The fluid motion in a vortex creates a dynamic pressure (in addition to any hydrostatic pressure) that is lowest in the core region, closest to the axis, and increases as one moves away from it, in accordance with Bernoulli's principle.  One can say that it is the gradient of this pressure that forces the fluid to follow a curved path around the axis.
In a rigid-body vortex flow of a fluid with constant density, the dynamic pressure is proportional to the square of the distance  from the axis. In a constant gravity field, the free surface of the liquid, if present, is a concave paraboloid.
In an irrotational vortex flow with constant fluid density and cylindrical symmetry, the dynamic pressure varies as , where  is the limiting pressure infinitely far from the axis.  This formula provides another constraint for the extent of the core, since the pressure cannot be negative. The free surface (if present) dips sharply near the axis line, with depth inversely proportional to .  The shape formed by the free surface is called a hyperboloid, or "Gabriel's Horn" (by Evangelista Torricelli).
The core of a vortex in air is sometimes visible because water vapor condenses as the low pressure of the core causes adiabatic cooling; the funnel of a tornado is an example. When a vortex line ends at a boundary surface, the reduced pressure may also draw matter from that surface into the core. For example, a dust devil is a column of dust picked up by the core of an air vortex attached to the ground. A vortex that ends at the free surface of a body of water (like the whirlpool that often forms over a bathtub drain) may draw a column of air down the core. The forward vortex extending from a jet engine of a parked airplane can suck water and small stones into the core and then into the engine.