This is a study of the spontaneous formation of electric current sheets in an incompressible viscous fluid with perfect electrical conductivity, governed by the magnetohydrodynamic Navier–Stokes equations. Numerical solutions to two initial value problems are presented for a three-dimensional, periodic, untwisted magnetic field evolving, with no change in magnetic topology under the frozen-in condition and at characteristic fluid Reynolds numbers of the order of 500, from a nonequilibrium initial state with the fluid at rest. The evolution converts magnetic free energy into kinetic energy to be all dissipated away by viscosity so that the field settles into a minimum-energy, static equilibrium. The solutions demonstrate that, as a consequence of the frozen-in condition, current sheets must form during the evolution despite the geometric simplicity of the prescribed initial fields. In addition to the current sheets associated with magnetic neutral points and field reversal layers, other sheets not associated with such magnetic features are also in evidence. These current sheets form on magnetic flux surfaces. This property is used to achieve a high degree of the frozen-in condition in the simulations, by describing the magnetic field entirely in terms of the advection of its flux surfaces and integrating the resulting governing equations with a customized version of a general-purpose high-resolution (viz., nonoscillatory) hydrodynamical simulation code EULAG [ J. M. Prusa et al., Comput. Fluids 37, 1193 (2008) ]. Incompressibility imposes the additional global constraint that the flux surfaces must evolve with no change in the spatial volumes they enclose. In this approach, current sheet formation is demonstrated graphically by the progressive pressing together of suitably selected flux surfaces until their separation has diminished below the minimal resolved distance on a fixed grid. The frozen-in condition then fails in the simulation as the field reconnects through an effecting numerical resistivity. The principal results are related to the Parker theory of current-sheet formation and dissipation in the solar corona.