question: Why are bubbles always spherical?
comment: The pressure inside the bubble is pushing with roughly the same force in all directions, and the material which is the edge of the bubble is resisting it again with roughly the same pressure. The result is it deforms the same amount in all directions and that shape is a sphere.
passages: At the stem of the bubble, the shape of the bubble is nearly cylindrical; consequently, either R or R is large while the other radius of curvature is small. As the stem of the bubble grows in length, it becomes more unstable as one of the radius grows and the other shrinks. At a certain point, the vertical length of the stem exceeds the circumference of the stem and due to the buoyancy forces the bubble separates and the process repeats.
  
  Section::::Stability.
  
  Section::::Stability.:Stabilization.
   Section::::Types of pattern.:Bubbles, foam.
  
  A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one.
   BULLET::::- It follows from isoperimetric inequality. These properties define the sphere uniquely and can be seen in soap bubbles: a soap bubble will enclose a fixed volume, and surface tension minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape).
  
  BULLET::::8. "The sphere has the smallest total mean curvature among all convex solids with a given surface area."
   The total area can thus be obtained by integration:
  
  The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
  
  The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as
  
  where is the density (the ratio of mass to volume).
   Deviations are due to the Marangoni effect and capillary pressure, which affect the assumption that the bubbles are spherical.
  
  For laplace pressure of a curved gas liquid interface, the two principal radii of curvature at a point are R and R. With a curved interface, the pressure in one phase is greater than the pressure in another phase. The capillary pressure P is given by the equation of:
   Since, the density of the gas is less than the density of the liquid the left hand side of the equation is always positive. Therefore, the inverse of R must be larger than the R. Meaning that from the top of the bubble to the bottom of the bubble the radius of curvature increases. Therefore, without neglecting gravity the bubbles cannot be spherical. In addition, as z increases, this causes the difference in R and R too, which means the bubble deviates more from its shape the larger it grows.
   The above explanation only holds for bubbles of one medium submerged in another medium (e.g. bubbles of gas in a soft drink); the volume of a membrane bubble (e.g. soap bubble) will not distort light very much, and one can only see a membrane bubble due to thin-film diffraction and reflection.
  
  Section::::Physics and chemistry.:Applications.
  
  Nucleation can be intentionally induced, for example to create a bubblegram in a solid.
  
  In medical ultrasound imaging, small encapsulated bubbles called contrast agent are used to enhance the contrast.
   where R and R are the radii of curvature and are set as positive. At the stem of the bubble, R and R are the radii of curvature also treated as positive. Here the hydrostatic pressure in the liquid has to take in account z, the distance from the top to the stem of the bubble. The new hydrostatic pressure at the stem of the bubble is "p"("ρ" − "ρ")"z". The hydrostatic pressure balances the capillary pressure, which is shown below:
  
  Finally, the difference in the top and bottom pressure equal the change in hydrostatic pressure:
   When two bubbles merge, they adopt a shape which makes the sum of their surface areas as small as possible, compatible with the volume of air each bubble encloses. If the bubbles are of equal size, their common wall is flat. If they aren't the same size, their common wall bulges into the larger bubble, since the smaller one has a higher internal pressure than the larger one, as predicted by the Young–Laplace equation.
   As the vapor bubbles distort and take the form of a more complex geometry than a simple sphere, the two principal radii of curvature and would be used in the following equation:
  
  As pressure grows inside the bubbles, the liquid lamellae shown in the figure above will forced to move toward plateau borders causing a collapse of the lamellae.
  
  Section::::Background.:Theory.:Gibbs adsorption isotherm.
   In sharp interface models, the interface is not diffused by advection. The model maintains a sharp interface. Naturally, this is only appropriate when the bubble size is at least on the order of a few cells.
  
  Section::::Phase change models.
   BULLET::::4. Balloon scene (lower right; heat constraints): "Why am I lighter in the water?" Dax asked after a recent swimming lesson. Dax, like balloons, floats since there are more particles pushing on the bottom than on the top, as in buoyancy.
  
  BULLET::::5. Contained Air Burst (N-body particle system, soft body dynamics): 32.068 vertices, 35.283 constraints. After a brief mushroom formation, the semi-spherical shockwaves propagate to the rectangular container walls, where they are reflected, eventually forming a wedge shape in the middle, quickly degrading to a half-sphere under the influence of gravity.
  
  Section::::See also.
  
  BULLET::::- Electrostatics
  
  BULLET::::- Game physics
  
  BULLET::::- Magnetism
   The shape of liquid water drops is generally spherical because of the effects of surface tension. Depending on the size of the drop the friction of the air flowing past a falling drop may squish the bottom on the drop so that it is slightly non-spherical. However, solid ice does not generally form into nice spherical shapes. Ice crystals have a preference to form hexagonal structures by deposition, but can form odd shapes in the presence riming or aggregation into graupel.
   BULLET::::- * For channel surfaces one sheet forms a curve and the other sheet is a surface
  
  BULLET::::- * For cones, cylinders, tori and cyclides both sheets form curves.
  
  BULLET::::- * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.
  
  BULLET::::6. "All geodesics of the sphere are closed curves."
   where "φ" is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center.
  
  The volume "V" of the sector is related to the area "A" of the cap by:
  
  Section::::Area.
  
  The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is
  
  It is also 
   Section::::Thermodynamics.:Thermodynamics of soap bubbles.
  
  The pressure inside an ideal (one surface) soap bubble can be derived from thermodynamic free energy considerations. At constant temperature and particle number, , the differential Helmholtz energy is given by
  
  where is the difference in pressure inside and outside of the bubble, and is the surface tension. In equilibrium, , and so,
  
  For a spherical bubble, the volume and surface area are given simply by
  
  and
  
  Substituting these relations into the previous expression, we find
   If the container contains particles of different sizes, the downward-moving region at the sides is often narrower than the largest particles. Thus, larger particles tend to become sorted to the top of such a mixture. This is one possible explanation of the Brazil nut effect.
  
  Section::::Convection mechanisms.:Solid-state convection in ice.
  
  Ice convection on Pluto is believed to occur in a soft mixture of nitrogen ice and carbon monoxide ice. It has also been proposed for Europa, and other bodies in the outer solar system.
  
  Section::::Convection mechanisms.:Thermomagnetic convection.
   Section::::In nature.
  
  Section::::In nature.:Bubbles.
  
  Like a droplet, bubbles are round because cohesive forces pull its molecules into the tightest possible grouping, a sphere. Due to the trapped air inside the bubble, it is impossible for the surface area to shrink to zero, hence the pressure inside the bubble is greater than outside, because if the pressures were equal, then the bubble would simply collapse. This pressure difference can be calculated from Laplace's pressure equation,
  
  For a soap bubble, there exists two boundary surfaces, internal and external, and therefore two contributions to the excess pressure and Laplace's formula doubles to
   Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong . 
   Section::::Quotes.:Astronomy.
  
  When the Director of the Astronomical Observatory asked Shen Kuo if the shapes of the sun and moon were round like balls or flat like fans, Shen Kuo explained his reasoning for the former:
   At lower scale than the bubble is the thickness of the film for metastable foams, which can be considered a network of interconnected films called lamellae. Ideally, the lamellae connect in triads and radiate 120° outward from the connection points, known as Plateau borders.
  
  An even lower scale is the liquid–air interface at the surface of the film. Most of the time this interface is stabilized by a layer of amphiphilic structure, often made of surfactants, particles (Pickering emulsion), or more complex associations.
  
  Section::::Formation.
   According to Plateau's laws, the minimum area shape that encloses any volume or set of volumes must take a form commonly seen in soap bubbles in which surfaces of constant mean curvature meet in threes, forming dihedral angles of 2/3. In a "standard double bubble", these surfaces are patches of spheres, and the curve where they meet is a circle. When the two enclosed volumes are different from each other, there are three spherical surfaces, two on the outside of the double bubble and one in the interior, separating the two volumes from each other; the radii of the spheres is inversely proportional to the pressure differences between the volumes they separate, according to the Young–Laplace equation. When the two volumes are equal, the middle surface is instead a flat disk, which can be interpreted as a patch of an infinite-radius sphere.
   For the Marangoni effect to occur, the foam must be indented as shown in the first picture. This indentation increases local surface area. Surfactants have a larger diffusion time than the bulk of the solution—so the surfactants are less concentrated in the indentation.
   Section::::Background.:Theory.
  
  Section::::Background.:Theory.:Young-Laplace equation.
  
  As vapor bubbles form in a liquid solvent, interfacial tension causes a pressure difference, Δ"p", across the surface given by the Young-Laplace equation. The pressure is greater on the concave side of the liquid lamellae (the inside of the bubble) with radius, R, dependent on the pressure differential. For spherical bubbles in a wet foam and standard surface tension , the equation for the change in pressure is as follows:
   Earth undulates as the shape of its lowest potential energy changes daily due to the gravity of the Sun and Moon as they move around with respect to the Earth. This is what causes tides in the oceans' water, which can flow freely along the changing potential.
  
  Section::::Cause.:Shapes of other bodies.
  
  The IAU definitions of planet and dwarf planet require that a Sun-orbiting body has undergone the rounding process to reach a roughly spherical shape, an achievement known as hydrostatic equilibrium. The same spheroidal shape can be seen from smaller rocky planets like Mars to gas giants like Jupiter.
  