Cross-talk between stagnation points in flow and orientation

What happens when you press two books horizontally against each other? The built-up pressure eventually will be released in an upward motion of the top pages and a downward motion of the bottom pages, like mountains that form from the earth's crust, but in two directions. However the question remains which way the center page goes. In theory, this is undecided, and soft matter physicists call this a stagnation point or defect. Also fluids have stagnation points, for example a whirl in a bathtub.