Editing 969: Delta-P

{{comic
| number    = 969
| date      = October 26, 2011
| title     = Delta-P
| image     = delta p.png
| titletext = If you fire a Portal gun through the door of the wardrobe, space and time knot together, which leads to a frustrated Aslan trying to impart Christian morality to the Space sphere.
}}

==Explanation==
''{{w|The Lion, the Witch and the Wardrobe}}'' is a book in which four children accidentally wander into a world known as {{w|Narnia}} through a wardrobe that only allows passage through to Narnia when you aren't looking for it. In the comic, someone connects an anchor to the wardrobe and throws it into the ocean. The formula describes the flow of water through the open doors when the wardrobe sits at the bottom of the ocean in 2 km depth, which means that a steady stream of water at an approximate velocity of 200 meters per second will flow into Narnia.

The evil {{w|White Witch}}, who has made it [http://www.goodreads.com/quotes/93155-always-winter-but-never-christmas "always winter, and never Christmas,"] could not have anticipated that a wardrobe portal would suddenly begin spewing approximately 400,000 liters of water per second into Narnia.{{Citation needed}}

Sea water freezes at low temperatures and flowing water freezes at even lower temperatures, depending how fast it is going. Water jetting out from this portal would be flowing very quickly indeed, approximately 200 meters per second (450 mph or 720 km/h) as the comic says; this is over half the speed of sound. And the water flow is approximately 400,000 liters per second, again, provided in the image above. The force of this water jet would be incredible. If you can imagine the force of a tractor trailer truck careening down the highway, the force of this jet is much much larger than that.

This water would not freeze. First it would devastate any forest trees or iron lamp posts in front of it until it eventually slowed down and fell to the ground. There it would create a rapidly expanding river of sea water. Narnia would not stay frozen for long. Snow would melt, ice would break apart and the valley would quickly flood.

Delta-P is a mathematical term for the difference in pressure. The shown formula is based on the {{w|Hagen–Poiseuille equation}} which can be applied to a flowing liquid in a long cylindrical pipe; thus the equation here results in an unphysically high flow rate because the opening is rectangular and too short for a {{w|Laminar flow|laminar flow}}. Using the ''Hagen–Poiseuille equation'' the maximum flow rate is given by:
:<math>Q_\max{} = \pi R^2 \sqrt\frac{2 \Delta P}{\rho}</math>
:<math>\pi R^2</math> is the pipe cross-sectional area (m<sup>2</sup>) and <math>\rho</math> is the fluid density (kg/m<sup>3</sup>).
From the {{w|Hydrostatics|hydrostatics}} of water the pressure difference depends on gravity and the height:
:<math>\Delta P = \rho g h,</math>
:<math>g</math> is the gravitational acceleration (m/s<sup>2</sup>) and <math>h</math> is the height (m).
Putting this together and changing the cross-sectional area to a rectangular area <math>A</math> we get the formula used by Randall:
:<math>Q_\max{} = A \sqrt{2 g h}</math>
Assuming the wardrobe is two meter high and one in width (''A = 2 m<sup>2</sup>'') and using the gravitational constant ''g = 9.81 m/s<sup>2</sup>'' the flow rate is 396 m<sup>3</sup> per second, or roughly 400,000 liters per second.
The water jet velocity ''v'' is based on {{w|Torricelli's law}}:
:<math>v=\sqrt{{2 g}{h}}</math>
It gives 198 <sup>m</sup>/<sub>s</sub> in this scenario.

The title text references the video game ''{{w|Portal (video game)|Portal}}'' in which you solve puzzles using a gun which projects portals onto certain surfaces. In the game you cannot shoot a portal through a portal, but [[Randall]] says that if you try to create a portal with the portal gun through the wardrobe, space and time knot together.   C. S. Lewis's ''{{w|The Chronicles of Narnia}}'' series (like most of his writing) is influenced by his views on Christianity and morality (he did not set out to write a "Christian story", but later accepted that there are many parallels between the ''Chronicles'' and his faith), with {{w|Aslan}} the lion representing Jesus Christ.  As such, Aslan often provides lessons and advice on morality and faith to the main characters; however, the Space Sphere (a minor character in the game ''{{w|Portal 2}}'') is a barely-sentient AI whose only preoccupation is going to space, and it would not be receptive to Aslan's teachings.  The Space Sphere might be more interested in Lewis's ''{{w|The Space Trilogy}}'', a trilogy of science-fiction books in which the main character travels through space and learns that the divine struggles between good and evil on Earth are also reflected elsewhere in the solar system.

=== Trivia ===

* In the dark fantasy light novel series ''{{w|Goblin Slayer}}'' (and its anime adaptation), the titular main character uses a portal scroll keyed to the bottom of the ocean to slay a powerful ogre with a water jet as described here.

==Transcript==
:[An open wardrobe, with a boat anchor attached to one corner, falling towards water.]
:[Below the water line a formula with its variables explained is shown:]
:Q = A * sqrt(2 * g * d)
:Q = flow rate
:A = area of opening
:d = ocean depth (2&nbsp;km)
:g = Earth gravity

:[And below the formula:]
:Flow: ~400,000 <sup>liters</sup>/<sub>s</sub>
:Water jet velocity: ~200 <sup>m</sup>/<sub>s</sub>

:[Caption below the frame:]
:The White Witch didn't know what hit her.

{{comic discussion}}
[[Category:Comics with color]]
[[Category:Physics]]
[[Category:Video games]]
[[Category:Fiction]]
[[Category:Chronicles of Narnia]]