\r\n\t2) The divergence between the levels of reliability required (twelve-9’s are not uncommon requirements) and the ability to identify or test failure modes that are increasingly unknown and unknowable
\r\n\t3) The divergence between the vulnerability of critical systems and the amount of damage that an individual ‘bad actor’ is able to inflict.
\r\n\t
\r\n\tThe book examines pioneering work to address these challenges and to ensure the timely arrival of antifragile critical systems into a world that currently sees humanity at the edge of a precipice.
Scale Relativity Theory (SRT) affirms that the laws of physics apply in all reference systems, whatever its state of motion and its scale. In consequence, SRT imply [1-3] the followings:
Particle movement on continuous and non-differentiable curve (or almost nowhere differentiable), that is explicitly scale dependent and its length tends to infinity, when the scale interval tends to zero.
Physical quantities will be expressed through fractal functions, namely through functions that are dependent both on coordinate field and resolution scale. The invariance of the physical quantities in relation with the resolution scale generates special types of transformations, called resolution scale transformations. In what follows we will explain the above statement.
Let
We can now say that
Let us now consider
situation in which we can consider the infinitesimal scale transformation as being
Such transformation in the case of function
respectively, if we limit ourselves to a first order approximation:
Moreover, let us notice that for an arbitrary but fixed
situation in which (5) can be written as:
Therefore, we can introduce the dilatation operator:
At the same time, relation (8) shows that the intrinsic variable of resolution is not
The fractal function is explicitly dependent on the resolution
where
in which case the differential equation (9) takes the form:
Hence by integration and substituting:
we obtain:
We can now generalize the previous result by considering that F is dependent on parameterization of the fractal curve. If p characterizes the position on the fractal curve then, following the same algorithm as above, the solution will be as a sum of two terms i.e. both classical and differentiable (depending only on position) and fractal, non-differentiable (depending on position and, divergently, on
where
The following particular cases are to be considered:
in asymptotic small scale regime
in the asymptotic big scale regime
Particularly, if
In this situation,
By differentiating these two parts we obtain:
where
There is infinity of fractal curves (geodesics) relating to any couple of points (or starting from any point) and applied for any scale. The phenomenon can be easily understood at the level of fractal surfaces, which, in their turn, can be described in terms of fractal distribution of conic points of positive and negative infinite curvature. As a consequence, we have replaced velocity on a particular geodesic by fractal velocity field of the whole infinite ensemble of geodesics. This representation is similar to that of fluid mechanics [4] where the motion of the fluid is described in terms of its velocity field
The local differential time invariance is broken, so the time-derivative of the fractal field Q can be written two-fold:
Both definitions are equivalent in the differentiable case
Notation “+” corresponds to the forward process, while “-” to the backward one.
We denote the average of these vectors by
Since, according to (19), we can write:
and it results:
The differential fractal part satisfies the fractal equation:
where
The local differential time reflection invariance is recovered by combining the two derivatives,
Applying this operator to the “position vector”, a complex velocity yields
with:
The real part, V, of the complex velocity
Let us now assume that curves describing particle movement (continuous but non-differentiable) are immersed in a 3-dimensional space, and that
where only the first three terms were used in Nottale’s theory (i.e. second order terms in the motion equation). Relations (29) are valid in any point both for the spatial manifold and for the points X on the fractal curve (selected in relations 29). Hence, the forward and backward average values of these relations take the form:
The following aspects should be mentioned: the mean value of function
or more, using equations (23) with characteristics (24),
Even if the average value of the fractal coordinate
Then, let us consider the averages
Then, equations (32) may be written as follows:
If we divide by
These relations also allow us to define the operator:
Under these circumstances, let us calculate
This relation also allows us to define the fractal operator:
Particularly, by choosing:
the fractal operator (39) takes the usual form:
We now apply the principle of scale covariance and postulate that the passage from classical (differentiable) to “fractal” mechanics can be implemented by replacing the standard time derivative operator,
This means that at any point on a fractal path, the local temporal
Newton\'s fundamental equation of dynamics in the dissipative approximation is:
where m is the mass,
is the fractal operator in the dissipative approximation.
In what follows, we study what happens with equation (44), in the free particle case (Φ = 0), if one considers the space-time where particles move changes from classical to nondifferentiable.
According to Nottale [11], the transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic). Separating the real and imaginary parts, (44) becomes:
where
Let us analyze in what follows, the second equation (46) which, one can see, may contain some interesting physics. If we compare it with Navier-Stokes equation, from fluid mechanics [12]
we can see the left side of (46) gives the rate at which V is transported through a \'fluid\' by means of the motion of \'fluid\' particles with the velocity U; the right hand side gives the diffusion of V, (D which is the amplitude of the fractal fluctuations, plays here the role of the \'cinematic viscosity\' of the \'fluid\'). One can notice, in those regions in which the right hand side of (47) is negligible, Dv/Dt = 0. This means that in inviscid flows, for instance, V∧ is frozen into the \'particles of the fluid\'. Physically this is due to the fact that in an inviscid \'fluid\' shear stresses are zero, so that there is no mechanism by which V∧ can be transferred from one \'fluid\' particle to another. This may be the case for the transport of V by U in the second equation (II.3).
If we consider the flow of V induced by a uniform translational motion of a plane spaced a distance Y above a stationary parallel plane (Fig. 1), and if the \'fluid\' velocity increases from zero (at the stationary plane) to U (at the moving plane) like in the case of simple Couette flow, or simple shear flow, then
Uniform translational motion of a plane spaced a distance Y above a stationary parallel plane.
For many fluids it is found that the magnitude of the shearing stress is related to the rate of shear proportionally:
Fluids which obey (49) in the above situation are known as Newtonian fluids, which have a very small coefficient of viscosity. When such \'fluids\' flow at reasonable velocities it is found that viscous effects appear only in thin layers on the surface of objects or surfaces over which the \'fluid\' flows. That is, if one continues the analogy, and questions how is V transported by the motion of \'fluid\' particles with the velocity U, in second equation (46), one can assume that the mechanism of transfer of V from one particle of \'fluid\' to another is achieved over small distances (in thin layers, as stated above).
We study an important case, of the one-dimensional flow along the Ox axis :
To resume, the model considered here consists in analyzing the transport of V, along a small elementary distance Λ, by the \'particles\' of a Newtonian fluid moving with velocity U, where the stress tensor obeys (49), i.e.
like in the case of simple Couette flow, or simple shear flow.
Consequently, the second eq. (46) reduces to the scalar equation
which is the time independent Schrödinger equation, and
with Λ and D having the significance of a small elementary distance and of the \'cinematic viscosity\' (or amplitude of the fractal fluctuations), respectively, and U(x) is the velocity of the \'Newtonian fluid\', which is nothing but the imaginary part of the complex velocity [13]. In what follows, we solve this equation accurately by means of the WKBJ approximation method with connection formulas.
Let us re-write (53) in the form
where we take
where
χ is a limit velocity and γ(x) a \'velocity potential\'.
Let us try a solution of the form ζ(x)=A exp ((i/ħ)S(x)). Substituting this solution into the time-independent Schrödinger equation (52) we get:
Assume that ħcan, in some sense, be regarded as a small quantity and that S(x) can be expanded in powers of ħ, S(x) = S0(x)+ ħ S1(x) +....
Then,
We assume that
We have used:
Therefore, for χ>γ(x)
In the classically allowed region
Similarly, in regions where χ<γ(x) we have:
For our first order expansion to be accurate we need that the magnitude of higher order terms decreases rapidly. We need
Near the classical turning points the WKBJ solutions become invalid, because k goes to zero here. We have to find a way to connect an oscillating solution to an exponential solution across a turning point if we want to solve barrier penetration problems or find bound states.
We want to find the velocity wave function in a given velocity potential well γ(x). Assuming that the limit velocity of the particle isχ and that the classical turning points are x1 and x2, x1<x2, i.e. we have a velocity potential well with two sloping sides (Fig. 2).
Bound state problem.
For x < x1 the velocity wave function is of the form:
For x > x2 the velocity wave function is of the form:
In the region between x1 and x2 it is of the form:
At x = x1 and x = x2 the velocity wave function ζ and its derivatives have to be continuous. Near x1 and x2 we expand the velocity potential well γ(x) in a Taylor series expansion in x and neglect all terms of order higher than 1. Near x1 we have
In the neighborhood of x1 the time-independent Schrödinger equation then becomes:
and in the neighborhood of x2 the time-independent Schrödinger equation becomes:
Let us define
which for large |z| has the asymptotic form
and
If the limit velocity χ is high enough, the linear approximation to the velocity potential well remains valid over many wavelengths. The Airy functions can therefore be the connecting velocity wave functions through the turning point at x1.
If we define
In the neighborhood of x1 we have
Therefore
Similarly
By comparing this with the asymptotic forms of the Airy functions we note that
must continue on the right side as
In the neighborhood of x2we similarly find that
must continue in region 2 as
Both expressions for ζ2(x) are approximations to the same eigenfunction. We therefore need
For (79) to be satisfied, the amplitudes of each side must have the same magnitude, and the phases must be the same modulo π :
Knowing that
This can be re-written as
with
Here
The factor of π/2 arises here due to the two phase changes of π/4 at x1 and x2. In case where only one of the walls is soft and the other is infinitely steep the factor of 1/2 is replaced by 1/4 in (81). If both walls are infinitely steep, the factor of 1/2 in (81) is replaced by 0.
WKBJ approximation is a semi classical approximation, since it is expected to be most useful in the nearly classical limit of large quantum numbers. The method will not be good for, say, lowest limit velocity states χ, so in order to overcome this shortcomings there is a need for a modified semi classical quantization condition. For oscillations between the two classical turning points x1 and x2, we obtain the semi classical quantization condition by requiring that the total phase during one period of oscillation to be an integral multiple of 2π; [15] such that
where ϕ1 is the phase loss due to reflection at the classical turning point x1 and ϕ2 is the phase loss due to reflection at x2. Taking ϕ1 and ϕ2 to be equal to π/2leads to the modified semiclassical quantization rule, i.e.
where mis the Maslov index [15], which denotes the total phase loss during one period in units of π/2. It contains contributions from the phase losses ϕ1and ϕ2 due to reflections at points x1 and x2, respectively. It is pertinent to note that taking ϕ1 = ϕ2 = π/2 and an integer Maslov index m = 2 in (85), we have the familiar semi classical quantization rule, i.e. (81).
Let us apply the constraint equation (81) to an harmonic oscillator. The condition then is (passing without loss of generality to the limits -a to +a)
where the energy Wc of the oscillator U(x) with the pulsation ωwrites
and we get the expression for the x dependence of the velocity term,
Theleft side term of (86) is an elementary integral and we find:
where
Dependence of the limit velocity χ on the linear velocity υ.
We try to estimate a value for the limit velocity χ. Let us expand the left side term of (88) in series and keep the first term. If we replace μ from (56) and take a = Λ (the Compton length), we get:
It is interesting to note that only the first velocity in (89) is less than the velocity of light, c.
Let us analyze now, one more bound state, the velocity wave function in a given velocity double well potential γ(x).
We begin by deriving a quantization condition for region 2 analogous to (81). Again, applying the boundary condition for region 1 leaves only the exponentially growing solution. Applying the connection formula at x1 then gives an expression for the velocity wave function in region 2 :
However, the solution in region 3 must have both growing and decaying solutions present. Considering the region 3 solutions in terms of x2 and letting BL and CL be the amplitudes of the decaying and growing solutions respectively, the connection formulas give another expression for the velocity wave function in region 2:
with
We equate the two expressions (90), (91) for the velocity function in region 2 and cancel common factors giving
Using trigonometric identities to simplify the right hand side, gives
where
The magnitude of the sin function must be equal, and the magnitude of the phases must be equal modulo π :
Simplifying and combining the integrals gives the quantization condition for region 2:
with n = 1, 2,....
A similar treatment for the turning point x3 yields the condition for region 4:
with m = 1, 2,... and ϕR given by:
where BR and CR are the amplitudes of the decaying and growing region 3 solutions in terms of x3.
We now have the quantization conditions (98, 99) for regions 2 and 4, but they contain the free parameters ϕL and ϕR. To eliminate these free parameters, we consider the WKBJ solution in region 3. The coefficients BL, CL, BR, CR define two expressions for solution, which must be equal:
Equations (101) and (102) each contain a term that grows exponentially with x and a term that decays exponentially with x. Equating the growing terms from each equation and the decaying term from each equation gives two constraints:
Combining the integrals in these constraints gives
with
The constraints (98, 99, 105) may be combined to give a single quantization condition for the allowed WKBJ velocity limits χ for a double-well velocity potential γ(x). Applying trigonometric identities to (95) and (100), and plugging into (105) gives
Equation (107) may be combined with (98) and (99) to give the WKBJ quantization condition for a double-well potential in terms of the phase integrals θ12 and θ34 :
confirming the results given in [16].
Equation (108) is a nonlinear constraint approximately determining the allowed velocity levels χ of a double-well velocity potential γ(x) (see Fig. 4) and can be written (taking ϕR\n\t\t\t\t\t= ϕL\n\t\t\t\t\t= π/4 in (98) and (99), i.e. the velocity quarter-wave shift in the connection formulas, which is known to optimize the tunneling effect between two oscillating waves [17] ) as :
Tunneling potential barrier.
In terms of the momentum Π we have :
where
since γ(x)>χfor the integration limits, i.e. region 3 (see Fig. 5). We get again a quantization of the complex velocityU(x), where the levels are equally spaced at a value of ħ ln2.
Schematic diagram of a double-well potential with three forbidden regions (1, 3, 5) and two allowed regions (2, 4).
In 1961, Landauer [18] discussed the limitation of the efficiency of a computer imposed by physical laws. In particular he argued that, according to the second law of thermodynamics, the erasure of one bit of information requires a minimal heat generation kBT ln2, where kBis Boltzmann’s constant and T is the temperature at which one erases. Its argument runs as follows. Since erasure is a logical function that does not have a single-valued inverse it must be associated with physical irreversibility and therefore requires heat dissipation. A bit has one degree of freedom and so the heat dissipation should be of order kBT. Now, since before erasure a bit can be in any of the two possible states and after erasure it can only be in one state, this implies a change in information entropy of an amount −kBln2.
The one-to-one dynamics of Hamiltonian systems [19] implies that when a bit is erased the information which it contains has to go somewhere. If the information goes into observable degrees of freedom of the computer, such as another bit, then it has not been erased but merely moved; but if it goes into unobservable degrees of freedom such as the microscopic motion of molecules it results in an increase of entropy of at least kBln2.
Inspired by such studies, a considerable amount of work has been made on the thermodynamics of information processing, which include Maxwell’s demon problem [20], reversible computation [21], the proposal of the algorithmic entropy [22] and so on.
Here, considering a double-well velocity potential γ(x) and the velocity quarter-wave shift in the connection formulas, a quanta of ħ ln2 for the complex velocityU(x) of the moving Newtonian \'fluid\' occurs. It can be argued that it can be put into a one-to-one correspondence to the quanta of information Landauer and other authors discussed about [23, 24].
Furthermore, one gets an interesting result when taking ϕR\n\t\t\t\t\t= ϕL\n\t\t\t\t\t= π/2, i.e. the velocity half-wave shift in the connection formulas, when singularities occur in (II.66). We try to solve this case by making use of the vortices theory. Benard in 1908 was the first to investigate the appearance of vortices behind a body moving in a fluid [12]. The body he used was a cylinder. He observed that at a high enough fluid velocity (or Reynolds number based on the cylinder diameter), which depends on the viscosity and width of the body, vortices start to shed behind the cylinder, alternatively from the top and the bottom of the cylinder.
Consequently, we write (109) in the form
where we use ctg(α + π/2) = - tg(α), take m = n, make the notations
Solving (112) one gets
where we assume l = Λ (the Compton length), U(x) = [γ(x) - χ]/2π2 and replace
The complex velocity U(x) singularities\' distribution along the Ox axis.
Usually, at some distance behind a body placed in a fluid, vortices are arranged at a definite distance l apart and with a definite separation h between the two rows. The senses of the rotation in the two rows are opposite (see Fig. 7).
Von Karman vortex streets.
In 1912 von Karman expounded a theory of such vortex streets and the drag which a cylinder would experience due to their formation [12]. Since we considered here the one-dimensional case, we get the solution of a single row of rectilinear vortices, which has already been referred to as characterizing a surface of discontinuity (see Fig. 8).
Single row of rectilinear vortices.
A typical bound state in a double-well velocity potential has two classically allowed regions, where the velocity potential γ(x)is less than the limit velocity χ. These regions are separated by a classically forbidden region, or barrier, where the velocity potential is larger than the limit velocity. As we can see, quantum mechanics predicts that a velocity wave ζ(x) travelling in such a potential is most likely to be found in the allowed regions. However, unlike classical mechanics, quantum mechanics predicts that this velocity wave can also be found in the forbidden region. This uniquely quantum mechanical behavior allows a velocity wave, initially localized in one potential well, to penetrate through the barrier, into the other well (as we will see in what follows).
We already know at the points where
Barrier to the right ( x = b turning point )
Barrier to the left ( x = a turning point )
The connection formulas enable us to obtain relationships between the solutions in a region at some distance to the right of the turning point with those in a region at some distance to the left [25-27].
One of the most important problems to which connection formulas apply is that of the penetration of a potential barrier. The barrier is shown in Fig. 4 and the limit velocity χ is such that the turning points are at x = a and x = b.
Suppose that the motion is incident from the left. Some waves will be reflected and some transmitted, so that in region III we will have:
The phase factor is included for convenience of applying the connection formulas.
In region II (using (114) and (115) on (118)) we have:
Now using
we can write
Again, using the connection formulas for the case barrier to the right (using (116) and (117) on (121)), we get for region I:
Hence
Having obtained the expression for ζ1inc(x) and ζ1ref(x) we are now in position to calculate the transmission coefficient using:
To summarize, for a barrier with large attenuation e-2α→0, the tunneling probability equals
The reflection coefficient is:
and also in the same large attenuation limit, we have:
One can see from (125) and (127) thatthe velocity wave ζ(x) on small distances, with the same order of magnitude as Λ, may be influenced by U(x),i.e. it can be transmitted, attenuated or reflected at this scale length. In other words, we get from the calculus, that the velocity field V is indeed transported by the motion of the \'Newtonian fluid\' particles with the velocity U(x) (the imaginary part of the complex velocity [13]).
In recent years, new and exciting advances in experimental techniques [28] prompted a great revival of interest in the Casimir effect, over fifty years after its theoretical discovery (for a recent review on both theoretical and experimental aspects of the Casimir effect, see Refs. [29-31]). As is well known, this phenomenon is a manifestation of the zero-point fluctuations of the electromagnetic field: it is a purely quantum effect and it constitutes one of the rare instances of quantum phenomena on a macroscopic scale.
In his famous paper, Casimir evaluated the force between two parallel, electrically neutral, perfectly reflecting plane mirrors, placed a distance L apart, and found it to be attractive and of a magnitude equal to:
Here, A is the area of the mirrors, which is supposed to be much larger than L2, so that edge effects become negligible. The associated energy EC
can be interpreted as representing the shift in the zero-point energy of the electromagnetic field, between the mirrors, when they are adiabatically moved towards each other starting from an infinite distance. The Casimir force is indeed the dominant interaction between neutral bodies at the micrometer or submicrometer scales, and by modern experimental techniques it has now been measured with an accuracy of a few percent (see [28] and references therein).
Since this effect arises from long-range correlations between the dipole moments of the atoms forming the walls of the cavity, that are induced by coupling with the fluctuating electromagnetic field, the Casimir energy depends in general on the geometric features of the cavity. For example, we see from (129) that, in the simple case of two parallel slabs, the Casimir energy ECis negative and is not proportional to the volume of the cavity, as would be the case for an extensive quantity, but actually depends separately on the area and distance of the slabs. Indeed, the dependence of ECon the geometry of the cavity can reach the point where it turns from negative to positive, leading to repulsive forces on the walls. For example [29], in the case of a cavity with the shape of a parallelepiped, the sign of ECdepends on the ratios among the sides, while in the case of a sphere it has long been thought to be positive. It is difficult to give a simple intuitive explanation of these shape effects, as they hinge on a delicate process of renormalization, in which the finite final value of the Casimir energy is typically expressed as a difference among infinite positive quantities. In fact, there exists a debate, in the current literature, whether some of these results are true or false, being artifacts resulting from an oversimplification in the treatment of the walls [33].
There are three well-known technical types of derivation of the Casimir force for different geometries including the simplest geometry of two parallel, uncharged, perfectly conducting plates firstly explored by Casimir. One modern method is the quantum field theoretical approach based on the appropriate Green\'s function of the geometry of problem [34]. The other technical type is the dimensional regularization method that involves the mathematical complications of the Riemann zeta function and the analytical continuation [34]. The last (the most elementary/the simplest) method is based on modes summation by using the Euler-Maclurian integral formula [35-37].
The problem of finding the Casimir force, not only for the simplest geometry of two plates or rectangular prism, that we want to study here, but also for other more complicated geometries, indispensably/automatically involves some infinities/irregularities; thus, one should regularize the calculation for arriving at the desired finite physical result(s). In the Green\' function method, one uses the subtraction of two terms (two Green\'s functions) to do the required regularization. In the dimensional regularization method, although there isn\'t an explicit subtraction for the regularization of the problem, as is clear from its name, the calculation is regularized dimensionally by going to a complex plane with a mathematically complicated/ambiguous approach. In the simplest method in which the Euler-Maclurian formula is used, the regularization is performed by the subtraction of the zero-point energy of the free space (no plates) from the energy expression under consideration/calculation (e.g. summation of the interior and exterior zero-point energies of the two parallel plates).
Navier-Stokes equations in scale relativity theory predict that the (vector) velocity field V and/or the (scalar) density field ρ, on small distances (the same magnitude as the Compton length) behave like a wave function and are transported by the motion of the Newtonian fluid with velocity U.
Furthermore, when considering vacuum from the Casimir cavity, a non-differentiable, Newtonian, 2D non-coherent quantum fluid whose entities (cvasi-particles) assimilated to vortex-type objects, initially non-coherent, become coherent (the coherence of the quantum fluid reduces to its ordering in vortex streets) due to the constraints induced by the presence of slabs. Casimir type forces are derived which are in good agreement with other theoretical results and experimental data, for both cases: two metallic slabs, parallel to each other, placed at a distance d apart, that constitute the plates of the cavity and a rectangle of sides d1, d.
In other words, non-differentiability and coherence of the quantum fluid due to constraints generate pressure along the Ox and Oy axis.
For viscous compressible fluids, Navier-Stokes equations
together with the equation of continuity
where ρ is the density, ν the velocity of the fluid, X the body force, p the pressure, μ the shear viscosity and
Let us see first, what happens with the set of equations (130) and (131), if one considers that the space-time, where particles move, changes from classical to non-differentiable.
We already know, according to Nottale [11], that a transition from classical (differentiable) mechanics to the scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic) and defining two fractal velocity fields which are fractal functions of the scale variable dt (the non-differentiability of space).
Consequently, replacing d/dt with the fractal operator (42) and solving for both real and imaginary parts, (130) and (131) become, in a stationary isotropic case, taking the body force X = 0 (constant gravitational field) and ∇U = 0 (assuming a constant density of states for the “fluid particles” moving with the velocity U – see further in this section):
and
where V represents the standard classical velocity, which does not depend on resolution, while the imaginary part, U, is a new quantity coming from resolution dependant fractal, υ = μ/ρ the kinematic viscosity and D =ħ/2m defines the amplitude of the fractal fluctuations.
The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination to be quantized, that is, that the strength of the field to be quantized at each point in space. Canonically, the field at each point in space is a simple harmonic oscillator, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, even the vacuum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of vacuum. The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, all of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy.
Let us consider here, vacuum, as a non-differentiable, Newtonian, 2D non-coherent quantum fluid whose entities (cvasi-particles) assimilate to vortex-type objects [38] (see Fig.9) and are described by the wave function Ψ [39, 40]
with
and K, K’ complete elliptic integrals of the first kind of modulus k [41], form a vortex lattice of constants a, b.
The vacuum from a Casimir cavity whose entities (cvasi-particles) are assimilated to vortex-type objects.
Applying in the complex plane [42], the formalism developed in [13] by means of the relation
with Γ the vortex constant. In the general case Γ = cΛ = ħ/m [38-40], the interaction scale being specified through Γ ’s value (Λ being considered as the Compton length).
Based on the complex potential (136), one defines the complex velocity field of the non-coherent quantum fluid, through the relation:
or explicitly, using the notations [41, 42]:
Having in view that
The equipotential curves G(xr,yr) = const., a) for vortex streets aligned with the Ox axis and b) for vortex streets aligned with the Oy axis.
Now, writing the Navier-Stokes equation (132a) and the equation of continuity (133a) in scale relativity theory for constant density (incompressible fluids) in two dimensions, one gets
where the shear viscosity υ is replaced by D since we are dealing here with a non-differentiable quantum fluid.
Then, after some rather long yet elementary calculus one gets from (140a,b) through the degenerations :
with
and
with
Here, ρis the quantum fluid’s density, M the mass of the quantum fluid entities, d and d1 are the elementary space intervals considered along the Oy and Ox axis, respectively.
In other words, non-differentiability and coherence of the quantum fluid due to constraints, generate pressure along the Ox and Oy axis.
Moreover, one can show that the equation of continuity (141) is identically satisfied for both cases of degeneration.
Let us consider a Casimir cavity consisting of the vacuum with the vortex lattice depicted above and two metallic slabs, that constitute the plates of the cavity, placed at a distance d apart, parallel to each other and to the xOz plane (see Fig. III.1). According to the analysis from the previous section, one can see that if the quantum fluid is placed in a potential well with infinite walls (the case of the Casimir cavity analyzed here, where the two plates are the constraints of the quantum fluid), along a direction perpendicular to the walls (the Oy axis here) a coherent structure, a vortex street forms (see Fig. III.2b). Consequently, by integrating (144a,b) with (145a-c) over αr and βr, and using the result in the quantization rule:
where d1 ~ m π a,d ~ n π b, with m, n = 1,2,...., one gets
where
Graphically this is presented in Fig. III.3a,b for different values of the parameters m, n = 1, 2,.... and r.
If the plates were in the yOz plane the constraints being along the Ox axis, vortex streets would form along this axis and the result in (142a,b) with (143a-c) would have been applied, i.e. the cases i) or ii) are identical, yet they depend on the geometry chosen.
Firstly, one can notice that the pressure py on the plates, given by (147a), stabilizes for great r values, is always negative and an attractive force results (see Fig. 11 a), as is the case of the Casimir force (128).
Secondly, the theory predicts, that besides the pressure py acting on the plates, there must be yet another pressure, px (see Fig. 11 b), acting along the Ox axis and given by (147b). One can see that this pressure annuls for great r values, and has a minimum for some values of the parameters m, n. This result is new and should be checked by experiments.
a) Plot of the pressure py on the plates, versus the parameter r for different values of parameters m, n; b) Plot of the pressure px versus the parameter r for different values of parameters m, n.
Moreover, if one tries to compute the order of magnitude of this force, and replaces in (144a) : ħ = 1.054 10-34 J.s, m = 9.1 10-31 kg, ρ~ 1021 cm-3, b = 1Ǻ (values specific to a bosonic gas, i.e. found in high-Tc superconductors [46]) and d ~ 5 b (the distance between the plates), gets a value for py\n\t\t\t\t≅6.18 1010 N m-2 the same order of magnitude as the value calculated using (128), FC≅ 2.08 1010 N m-2.
As a final test, let us study the case of a Casimir cavity, as a rectangle of sides d1, d. Now, the plates induce constraints along both Ox and Oy axis, thus correlations (vortex streets) form along these directions and one should use the degenerations i) and ii), simultaneously. Consequently, from (142a,b) with (143a-c) and (144a,b) with (145a-c) one gets
with
and
with
At every point (x, y) there is a pressure formed of the two constraints. Consequently, adding the pressures in (149) and (151) and using again the result in (146) (i.e. d1 ~ m π a,d ~ n π b, where m, n = 1,2,....) one gets:
where
Furthermore, we integrate (153) over xr and yr, respectively, in order to find a value of the pressure acting on the sides of the rectangular enclosure. After some long, yet elementary calculus, one finds:
Plots of (155) for various values of parameters m, n = 1, 2,.... and r are depicted in Fig. III.4a,b.
One can notice that if the two parameters m and n have close values, the force acting on the Casimir rectangle is always negative and decreases exponentially for increasing r. For parameters m and n (1,5 and 5,1, i.e. very asymmetric) the force has negative and positive domains (see Fig. 12 b) and increases exponentially for increasing r. Moreover, if one tries to find the positive and negative domains, and solve (155) for m = 5, n = 5 finds prect< 0 for 0.45753 ≤r≤ 2.18565 and prect> 0 for r> 2.18565 and r< 0.45753. This result is in agreement with the calculus of regularization using the Abel-Plana formula where E< 0 for 0.36537 ≤L/l≤ 2.73686 and E> 0 for L/l> 2.73686 and L/l< 0.36537 [47].
a) Plots of prect versus the parameter r for various values of parameters m, n = 1,2,...; b) the same plot, yet we present here a magnification of the domain of r for highly asymmetric values of m, n (1,5 and 5,1).
Polymer matrices can be produced in one of the following forms: micro/nano-particles, micro/nano capsules, hydro gels, films, patches.Our new approach considers the entire system (drug loaded polymer matrix in the release environment) as a type of “fluid” totally lacking interaction or neglecting physical interactions among particles. At the same time, the induced complexity is replaced by fractality. This will lead to particles moving on certain trajectories called geodesics within fractal space. This assumption represents the basis of the fractal approximation of motion in Scale Relativity Theory (SRT) [1, 2], leading to a generalized fractal “diffusion” equation that can be analyzed in terms of two approximations (dissipative and dispersive).
In the dissipative approximation the fractal operator (42) takes the form [48, 49]:
As a consequence, we are now able to write the fractal “diffusion” type equation in its covariant form:
Separating the real and imaginary parts in (157), i.e.
we can add these two equations and obtain a generalized “diffusion” type law in the form:
The standard “diffusion” law, i.e.:
results from (159) on the following assertions:
the diffusion path are the fractal curves of Peano’s type. This means that the fractal dimension of the fractal curves is DF\n\t\t\t\t\t\t\t\t= 2.\n\t\t\t\t\t\t\t
the movements at differentiable and non-differentiable scales are synchronous, i.e.
the structure coefficient
The anomalous diffusion law results from (IV.4) on the following assumptions:
the diffusion path are fractal curves with fractal dimension
the time resolution, δt, is identified with the differential element dt, i.e. the substitution principle can be applied also, in this case;
the movements at differentiable and non-differentiable scales are synchronous, i.e.
Then, the equation (IV.4) can be written:
In one-dimensional case, applying the variable separation method [50]
with the standard initial and boundary conditions:
implies:
where L is a system characteristic length, m a separation constant, dependent on diffusion order n.
Accepting the viability of the substitution principle, from (164), through integration, results:
Taking into consideration some results of the fractional integro-differential calculus [51, 52], (165) becomes:
Moreover, (166a,b) can be written under the form:
The relative variation of concentrations, time dependent, is defined as:
where
From (167) and (168) results:
equation similar to Weibull relation
We observe that both constants, a and b, are functions of the fractal dimension of the curves on which drug release mechanism take place, dimension that is a measure of the complexity and nonlinear dynamics of the system. Moreover, constant a depends, also, on the “diffusion” order n.
The experimental and Weibull curves for HS (starch based hydrogels loaded with levofloxacin) and GA (GEL-PVA microparticles loaded with chloramphenicol) samples are plotted in Fig. 13.
Experimental and Weibull curves for HS (left plot) and GA samples (right plot).
The experimental data allowed to determine the values of Weibull parameters (a and b), and implicitly, the value of the fractal dimension from the curve on which release takes place [55].
These values confirmed that the complexity of the phenomena determines, also, naturally, a complex trajectory for the drug particles. Most values are between 1 and 3, in agreement with the values usually accepted for fractal process; higher values denotes the fact that, either fractal dimension must be redefined as function of structure “classes”, or the drug release process is complex, involving many freedom degrees in the phase space [56]. Another observation that can be made based on this results is that the samples with
Let us now consider that, in comparison with dissipative processes, convective and dispersive processes are dominant ones. In these conditions, the fractal operator (42) takes the form:
Consequently, we are now able to write the diffusion equation in its covariant form, as a Korteweg de Vries type equation:
If we separate the real and imaginary parts from Eq. (172), we shall obtain:
By adding them, the fractal diffusion equation is:
From Eq. (173b) we see that, at fractal scale, there will be no Q field gradient.
Assuming that
and normalizing conditions:
take the form:
In relations (175a,b,c) and (176)
Through substitutions:
eq.(177), by double integration, becomes:
with g, h two integration constants and u the normalized phase velocity. If
where cn is Jacobi’s elliptic function of s modulus [41],
are the complete elliptic integrals [41].
Parameter s represents measure characterizing the degree of nonlinearity in the system. Therefore, the solution (180) contains (as subsequences for
The three dimensional plot of solution (180) shows one-dimensional cnoidal oscillation modes of the concentration field, generated by similar trajectories of the drug particles (see Fig. 14). We mention that cnoidal oscillations are nonlinear ones, being described by the elliptic function cn, hence the name (cnoidal).
It is known that in nonlinear dynamics, cnoidal oscillation modes are associated with nonlinear lattice of oscillators (the Toda lattice [56]). Consequently, large time scale drug particle ensembles can be compared to a lattice of nonlinear oscillators which facilitates drug release process.
One-dimensional cnoidal oscillation modes of the field Φ
In what follows we identify the field
For best correlation between experimental data and the theoretical model (for each sample) we used a planar intersection of the graph in Fig. 14 [57], in order to obtain two-dimensional plots.
The highest value of the correlation coefficient (for two data sets: one obtained from the planar intersection, the other from experimental data) will represent the best approximation of experimental data with the theoretical model.
Our goal was to find the right correlation coefficient which should be higher than
The best correlations among experimental and theoretical curves (blue line – experimental curve, red line – theoretical curve).
Scale relativistic framework is implemented by passing to a fluid-like description (the fractality of space), considering the velocity field a fractal function explicitly depending on a scale variable (the fractal geometry of each geodesic) and defining two fractal velocity fields which are fractal functions of the scale variable dt (the non-differentiability of space).
An application of these principles to the motion equation of free particles leads to the occurence of a supplementary TISE (time independent, Schrödinger-type equation) and the following interesting results :
ζ(x) behaves like a wave function on small distances (the same magnitude as the Compton length);
for γ(x) a velocity potential well, U(x) is quantified;
for the harmonic oscillator case, the limit velocity χ has discrete values, and only the first value is less than the velocity of light, c;
in the double-well velocity potential, the complex velocity U(x) is again quantized, this time the levels are equally spaced at a value of ħ ln2;
if one takes ϕR\n\t\t\t\t\t\t\t\t= ϕL\n\t\t\t\t\t\t\t\t= π/2, singularities are obtained for x - x0\n\t\t\t\t\t\t\t\t= Λ/2 and for x - x0\n\t\t\t\t\t\t\t\t= Λ/4 one gets minima for U(x)= c in a double-well velocity potential;
since we considered here the one-dimensional case we get the solution of a single row of rectilinear vortices, which has already been referred to as characterizing a surface of discontinuity;
a typical bound state in a double-well has two classically allowed regions, where the velocity potential is less than the limit velocity; these regions are separated by a classically forbidden region, or barrier, where the velocity potential is larger than the limit velocity;
for tunneling case, there is a nonzero transmission, reflection coefficient, which leads to the proof of the transport of the V field by the motion of the Newtonian fluid with velocity U(x), on small distances (of the order of magnitude of Compton length).
We analyzed vacuum from the Casimir cavity, considered a non-differentiable, Newtonian, 2D non-coherent quantum fluid, by writing the Navier-Stokes equations in scale relativity theory’s framework. As a result the following results may be extracted:
the (vector) velocity field V and/or the (scalar) density field ρ behave like a wave function on small distances (the same magnitude as the Compton length);
the (vector) velocity field V and/or the (scalar) density field ρ are transported by the motion of the Newtonian fluid with velocity U, on small distances (the same magnitude as the Compton length);
Also, the entities assimilated to vortex-type objects from the Casimir cavity, initially non-coherent, become coherent due to constraints induced by the presence of walls and generate pressure along the Ox and Oy axis, thus one can stress out :
the pressure py on the plates, is negative and an attractive force results, as is the case of the Casimir force;
besides the pressure py acting on the plates, there must be yet another pressure, px, acting along the Ox axis;
the order of magnitude of this force, py ≅ 6.18 1010 N m-2 is the same with the value of the classical Casimir force calculation, FC ≅ 2.08 1010 N m-2;
in the case of the Casimir cavity from inside a rectangular enclosure of sides d1, d, the plates induce constraints along both Ox and Oy axis, and one can notice that if the two parameters m and n have close values, the force acting on the Casimir rectangle is always negative and for parameters m and n very asymmetric the force has negative and positive domains, in agreement with the calculus of regularization using the Abel-Plana formula.
Using fractional calculus, the fractal “diffusion” equation give rise to Weibull relation, a statistical distribution function of wide applicability, inclusively in drug release studies. In this approach, we consider all the simultaneous phenomena involved, equivalent with complexity and fractality, offering, in this way, a physical base to this equation and for its parameters. They are functions of fractal dimension of the curves on which drug release mechanism takes place, dimension that is a measure of the complexity and nonlinear dynamics of the system, dependent on the diffusion order.
This theory offers new alternatives for the theoretical study of drug release process (on large time scale) in the presence of all phenomena and considering a highly complex and implicitly, non linear system. Consequently, the concentration field has cnoidal oscillation modes, generated by similar trajectories of drug particles. This means that the drug particle ensemble (at time large scale) works in a network of non linear oscillators, with oscillations around release boundary. Moreover, the normalized concentration field simultaneously depends on normalized time non linear system (through s parameter).
In electro rheological (ER fluids) the additive particles are kept in suspension in a dielectric fluid which is non-conducting. The Dielectric fluid, i.e., the Carrier fluid has high electrical resistivity and has a low viscosity like silicon oil, olive oil, hydrocarbons, etc. The additive particles which are mixed in the carrier fluids are mainly polymers, alumina silicates, metal oxides silica, etc. These additive particles commonly have low particles size which allows the carrier fluid to maintain low viscosity when the external electric field is not applied. In ER fluid the additive particles size range remains in 0.1–100 μm in the carrier fluid. Without any external electric field these fluids stays in liquid condition as soon as the external electric field is applied the ER fluid changes from liquid to solid by viscosity change of the fluid. In Electro rheological (ER) fluids a suspension of particles are present in a non-conducting fluid. The commonly used liquid i.e. hydrocarbon or silicon oil for suspension are low viscous and have high resistivity. Suspension particles are mainly polymers, alumina, silicates, metal oxides etc. These particles are present is very low concentration so that the viscosity of the suspending fluid remains low without application of the applied electric field. The suspension particles are dielectrics of size 0.1–100 μm. In absence of the electric field the particles exhibits properties like fluid and as the electric field is applied the particles behaves like solid. These fluids which change its physical properties like viscosity due to application of electric field are called electro rheological (ER) fluids or smart fluids. Types of ER or Smart fluids: (a) Electro Rheological (ER) Fluids—electric field changes the physical properties of the fluid, (b) Magneto Rheological (MR) Fluids—magnetic fields changes the physical properties of the fluid, (c) Positive Electro Rheological (ER) Fluids—by application of the electric field the viscosity increases and (d) Negative Electro Rheological studied by Ko et al. [1] (ER) Fluids—by application of electric field the viscosity decreases. These ER fluids are one kind of smarts fluids. One of the most easily made ER fluid is adding corn flour in silicon oil or vegetable oil.
\nWhen the electric field is applied on the ER fluid the suspension particles gets polarized and form a thick chain which is parallel to the electric field between the two electrodes. The thickness of the polarized suspension particles between the two electrodes is directly proportional to the intensity of the electric field. The rheological properties of the suspension depend on its change in structure. The more yield stress of the fluid is obtained from the particle columnar structure. When the electric field is removed the suspension particles polarization gets lost and the loose there structure and roam freely in the fluid which in turn reduces the viscosity. The period of returning from the solid state to the liquid state is few milliseconds upon removing the electric field. The material for electrorheological fluid is a superfine suspension of dielectric small particles which react to the applied electric field resulting in changing in the rheological properties of the ER fluid. There are three operational modes of the ER fluid which are as follows: (a) Flow mode—in this mode the electrodes are mounted and fixed and by controlling the motion of the flow the vibrational control is achieved, (b) Shear Mode—in this mode the vibrational control is achieved by varying the shear force here one electrode is fixed and the other is free for rotation and (c) Squeeze Mode—in this mode the space between the electrodes is changed which presses the ER fluid results with a normal force.
\nIn electro rheological fluids there is a large reversible change in the colloidal suspension rheological properties when subjected to the external electric field. Lots of studies are present in which the principle and the uses of the electrorheological fluid are presented by many researchers across the globe. Another property of the ER fluids is that the response time of the ER fluid is very quick for the applied electric fields so the band width is thick. \nFigure 1\n represents the effect of ER fluid particles when application of electric field. For this interesting property the ER fluid has more demand is carious technological applications like smart structure, shock absorbers, engine mount and machine mount. The yield stress of the ER fluid can also be varied by introduction of the external electric field that is why it is also known as functional fluid. Winslow [2] patented the invention of the ER fluid. This ER effect is introduced in state of art automobile. The ER effect was first invented in 1942 by Winslow [2] after that the details understanding of the EF effect took lots of time and then to find the suitable solution for the ER fluid effect took further more time. The properties which delays and stops the ER fluid in few application fields are temperature stability, yield stress and power consumption. Particles size, carrier fluid properties, density, temperature and additives of the ER fluids plays a vital role for most of the properties changes of the ER fluids.
\n(a) Dispersing particles without electric field, (b) dispersing particles with electric field.
There is a limit up to which the dispersing particles can be mixed with the fluid because by increasing the concentration of the dispersing particles volume fraction the electrorheological effect of the solution increases which also causes few problems. As increasing the concentration of the dispersing particle after a certain concentration limit the particles started settling down which cause a problem another problem which arises is the zero field viscosity increment. The viscosity is linked with the temperature i.e. the viscosity decreases when the temperature is increased. Temperature also decreases the dynamic yield strength. Mainly the change in the yield strength occurs due to relative permittivity and the conductivity of particle and also the chemical components of the fluid. Less amount of voltage approx. 1–4 KV/mm is needed for producing ER effect in the solution. 10–6 to 10–3 amp/cm2 is the minimum needed current density for the ER effect. For calculating the power consumption of the suitable ER fluid the measurement of the current density are needed. Dynamic yield stress is one of the important ER fluid property, this stress is the maximum amount of stress required to flow the liquid when the electric field is applied. 100 Pa to 3 KPa is the range of the dynamic yield stress in current ER fluid. The comparison of the various ER fluids are still now difficult as because the standard testing procedure and the state for the fluid is not yet available properly and due to the dependency of the ER fluids on its dispersing particles and the fluid used combinations. For practical applications of the ER fluid the fluid must meet the desired criteria which are (a) Current density 4.0 KV/mm DC less than 10 μA/cm2, (b) dynamic yield stress 4.0 KV/mm <3.0 KPa, (c) Zero field viscosity 0.1–0.3 Pas, i.e., 1–3 Poise, (d) Operational temp range −25°C to +125°C, (e) dielectric breakdown strength >50 KV/mm2, (f) particle size 10 μm, (g) response time < millimeter, (h) Density 1–2 g/cm3, (i) maximum energy density 0.001 Joule/cm3, (j) power supply 2–5 KV@ 1–10 mA, (k) Any conductive surface material, (l) any opaque or transparent, and (m) physically and chemically stable with low conductivity and high breakdown voltage.
\nFor shear loading state applications usually the ER materials are used. The relationship between the ER material and the share are shown in the \nFigure 2\n. In the year 1949 Winslow [2] invented the post-yield appearance of the ER effect. During that time the materials which behave like changing in viscosity were called electro-viscous fluids as their effective or actual viscosity changes were noticeable macroscopically. Many years after it was investigated that with the change in the applied electric field the apparent or the effective viscosity ʋ remains constant, only the noticeable change was found out was the yield stress of the Bingham plastic suspension. This is shown in \nFigure 2\n. Ideal plastic fluids are also another name given to the Bingham plastics, i.e., this fluid does not have viscosity (zero viscosity). A formula representing the shear stress exceeds the yield stress of the material is given by τ = τy + ϑγ, where τ represents Shear stress, τy represents Yield Stress and ϑγ represents Shear Strain. The behavior of the ER material the comparison of the post yield behavior still not investigated. With increasing in the electric field the shear yield stress increases while the yield strain remains 1% for almost all fields. The reaction of the ER fluid on electric field is shown in \nFigure 3\n.
\nSmart fluid characterization (a) without electric field and (b) with electric field.
Reaction of the ER fluid when external electric field is applied.
The ER fluids which are available in the markets are very costly so here are few lists of combinations of the additive particles with the fluid to prepare the cost effective ER fluid. With suitable proportions and amount of the additive particle we can achieve the desired ER fluid as per our need. Various carrier fluids are aldehyde, grease, ketones, kerosene, aroclor, castor oil, chloroform, mineral oil, olefins, olive oil, dielectric oil, diphenyl sebacate, various ethers, resin oil, transformer oil, silicon oil etc. Various additive particles for the ER fluid are alfa silica, alginic acid, alumina, alfa methylacrylate, mannitol, boron, macrocel-C, carbon, cellulose, charcoal, chlorides, dyes, gypsum, micronized mica, nylon powder, olefins, porhin, pyrogenic silica, quartz, rubber, silica gel, etc. [3].
\nER fluid preparation procedure are very simple and mostly all the ER fluids are prepared by this manner the following procedure is used for preparing the ER fluids: (a) The desired powder is chosen and same particle powder size particles are required for the ER fluid dispensing particle, (b) the chosen powder must be passed through size sieve for all the particles same and must be weighted on the weighing machine, (c) the powder is poured in glass container and desired amount of the ER fluid is poured in the glass container which contains the powder of uniform size and are stirred continuously until the powder mixed with the fluid completely, (d) the mixture of the powder and the fluid are stirred for 2 h by glass rod or magnetic starrer at a constant RPM to get a uniform homogenous mixture, (e) the mixed solution is passed to a vane pump five times to get a good result homogenous solution and (f) this process should be followed for other ER solution preparation [3].
\nThe testing of the ER fluid is necessary for selecting of the desired ER fluid for the desired application. The following tests are mainly used (a) Temperature test, (b) breakdown test, (c) viscosity test and (d) sedimentation test.
\nThe electrorheological fluids which are totally dependent on the applied electric fields are used in resistive force creation and damping. Examples of applications are active vibration suppression and motion control. Wang et al. [4] have presented the uses of ER fluids in microfluidics [5]. Various industries like automobiles industries are demanding modified ER fluids with more efficiency Gurka et al. [6] introduced ER-Fluid RheOil®3.0 which improves the sedimentation and re-dispersing behavior. Brennan et al. [7] studied and distinguished the two classes of the ER dampers, first one acts by shearing the stationary fluid and the second one acts by pumping the ER fluid [5]. The two classes are described in details below. Most of the dampers of smart fluids have three common components, i.e., a cylinder, cylinder valve housing and a piston. The vibrating structure kinetic energy can be controlled and dissipated by providing either electric or magnetic field in the valve. In the ER damping process two types of frictions are used they are viscous and coulomb friction [8]. The columbic force denotes the friction acting when two surfaces comes in contact to each other like friction of bearing and hinges friction. Friction is independent to the body velocity, i.e., it is constant. To push fluids through narrow obstructive passage viscous friction comes into play these exists in valves and orifices and is body velocity dependent. The viscous friction and the columbic friction summation is the actuation friction which is denoted in \nFigure 4\n. These frictions have good effects also in the damping machines. The transmission of the vibration to the device is possible by dry sealing friction. For sensitive instruments small vibrations can cause poor accuracy [9]. Bad effect of the friction is also present in the system when the force applied is near to overwhelm the static friction this is known as motion of stick–slip.
\nActuator friction (a) friction columbic, (b) friction viscous, and (c) total friction.
At a near to zero velocity the stick–slip motion happens like an unexpected motion of jerking. Naturally, kinetic friction coefficient in between the two surfaces is smaller than the static friction coefficient. When the given force is more to overwhelm than the static friction then the friction decreases from static to dynamic. Because of this sudden decrease of the friction there will be a sudden velocity jump movement. To show this effect the system of two degree of freedom is taken.
\nIn this type of mode of ER damper there are one or two parallel electrodes which can move parallel to each other and is always perpendicular to the electric field applied so that the fluid can have uniform shear and the ER fluid is present in between the two electrodes. From \nFigure 5a\n c and l are the breath and length of the electrode and j is the electrodes gap. Here E is given voltage, F is net damping force and V is the relative velocity of the electrodes. Two forces are acting in this ER damper (a) Active force Fc because of ER effect and (b) Passive force Fy due to the fluid viscosity. Fy, i.e., the passive force is always present and directly linked with the viscosity of the fluid as well as the damper geometric properties. During application of the electric field a force Fc (because of creation of particles suspension lining up between the electrodes) i.e. static force which is needed to overwhelmed so that the motion can occur [10]. The force Fc is product of area of electrode and the yield strength of the fluid and does not depend on the electrode plate velocity. The net force F of damping of this ER damper is the sum of two components of force. The main aim of this ER damper is to give large ratio of off-field to on-field damping by force ratios Fy and Fc. Because of this large ratio gives various responses by ER unit with changing voltage.
\nModes of operation: (a) shear, (b) valve, and (c) squeeze.
In this type of mode the ER fluid is pressed between the two electrodes as given in \nFigure 5b\n. Because of this the ER fluid is exposed to tensile, compression as wells as shear. In the absence of the given electric field if the ER fluid is pressed it behaves like Newtonian fluid. There is a pressure drop AP occurs at flow rate volume Q. This pressure change in between the valve is because of the velocity of the ER fluid. Moreover, during the presence of the electric field, yield stress is generated by the ER fluid which results more pressure drop between the electrodes plates length. The net damping force is summation of two force components of this type of ER damper. In this type of mode the device effectiveness is the across valves pressure drop with or without the effect of ER [10].
\nThe electrorheological fluids which are totally dependent on the applied electric fields are used in resistive force creation and damping. Examples of applications are active vibration suppression and motion control. L. Wang et al. [4] have presented the uses of ER fluids in microfluidics. Various industries like automobiles industries are demanding modified ER fluids with more efficiency Gurka et.al [6] introduced ER-Fluid RheOil®3.0 which improves the sedimentation and re-dispersing behavior. Brennan et al. [7] studied and distinguished the two classes of the ER dampers, first one acts by shearing the stationary fluid and the second one acts by pumping the ER fluid. The two classes are described in details below.
\nIn this mode the gap between the electrodes are changed and the ER fluid is pressed or squeezed by the force acting normally. \nFigure 5c\n represents the squeezing mode of the ER fluid.
\nER fluids have wide applicability, economic benefit, social benefit high performance for these advantages these smart fluids will find path in various engineering applications in various technological fields. Without any doubt we can say in the future ER technology is going to rule various applications in engineering technological fields. As soon as this technology is accepted then it will be a revolution in both economy and society. From all these advantages of the ER fluids we can predict that in the near future the ER fluids will be used in various technological fields as given below.
\nScientists and Engineers can develop new kind of parts that can easily fulfill the needs of the motor vehicles using the technology of ER. Like for example ER technology used for cooling engine i.e. speed fan clutch of the motor vehicle, shock absorber, brake having break torque controlled, system for suspensions by damping controlled etc., These components using ER technologies will have less wear and tear, more performance, less cost, prolong life service, controlled easily, easy to produce by microcomputer, fast response, high sensitivity.
\nThe valves which are used nowadays for control of pressure and flow rate control can be replaced by ER technology in the future. Because ER technology valves will have no or less movable parts, simple easy structure, low cost, prolong service life, no mechanical processing, minimal tear and wear and electronical control of pressure and rate of flow. For this reasons ER technology will rule the hydraulic industry in the near future.
\nBy utilizing the benefits of the ER technology engineers can produce new type of rotational sealing controlled devices for face the challenges of the magnetic fluid sealing and rubber fluid sealing. Because of the pros like good effect of sealing, minimal tear and wear, less magnetic field and prolong life of service.
\nIn robotic industries nowadays for flexible joints are being controlled by hydro-electric control devises instead of ER fluidic joints technology which can perform much better function than the hydraulic-electric control. Engineers are designing and manufacture flexible joints which will have less volume, fast response time, minimal wear as well as tear, nimble, and which can be easily controlled by micro-computers. ER fluids can provide all these advantages over the hydraulic-electric controls.
\nThere are various commercial uses of the ER fluids and many uses are still undiscovered, in automotive industries the ER fluids are used in clutches, seat dampers, shock absorber, engine mount etc. Many other applications of the ER fluids are listed as follows: (a) Fluid flow via thin channel, (b) for friction instruments clutches, (c) servomechanism for impact and vibrator instruments, (d) pick-pick applications, (e) damping isolator, (f) automobile damping, (g) mounts for engine, (h) power transmission in robots, (i) machine tool artificial intelligence, etc. This list is not the final list because still now many uses of the ER fluid in various fields are yet to discover.
\nRheological characterization is done to identify the change in viscosity of the ER fluid with respect to the shear rate at various electric fields. Garcia et al. [11] have studied the rheological properties of the ER fluid by using ARES rheometer by using parallel plate diameter 50 mm diameter electrode with 1 mm gap between them. High voltage amplifier was used to supply the DC voltage.
\nTo study the permittivity and the power factor of the ER fluid the dielectric properties characterization are done. Rejon et al. [12] describes the method of measuring the dielectric properties of the ER fluid. They used guard ring capacitors and high resistor meter. DC high voltages were used for the test.
\nThe structural changes of the ER field during and before the DC voltage was studied by Rejon et al. [13]. The studied the microscopic structure of the ER fluid by microscope. They studied the microstructural changes of the ER fluid at different DC applied voltages from 0.5 to 2.5 KV/mm.
\nER fluids have lots of interesting properties which attracts them in various applications fields among the various important properties of the ER fluid lies fast reaction, precise controllability and easy boundary between the electrical and mechanical input output power. Because of these interesting properties of the ER fluid the ER fluid is used in motion control and will be used in various applications fields in the near years to come. ER fluids characteristics in most advanced way is briefly described below as given in latest reports: (a) When external electric field is given ER effect is seen by change in viscosity of the carrier fluid from liquid to solid as the viscosity of the liquid increases and after removal of the electric field solid to liquid viscosity decreases making the liquid less thick like the initial state, (b) the process in which the ER fluid changes its state from liquid to solid upon application of the electric field must be reversible, i.e., it should return back to its original state (liquid state) as soon as the external electric field is removed. Viscosity change must be less step, (c) upon application of the electric field the transition of the liquid state to the solid state must be very fast, i.e., 5–10 s, (d) and liquid to solid transition must be only possible by electric field only and not by any other means. By all these characteristics of the ER fluid the ER fluid can be connected with the modern technological applications. This technology is one newly type of future challenge as its attractive properties are being used broadly, which can bring a big change in industries. The main component of the ER technology is the ER fluid which should bring in the technological applications like dampers of ER fluids which is a best solution for control of vibrations.
\nSupporting women in scientific research and encouraging more women to pursue careers in STEM fields has been an issue on the global agenda for many years. But there is still much to be done. And IntechOpen wants to help.
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