Two-Dimensional Problem

a type of problem in mathematical physics. It arises when (1) the pattern of the phenomenon under study is the same in all planes parallel to a given plane or (2) as a result of disregarding one of three dimensions, a problem is reduced to a two-dimensional problem.

Areas where two-dimensional problems are encountered include the theory of elasticity, hydromechanics, aeromechanics, the theory of electricity, and the theory of heat conduction. In elasticity theory, for example, the two-dimensional problem arises when the lateral surface of an infinitely long cylinder is subjected to a load that is constant along each generatrix of the cylinder. To study phenomena inside the cylinder it is sufficient to study the phenomena in any of the planes perpendicular to its generatrices. If the phenomenon under study is steady-state— that is, if the pattern of the phenomenon does not vary with time —then the solution of the two-dimensional problem is in many cases associated with the theory of the functions of a complex variable and is carried out by the methods of conformal mapping. An example is the problem of the lines of flow around objects with different profiles in hydrodynamics and aerodynamics.