I would like to hear how many people are familiar with doublet flow ?

Doublet flow in fluid dynamics refers to the flow pattern created by combining a source and a sink (an inflow point) placed very close to each other. When these two singularities are brought infinitesimally close while maintaining a constant product of their strength and separation distance, they form a doublet. This configuration creates streamlines that resemble the flow around a cylinder in two dimensions or a sphere in three dimensions. 

When two spheres move close to each other inside a fluid, the flow pattern between them can be approximated by a Rankine oval. This is because the combination of source and sink flows, induced by the moving spheres, can be superposed to create a closed streamline shape resembling an oval. The shape and characteristics of this "Rankine oval" are influenced by the relative size and separation distance of the spheres, as well as the fluid's velocity. 

The statement "sphere moving inside  fluid is reversed bernoulli pipe" refers to a thought experiment or a way to visualize the relationship between fluid pressure and velocity, as described by Bernoulli's principle. It's not a literal reversed Bernoulli pipe, but rather a conceptual analogy. In a typical Bernoulli pipe, a fluid speeds up in a narrower section, causing a pressure drop. In this conceptualization, a sphere  is used to create a local constriction in a fluid flow, similar to how a narrower pipe section would. This constriction, according to Bernoulli's principle, would cause a localized increase in fluid velocity and a corresponding decrease in pressure. The "reversal" refers to the fact that the sphere's presence is causing the pressure drop, not the pipe's geometry itself. 

In a fluid system, the fluid's speed is fastest when flowing through the smallest diameter section. This is due to the principle of continuity, which states that the volumetric flow rate (volume of fluid passing a point per unit time) must remain constant in a closed system if the fluid is incompressible. 

Two spheres (white circles) in the center in a picture above are positioned in wind tunnel

The flow of wind(fluid ) is marked lines with arrows.

The space in between spheres is acting like source and sink- and doublet flow is created - the wind travels in concentric circles in direction opposite of main flow.

These concentric circles have different speed and consequently pressure is different.

This can be called waves .

The flow of waves can be described with wave function .

The point where main flow “wraps” around waves(concentric circles) is called stagnation

Point.

The main flow traveling around stagnation point “circle “ is layered and has different speed (and pressure ) in each layer .

This is perceived as distortion of fluid field .

The attractive forces crated between two spheres are product of static and dynamic pressures of wind (potential and kinematic energy of moving fluid (wind).

Opinions ?

In these types of fluid flow representations, the common understanding is that the closer together the streamlines are, the faster the fluid is moving.
The application of Bernoulli then implies that cylindrical or spherical objects placed in a moving fluid, will be drawn together until the distance between source and sink of the dipole flow is reduced to zero.

This is interesting, but was there something you wished to discuss with regards to dipole flow ?

Yes

The force of attraction between spheres is F.

What if distance of source and sink is constant for one pair of spheres and distance between spheres is L1

Then we add one more pair of spheres(same L1 )

But they are positioned at L 2 distance from each other (side by side ) at distance L2 .

Is there the possibility the flow of fluid (wind ) create bigger doublet flow around these pairs but the attracting force between them F1 is smaller ?

Here is a bit better explanation of what I would like to hear your opinion about.

The shortest distance between spheres (picture to the right ,above) is point where distance between source and sink is shortest.

The area above the line to the top of spheres is source of flow and sink is area from the line to the bottom of the spheres.

The distance between source and sink is virtually 0m(or maybe the size of atom or two of material the sphere is made of).

Instead of spheres let’s use cylinders to observe the doublet flow in 2D only.

This is the “top”view of flow.

The doublet flow around lower cylinder moves in CW direction .The doublet flow around upper cylinder moves in CCW direction .

The fluid inside doublet flow around spheres is partially flowing in concentric circles (static wave)and partially going from source through the sink and out to uniform flow(wind).

The uniformed flow of wind flows around this new “single cylinder “ with bigger diameter .

The diameter of this “bigger “ cylinder is created by static wave like movement of fluid inside concentric circles of doublet flow.

For this to happen the wind is non compressible and friction.

The stagnation point for single cylinder is the point where speed is 0m/s.

For two cylinders - newly created sphere is “hollow”in the middle and flow is not totally stopped.

The split in doublet flow happens because speed at stagnation point of pressure between two spheres is not 0.

The stagnation pressure formula

Po=P+0.5xDxV^2

The speed between spheres is Vmax and pressure Po=10^5Pa

The density of air id D=1.2kg/m3

Later comes a practical example .

Here is theoretical example :

Two cylinders with diameter of d=1m

And height h=1m are positioned at distance of r1=1m.

Speed of wind moving against the cylinders is V2=0.4m/s

At this distance measured speed in between cylinders is V1=3m/s

At distance r2= 2.75 m between cylinders measured speed V2=0.4m/s

Doublet flow has two concentric circles  moving around each cylinder.

In cylinder to the left the direction is CWand for cylinder to the right the direction is CCW.

The radius of first concentric circles is

Cr1=1m(from the center of cylinders)

Cr2=1.875m

The   Average angular speed of fluid traveling in Cr1 is

Av1=V1x Cr1

Av1=3x1‎ = 3rad/s

Av2=V2xCr2

Av2=0.4x1.875‎ = 0.75rad /s

The stagnation pressure - the speed in “hollow “sphere is not 0,so we can calm it pressure P1 is calculated :

Po=Ps+0.5 x D x V^2

D=1kg/m3     density of air

V=V2=3m/s

Po=10^5Pa

P1=100000-4.5=99995.5

This pressure is uniformed through Cr1

The pressure P2

P2=100000-0.16=99999.84Pa

This pressure is uniformed  in area around Cr2

From difference between Po,P1 and P2 (potential energy of fluid)we can  calculate force difference between the   Force at distance r1  compared to force at distance r2.

The cylinders are balloons with plastic skin of negligible mass filled with air .

The mass of cylinder is

Mc1=Mc2=v(volume )cad

Mc1=3.14kg=Mc2

The wind tunnel is positioned inside the diving aircraft so gravitational force of earth is not involved.

The pressure difference P3:

P3=P2-P1=E

E=99999.84-99995.5

E=4.34J

E-energy(work ) which creats the force difference between Cr1 and Cr2.

E=Fxs

S=2.75-1=1.75m

F=E /s=4.34/1.75‎ = 2.48N

This is the force which can be created by different potential energy of moving fluid between Cr1 and Cr2.

Negative acceleration

a=(V1-V2)/t

F=mxa

F=Mc1x(V1-V2)/t

t=F/Mc1x(V1-V2)

t=2.48/2.6=0.95 sec

The force needed to “push” cylinders apart has to be minimum 2.48N and it would push the cylinders apart in 0.95sec.

Now if we put  two of these “new bigger hollow cylinders “  at distance Lx

The same doublet flow will happen

And we can call it fractal nature of doublet flow .

Or dipole flow .