An approximate method for estimation of fluid motion in a lake containing islands

The physical phenomenon considered in the paper deals with flow in a lake, which contains one or several islands. The flow is generated by rivers, their inlets and outlets being distributed on the shoreline of the lake. The problem to be solved consists in determination of the velocity field - in the domain bounded by the shorelines of the lake and the islands. In fact, it is attempted to arrive solely at an estimation of the field. Consequently, a rather simple physical model of the phenomenon, as well as its simple mathematical description has been applied. In particular, plane, irrotational and steady flow of ideal liquid has been introduced, the inlets and outlets of the rivers being simulated by sources and sinks. Hence, the problem reduces to determination of a complex function, representing the complex velocity field, which satisfies the impermeability condition on all contours representing the shorelines, the field being generated by the singularities already mentioned. Unfortunately, the so formulated problem is "overconditioned" or "too stiff", what means that the impermeability condition on the outer contour cannot be satisfied. Nevertheless, we arrived at a simple method for circumventing this obstacle, the payoff consisting in some modification of this contour. We had this particular modification in mind, applying the word "approximate" in the title of the paper. The paper contains results - in the form of streamline patterns - for lakes containing from 1 to 3 islands. In the relevant figures the distances between the given and the modified exterior contours can be seen distinctly - allowing the reader to draw conclusions, whether the errors due to the modifications are admissible or not. Of course, it depends anyway on the point of view of the user of the results

A method for determination of flow at the plane, horizontal interface separating streams of two different fluids

The paper deals with the theoretical investigation of the phenomenon, which consists in generation - by wind - of the movement of water, the bodies of air and of water being separated by a horizontal, plane interface, which is identical with the free surface of water. A set of assumptions defining a physical model of the phenomenon, borrowed from Lock (1951), is introduced, as well as the mathematical description of this model. It reduces to a composite ordinary differential problem, containing two non-linear equations of the third order, which have to satisfy some boundary conditions. A novel method of solution of the differential problem just mentioned is presented in the paper. In the method use is made of exact formulae for coefficients of series representing the solution. The method seems to be competitive with the one given in the paper by Lock (1951)

Transformation of the Vistula Lagoon onto a canonical domain

The paper deals with the conformal mapping of finite, plane, simply connected domains, representing oceans, lakes, estuaries, bays, lagoons, and other natural water bodies of this kind. As a rule, they are bounded by geometrically complex shorelines. The partial differential problems investigated in Oceanology and posed in such domains have turned out to be difficult to solve for at least three reasons. They follow on from the mathematical properties of the differential equations governing such problems, from the just-mentioned geometrical complexity of the domains of solution, and from the sensitivity of the solutions to boundary conditions. In view of the last reason the contours admitted as boundaries of the domains of the solution ought to be as close to the real shorelines as possible. The obviously inaccurate approximation of the shorelines by "staircases", which appears rather often (cf. Catewicz & Jankowski 1983, Lin & Chandler-Wilde 1996) as a consequence of applying finite difference methods to the solution of the partial differential problems, raises serious doubts from the point of view of Numerical Fluid Mechanics. It is recalled in the paper that such inaccuracies are not unavoidable: that complicated plane domains can be transformed accurately by means of properly applied conformal mapping onto regular, canonical domains - in particular, onto discs or squares. Such a transformation is demonstrated on the rather difficult example of the Vistula Lagoon. The transformation begins with the decomposition of the domain into five plane subdomains, each one of which is eventually transformed onto a disc. Every such result is arrived at quite independently of the remaining subdomains, by means of a set of properly selected consecutive mappings. Hence, the final canonical domain consists in this case of a system of five discs which, however, within the framework of this differential problem, have to be treated as interconnected. The interconnections involve images of four segments of straight lines, separating the original subdomains. The transformations and the resulting canonical domain presented in the paper are intended to be applied to the solution of certain hydrodynamical problems concerning the Vistula Lagoon, which will be published elsewhere