Whether the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density

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You asked“can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time.”

Talkie was asked“Make a slide deck about whether the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density.”

Talkie's reply was started withset-background(color=" (the worked example's first words, so that it writes a deck rather than an answer)

ModelFull model

Seed1930

Temperature0.35

Writing took17 min 3 s

Valid commands27 of 27

Ended itselfYes, with end-deck()

set-background(color="black") Slide 1: the background is now #000000.
set-text(text="The Navier-Stokes equations for a fluid at rest are dV=0, and dQ=0.") Slide 1: text added.
next-slide() Slide 1 is finished. Now working on slide 2.
find-image(search-term="spherical shell", filename="A spherical shell.jpg") Saved "A spherical shell.jpg": Title: Lt. Porter tries spherical case, and shell Abstract/medium: 1 drawing on tan paper : pencil and Chinese white ; 10.0 x 26.2 cm. (sheet). (1864)
set-image(filename="A spherical shell.") Slide 2: "A spherical shell." will be found when the slide is finished, to suit its words; it will be captioned "A spherical shell".
set-text(text="The shell is bounded by two similar and equal surfaces,") Slide 2: text added.
next-slide() Saved "A spherical shell.": Title: Lt. Porter tries spherical case, and shell Abstract/medium: 1 drawing on tan paper : pencil and Chinese white ; 10.0 x 26.2 cm. (sheet). (1864) Slide 2 is finished. Now working on slide 3.
find-image(search-term="spherical cavity", filename="A spherical cavity.jpg") Saved "A spherical cavity.jpg": A complete lead or lead alloy pistol shot of the Post Medieval period, dating from c. AD 1600 - c. AD 1800. The ball is broadly spherical in plan and is... (between 1600 and 1800)
set-image(filename="A spherical cavity.") Slide 3: "A spherical cavity." will be found when the slide is finished, to suit its words; it will be captioned "A spherical cavity".
set-text(text="and has a volume equal to one-third of the volume of the shell.") Slide 3: text added.
next-slide() Saved "A spherical cavity.": A complete lead or lead alloy pistol shot of the Post Medieval period, dating from c. AD 1600 - c. AD 1800. The ball is broadly spherical in plan and is... (between 1600 and 1800) Slide 3 is finished. Now working on slide 4.
find-image(search-term="spherical volume", filename="A spherical volume.jpg") Saved "A spherical volume.jpg": With spherical coordinates volume of a sphere P : r,zr,xy -> 1 ar : zr,xy -> 0 [0..oo] br : zr,xy -> 1 azr : xy -> 0 [0..PI] bzr : xy -> PI axy : -> 0 [0..2PI]... (2024-02-22)
set-image(filename="A spherical volume.") Slide 4: "A spherical volume." will be found when the slide is finished, to suit its words; it will be captioned "A spherical volume".
set-text(text="The volume of the shell is two-thirds of the volume of the cavity.") Slide 4: text added.
next-slide() Saved "A spherical volume.": Volume element in spherical coordinates (11/14/08) Slide 4 is finished. Now working on slide 5.
find-image(search-term="spherical density", filename="Density of a spherical shell.jpg") Saved "Density of a spherical shell.jpg": Schrödinger 3D Spherical Harmonic Orbital solutions in 3D Density plots with Mathematica(TM) source code snippet at the top (2013-05-26)
set-image(filename="Density of a spherical shell.") Slide 5: "Density of a spherical shell." will be found when the slide is finished, to suit its words; it will be captioned "Density of a spherical shell".
set-text(text="The density of the shell is two-thirds of the density of the cavity.") Slide 5: text added.
next-slide() Saved "Density of a spherical shell.": This is a CROPPED PART(!) of a figure from the study. Screenshot was made via Okular & Spectacle. Description for the complete(!) figure below. See the source... (2021-09-20) Slide 5 is finished. Now working on slide 6.
find-image(search-term="spherical pressure", filename="Pressure on a spherical shell.jpg") Saved "Pressure on a spherical shell.jpg": مخازن تحت فشار کروی، به دلیل ساختار مستحکم برای سیالات پرفشار ایده آل هستند، اما ساخت آن‌ها دشوار و پرهزینه است. تنش داخلی و خارجی به طور منظم روی سطح کره... (2024-01-03)
set-image(filename="Pressure on a spherical shell.") Slide 6: "Pressure on a spherical shell." will be found when the slide is finished, to suit its words; it will be captioned "Pressure on a spherical shell".
set-text(text="The pressure on the shell is two-thirds of the pressure in the cavity.") Slide 6: text added.
next-slide() Saved "Pressure on a spherical shell.": Caldwell 90 Slide 6 is finished. Now working on slide 7.
find-image(search-term="spherical velocity", filename="Velocity of a spherical shell.jpg") Saved "Velocity of a spherical shell.jpg": The linear model derived from Stokes’ Law is accurate for small particles with low Reynolds numbers. Parameterized terminal velocity from Clift, Grace & Weber... (2023-04-27)
set-image(filename="Velocity of a spherical shell.") Slide 7: "Velocity of a spherical shell." will be found when the slide is finished, to suit its words; it will be captioned "Velocity of a spherical shell".
set-text(text="The velocity of the shell is two-thirds of the velocity in the cavity.") Slide 7: text added.
end-deck() The deck is finished (7 slides in all).

Found by description on Wikimedia Commons and Openverse: public domain, CC0, CC BY and CC BY-SA only. Neither Talkie nor the deck runner sees them; each is captioned with the filename its author chose.

A spherical shell.jpg Lt. Porter tries spherical case, and shell LCCN2004660061.jpg Wikimedia Commons · Waud, Alfred R. (Alfred Rudolph), 1828-1891, artist · 1864 · Public domain
A spherical shell. Lt. Porter tries spherical case, and shell LCCN2004660061.jpg Wikimedia Commons · Waud, Alfred R. (Alfred Rudolph), 1828-1891, artist · 1864 · Public domain
A spherical cavity.jpg Post Medieval, Pistol shot (with casting cavity) (FindID 850538).jpg Wikimedia Commons · Birmingham Museums Trust, Helen Glenn, 2017-06-01 12:29:41 · between 1600 and 1800 · CC BY 2.0
A spherical cavity. Post Medieval, Pistol shot (with casting cavity) (FindID 850538).jpg Wikimedia Commons · Birmingham Museums Trust, Helen Glenn, 2017-06-01 12:29:41 · between 1600 and 1800 · CC BY 2.0
A spherical volume.jpg My spherical coordinates.png Wikimedia Commons · Xhungab · 2024-02-22 · CC0
A spherical volume. Volume element spherical coordinates.JPG Wikimedia Commons · Brews ohare · 11/14/08 · CC BY-SA 3.0
Density of a spherical shell.jpg 3D Spherical Harmonic Orbitals.png Wikimedia Commons · Jgmoxness · 2013-05-26 · CC BY-SA 4.0
Density of a spherical shell. Side-on 3D view of the Per-Tau Shell, giant structure forming star-forming molecular clouds (with Sun).png Wikimedia Commons · Authors of the study: Shmuel Bialy, Catherine Zucker, Alyssa Goodman, Michael M. Foley, João Alves, Vadim A. Semenov, Robert Benjamin, Reimar Leike, Torsten Enßlin · 2021-09-20 · CC BY 4.0
Pressure on a spherical shell.jpg Spherical pressure vessels damirchi.webp Wikimedia Commons · Mrdarkviolet · 2024-01-03 · CC BY-SA 4.0
Pressure on a spherical shell. Caldwell 90 Openverse (flickr) · NASA Hubble · CC BY 2.0
Velocity of a spherical shell.jpg Terminal Velocity of a Spherical Particle in 20C Water.png Wikimedia Commons · MMenczer · 2023-04-27 · CC BY-SA 4.0
Velocity of a spherical shell. Retardierte.Gravitationspotentiale.png Wikimedia Commons · Bautsch · 2024-02-22 · CC0

Everything Talkie was told, and its replies

# Transcript: 1 call to Talkie

## Call 1: pytorch, max_tokens 1600, temperature 0.35, seed 1930 (1022.5 s)

**User**

````text
You are to make a slide deck by writing commands, one on each line. These are the commands:
find-image(search-term="WORDS", filename="NAME.jpg") finds a picture of WORDS and keeps it under the name NAME.jpg.
set-background(color="COLOUR") colours the whole slide, for example "navy", "ivory", "black" or "crimson".
set-image(filename="NAME.jpg") shows a picture you have already found.
set-text(text="WORDS") writes words at the bottom of the slide; set-text(text="") removes them.
next-slide() finishes the slide and begins a new, empty one.
end-deck() ends the deck; write it once, after the last slide.
All the text messages must be meaningful when strung together: read one after another, from the first slide to the last, they should make one connected account.
Write only commands.

Make a slide deck about the sea.
````

**Talkie**

````text
set-background(color="navy")
set-text(text="The sea covers the greater part of the world.")
next-slide()
find-image(search-term="ocean waves", filename="Ocean waves.jpg")
set-background(color="black")
set-image(filename="Ocean waves.jpg")
set-text(text="Its waves are raised by the wind,")
next-slide()
find-image(search-term="tide on the shore", filename="The tide.jpg")
set-background(color="black")
set-image(filename="The tide.jpg")
set-text(text="and twice each day its tide rises and falls upon the shore.")
next-slide()
find-image(search-term="sailing ship", filename="A sailing ship.jpg")
set-background(color="black")
set-image(filename="A sailing ship.jpg")
set-text(text="Upon its waters men have sailed ships since ancient times,")
next-slide()
find-image(search-term="lighthouse", filename="A lighthouse.jpg")
set-background(color="black")
set-image(filename="A lighthouse.jpg")
set-text(text="guided at night by lighthouses upon the rocks.")
next-slide()
find-image(search-term="fishing boats", filename="Fishing boats.jpg")
set-background(color="black")
set-image(filename="Fishing boats.jpg")
set-text(text="Fishermen draw from it a great harvest of fish,")
next-slide()
find-image(search-term="whale", filename="A whale.jpg")
set-background(color="black")
set-image(filename="A whale.jpg")
set-text(text="and in its depths swims the whale, the largest of all creatures.")
next-slide()
find-image(search-term="storm at sea", filename="A storm at sea.jpg")
set-background(color="navy")
set-image(filename="A storm at sea.jpg")
set-text(text="Yet for all our ships, the sea remains as wide and wild as ever.")
end-deck()
````

**User**

````text
Make a slide deck about whether the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density.
````

**Talkie (given, continued below)**

````text
set-background(color="
````

**Talkie's reply**

````text
black")
set-text(text="The Navier-Stokes equations for a fluid at rest are dV=0, and dQ=0.")
next-slide()
find-image(search-term="spherical shell", filename="A spherical shell.jpg")
set-image(filename="A spherical shell.")
set-text(text="The shell is bounded by two similar and equal surfaces,")
next-slide()
find-image(search-term="spherical cavity", filename="A spherical cavity.jpg")
set-image(filename="A spherical cavity.")
set-text(text="and has a volume equal to one-third of the volume of the shell.")
next-slide()
find-image(search-term="spherical volume", filename="A spherical volume.jpg")
set-image(filename="A spherical volume.")
set-text(text="The volume of the shell is two-thirds of the volume of the cavity.")
next-slide()
find-image(search-term="spherical density", filename="Density of a spherical shell.jpg")
set-image(filename="Density of a spherical shell.")
set-text(text="The density of the shell is two-thirds of the density of the cavity.")
next-slide()
find-image(search-term="spherical pressure", filename="Pressure on a spherical shell.jpg")
set-image(filename="Pressure on a spherical shell.")
set-text(text="The pressure on the shell is two-thirds of the pressure in the cavity.")
next-slide()
find-image(search-term="spherical velocity", filename="Velocity of a spherical shell.jpg")
set-image(filename="Velocity of a spherical shell.")
set-text(text="The velocity of the shell is two-thirds of the velocity in the cavity.")
end-deck()
````

Whether the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density

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