From fluid mechanics,
the equation of continuity (which is a subset of mass conservation)
demands that when you pass a flowing fluid from a wider orifice
into a narrow orifice, its velocity must go up.
That is to say, Av =
constant, where A: x-section area, v: fluid velocity.
So while the ketchup
may go up the dispenser tube at a relatively low velocity, if the
cross-sectional area of the dispenser exit is smaller than the
dispenser tube's cross-sectional area, the velocity has to go up.
And that's why the ketchup splashes out.
Will this explanation
suffice, or do you want to take into account the non-Newtonian
characteristics of the ketchup and pressure loss through pipe
friction?
What happened was that when TS pressed too fast, the pressure
increased in the dispensing pipe as well. The force of the ketchup
on the inner walls of the pipes became higher than atmospheric
pressure.
When the ketchup exits the pipe, there no more inner wall to
exert the force back on the ketchup, and hence, the resultant force
will cause the ketchup to spurt outwards, hence staining TS's
shirt.
To summarize, TS didn't have sufficient common sense to press
slowly.
