one of which uses hypergeometric functions while the other uses Cauchy's theorem. \n A. Brown\en [Bull. Austral. Math. Soc. 10 (1974), 91--94; MR0338335 (49 \#3100)] deduced the result from relationships between hypergeometric functions. The author gives an elementary and short proof of this result which uses the Schwarz reflection principle and a Laurent expansion in a neighborhood of infinity.

From the text: "Suppose that a Newtonian fluid is injected at a given point into the narrow gap between two plane parallel surfaces, and consider the growth of the plan-view of the resultant blob of fluid when it is restricted by barriers placed within the gap. An analytic description of this growth is obtained when it is confined, firstly, to an infinite strip bounded by two infinite parallel lines and, secondly, to a semi-infinite strip bounded by two semi-infinite parallel lines and a finite line segment perpendicular to them. The solution for expansion within the infinite strip is generally applicable; that for expansion within the semi-infinite strip is applicable when the injection point is so placed that the portion of the semi-infinite strip not occupied by the blob remains connected throughout the motion---that is, in physical terms, when no air is trapped in the corners.

"The solutions obtained require the construction of doubly periodic functions of a complex variable with certain specified poles. We work with the Jacobian elliptic functions."

Author's summary: "We consider a blob of Newtonian fluid sandwiched in the narrow gap between two plane parallel surfaces. At some initial instant, its plan-view occupies a given, simply connected domain, and its growth as further fluid is injected at a number of injection points in its interior is to be determined. It is shown that certain functionals of the domain, of a purely geometric character, infinite in number, evolve in a predictable manner, and that these may be exploited in some cases of interest to yield a complete description of the motion.

"By invoking images, these results may be used to solve certain problems involving the growth of a blob in a gap containing barriers. Injection at a point in a half plane bounded by a straight line, with an initially empty gap, is shown to lead to a blob whose outline is part of an elliptic lemniscate of Booth for which there is a simple geometric construction. Injection into a quarter plane is also considered in some detail when conditions are such that the image domain involved is simply connected."

From the text: "It is the purpose of this note to draw attention to some connections between the topics mentioned in the title. Consider a point (or rectilinear) vortex in a two-dimensional, irrotational flow taking place in some bounded, simply connected domain $D$, the motion being generated by the vortex alone. The stream function of the flow is essentially the harmonic Green function in $D$ with pole at the vortex. If the vortex is free it will, in general, move under the influence of this self-induced flow field. If $D$ is such that the Green function can be constructed by any of the classical methods involving images, conformal mapping, separation of variables, etc., then the vortex paths may be determined explicitly.

"The above approach leads to the general conclusion that the motion of the vortex itself may be described by means of a stream function $\varphi$ defined in $D$, in the sense that the vortex moves along the level lines of $\varphi$, its velocity at any point being proportional to the magnitude of $\nabla\varphi$ at that point. Moreover, one finds that this stream function $\varphi$, suitably normalized, satisfies Liouville's equation $\nabla^2\varphi=e^\varphi$ in $D$ and is such that $\varphi\rightarrow+\infty$ as the boundary is approached.

"The Bergman kernel function is defined in Section 4. For present purposes we need note only that it is a function of four real variables or, equivalently, of two complex variables, defined on $D\times D$, which can be constructed to arbitrary accuracy using only the familiar Gram-Schmidt orthonormalization procedure for any bounded domain $D$. Equating the two complex variables in the kernel function we obtain a function of a single complex variable or, equivalently, of two real variables, defined in $D$, which may be identified with $(1/8\pi)e^\varphi$, where $\varphi$ is the solution of Liouville's equation mentioned in the previous paragraph. The vortex paths are thus also the level lines of a function which is easily obtained from the Bergman kernel function associated with the domain $D$."

The author considers Blasius's equation $(*)$ $y"'+yy"=0 (z\geq 0,{}'=d/dz)$, subject to the initial conditions $y(0)=y'(0)=0$, $y"(0)=1$. By introducing a new independent variable $p=y'(z)$ and dependent variable $z(p)$, $(*)$ may be written as $z"(p)=[z'(p)]^2\int_0^ppz'(p)\,dp$, where primes now denote differentiation with respect to $p$. A series solution may be found in the form $z'(p)=\sum_{n=0}^\infty b_np^{3n}(0\leq p<\beta=y'(\infty))$, where the initial conditions require $\tau_0=1$. This series has the advantage that it is valid over the whole range of physical interest, while a power series expansion of $y(z)$ has a limited radius of convergence and is accordingly not valid for all $z\geq 0$. Upper and lower bounds are found for $\alpha=[y'(\infty)]^{-3/2}$, which is a parameter of physical interest.

In an earlier paper A. J. Callegari and M. B. Friedman [J. Math. Anal. Appl. 21 (1968), 510--529; MR0224331 (36 \#7375)] gave the results of the present paper as well as other analytical and numerical results concerning the solutions of $(*)$. Their analysis was based on a transformation of $(*)$ to Crocco variables.

From the author's summary: "The representation of a biharmonic function in terms of analytic functions [cf., e.g., N. I. Musheli\v svili, Some basic problems of the mathematical theory of elasticity\/ (Russian), fifth edition, Izdat. "Nauka" Moscow, 1966; MR0202367 (34 \#2238)] is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory."