D A T A    O N    B U B B L E   F O R C E F I E L D S

By Henryk Szubinski

data  ON SPECIAL VALUE RELATIVITY WITH THE DISPLACEMENTS OF PROJECTIVE BUBBLES AND THEIR FLAT STATE ASTROPHYSICS AS IMPLIED IN LEVELING OF GRAVITY..

For a plane curve given parametrically as c(t) = (x(t),y(t)), the curvature is

\kappa = \frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}},

and the signed curvature k is

k = \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}.

For the less general case of a plane curve given explicitly as y = f(x) the curvature is

\kappa = \frac{|y''|}{(1+y'^2)^{3/2}}.

Slightly abusing notation, the signed curvature may also be written in this way as

k=\frac{y''}{(1+y'^2)^{3/2}}

with the understanding that the curve is traversed in the direction of increasing x.

This quantity is common in physics and engineering; for example, in the equations of bending in beams, the 1D vibration of a tense string, approximations to the fluid flow around surfaces (in aeronautics), and the free surface boundary conditions in ocean waves. In such applications, the assumption is almost always made that the slope is small compared with unity, so that the approximation:

\kappa \approx \left|\frac{d^2y}{dx^2}\right|

may be used. This approximation yields a straightforward linear equation describing the phenomenon, which would otherwise remain intractable.

If a curve is defined in polar coordinates as r(θ), then its curvature is

\kappa(\theta) = \frac{|r^2 + 2r'^2 - r r''|}{\left(r^2+r'^2 \right)^{3/2}}

 

The function

f(x) = \begin{cases}x & \mbox{if }x \ge 0, \\ 0 &\mbox{if }x < 0\end{cases}

is continuous, but not differentiable at x = 0, so it is of class C0 but not of class C1.

The function

f(x) = \begin{cases}x^2\sin{(1/x)} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0\end{cases}

is differentiable, with derivative

f'(x) = \begin{cases}-\mathord{\cos(1/x)} + 2x\sin{(1/x)} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0.\end{cases}

Because cos(1/x) oscillates as x approaches zero, f ’(x) is not continuous at zero. Therefore, this function is differentiable but not of class C1. Moreover, if one takes f(x) = x3/2 sin(1/x) (x ≠ 0) in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a compact set and, therefore, that a differentiable function on a compact set may not be locally Lipschitz continuous.

The exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined.

The function

f(x) = \begin{cases}e^{-1/(1-x^2)} & \mbox{ if } |x| < 1, \\ 0 &\mbox{ otherwise }\end{cases}

is smooth, so of class C∞, but it is not analytic at x=\pm 1, so it is not of class Cω. The function f is an example of a smooth function with compact support.

This 84-million-year-old air bubble lies trapped in amber (fossilized tree sap). Using a quadrupole mass spectrometer, scientists can learn what the atmosphere was like when the dinosaurs roamed the earth. Source: USGS

what im trying to illustrate here is ; what happens when a bubble like the mean curvature graph

alters its containment state when a mass drags over it and punctures a hole in the surface that FLOWS ONTO THE MAIN SURFACE AREA AND contains it..BY THE FOLLOWING;

and distributes the fluid onto the surface being pressed……as a example of what can happen when such

displacements with puntures ARE resultant in fluid release can be utilised in dynamical systems by using a

format cylinder that maintains the fluid dynamical levels on the same planar pressure zone as the flow and distribution of fluid into the bust MEAN bubble and back into pressure state leveling where it is open for continued flow by gravity….what happens when theese processess are accellerated on the field of the whole surface is termed as dissipations in projective hyper responses made by a connective level height and the usage of the fluid bubble matterial in fluxuations of convex /concave occilation….to a resultant mass. 

 

The focal length of a lens in air can be calculated from the lensmaker’s equation:[9]

\frac{1}{f} = (n-1) \left[ \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{n R_1 R_2} \right],

where

f is the focal length of the lens,
n is the refractive index of the lens material,
R1 is the radius of curvature of the lens surface closest to the light source,
R2 is the radius of curvature of the lens surface farthest from the light source, and
d is the thickness of the lens (the distance along the lens axis between the two surface vertices).

what can happen at the angle of flattening………………………………………………………………………………………

what happens when the cylindrical pressure fluid delivery system is used with multiple cylinders……………….

Formally, a real-valued function f defined on an interval (or on any convex set C of some vector space) is called concave, if for any two points x and y in its domain C and any t in [0,1], we have

f(tx+(1-t)y)\geq t f(x)+(1-t)f(y).

Also, f(x) is concave on [a, b] if and only if the function −f(x) is convex on [a, b].

A function is called strictly concave if

f(tx + (1-t)y) > t f(x) + (1-t)f(y)\,

for any t in (0,1) and x ≠ y.

This definition merely states that for every z between x and y, the point (z, f(z) ) on the graph of f is above the straight line joining the points (x, f(x) ) and (y, f(y) ).

 

 

 

 

 

 

 

 

  disregardless of the rest as a planar involvement while at the same time being a multi dimensional effect  of interactions..Richardson’s notion of turbulence was that a turbulent flow is composed by “eddies” of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.

In his original theory of 1941, Kolmogorov postulated that for very high Reynolds number, the small scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as L). Kolmogorov’s idea was that in the Richardson’s energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number is sufficiently high.

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the viscosity (ν) and the rate of energy dissipation (\varepsilon). With only these two parameters, the unique length that can be formed by dimensional analysis is

\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}.

This is today known as the Kolmogorov length scale (see Kolmogorov microscales).

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length η, while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. \eta \ll r \ll L). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called “inertial range”).

Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range \eta \ll r \ll L are universally and uniquely determined by the scale r and the rate of energy dissipation \varepsilon.

The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the energy spectrum function E(k), where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u(x):

\mathbf{u}(\mathbf{x}) = \iiint_{\mathbb{R}^3} \widehat{\mathbf{u}}(\mathbf{k})e^{i \mathbf{k \cdot x}} \mathrm{d}^3\mathbf{k},

where û(k) is the Fourier transform of the velocity field. Thus, E(k)dk represents the contribution to the kinetic energy from all the Fourier modes with k < |k| < k + dk, and therefore,

\mathrm{Total\,\, kinetic\,\, energy} = \int_{0}^{\infty}E(k)\mathrm{d}k.

The wavenumber k corresponding to length scale r is k = 2π / r. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov’s hypothesis is

E(k) = C \varepsilon^{2/3} k^{-5/3} ,

where C would be a universal constant. This is one of the most famous results of Kolmogorov 1941 theory, and considerable experimental evidence has accumulated that supports it[4].

In spite of this success, Kolmogorov theory is at present under revision. This theory implicitly assumes that the turbulence is statistically self-similar at different scales. This essentially means that the statistics are scale-invariant in the inertial range. A usual way of studying turbulent velocity fields is by means of velocity increments:

\delta \mathbf{u}(r) = \mathbf{u}(\mathbf{x} + \mathbf{r}) - \mathbf{u}(\mathbf{x});

that is, the difference in velocity between points separated by a vector r (since the turbulence is assumed isotropic, the velocity increment depends only on the modulus of r). Velocity increments are useful because they emphasize the effects of scales of the order of the separation r when statistics are computed. The statistical scale-invariance implies that the scaling of velocity increments should occur with a unique scaling exponent β, so that when r is scaled by a factor λ,

\delta \mathbf{u}(\lambda r)

should have the same statistical distribution as

\lambda^{\beta}\delta \mathbf{u}(r),

with β independent of the scale r. From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the velocity increments (known as structure functions in turbulence) should scale as

\langle [\delta \mathbf{u}(r)]^n \rangle = C_n \varepsilon^{n/3} r^{n/3},

where the brackets denote the statistical average, and the Cn would be universal constants.

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the n/3 value predicted by the theory, becoming a non-linear function of the order n of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov n/3 value is very small, which explain the success of Kolmogorov theory in regards to low order statistical moments. In particular, it can be shown that when the energy spectrum follows a power law

E(k) \propto k^{-p},

with 1 < p < 3, the second order structure function has also a power law, with the form

\langle [\delta \mathbf{u}(r)]^2 \rangle \propto r^{p-1} .

Since the experimental values obtained for the second order structure function only deviate slightly from the 2/3 value predicted by Kolmogorov theory, the value for p is very near to 5/3 (differences are about 2%[5]). Thus the “Kolmogorov -5/3 spectrum” is generally observed in turbulence. However, for high order structure functions the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear. This behavior, and the lack of universality of the Cn constants, are related with the phenomenon of intermittency in turbulence. This is an important area of research in this field, and a major goal of the modern theory of turbulence is to understand what is really universal in the inertial range.