Portals in Ocean and Sky

The video game Portals is centered around the use of the Aperture Science Handheld Portal Device, or portal gun. The gun can fire projectiles that upon impact on a flat surface expand into either an orange or a blue portal. By entering one portal, the player can exit from the other.

For this blog post, I am taking inspiration from two other weeks. In this post, I will be looking at the flow rate of water if the portal gun created a portal at the bottom of the Mariana Trench and then, taking a flight up to the stratosphere like Felix Baumgartner did with Red Bull, fired another portal onto the balloon.

Intuitively, we know that the pressure at that height is very low whereas the pressure at that depth is very large. Thus, the water should flow from the trench portal out of the stratosphere portal.
To understand this, we will use Poiseuille's equation.

Since the portals make it so that a path from the ocean to the sky is not a straight line from a to b the idea of height, y, is made strange. One idea is that the portal is worm-hole-like. The two portals may be connected by an infinitesimally small tunnel of near zero length L.  So, the flow rate would be infinite. For the sake of argument, let us assume the tunnel is 10^-6 m long.

The portal can accommodate a young woman and have about 2 feet in height left. The average height of a 20+ American femaile is 5 ft 4 in according to Wikipedia. We will assume the portal can be approximated as a circle (it's really an ellipse) and use 1.9 meters as the diameter. The radius shall be 0.95 meters. The viscosity of water near 0 degrees Celsius will be 1.8*10^-3 Pa*s.

The Mariana Trench has a pressure of about 1086 bars, or 108,600,000 Pa according to Wikipedia. The pressure at that height is very low and we shall assume it is zero.Setting up the equation,

The resultant answer is 6.8 x 10^16 m/s . This is overly large due to the portal's small length and our assumption that the flow is not turbulent. The volume flow rate is about 1.9 x 10^16 m^3.