Fluid models have been used as toy-models of event horizons in general relativity and its generalizations to more than four spacetime dimensions. We study here a fluid model for the Gregory-Laflamme instability in black strings. With consider a Newtonian, incompressible, static, axially-symmetric fluid with surface tension, in n dimensions plus one periodic dimension. The fluid configurations are those that minimize the fluid surface area for fixed volume. Homogeneous fluid configurations are known to be stationary solutions of this functional, and they are stable in a dynamical sense above a critical value of the fluid volume. Below that value Plateau-Rayleigh instabilities occur. We show in this article that at this critical value of the volume there is a pitchfork bifurcation point. We prove that there are infinitely many other pitchfork bifurcation points at smaller values of the fluid volume. Each bifurcation solution represents a non-homogeneous static fluid configuration and its stability depends on the space dimension. By stability we mean in the sense of minimum of the above functional. We show that the non-homogeneous configurations are all unstable if n less or equal 10, and they all become stable if n bigger or equal 11. This stability inversion for high space dimensions could be of interest in gravitational theories in more than four dimensions and in string theory.