On the Relationship between Fluid Velocity and de Broglie s Wave Function and the Implications to the Navier Stokes Equation

Transcription

1 International Journal of Fluid Mechanics Research, Vol. 29, No. 1, 22 On the Relationship between Fluid Velocity and de Broglie s Wave Function and the Implications to the Navier Stokes Equation V. V. Kulish School of Mechanical & Production Engineering, Nanyang Technological University, 5 Nanyang Ave., Singapore, Tel.: +65) ; Fax: +65) ; mvvkulish@ntu.edu.sg J. L. Lage Mechanical Engineering Department, Box 75337, School of Engineering, Southern Methodist University, Dallas, TX, USA, Tel.: ) ; Fax: ) ; jll@engr.smu.edu By exploring the relationship between the group velocity of the de Broglie s waves and a particle velocity we can demonstrate the existence of a close relationship between the continuity equation and the Schrödinger s equation. This relationship leads to the proportionality between the fluid velocity v and the corresponding de Broglie s wave s phase at the same location. That is, the existence of a scalar function θ proportional to the phase of the de Broglie s wave, such that v = θ can be proven without reference to the flow being inviscid. We then proceed to show that the Navier Stokes equation in the case of constant viscosity incompressible fluid is equivalent to a reaction diffusion equation for the wave function of the de Broglie s wave associated to the moving fluid element. A general solution to this equation, written in terms of the Green s functions, and the criterion for the solution to be stable is presented. Finally, in order to provide an example, the procedure is applied to obtain the solution for the simplest case of the Burgers equation. * * * Introduction The present article is an attempt to look at the problem of solving the governing equation of fluid motion in the most general way, considering, whenever possible, only the fundamental principles of scientific knowledge. The Navier Stokes equation, together with the continuity equation, is the governing equation Received ISSN c 22 Begell House, Inc. 4

2 of fluid motion valid as long as the continuum hypothesis holds. For a wide variety of flow phenomena encountered in engineering, the assumption of constant properties is adequate. However, even when simplified by this assumption the Navier Stokes equation still presents a challenge, namely the nonlinear convective term, which makes the solving procedure very cumbersome, if possible at all. Closed form solutions of the Navier Stokes equation become possible after much more significant simplifications and are few, e. g., Landau & Lifschiz 1988) [1] and White 1991) [2]. Even from the point-of-view of numerical flow simulation, the task of finding an exact solution to the Navier Stokes equation is often impossible to be accomplished because of the necessary computer resources to take into account the very large range of time and spatial scales) to avoid spurious effects caused by the non-linear nature of the equation, Anderson 1995) [3]. Surprisingly enough, it becomes possible to transform the Navier Stokes equation, written for the case of constant and uniform dynamic viscosity and uniform density, into the linear so called reaction diffusion equation. The transformation is accomplished by rewriting the convective and diffusion terms in such a way as to eliminate the convective part. This work underlines the belief that only very few basic principles such as conservation of energy or the least action) are necessary to model any physical process and, therefore, differential equations describing different processes must be kin through following these principles. Another idea underlying this work is the concept according to which any fundamental equation must comprise all the fundamental principles in itself. It is not possible, for example, for a fundamental equation to satisfy the energy conservation principle and to not satisfy the least action one. It is commonly believed that the Navier Stokes equation indeed governs any fluid motion, including turbulence. If it is so, no additional hypothesis should be made in order to describe turbulence by means of the Navier Stokes equation. However, since no general method for solving the Navier Stokes equation exists nowadays, it has become usual to resort to various models that often are inconsistent with the phenomenon in question. Thus, for example, the turbulence concept of eddy viscosity emerges, although it is independent of physical properties of the fluid. The advantage of splitting the velocity components into time averaged and fluctuating parts is doubtful, for it leads to the closure problem of the Navier Stokes equation. Besides, it is not completely clear how the averaging should take place in reality, for depending on the scale of the time step used for averaging the result can differ significantly. This is so because any turbulent process is by definition a fractional Brownian process, Koulich 1999) [4]. In fact this problem is similar, if not the same, to the problem of measuring the length of the coast of Britain, Mandelbrot 1982) [5], where the result depends on the measuring scale. All such practices in studying turbulence became so common that they shadow the real problem of tackling the Navier Stokes equation, trying to recover the information of turbulence encoded in it. In what follows we provide a fresh look at the Navier Stokes equation without having recourse to any additional model or hypothesis but considering this famous equation as it is. 1. Wave Function Any freely moving particle, which has energy E and momentum p, can be associated with a flat wave ψr, t) = C exp[iωt kr)], 1) 41

3 where r is the radius-vector of an arbitrary point in the space, C is the wave amplitude, t is the time, de Broglie 1923) [6]. The frequency ω of this wave and its wave number k are related to the particle energy E and momentum p by the same equations which are valid for a photon, i. e., E = hω, 2) p = hk, 3) where h is the Planck s constant. Eqs. 2) and 3) are called the de Broglie s equations. By substituting Eqs. 2) and 3) into Eq. 1), the latter equation can be written as [ Et ψr, t) = C exp i h pr )]. 4) h Such a wave is called the de Broglie s wave. The function ψ defined by Eq. 4) is called the wave function. The wave function determines the probability with which the particle can be found in a given point in space at a given moment of time. In fact, the probability to find the particle in an infinitesimally small volume d 3 r around the point r is proportional to ψr, t) 2 d 3 r. Thus, the probability density is proportional to the squared modulus of the wave function. A given wave function corresponds to a certain motion of the particle. It is worth to emphasize that if ψr, t) is possible for a certain state of the particle, then the wave function ψ 1 r, t) = ψr, t) expiθ) is also possible provided θ is a real quantity, Blokhintsev 1976) [7]. It is necessary that the probability densities determined by both functions ψ and ψ 1 be equal, so that these two functions describe the same state of the particle motion. 2. Group Velocity In principle, there seems to be no relationship between the wave movement described by Eq. 1) and the mechanical laws of the particle motion. However, this is not so. Consider, for simplicity, a one-dimensional de Broglie s wave propagating in the direction that coincides with the x-axis of a Cartesian co-ordinate system, i. e., ψx, t) = C exp[iωt kx)]. 5) The quantity ωt kx) is the phase of the wave. Consider now a certain point x, where the phase has a given value α. The location of this point can be found from the equation α = ωt kx. One can see that the phase value α moves in the course of time along the x axis with the velocity u the phase velocity) whose value can be determined by differentiating α = ωt kx with respect to time, and equating the result to zero. Hence, dα dt = ω ku = u = ω k. 6) Consider now a group of the de Broglie s waves, i. e., a superposition of waves whose wave lengths belong to a very narrow interval. The wave function of this group of waves is ψx, t) = k + k Ck) exp[iωt kx)]dk, 7) k k 42

4 where k is the wave number, around which the wave numbers of other waves, forming the group, are located. The value k is very small. Now, for a non-relativistic particle, the particle energy can be written as see Blokhintsev 1976) [7]) E = [ m 2 c 4 + p 2 c 2] 1/2 = m c 2 + p2 2m +..., 8) where m is the rest-mass of the particle, and c is the speed of light. By substituting Eq. 8) into Eq. 2) and expressing p 2 through k 2, using Eq. 3), one obtains ω = m c 2 h + hk2 2m ) Expanding the frequency ω as a function of k in series around k : ) dω ω = ω + k k ) +..., k = k + k k ). 1) dk Now, introducing a new integration variable ξ = k k and assuming ck) a slowly varying function of k, one finds: ψx, t) = Ck ) exp[iω t k x)] k k exp {i[dω/dk) t x]ξ} dξ, 11) where Ck ) is the mean between Ck k) and Ck + k) and, therefore, a constant. The integration yields ψx, t) = 2Ck ) sin {[dω/dk) t x] k} dω/dk) t x = Cx, t) exp[iω t k x)]. exp[iω t k x)] 12) Observe that Ck ) and Cx, t) both denote the same quantity the wave amplitude however, expressed in two of the three possible co-ordinate systems k, ω), x, t), or E, p). Since k in the sine term of Eq. 12) is small, then Cx, t) changes slowly with time and x. Therefore, one can consider Cx, t) as the amplitude of an almost monochromatic wave whose phase is ωt k x). It is obvious that the amplitude has its maximum value at x = dω/dk) t. This point is called the center of the wave group. This center moves with the velocity U = dω/dk) obtained by differentiating the expression for x with respect to time. The velocity U is called the group velocity. The group velocity U can be found by using Eq. 9) as U = dω/dk = hk/m. On the other hand, according to Eq. 3), hk = p = m v, where v is the particle velocity. Therefore, one comes to an important conclusion: the group velocity of the de Broglie s waves U equals to the velocity v of the particle corresponding to this group of waves, i. e., U = v. The above conclusion can be easily drawn in a multi-dimensional case, Blokhintsev 1976) [7], having or, in a vector form, U x = ω k x = E p x, U y = ω k y = E p y, U z = ω k z = E p z 13) U = k ω = p E = v. 14) 43

5 3. Schrödinger s Equation and Continuity Equation The Schrödinger s equation forms the basis of quantum mechanics. This equation determines the wave function of a particle of mass m moving in a potential field Φx, t). It can be written in the form, Blokhintsev 1976) [7], i h ψ t = h2 2m 2 ψ + Φx, t)ψ. 15) The most important particularity of the Schrödinger s equation is the presence of the imaginary unity i in front of the time derivative. In classical physics, equations of first order in time have no periodic solutions and, therefore, describe irreversible processes, e. g., mass diffusion, heat conduction, etc. Due to the imaginary coefficient, the Schrödinger s equation, although first order in time, allows periodic solutions. It will be shown now that the Schrödinger s equation also allows to derive the continuity equation in its most general form. In order to do so, consider the Schrödinger s equation written for the conjugate wave function ψ = C exp iθ), namely: i h ψ t = h2 2m 2 ψ + Φx, t)ψ. 16) By multiplying Eq. 15) by ψ and Eq. 16) by ψ, and subtracting the resulting equations, one obtains: i h ψ ψ ) t + ψ ψ = h2 t 2m ψ 2 ψ ψ 2 ψ ). 17) where Eq. 17) can be re-written as w t + divj =, 18) divj = i h ψ 2 ψ ψ 2 ψ ). 2m Observe also that ψψ =w by definition. Indeed, ψ =C expiθ) and its conjugate ψ =C exp iθ) lead to ψψ = C 2 = w, i. e., the probability density to find the particle in the state described by ψ. This is stated in, for instance, Blatt 1992) [8]. Since mass itself is a form of energy and the wave function ψ represents the wave field of a particle of mass m, then ψψ may be thought of as an energy density associated with that mass. Accordingly, the product ψψ = ψ 2 = w is to be interpreted as a probability density. If one notes that w can be also considered as the density of the particles, then Eq. 18) is the equation describing conservation of the amount of particles. Hence, j should be interpreted simply as the flux of particles. It is worth to emphasize that after multiplying Eq. 18) by the mass of a particle m, one obtains ρ m t + divj m =, 19) where ρ m is the particle density and j m is the density flux. Therefore, Eq. 19) is nothing else but the continuity equation. A more elaborate procedure to reach the same conclusion is presented in the Appendix. 44

6 Observe that the form Eq. 18), derived from the Schrödinger s equation, is the most general form of the continuity conservation) equation. Thus, after multiplying Eq. 18) by the electric charge of the particle, one obtains the equation describing conservation of the electric charge, i. e., ρ e t + divj e =, 2) where ρ e is the density of the charge and j e is the density of the electric current. Recall now the representation of the wave function given by Eq. 5). This equation can be written in a more compact form ψx, t) = C expiθ), 21) where C is the real amplitude and θ is the real phase of the wave function. By substituting Eq. 21) into the explicit expression for j, one obtains j = i h 2m ψ ψ ψ ψ), 22) j = C 2 h θ. 23) m Since C 2 = w, then the quantity h/m) θ must be the velocity at the point x. The conclusion, drawn above, is of great importance and is hard to overestimate. It will be extensively used in the course of further derivations. Indeed, velocity at a certain location x is proportional to the gradient of the corresponding de Broglie s wave s phase in the same location, i. e., vx) = θ = h m θ = i h ψ m ψ, 24) where the value h/m)θ = θ is the velocity potential. 4. Navier Stokes and Diffusion Equations Consider the Navier Stokes equation written for the case of a constant viscosity fluid, Landau & Lifschiz 1988) [1]: [ ] v ρ t + v )v = p + µ 2 v + λ + µ ) v), 25) 3 where ρ, λ, and µ are density, second viscosity, and dynamic viscosity of the fluid respectively, p is the pressure, and v is the fluid velocity. In the case of an incompressible fluid, Eq. 25) can be written in a simpler form, since v =, i. e., v t + v )v = 1 ρ p + ν 2 v, 26) where ν = µ/ρ is called kinematic viscosity. Now, using the result expressed by Eq. 24), the Navier Stokes equation can be written in terms of the phase of the de Broglie s wave corresponding to the moving fluid particle, taking into account that v )v = θ ) θ = 1 2 θ θ) θ θ) = 1 2 θ θ), 45

7 and since : 2 v = v) v) = θ) θ) = 2 θ), [ θ ] t + 1 [ 2 θ θ) = p ] ρ + ν 2 θ, 27) or θ t θ θ) = p ρ + ν 2 θ, 28) where p is the difference between the actual pressure p and a certain reference pressure p. Now, substituting θ = 2ν ln ψ, the Navier Stokes equation becomes ψ t p ν 2 ψ = 2µ ψ, 29) which is the reaction diffusion equation written in terms of the scalar function ψ. From Eq. 24), it follows that θ = i h/m) ln ψ. On the other hand, θ = 2ν ln ψ was used in order to transform the Navier Stokes equation. It is obvious that ψ and ψ are related as ψ = ψ expi h/2mν). It was already mentioned in Section 1 that both ψ and ψ represent the same state of the particle, for their amplitudes are equal and, therefore, the probability densities determined by them are the same. Thus, it is no wonder that ψ = ψ expi h/2mν) being substituted into Eq. 29) leads to ψ t ν 2 ψ = p 2µ ψ 3) which is the reaction diffusion equation for the wave function corresponding to the de Broglie s wave of a fluid particle moving with velocity v. Thus, it was so far shown that the Navier Stokes equation for an incompressible fluid of constant viscosity yields the reaction diffusion equation for the wave function of the de Broglie s wave associated with the moving fluid element. That is, the Navier Stokes equation allows only an irrotational solution because of Eq. 24)). Otherwise, in case of a rotational flow, the solution becomes a set of chaotic values and, in fact, does not exist in the sense understood outside the chaos theory. Eq. 3) is a particular case of the Einstein Kolmogorov differential equation whose derivation is given in Tikhonov & Samarskii 1963) [9]. The general case of the equation describes a random motion, so called Brownian motion, of a microscopic particle, existing in free suspension in a medium. 5. Initial and Boundary Conditions From the introduction of the wave function ψ, it is seen that this function and velocity are related as follows: v = 2ν ψ ψ. 31) In order to be able to solve Eq. 3), one has to re-write the initial and boundary conditions, which are given in terms of velocity, into conditions written in terms of the wave function. However, from Eq. 31) the way for converting these conditions cannot be found just as: ψ = exp 1 ) vdx 32) 2ν 46

8 for it is impossible to determine the wave function precisely but to a certain constant coefficient only. From Eq. 24), it follows that θ = m h vdx 33) and, therefore, ψx, t) = C exp i m h ) vdx. 34) In particular, one can immediately see that the no-slip condition becomes equivalent to the wave function being a certain non-zero constant value in fact, the wave function value can never be zero to avoid degenerating the solution into a trivial one). One should not be concerned with the fact that the wave function can only be determined to a constant multiplier, for, looking at Eq. 31), it becomes clear that the velocity is uniquely determined even though the wave function contains an arbitrary non-zero coefficient. According to Eq. 34), the initial condition for the Navier Stokes equation can be written in terms of ψ-function as ψx, ) = C exp i m h ) vx, )dx. 35) 6. General Solution The general solution of Eq. 3) can now be written in terms of the Green s function of the non-homogeneous diffusion equation, Haberman 1987) [1, p. 49]: ψx, t) = t t +ν p x, t; ξ, τ) 2µ ψx, t)d3 ξdτ + x, t; ξ, )ψξ, )d 3 ξ [ x, t; ξ, τ) ξ ψ ψξ, τ) ξ x, t; ξ, τ)] n ds dτ, 36) where x, t; ξ, τ) is the Green s function of the diffusion equation for the domain of interest. Eq. 36) is a linear integral equation for ψx, t). One can show that ψx, t) is a solution of Eq. 3) if, and only if, ψx, t) is a solution of Eq. 36), Logan 1994) [11]. Observe that Eq. 36) has the form ψ = Υψ), where Υ is the mapping defined on the set of bounded continuous functions. Thus one can define a fixed-point, iterative process by ψ m+1 = Υψ m ) with ψ x, t) = x, t; ξ, )ψξ, )d 3 ξ. The initial approximation ψ x, t) is just the second term on the right side of Eq. 36), and it is the solution to the linear, homogeneous diffusion equation with initial condition ψx, ) = ψ x). Therefore, the solution of Eq. 3) can be found from where ψx, t) is determined by Eq. 36). v = 2ν ψ ψ, 37) 47

9 Consider now the influence of the domain boundary on the solution, i. e., the third term in Eq. 36). Both ψ and its normal derivative seem to be needed on the boundary. In most fluid mechanics problems, however, the velocity value, but not its normal derivative, is given as the boundary condition. Hence, ψ B x, t) can be found by using Eq. 34). The Green s function satisfies the related homogeneous boundary conditions, in this case x, t; ξ, τ) = along the boundary. Thus, the effect of this imposed wave function distribution is t ν [ψξ, τ) ξ x, t; ξ, τ)] n ds dτ for those problems in fluid mechanics in which the velocity value is given as the only boundary condition. It may be helpful to illustrate the modifications necessary for one-dimensional problems. Volume integrals in Eq. 36) become one-dimensional integrals. Boundary contributions on the closed surface become contributions at the two ends x = x 1 and x = x 2. Further simplifications can be made in the case of zero pressure gradient, i. e., if p is equal to a constant. In this special case, Eq. 3), by the substitution ψ = ˆψ exp[ p/2µ))t], reduces to ˆψ t = ν 2 ˆψ 38) with the same initial condition and with the boundary condition ˆψ = ψ exp[ p/2µ))t]. The solution of Eq. 38) is not a mapping anymore and can be written as: ˆψx, t) = x, t; ξ, ) ˆψξ, )d 3 ξ t +ν [ ] x, t; ξ, τ) ξ ˆψ ˆψξ, τ) ξ x, t; ξ, τ) n ds dτ. 39) 7. Stability of the Solution. Transition to Chaos Consider now the mapping ψ m+1 = Υψ m ), in general non-linear, to draw a conclusion about stability of the solution to Eq. 3). It was pointed out by Landau & Lifschiz 1988) [1] that transition to turbulence is related to the flow losing stability and can be treated as a bifurcation process, i. e., period doubling which follows the Feigenbaum s scenario, Glendinning 1995) [12]. Under a certain condition the solution 36) becomes non-periodic chaotic ) but bounded and nearby solutions separate rapidly in time. This latter property, called sensitive dependence upon the initial condition, can be thought as a loss of memory of the flow of the past history. It implies that long term predictions of the flow are almost impossible despite the deterministic nature of the Navier Stokes equation. The solution of the mapping ψ m+1 = Υψ m ) becomes chaotic, if the following condition holds: ψ m ψ m 1 lim = λ = ) m ψ m 1 ψ m 2 The constant λ = , called Feigenbaum s constant, is universal and marks the transition to chaos, Landau & Lifschiz 1988) [1], Peitgen, Jügens & Saupe 1992) [13]. 48

10 Turbulence becomes fully developed, if the solution of Eq. 36) satisfies condition 4). In the flow region, for which the condition in question is not satisfied, either transient or laminar regimes take place depending on the limit value given by the left-hand side of Eq. 4). 8. Analytic Solution of the Burgers Equation In order to illustrate feasibility of the described way of solving the Navier Stokes equation, consider a simple case of the Burgers equation. The Burgers equation is a simplified form of the Navier Stokes equation in the one-dimensional, Cartesian, time-dependent, compressible, Newtonian case. This equation has been used in studying the decay of free turbulence. The Burgers equation is a simple equation showing the complicated interplay between the non-linear growth and decay of a wave. Consider the initial value problem for the Burgers equation, Landau & Lifschiz 1988) [1]: u t + u u x = ν 2 u x 2, 41) where v is uniform throughout the domain, with the initial condition ux, ) = u x). By using the substitution Eq. 41) reduces to the diffusion equation u = 2νψ ψ x, 42) ψ t ν 2 ψ =. 43) x2 Also, via Eq. 42), the initial condition on u transforms into an initial condition on ψ, i. e., Integrating both sides of Eq. 44) yields ux, ) = u x) = 2ν ψ xx, ) ψx, ). 44) ψx, ) = ψ x) = exp 1 2ν x u ξ)dξ. 45) Now, the solution to the initial value problem for the diffusion Eq. 43) is ] 1 x ξ)2 ψx, t) = ψ 4πνt) 1/2 x) exp [ dξ. 46) 4νt Therefore, ψ x = 1 4πνt) 1/2 ψ x) x ξ ] [ 2νt exp x ξ)2 dξ. 47) 4νt Consequently, from Eq. 44), we obtain [x ξ)/t] ψ ξ) exp[ x ξ) 2 /4νt)] dξ ux, t) =, 48) ψ ξ) exp[ x ξ) 2 /4νt)] dξ 49

11 where ψ ξ) is given by Eq. 45). It is now straightforward to see that the solution 48) can be written as ux, t) = [x ξ)/t] exp[ Gξ, x, t)/2ν)] dξ exp[ Gξ, x, t)/2ν)] dξ. 49) Here, Gξ, x, t) = x ξ)2 2t ξ + u ξ)d ξ. 5) Observe that Eq. 49) can be derived from the general solution listed in Eq. 36), by realizing that p is constant because p = in the Burgers equation), so one can re-define ψ to transform Eq. 3) into a homogeneous equation this is equivalent of setting the p of Eq. 36) as equal to zero. Moreover, in infinite unidirectional space the surface integral in Eq. 36) disappears, and the last term reduces to a single integral in x. Concluding Remarks It is important to point out a distinction between the approach described so far and the one following from the inviscid flow assumption. It is usual in fluid mechanics to invoke the inviscid condition and from it to demonstrate that the flow is irrotational, i. e., ϖ = v =, where ϖ is the vorticity function. As a consequence, a potential function φ must exist, such that φ = v because φ = for any scalar function θ. In the present case, the existence of the scalar function θ, proportional to the phase of the de Broglie s wave, such that v = θ, was shown without requiring the flow to be inviscid. Moreover, turbulence is not precluded in this case, even though the flow is irrotational, because the viscous effect necessary for turbulence is still present in the momentum transport equation. The similarity between the diffusion form of the Navier Stokes equation and the Schrödinger s equation is by no means accidental but reflects the fundamental connection of the diffusion process to that one of random walks. The transformation performed for the Navier Stokes equation stretches a bridge between quantum mechanics of the micro-world and the continuous world of fluid motion. Consider the Schrödinger s equation Appendix i h ψ t = h2 2m 2 ψ + Uψ, where the wave function ψ can be defined as [ ψ = C expiθ) = C exp ī )] Et pr = C exp i S h ), A2) h where S is called action, and S/m = p/m = v is the classical velocity of the particle. Since ψ C t = t + ī h C S ) exp i S h ) A3) t A1) 5

12 and 2 ψ = [ 2 C C h 2 S)2 + ī h 2 S + 2 S C)] exp i S h ). A4) On substituting Eqs. A3) and A4) into Eq. A1), the latter becomes C S t h2 2m 2 C + C [ C 2m S)2 + UC i h t + 1 ] C S C + m 2m 2 S =. A5) Eq. A5) consists of the pure real and the pure imaginery parts and, thus, and must hold simultaneously. C S t h2 2m 2 C + C 2m S)2 + UC = C t + 1 C S C + m 2m 2 S = If the term containing h 2 in Eq. A6) is neglected, this equation becomes nothing else but the classical Hamilton Jacobi equation for the particle s action. This is, by the way, a good illustration of the fact that as h the classical mechanics is valid up to the values of the first and not zeroth as is the common belief) order of h inclusive. Now, Eq. A7), on being multiplied by 2C, becomes A6) A7) which can be re-written as Since C 2 = w = ψψ and 2C C t + 2C 2C2 S C + m m 2 S = C 2 t A8) + C 2 S ) =. A9) m where C 2 S m = wv = j = ψ ψ ψ ψ, ) ψ = C exp iθ = C exp i h ) Et pr) = C exp i S h ) denotes the complex conjugate wave function, Eq. A9) can be written as w t + j = A1) which is the continuity equation written in its most general form. The vector j in Eq. A1) can be called the probability flux. REFERENCES 1. Landau, L. D., and Lifschiz, E. M., Hydrodynamics, Nauka, Moscow, 1988 [in Russian]. 2. White, F. M., Viscous Fluid Flow, McGraw Hill, New York, Anderson, J. D., jr., Computational Fluid Dynamics: the Basics with Applications, McGraw- Hill, New York,

13 4. Koulich, V. V., Heat and Mass Diffusion in Microscale: Fractals, Brownian Motion and Fractional Calculus Ph. D. Dissertation), Southern Methodist University, Mech. Engng Dept, Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman, San Francisco, de Broglie, L. V., Ondes et Quanta, C.R., 1923, 177, pp Blokhintsev, D. I., Elementary Quantum Mechanics, Nauka, Moscow, 1976 [in Russian]. 8. Blatt, F. J., Modern Physics, McGraw-Hill, New York, 1992, pp Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, Dover Publications, New York, Haberman, R., Elementary Applied Partial Differential Equations, Prentice Hall, Englewood Ciffs, NJ, Logan, J. D., An Introduction to Non-Linear Partial Differential Equations, John Wiley & Sons, New York, Glendinning, P., Stability, Instability and Chaos: an Introduction to the Theory of Non-Linear Differential Equations, Cambridge University Press, Cambridge, Peitgen, H.-O., Jürgens, H., and Saupe, D., Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York,