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Fluid Dynamics and the Navier-Stokes Equations

Formation of Spherical Body

Deformation of Solid Body Geometric Shape Solid body does not flow like fluid, the Navier-Stokes Equations doesn't seem to be applicable in such case. However, the equation can be used to calculate the critical mass for the self-gravity of a solid body to overcome its resistant forces so that it assumes a hydrostatic equilibrium (nearly round) shape. Solid body actually has

Figure 21 Deformation of Solid Body

Figure 22 Geometric Shape [view large image]

 elastic property, that makes it to deform in response to the external force as shown by the deformation of the moon by the gravitational tug of a planet in Figure 21.

Since there is no flow in solid body u 0 in the Navier-Stokes Equation (2), which then becomes:
F = p / ---------- (27).
The external force per unit mass is just the gravitational force (per unit mass):
F = GM / r2 ---------- (28).
The pressure gradient in terms of the spherical coordinate r is:
p = dp / dr = B / r ---------- (29)
where B = stress/(% volume compression) = dp/(V/V) is the Bulk modulus, the value of which ranges from 0.02 x 1012 dynes/cm2 for water to about 4.4 x 1012 dynes/cm2 for diamond. It lumps all the microscopic forces into one macroscopic measurement.

Substituting Eq.(28) and (29) into Eq.(27) yields:
GM / r2 = B / r. ---------- (30)

There are 3 possible cases for the development of geometric shape :
Case 1: GM / r2 B / r, the solid body does not has enough gravitational pull to overcome the resistance, its geometric shape would be irregular as shown by the Itokawa asteroid in Figure 22.
Case 2: GM / r2 = B / r, the solid body just has enough gravitational pull to overcome the resistance, its geometric shape would be round as shown by the Earth/Moon in Figure 22.
Case 3: GM / r2 B / r, the solid body has more than enough gravitational pull to overcome the resistance. The object would shrink until equilibrium is achieved.

Eq.(30) can be simplified further, if we assume uniform distribution of matter: = M/(4r3/3), thus it can be rewritten as:

B = GM2/(4/3)r4 = 1.6x10-8(M2/r4) ---------- (31)

Table 01 lists some solid objects have (or have not) sufficient gravitational strength to develop a spherical shape (to overcome the elastic resistance of the material).

Object Type Mass M (gm) Radius or Size r (cm) B (1012dynes/cm2) Geometric Shape
Neutron Star Star 0.2 - 6.0x1033 ~1.0x106 ~ 1022 - Super-hard Spherical
Mercury Planet 3.6x1026 1.1x108 14.4 Spherical
Venus Planet 5.7x1027 5.7x108 5.2 Spherical
Earth Planet 6.0x1027 6.0x108 4.5 Spherical
Moon Satellite 7.2x1025 7.8x107 2.3 Spherical
Mars Planet 6.6x1026 3.0x108 0.9 Spherical
Pluto Dwarf Planet 1.2x1025 1.2x108 0.01 Spherical
Phobos Satellite of Mars 1.0x1019 2.7x106 N/A Irregular
Deimos Satellite of Mars 1.8x1018 1.5x106 N/A Irregular
Itokawa Asteroid 3x1013 5.4x104 N/A Irregular
Comet Planetesimal ~6x1015 ~6x104 N/A Irregular
Human Living Organism ~105 ~2x102 N/A Definitely not Spherical
Neutron Composite Particle 1.67x10-24 ~10-13 N/A Unknown
Electron Elementary Particle 9.1x10-28 ~10-16 N/A Unknown

Table 01 Geometric Shape of Solid Body

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