TO NATURE Kugel ball as an interesting application of designing the hydrosphere

Analytical solutions are not available for the partial hemispherical hydrosphere which called as the Kugel ball fountain or the Kugel ball. However, this study offers a comprehensive idea about this phenomenon presenting a design map that gives a panoramic sight enabling the designers to easily choose whatever specifications needed for their fountain. Through simplifying the author previous formulae for this type of bearings, this paper removes the mystery of the Kugel ball phenomenon and shows that no complicated mathematic or physics are needed, as believed, to be grasped for producing such fountains. A new simple design technique is used and the most two famous fountains (at the House of Science in Patras, Greece and the largest at the Science Museum of Virginia, Richmond, USA.) are checked as an application of this design. One of the most important side results of this study is finding the equilibrium point, discovered in the author previous papers, which was considered as the equilibrium point between the forces of centripetal inertia, viscosity and friction due to the surface roughness. It becomes clear that this point is a natural characteristic of this type of bearings.


Introduction
Kugel ball, as defined in the fountain Wikipedia, is a water feature or sculpture where a sphere sits in a fitted hollow in a pedestal, and is supported by aquaplaning on a thin film of water. Pressurized water flows between the sphere and socket, creating a mechanical hydrostatic bearing that is nearly frictionless. The sphere can weigh thousands of kilograms, but the efficient bearing allows it to be spun by the force of a hand. The sphere does not float, being denser than water; it is often made from granite. The hydraulics of the fountain can be controlled so that the axis of rotation of the sphere changes continually [1]. To understand the Kugel ball phenomenon it is necessary to go back to the first hydrosphere found by Shaw and Strang [2] where it is suggested through theoretical and experimental work that the lubricant inertia could explain the performance of the bearing. Block and Cameron [3] disputed Shaw and Strang suggestion putting forward approximate calculations to support their view. Dowson and Taylor [4][5] offered a more complete theoretical analysis by starting with the equations of motion and reducing them to a usable form retaining only the centripetal inertia terms. Yacout [6][7][8][9][10][11] developed Dowson form to be able to handle this type of bearings with its different configurations, in presence of the surface roughness centripetal inertia and the lubricant variable viscosity, investigating its performance in details and offering a single equation covers this type and a design for restricted and self-restriction fitted bearing ,which the Kugel ball is to be considered a partially hemispherical configuration of the hydrosphere. Snoeijer and Weele [12] offered mathematical analysis for the Kugel ball and concluded that, as a matter of fact, the kugel fountain can be thought of as a giant ball bearing. D'Alessio and Pascal [13] offered analytical investigation of the steady flow of a thin fluid layer over a sphere resulting from a constant discharge from a small hole at the top of the sphere concluding that the technique and approach adopted can be used to model other thin flows that occur in similar settings. Inasmuch as the hydrosphere has been found, the arguments about this type of bearings have not been stopped; however, the present study mainly offers a new technique for designing the Kugel ball as a partially hemispherical hydrostatic thrust spherical bearing.

DESIGN PROCEDURES
The ball diameter is (D) and its specific weight is ) (   (4) where (Jf =1/W) the geometry factor could be known mathematically or graphically as: Jf =3.1868 From equation (8):

Checking Virginia Museum fountain in Richmond, USA
The main difference between this example and the previous one is the ball stagnation.
The fountain specifications It has the same previous specifications excep the ball radius. Ball radius = 1.5 m Starting with the selection of:  i  2.31deg, mor = 1,  =2/3 and with aid of subfigure (3-b) select Ke=1/1750, which gives the minimum ) ( that satisfies the condition of (m = 0), i.e., no rotation, the figures (8-10) could be drawn and all needed information could be got as before either mathematically or graphically.

RESULTS
The Kugel ball is treated as an un-recessed fitted self-restriction hydrosphere trough a new design technique simplifying the hydrosphere formulae and developing two especial simple characteristic equations governing the kugel ball design. A side result of this study is finding the partially hemispherical hydrosphere, equilibrium point which was believed to be as the point of balance between the centripetal inertia, the viscosity and the surface roughness friction forces.

DESCUTION
The new provided design technique simplifies the fitted hydrosphere phormaulae deriving the two characteristic equations (8,9) especially for this pattern of the fitted hydrosphere. Equation (8) simply calclates the pressure supply through multiplying the the ball diameter by its specific weight and the bearing geometry factor. Equation (10) is a master govering relation where it could be really defined as the bearing or the ball design parameter (bdp); it cobines the geometry factor with the dimensionless flow rate in a simple form yeilding a constant value. Figures (3, 4) show the ball behavior at different partial seats and different eccentricities. Subfigure (3-a) shows that the dimensionless rotational torque ) and with (jfqc) curve to determine ) ( , Subfigures (4-d) show a design map that gives a panoramic sight on the ball behavior to enable the designer to quickly choose the area of his work. Figures (5, 6 and 7) represent the first design example procedures and calculations. 6 6 Subfigures (6-c,d), represent the supply pressure and the flow rate where (ps and q) could be easily determined graphically. Figure (7) represent the lubricant film thicknesses at its inlet and outlet where the thickness could be graphically determined.

Checking the first example
Looking at the few data mentioned by Snoeijer and van der Weele [12] about this example, it could be realized the agreement with the present results. However, despite the lack of information about the specifications of this example, the results reveal that the Kugel ball is a self-restriction hydrosphere Yacout [11]. Figures (8-10) represent the second design example procedures and calculations.

Checking the second example
In this example the ball is stagnant, hence, figure (3-b) is nessary to get the idea about the parameter ) ( that could result in the zero moment then the same procedures are to be applied as before to get the results mathematically or graphically as shown in the first example. Due to the absence of data about this example, the present results could be considered as a temporary guide for the future investigations.

The side result
The side result got in this study is the point of minimum moment, subfigures (3-b); this point appears in yacout [6][7][8] which considered as the equilibrium point between the forces of the centripetal inertia, viscosity and surface roughness friction. This point of minimum moment appears around ) ( value of (0.75). From subfigures (5-b) and (8-b), the geometry factor at this point is (2.406 -2.411) and its load carrying capacity (W ~ 0.416) which coincides with the equilibrium point of the fitted hydrosphere in yacout [6]. It becomes clearly now that the name of this point and the reason of its appearance is not the balance between the aforementioned forces and it could be assured that the point's name is the minimum moment and the reason is the minimum difference between (mo and mi) where the centripetal inertia will not affect this difference despite its effect on the load carrying capacity (W).

CONCLUSION
The new design technique has many advantages where it:  Results in deriving two simple characteristic relations where the first is concerned with the supply pressure and the second with the flow rate. While the first one is especially for the Kugel ball, the second one could be generally applied to the hydrosphere where no hydrosphere could be outside this relation.  Proves that the Kugel ball is just a simple hydrosphere based on simple mathematics not as believed before.  Offers the mathematical reason of the ball rotation calculating its rotational speed whilst the concerned references treated the rotation or (spin as said in these references) as to be a fact.  Proves that the lubricant film is not a sheet as believed and mentioned in the references but a convergent shape with different thicknesses.  Finds, as a side result, the real reason of what is called the forces equilibrium point of the hydrosphere.  Offers a design map of panoramic sight on the ball characteristics to help the designer select the needed specifications.
 Show clearly, through the two design examples, that the increase in the seat arc length decreases the supply pressure which leads to levitate the ball by low pressure.

FUTURE WORK
Based on the conception of this especial hydrosphere, a general new design technique will be offered to simply design the externally pressurized thrust spherical hydrostatic bearings avoiding the difficulties in the previous designs and improving the method of selecting the proper bearing.

APPENDIX
All mathematical equations related to the hydrosphere could be found in Yacout [6][7][8][9][10][11] and the necessary ones which serve this design are listed.