Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/~tarasov/PDF/CSF2014.pdf
Дата изменения: Fri Dec 5 15:55:49 2014
Дата индексирования: Sun Apr 10 00:07:23 2016
Кодировка:
Поисковые слова: annular

Chaos, Solitons & Fractals 67 (2014) 2637

Contents lists available at ScienceDirect

Chaos, Solitons & Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal h omepage: www.else vier.c om/loc ate/chaos

Flow of fractal fluid in pipes: Non-integer dimensional space approach
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

article

info

abstract
Using a generalization of vector calculus for the case of non-integer dimensional space we consider a Poiseuille flow of an incompressible viscous fractal fluid in the pipe. Fractal fluid is described as a continuum in non-integer dimensional space. A generalization of the NavierStokes equations for non-integer dimensional space, its solution for steady flow of fractal fluid in a pipe and corresponding fractal fluid discharge are suggested. с 2014 Elsevier Ltd. All rights reserved.

Article history: Received 1 May 2014 Accepted 21 June 2014

1. Introduction A cornerstone of fractal fluids is the non-integer dimension [13]. The mass of fractal fluid satisfies a power law relation M $ RD , where M is the mass of the ball region with radius R, and D is the mass dimension [4]. Fractal fluid can be described by four different approaches: (1) Using the methods of ``Analysis on fractals'' [510] it is possible to describe fractal media; (2) An application of fractionaldifferential continuum models suggested in [11,12], and then developed in [1318], where so-called local fractional derivatives [19] are used; (3) Applying fractional-integral continuum models suggested in [4,2023] (see also [24 32]), where integrations of non-integer orders and a notion of density of states [4] are used; (4) Fractal media can be described by using the theory of integration and differentiation for a non-integer dimensional space [3335]. Let us note that main difference of the continuum models with non-integer dimensional spaces form the fractional continuum models suggested in [4,2023] may be reduced to the following. (a) Arbitrariness in the choice of the numerical factor in the density of states is fixed by the equation of the volume of non-integer dimensional ball region. (b) In the fractional continuum models suggested in
Tel.: +7 4959395989.
E-mail address: tarasov@theory.sinp.msu.ru http://dx.doi.org/10.1016/j.chaos.2014.06.008 0960-0779/с 2014 Elsevier Ltd. All rights reserved.

[4,20,21], the differentiations are integer orders whereas the integrations are non-integer orders. In the continuum models with non-integer dimensional spaces the integrations and differentiations are defined for the spaces with non-integer dimensions. In this paper, we consider approach based on the noninteger dimensional space. The power law M $ RD can be naturally derived by using the integrations in non-integer dimensional space [33], where the mass dimension of fractal fluid is connected with the dimension of this space. A vector calculus for non-integer dimensional space proposed in this paper allows us to use continuum models with non-integer dimensional spaces to describe for fractal fluids. This is due to the fact that although the non-integer dimension does not reflect completely the geometric properties of the fractal media, it nevertheless permits a number of important conclusions about the behavior of fractal structures. Therefore continuum models with noninteger dimensional spaces can be successfully used to describe fractal fluids. Integration over non-integer dimensional spaces are actively used in the theory of critical phenomena and phase transitions in statistical physics [36,37], and in the dimensional regularization of ultraviolet divergences in quantum field theory [33,38,39]. The axioms for integrations in non-integer dimensional space are proposed in [34,40] and this type of integration is considered in the

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

27

book by Collins [33] for rotationally covariant functions. In the paper [34] a mathematical basis of integration on non-integer dimensional space is given. Stillinger [34] suggested a generalization of the Laplace operator for noninteger dimensional spaces also. Using a product measure approach, the Stillinger's methods [34] has been generalized by Palmer and Stavrinou [35] for multiple variables case with different degrees of confinement in orthogonal directions. The scalar Laplace operators suggested by Stillinger [34] and Palmer, Stavrinou [35] for non-integer dimensional spaces, have successfully been used for effective descriptions in physics and mechanics. The Stillinger's form of Laplacian for the SchrЖdinger equation in non-integer dimensional space is used by He [4143] to describe a measure of the anisotropy and confinement by the effective non-integer dimensions. Quantum mechanical models with non-integer (fractional) dimensional space have been discussed in [34,35,4452]. Recent progress in non-integer dimensional space approach also includes description of the fractional diffusion processes in non-integer dimensional space in [53], and the electromagnetic fields in non-integer dimensional space in [5461]. Unfortunately, [34,35] proposed only the second order differential operators for scalar fields in the form of the scalar Laplacian in the non-integer dimensional space. A generalization of the vector Laplacian [62] for the noninteger dimensional space is not suggested in [34,35]. The first order operators such as gradient, divergence, curl operators are not considered in [34,35] also. In the work [61] the gradient, divergence, and curl operators are suggested only as approximations of the square of the Laplace operator. Consideration only the scalar Laplacian in the non-integer dimensional space approach greatly restricts us in application of continuum models with non-integer dimensional spaces for fractal fluids and material. For example, we cannot use the Stillinger's form of Laplacian for vector field v Пr; t ч in hydrodynamics of fractal fluids, in fractal theory of elasticity and thermoelasticity, in electromagnetic theory of fractal media to describe processes in the framework non-integer dimensional space approach. In this paper, we propose to use a vector calculus for non-integer dimensional space, and we define the first and second orders differential vector operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space. In order to derive the vector differential operators in non-integer dimensional space we use the method of analytic continuation in dimension. For simplification we consider rotationally covariant scalar and vector functions that are independent of angles. It allows us to reduce differential equations in non-integer dimensional space to ordinary differential equations with respect to r. The proposed operators allow us to describe fractal media to describe processes in the framework of continuum models with non-integer dimensional spaces. In this paper we describe a Poiseuille flow of an incompressible viscous fractal fluid in the pipe. A generalization of the NavierStokes equation for non-integer dimensional space to describe for fractal fluid are suggested. A solution of this equation for steady flow of fractal fluid in a pipe and corresponding fractal fluid discharge are derived.

2. Fractal fluids A basic characteristic of fractal fluids is the non-integer dimensions such as mass or ``particle'' dimensions [4]. For fractal fluids the number of particles N D ПW ч or mass MD ПW ч in any region W & R3 of this fluid increase more slowly than the 3-dimensional volume V 3 ПW ч of this region. For the ball region W with radius R in an isotropic fractal fluid, this property can be described by the relation between the number of particles N D ПW ч in the region W of fractal fluid, and the radius R in the form

ND ПW ч ј N0 ПR=R0 чD ;

R=R0 ) 1;

П 1ч

where R0 is the characteristic size of fractal fluid such as a minimal scale of self-similarity of a considered fractal fluid. The number D is called the ``particle'' dimension. It is a measure of how the fluid particles fill the space. The parameter D does not depend on the shape of the region W. Therefore fractal fluids can be considered as fluid with non-integer ``particle'' or mass dimension. If the fractal fluid consists of particles with identical masses m0 , then relation (1) gives

M D ПW ч ј M 0 ПR=R0 чD ;

R=R0 ) 1;

П 2ч

where M 0 ј m0 N 0 . In this case, the mass dimension coincides with the ``particle'' dimension. As the basic mathematical tool for continuum models of fractal fluids, we propose to use the integration and differentiation in non-integer dimensional spaces. In Section 7, we will show that the power-law relation (2) for an isotropic fractal fluid can be naturally derived by using the integration over non-integer dimensional space, where the space dimension is equal to the mass dimension of fractal fluid. In order to describe fractal fluid by continuum models with non-integer dimensional spaces, we use the concepts of density of states c3 ПD; rч that describes how closely packed permitted places (states) in the space R3 , where the fractal fluid is distributed. The expression dV D Пrч ј c3 ПD; rчdV 3 is equal to the number of permitted places (states) between V 3 and V 3 Ч dV 3 in R3 . The notaD tion d r also will be used instead of dV D Пrч. Note that density of states and distribution function are different concepts, and it is impossible to describe all properties of fractal fluids by the distribution function only. For fractal fluids, we can use the equation

dND ПW ч ј nПrч dV D Пrч;

П 3ч

where nПrч is a concentration of particles that describes a distribution of number of particles on a set of permitted places (possible states). The density of states is chosen such that dV D Пrч ј c3 ПD; rч dV 3 describes the number of permitted states in dV 3 . The form of the function c3 ПD; rч is defined by symmetries of considered problem and properties of the described fractal fluid. A general property of density of states for fractal fluids is a power-law type of these functions that reflects a scaling property (fractality) of the fractal fluid. To simplify our consideration in this paper we will consider only isotropic fractal fluids with density of states that

28

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

is independent of angles. In this case, the form of density of states is defined such that dV D is an elementary volume of the non-integer dimensional space. In the continuum models of fractal fluids, we should work with the dimensionless variables x=R0 ! x; y=R0 ! x; z=R0 ! x, r=R0 ! r, in order to physical quantities of fractal fluids have correct physical dimensions. 3. Vector differential operators in non-integer dimensional space To derive equations for vector differential operators in non-integer dimensional space, we use equations for the differential operators in the spherical (and cylindrical) coordinates in Rn for arbitrary n to highlight the explicit relations with dimension n. Then the vector differential operators for non-integer dimension D can be defined by continuation in dimension from integer n to non-integer D. To simplify we will consider only scalar fields u and vector fields v that are independent of angles

The vector Laplacian in non-integer dimensional space for the vector field v ј v Пr ч er is
V

D

D r

v

ј Gradr Div
2

D

D r

v v
r

ј

@ v r D ю 1 @v r D ю 1 Ч ю2 r r @ r2 @r

! er : П7ч

If D ј n, Eqs. (4)(7) give the well-known formulas for integer dimensional space Rn . Let us consider a case of axial symmetry of the fluid, where the fields uПrч and v Пr ч ј v r Пrч er are also axially symmetric. We will direct the Z-axis along the axis of symmetry. Therefore we use a cylindrical coordinate system. The divergence in non-integer dimensional space for the vector field v ј v Пrч is

Divr

D

v

ј

@v r D ю 2 Ч r @r

v

r

:

П8ч

The gradient in non-integer dimensional space for the scalar field u ј uПrч is

uПrч ј uПrч; vПrч ј vПrч ј v r er ;
where r ј jrj is the radial distance, er ј r=r is the local orthogonal unit vector in the directions of increasing r, and v r ј v r Пr ч is the radial component of v . We will work with rotationally covariant functions only. This simplification is analogous to the simplification for definition of integration over non-integer dimensional space described in Section 4 of the book [33]. 3.1. Vector differential operators for spherical and cylindrical cases Using the continuation from integer n to arbitrary noninteger D, we can get explicit definitions of differential operators for non-integer dimensional space in the following forms. Note that the same expressions can be obtained by using the integration in non-integer dimensional space and the correspondent Gauss's theorem. Let us define the differential vector operations such as gradient, divergence, the scalar and vector Laplacian for non-integer dimensional space. For simplifications, we assume that the vector field v ј v Пrч be radially directed and the scalar and vector fields uПrч; v Пrч are not dependent on the angles. The divergence in non-integer dimensional space for the vector field v ј v Пrч is

Gradr u ј

D

@u er : @r

П9ч

The scalar Laplacian in non-integer dimensional space for the scalar field u ј uПrч is
S

DD u ј r

@2u D ю 2 @u Ч : r @ r2 @r

П10ч

The vector Laplacian in non-integer dimensional space for the vector field v ј v Пr ч er is
V

D

D r

v

ј

@2v r D ю 2 @v r D ю 2 Ч ю2 r r @ r2 @r

!

v

r

er :

П11ч

Eqs. (8)(11) can be easy generalized for the case

u ј uПr; zч and v Пr; zч ј v r Пr; zч er Ч v r Пr; zч ez . In this case
the curl operator for

Curlr

D

v

@v r @v z eh : ј ю @z @r

v

Пr ; zч is different from zero, and

П12ч

Divr

D

v

ј

@v r D ю 1 Ч r @r

v

r

:

П4ч

For D ј 3 Eqs. (4)(11) give the well-known expressions for the gradient, divergence, curl operator, scalar and vector Laplacian operators The suggested operators for 0 < D < 3 allow us to reduce D-dimensional vector differentiations (4)(7) and (8)(11) to derivatives with respect to r ј jrj. It allows us to reduce partial differential equations for fields in noninteger dimensional space to ordinary differential equations with respect to r. 3.2. Stillinger's Laplacian for non-integer dimensional space For a function u ј uПr ; hч of radial related angle h measured relative to through the origin, the scalar Laplacian dimensional space proposed by Stillinger
St

The gradient in non-integer dimensional space for the scalar field u ј uПrч is

Gradr u ј

D

@u er : @r

П5ч

distance r and an axis passing in a non-integer [34] is

The scalar Laplacian in non-integer dimensional space for the scalar field u ј uПrч is
S

DD ј

DD u ј Divr Gradr u ј r

D

D

@2u D ю 1 @u Ч : r @ r2 @r

П6ч

@ r Dю1 r Dю1 @ r @ Dю2 sin б @h 1

@ 1 Ч Dю2 @r r 2 sin h @ ; h @h

П13ч

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

29

where D is the dimension of space (0 < D < 3), and the 2 variables r P 0; 0 6 h 6 p. Note that ПSt DD ч St D2D . If the function depends on radial distance r only (u ј uПrч), then
St

Grad

D;d r

uј

CПar =2ч @ uПr ч

p

ar =2 r ar ю1

@r

er :

П22ч

DD uПr ч ј
ј

1 r
Dю1

@ r @r

Dю1

@ uПr ч @r

For ar ј 1, Eq. (22) gives (5). Using the operators (22) and (19) for the fields u ј uПr ч and v ј v Пrч er , we can get the scalar and vector Laplace operators for the case d D ю 1 by
S

@ 2 uПr ч D ю 1 @ uПr ч Ч : @ r2 r @r

П14ч

DD;d u ј Div r

D;d r

Gradr u;

D ;d

V

D

D;d r

v

ј Gradr Div

D ;d

D ;d r

v

:

П23ч

It is easy to see that the Stillinger's form of Laplacian St DD for radial scalar functions uПrч ј uПrч coincides with the scalar Laplacian S DD defined by (6), i.e., r
St

The scalar Laplacian for d D ю 1 for the field u ј uПr ч is
S

DD;d u ј AПd; ar ч r

1 r
2ar ю2

D uПr ч ј D uПrч:

D

S

D

П15ч
where

! @ 2 u d Ч 1 ю ar @ u ; Ч @ r2 r 2ar ю1 @r

П24ч

The Stillinger's Laplacian can be applied for scalar fields only. It cannot be used to describe vector fields v ј v r Пrч er because this Laplacian for D ј 3 is not equal to the usual vector Laplacian for R3 ,
St

AПd; ar ч ј

CППd Ч ar ч=2ч CПar =2ч : par ю1=2 CППd Ч 1ч=2ч

П25ч

D

3

v

Пr ч Dv Пrч ј

@2v r 2 @v r 2 Ч ю2 r @r r @ r2

!

v

r

er :

П16ч

v
V

The vector Laplacian for d D ю 1 for the field ј v Пr ч er is
D ;d r

The gradient, divergence, curl operator and vector Laplacian are not considered by Stillinger in paper [34]. 3.3. Differential operators for d D ю 1 Let us consider a ball region BD in the fractal fluids with the boundary Sd ј @ BD with dimensions

D

v

ј AПd; ar ч

1 r
2ar ю2

@ 2 v r d Ч 1 ю ar @ v r d ar Ч 2ar ю1 ю @ r2 r @ r r2ar

!

v

r

er : П26ч

The vector differential operators (22), (19), (24) and (26) allow us to describe complex fractal fluids with the boundary dimensions d D ю 1 by continuum models with non-integer dimensional spaces. 4. NavierStokes equations in non-integer dimensional space for fractal fluid A motion of an incompressible viscous fractal fluid in the framework of continuum model with non-integer dimensional space is described by the equations

dimПBD ч ј D;

dimПSd ч ј d:

П17ч

Eqs. (4)(11) define differential operators for spaces with non-integer dimension D and boundary dimensions d ј D ю 1. In general, the dimension D of the region of a fractal fluid and the dimension d of boundary of this region are not related by the equation d ј D ю 1, i.e.,

dimП@ BD ч D ю 1:

П18ч

Div

D r

v

ј 0;

П27ч

Using the integration in non-integer dimensional space and the correspondent Gauss's theorem, we can define the divergence for the case d D ю 1 by the equation

dv 1 D ј f ю Gradr p Ч dt q

m D v;
V

D r

П28ч

Div

D;d r

v

ј

p

ПdЧ1юDч=2

CПD=2ч

r

1
Dю1юd

@

v

CППd Ч 1ч=2ч

Пr ч d Ч Dю @r r
r

d

v

r

Пr ч : П19ч

where f is the vector field of mass forces, m is the kinematic viscosity is the ratio of the dynamic viscosity l to the density of the fluid q, and d=dt is the material derivative

For d ј D ю 1, we get (4). We can define the parameter

dv @ v D ј Ч v ; Gradr v : dt @t
D V

П29ч
D

ar ј D ю d;

П20ч

In Eqs. (27)(29) the gradient Gradr , the divergence Divr , are defined by equations and the vector Laplacian (5), (4), (7) for spherical symmetry. For cylindrical symmetry, these operators are defined by equations (9), (8), (11). If the dimension D of the region of a fractal fluid and the dimension d of boundary of this region are not related by the relation d ј D ю 1, i.e., ar ј D ю d 1, then we should use the equations

DD r

that can be interpreted as a dimension of fractal fluid along the radial direction er . Using (20), Eq. (19) can be rewritten in the form

Div

D;d r

v

ј

p

П1юar ч=2

CППd Ч ar ч=2ч 1 @ v r Пr ч d Ч ar r CППd Ч 1ч=2ч r ar ю1 @ r

v

r

Пr ч : П21ч

This is divergence operator for non-integer boundary dimension d D ю 1. For ar ј 1, we have d ј D ю 1, and then equations (19), (21) give (4). The gradient for the scalar field uПrч ј uПr ч and the radial dimension ar 1 is defined by

Div

D;d r

v

ј 0;
D ;d r

П30ч

dv 1 D;d ј f ю Gradr p Ч m V D dt q dv @ v D;d ј Ч v ; Gradr v ; dt @t

v

;

П31ч П32ч

30
D;d

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637
D;d

where the gradient Gradr , the divergence Divr , and the vector Laplacian V DD;d are defined by equations (22), (21) r and (26). Eqs. (28) and (31) can be called the NavierStokes equations for non-integer dimensional space. It is convenient to work in the dimensionless space variables x=R0 ! x; y=R0 ! x; z=R0 ! x; r =R0 ! r , that yields dimensionless integration and dimensionless differentiation in non-integer dimensional space. Here R0 is the characteristic size of a fractal fluid, which is always finite. For example, R0 can be the minimal scale of self-similarity of a considered fractal fluid. Then the density is properly scaled such that the mass Q of fractal fluid and the fields v ; p; f have correct physical dimensions. The NavierStokes equations (28) and (31) describe dynamics of fractal fluids in the framework of continuum models with non-integer dimensional spaces. These equations allow us to describe the isotropic fractal fluid only when the presence of spherical or cylindrical symmetry. Eqs. (28) and (31) can be used, when the fields p; v ; v have the form p ј pПr ч and v ј v r Пrч er , f ч ј fr Пrч er does not depend on the angles. As an example of application of the NavierStokes equations (28) and (31), we consider a steady flow of fractal fluid in a pipe, and fractal fluid discharge in the next sections. In the next sections, we derive the Poiseuille equation for fractal fluids from the NavierStokes equations (28) and (31). Equations can be used for any other problems of hydrostatics and hydrodynamics of fractal fluids within the case of spherical and cylindrical symmetries. To consider anisotropic fractal fluids, and problems without spherical and cylindrical symmetries, we cannot use the NavierStokes equations (28) and (31). In this case, we should apply a product measure approach. The product measure approach to describe fractal properties of space time has been considered in [63]. The product measure approach for the fractional spaces has been suggested in [64,65], where fractional phase space is considered with its interpretation as a non-integer (fractional) dimensional space. For non-integer dimensional spaces the product measure approach is suggested in [35], where each orthogonal coordinates has own dimension. The product measure approach for the fractional-integral continuum models has been considered in [2528]. 5. Steady flow of fractal fluid in a pipe In this section, we derive the Poiseuille equation for fractal fluids from the NavierStokes equation (28). Let us consider a simple problem of motion of an incompressible viscous fractal fluid. Using the continuum models with non-integer dimensional space, we describe a steady flow of fractal fluid in a pipe with circular cross-section. We take the axis of the pipe as the X-axis. The velocity for laminar of fractal fluid is along the X-axis at all points, and is a function of r only

Z-axis give that the pressure is constant over the crosssection of the pipe. We shall solve the equation for a pipe with circular cross-section. Taking the origin at the center of the circle we can use cylindrical symmetry v ј v Пrч ј v x Пr ч ex . Using the NavierStokes equation (28), we have
S

D

D r

v

x

Пr ч ј

1 dp

l dx

;

П34ч

where l ј q m, and dp=dx is a constant. The pressure gradient dp=dx may be written юDp=l, where Dp is the pressure difference between the ends of the pipe and l is its length. Using the scalar Laplacian for non-integer dimensional space the NavierStokes equation (34) takes the form

@

2

v

@r

x Пr 2

ч

Ч

D ю 2 @ v x 1 dp ю ј 0: r @r l dx

П35ч

For 1 < D < 3 and 0 < D < 1, the general solution of (35) is

v

x

Пr ч ј C 1 r

3юD

Ч C2 Ч

1 2 ПD ю 1ч D 1ч:

dp

l dx

r

2

П 0 < D < 3;
For D ј 3, we have

П36ч

v

x

Пrч ј C 1 lnПr чЧ C 2 Ч

1 dp 2 r: 4 l dx

П37ч

For D ј 1, we get the general solution

v

x

Пr ч ј C 1 r 2 Ч C 2 Ч

1 dp 2 r П2 lnПr чю 1ч: 4 l dx

П38ч

It should be noted that dimensions D ј 1 of the fractal fluid do not correspond to the distribution of particles along the line. The fractal media with D ј 1 describe a distribution of fluid particles in 3-dimensional space such that the mass dimension of the distribution is equal to D ј 1. Let us determine a flow of fractal fluid in a pipe of annular cross-section with the internal radius R1 and external radius R2 . The constants C 1 and C 2 in the general solution (36) are determined from the boundary conditions

v

x

ПR1 ч ј

v

x

ПR2 ч ј 0:

П39ч

Using (36), these conditions have the form

C1 R C1 R

3юD 1 3юD 2

1 2 П D ю 1ч 1 Ч C2 Ч 2 П D ю 1ч Ч C2 Ч

dp

l dx
dp

R2 ј 0; 1 R2 ј 0: 2

П40ч П41ч

l dx

Then the constants are

C1 ј C2 ј

1 2 ПD ю 1ч 1 2 ПD ю 1ч

l l

dp R2 ю R 2 dx R3юD ю R 1 dp dx

2 1 ; 3юD 2 R3юD R2 ю R3юD 1 2 2 R3юD ю R3юD 1 2

П42ч R
2 1

:

П43ч

Substitution of (42) and (43) into (36) gives

v

ј v Пr ч ј

v

x

Пr ч ex :

П33ч

v

x

1 dp R2 ю R 2 Пrчј 3юD 2 ПD ю 1ч l dx R1 ю R

2 1 r 3юD 3юD 2

Ч

R

3юD 2 R2 1 R3юD 1

юR юR

3юD 2 3юD 2

R

2 1

! Чr
2

;

The equation of continuity is satisfied identically. The components of the NavierStokes equation for Y-axis and

П44ч

where 0 < D < 1 and 1 < D < 3.

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

31

Using the variables

v

x

Пr ч ј ю

xј

R2 ; R1

yј

r ; R1 dp 1 ю x2 ю y dx 1 ю x3 ю D Ч y2 : П45ч

1 2 П D ю 1ч

l

dp 2 r 3 R dx R D 1ч:

юD

ю

r 2 R П46ч

П0 < D 6 3;

Eq. (44) can be represented in the form

v

Пx; yч ј

R2 1 2 П D ю 1ч

3юD

l

Ч

x2 ю x3 ю D 1 ю x3юD

Eq. (46) can be called the Poiseuille equation for flow of fractal fluid. For the case of non-fractal fluid (D ј 3), Eq. (46) gives the well-known Poiseuille equation

v

x

Пr ч ј ю

To demonstrate some properties of the velocity v x Пr ч defined by (44), we can visualize the function (45), for x 2 Ѕ1; 100; y 2 Ѕ1; 100 and different values of dimensions D ј 2:9; D ј 2:7; D ј 2:0; D ј 1:1. The plots of function (45) are presented by Figs. 14, where l ј 1; R1 ј 1 and dp=dx ј ю1. The flow in a pipe of annular cross-section with the radius R, i.e. R1 ј 0 and R2 ј R, we have

r 2 1 dp 2 : R 1ю 4 l dx R

П47ч

Thus the velocity distribution across the pipe is parabolic for the non-fractal fluids. For the fractal fluids, we have non-integer power-law (46). Note that suggested Poiseuille equation for fractal fluids, which are derived from the proposed NavierStokes equation for non-integer dimensional space can be used only to describe flow of fractal fluid in pipes. To describe

30000

20000

Z
10000

0

-10000 100 80 60 80 100

Y

40 20 00 20

60 40

X

Fig. 1. Plot of the velocity function z ј v Пx; yч defined by (45) for the ranges x 2 Ѕ1; 100, y 2 Ѕ1; 100, and D ј 2:9.

10000

5000

Z

0

-5000

100 80 60 80 100

Y

40 20 00 20

60 40

X

Fig. 2. Plot of the velocity function z ј v Пx; yч defined by (45) for the ranges x 2 Ѕ1; 100, y 2 Ѕ1; 100, and D ј 2:7.

32

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

0

Z
-2000

-4000 100 80 60 80 100

Y

40 20 00 20

60 40

X

Fig. 3. Plot of the velocity function z ј v Пx; yч defined by (45) for the ranges x 2 Ѕ1; 100, y 2 Ѕ1; 100, and D ј 2:0.

0

-50

Z
-100

100 80 60 80 100

Y

40 20 00 20

60 40

X

Fig. 4. Plot of the velocity function z ј v Пx; yч defined by (45) for the ranges x 2 Ѕ1; 100, y 2 Ѕ1; 100, and D ј 1:1.

flow of fractal fluid between parallel planes, we should define new differential operators and the NavierStokes equations (28) and (31), should be modified by using the product measure approach [35,6365]. 6. Fractal fluid with ar 6ј 1 Let us derive the Poiseuille equation for fractal fluids from the NavierStokes equation (31). The NavierStokes equations for fractal fluid with ar ј D ю d 1 has the form

and (24) where D ! Dx ј D ю ax and d ! dx ј d ю ax to get (48). Eq. (48) with ar ј ax ј 1 gives (35). For 1 < D < 3 and 0 < D < 1, the general solution of (48) is

v

x

Пr ч ј C 1 r

ar юd

x

Ч C2 Ч

1 2 Пdx Ч ar ч ar AПdx ; ar ч D 1ч:

dp

l dx

r

2ar

П 0 < D < 3;

П49ч

AПdx ; ar ч ю

1 r
2ar ю2

@

2

v

@

x Пr r2

ч

! dx Ч 1 ю ar @ v x Пr ч Ч @r r 2ar ю1

For ar ј ax ј 1 Eq. (49) gives (36). The constants C 1 and C 2 in the general solution (49) are determined by the boundary conditions

v
П48ч

x

ПR1 ч ј

v

x

ПR2 ч ј 0:

П50ч

1 dp ј 0; l dx

These conditions give the equations

C1 R C1 R

ar юdx
1

where AПdx ; ar ч is defined by (25), dx ј d ю ax , and ax is dimension along the X-axis. Using v x Пr ч as an effective scalar field ueff Пr ч ј v x Пr ч, we can apply Eqs. (22), (21)

ar юdx
2

1 2 Пdx Ч ar ч ar AПdx ; ar ч 1 Ч C2 Ч 2 Пdx Ч ar ч ar AПdx ; ar ч Ч C2 Ч

dp

l dx
dp

R R

2ar 1 2ar 2

ј 0; ј 0:

П51ч П52ч

l dx

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

33

Then the coefficients are

C1 ј ю C3 ј ю

1 2 Пdx Ч ar ч ar AПdx ; ar ч 1 2 Пdx Ч ar ч ar AПdx ; ar ч

dp

l dx l
dp dx

R2ar ю R 1 ar юdx R1 юR 2ar ar юdx R2 R1 Rar юdx 1

2ar 2 ar юd 2

x

;

П53ч ;

юR юR

2ar ar юdx 1 R2 ar юdx 2

fractal fluid is based on the idea of how the mass of a fluid region scales with the region size, if we assume unchanged density. For many cases, we can write the asymptotic form for the relation between the mass M D ПW ч of a ball region W of fluid, and the radius R containing this mass as follows:

П54ч
Substitution of (53) and (54) into (49) gives

M D ПW ч ј M

0

R R0

D

П57ч

v

x

Пr ч ј ю Ч

1 2 Пdx Ч ar ч ar AПdx ; ar ч R
2ar 2

l

R R

ar юdx
1

ю R2ar R 1 ю Ra 2
r

ar юd
2

x

dp R2ar ю R2ar 1 2 r ar юdx dx R1 ю Rar юdx 2 !
2ar

ar юdx

ar юdx
1

юd

x

юr

П55ч

for 0 < D < 3, where D 1. If R1 ј 0 and R2 ј R, then Eq. (55) has the form

v

x

Пr чјю

dp 2 R 2 Пdx Ч ar ч ar AПdx ; ar ч l dx

1

ar

r R

ar юdx

ю

r 2ar : R П56ч

For ar ј ax ј 1 Eq. (56) gives (46). The cases ar < 1 and/or ax < 1 corresponds to fractal fluid. We can assume that ax > 1 can be used to describe fractal on the curve, ax ј ln turbulent flow in pipe. This assumption is based fact that trajectories of the fluid particles are fractal then ax > 1 (for example, the Koch curve with П4ч= lnП3ч % 1:262).

for R=R0 ) 1. The constant M 0 depends on how the spheres of radius R0 are packed. The parameter D, which is interpreted as a dimension, does not depend on the shape of the region W, or on whether the packing of spheres of radius R0 is close packing, a random packing or a porous packing with a uniform distribution of holes. The non-integer mass dimension D of fractal fluid is a measure of how the fluid fills the integer n-dimensional Euclidean space it occupies. Note that the fact that a fluid is random or contains cavities does not necessarily imply that the fluid is fractal. Using the non-integer dimensional space approach, we can calculate the mass of fractal homogeneous fluids. Scaling law (57) is obtained naturally in the framework of this approach. We can use the integration in a non-integer dimensional space [34] that is described by the equation

Z

R

D

d r uПrч ј

D

2pПDю1ч=2 CППD ю 1ч=2ч Z1 Zp D б dr r Dю1 dh sin
0 0

ю2

h uПr; hч;

П58ч

7. Fractal fluid discharge In general, fractal fluids cannot be considered as a fluid on fractal. Real fractal fluids have a characteristic smallest length scale such as the radius, R0 , of a particle (for example, an atom or molecule). In real fluids the fractal structure cannot be observed on all scales but only those for which R > R0 , where R0 is the characteristic scale of the particles. The concept of non-integer mass dimension of

where d r represent the volume element in the non-integer dimensional space. Using (58) with uПr ; hч ј 1, and

D

Zp
0

dh sin

Dю2

hј

p1=2 CПD=2 ю 1ч
CПD=2ч

;

П59ч

we get the volume of D-dimensional ball with radius R in the form

VD ј

pD

=2

CПD=2 Ч 1ч

RD :

П60ч

0.4

0.2

Q

0 0

0.5

R
12

3 2.5

D

Fig. 5. Plot of the discharge function Q ј Q ПR; Dч defined by (64) for the range R 2 Ѕ0; 1 and D 2 Ѕ2; 3.

34

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

The mass of fluid in W is described by the integral

M D ПW ч ј

Z
W

qПrч dD r;

П61ч

There are areas unoccupied by particles of fractal fluid. In continuum model of fractal fluids, we take into account this fact by using the integration in space with non-integer dimension

where r is dimensionless vector variable. For a ball with radius R and constant density qПrч ј q ј const, we get

Q јq

2 pd=2 CПd=2ч

Z
0

R

v

x

Пr ч r

dю1

dr ;

П63ч

M D ПW ч ј q V D ј

pD=2 q
CПD=2 Ч 1ч

RD :

П62ч

This equation define the mass of the fractal homogeneous ball region of fluid with volume V D . For D ј 3, Eq. (62) gives the well-known equation for mass of non-fractal ball pffiffiffiffi region M3 ј П4qp=3ч R3 because CП3=2ч ј p=2 and CПz Ч 1ч ј z CПzч. Let us determine the mass Q of fluid passing per unit time through any cross-section of the pipe (called the discharge). Not all pipe volume is occupied by fractal fluid.

where d ј D ю 1 is non-integer dimension of the cross-section, q is a constant density, and v x Пrч is defined by Eq. (46). Substitution of (46) into (63) gives

Q јю

dp D R 2 ПD Ч 1ч CППD ю 1ч=2ч l dx

qp

ПDю1ч=2

Ч1

П0 < D 6 3 D 1ч: П64ч

Note that q has physical dimension of mass (for example, kilogram). The mass of fractal fluid is thus proportional to ПD Ч 1ч-power of the radius of the pipe.

0.08

0.06

Q
0.04

0.02

0 0

0.5

R
11

2 1.5

D

Fig. 6. Plot of the discharge function Q ј Q ПR; Dч defined by (64) for the range R 2 Ѕ0; 1 and D 2 Ѕ1; 2.

0.03

0.02

0.01

Q
0

-0.01 0

0.5

R
1 0.5

1.5 1

D

Fig. 7. Plot of the discharge function Q ј Q ПR; Dч defined by (64) for the range R 2 Ѕ0; 1 and D 2 Ѕ0:5; 1:5.

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637

35

0.4

0.2

Q

0 0

0.5

R
10 1

3 2

D

Fig. 8. Plot of the discharge function Q ј Q ПR; Dч defined by (64) for the range R 2 Ѕ0; 1 and D 2 Ѕ0; 3.

For D ј 3, Eq. (64) gives the well-known equation

8. Conclusion

Q јю

q p dp 4 R: 8 l dx

П65ч

The mass of non-fractal fluid is proportional to the fourth power of the radius of the pipe. The dependence of Q on dp=dx and R given by formula (65) was established empirically by G. Hagen in 1839 and J. L. M. Poiseuille in 1840, and theoretically justified by G. G. Stokes in 1845. Eq. (64) can be rewritten in the form

Q јю

8

q p dp R leff dx

DЧ1

П66ч

with the effective dynamic viscosity

leff ј

DЧ1 4

p

П3юDч=2

CППD ю 1ч=2ч l:

П67ч

The mass (discharge) of fractal fluid is proportional to the non-integer power ПD Ч 1ч < 4 of the radius of the pipe, and the dynamic viscosity is effectively changed. For D ј 3, Eq. (67) gives leff ј l. For 1 < D < 3, we have leff > l. If 0 < D < 1, then leff < 0. The effective dynamic viscosity of fractal fluid with D 2 П1; 3ч increases with increasing a deviation of the dimension D from three. We can see an interesting effect of a negative effective dynamic viscosity for fractal fluid with dimension D 2 П0; 1ч. The strong fractality of the fluid, which is caused by small dimension, leads to an increased fluid flow compared with conventional medium. This is probably due to an increase in freedom of particles motion for the fractal fluid similar to 3D Cantor dust. The discharge function Q ј Q ПR; Dч defined by (64) for the different values of dimensions 0 < D < 3 and the range R 2 Ѕ0; 1 are present on Figs. 58, where q ј 1; l ј 1, and dp=dx ј ю1.

In this paper, we propose a generalization of the Navier Stokes equations to describe fractal fluids in the framework of continuum models with non-integer dimensional spaces. These equations contain generalized differential vector operators for non-integer dimensional space. As an example of application of the suggested NavierStokes equations for fractal fluids, we consider a Poiseuille flow of an incompressible viscous fractal fluid in the pipe. The solution for steady flow of fractal fluid in a pipe and corresponding fractal fluid discharge have been derived. In this paper fractal fluid is described as a continuum in non-integer dimensional space. We assume that suggested continuum models with non-integer dimensional spaces and the correspondent NavierStokes equations for fractal fluids may be important for fractal theory of different type of media. As the main object for application of the proposed continuum models is a two-component medium, where distribution of one component (gas, liquid, solid) into another component (fluid, gas or empty space) can be characterized by non-integer mass or ``particle'' dimension. This non-integer dimensional component can be considered as a fractal fluid. One of the possible experimental methods for determining the presence of fractal properties of the two-component medium may be to use labels with radioactive isotopes for particles of component that is assumed a fractal. A basic idealized model of fractal fluid is a liquid distributed in empty space R3 with non-integer mass dimension D < 3. In some sense the fractal fluid is considered as a liquid analog of fractal porous solid material. Fractal fluid can also be viewed as a two-phase medium consisting of a liquid and a discharged gas instead of empty space, where the liquid is characterized by fractal mass dimension. As an object of study, we also can consider an emulsion, when both the dispersed and the continuous phase are liquids, and the dispersed phase is fractally distributed in

36

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637 [12] Carpinteri A, Cornetti P. A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos Solitons Fract 2002;13(1):8594. [13] Carpinteri A, Cornetti P, Kolwankar KM. Calculation of the tensile and flexural strength of disordered materials using fractional calculus. Chaos Solitons Fract 2004;21(3):62332. [14] Carpinteri A, Chiaia B, Cornetti P. A disordered microstructure material model based on fractal geometry and fractional calculus. Z Angew Math Mech 2004;84(2):12835. [15] Carpinteri A, Cornetti P, Sapora A, Di Paola M, Zingales M. Fractional calculus in solid mechanics: local versus non-local approach. Phys Scr 2009;T136:14003. [16] Carpinteri A, Cornetti P, Sapora A. Static-kinematic fractional operators for fractal and non-local solids. Z Angew Math Mech 2009;89(3):20717. [17] Yang X-J, Srivastava HM, Hed J-H, Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys Lett A 2013;377(2830):1696700. [18] Yang X-J, Baleanu D, Tenreiro Machado JA. Systems of NavierStokes equations on Cantor sets. Math Prob Eng 2013;2013:769724. [19] Yang X-J. Advanced local fractional calculus and its applications. New York: World Science Publisher; 2012. [20] Tarasov VE. Continuous medium model for fractal media. Phys Lett A 2005;336(23):16774. Available from: . [21] Tarasov VE. Fractional hydrodynamic equations for fractal media. Ann Phys 2005;318(2):286307 [physics/0602096]. [22] Tarasov VE. Dynamics of fractal solid. Int J Mod Phys B 2005;19(27):410314. Available from: . [23] Tarasov VE. Wave equation for fractal solid string. Mod Phys Lett B 2005;19(15):7218 [physics/0605006]. [24] Ostoja-Starzewski M, Li J, materials Fractal. beams and fracture mechanics. Z Angew Math Phys 2009;60(6):1194205. [25] Li J, Ostoja-Starzewski M, solids Fractal. product measures and fractional wave equations. Proc R Soc A 2009;465(2108):252136; Li J, Ostoja-Starzewski M. Correction to Li and Ostoja-Starzewski 465 (2108) 2521. Proc R Soc A 2011;467(228):1214. 1p. [26] Demmie PN, Ostoja-Starzewski M. Waves in fractal media. J Elast 2011;104(12):187204. [27] Li J, Ostoja-Starzewski M. Micropolar continuum mechanics of fractal media. Int J Eng Sci 2011;49(12):130210. [28] Ostoja-Starzewski M, Li J, Joumaa H, Demmie PN. From fractal media to continuum mechanics. Z Angew Math Mech (J Appl Math Mech) 2014;94(5):373401. [29] Balankin AS, Elizarraraz BE. Hydrodynamics of fractal continuum flow. Phys Rev E 2012;85(2):025302. [30] Balankin AS, Elizarraraz BE. Map of fluid flow in fractal porous medium into fractal continuum flow. Phys Rev E 2012;85(5):056314. [31] Li J, Ostoja-Starzewski M. Comment on hydrodynamics of fractal continuum flow and map of fluid flow in fractal porous medium into fractal continuum flow. Phys Rev E 2013;88(5):057001. [32] Balankin AS, Elizarraraz BE. Reply to comment on hydrodynamics of fractal continuum flow and map of fluid flow in fractal porous medium into fractal continuum flow. Phys Rev E 2013;88(5):057002. [33] Collins JC. Renormalization. Cambridge: Cambridge University Press; 1984. [34] Stillinger FH. Axiomatic basis for spaces with noninteger dimensions. J Math Phys 1977;18(6):122434. [35] Palmer C, Stavrinou PN. Equations of motion in a non-integerdimensional space. J Phys A 2004;37(27):69877003. [36] Wilson KG, Fisher ME. Critical exponents in 3.99 dimensions. Phys Rev Lett 1972;28(4):2403. [37] Wilson KG, Kogut J. The renormalization group and the expansion. Phys Rep 1974;12(2):75199. [38] 't Hooft G, Veltman M. Regularization and renormalization of gauge fields. Nucl Phys B 1972;44(1):189213. [39] Leibbrandt G. Introduction to the technique of dimensional regularization. Rev Mod Phys 1975;47(4):84976. [40] Wilson KG. Quantum field theory models in less than 4 dimensions. Phys Rev D 1973;7(10):291126. [41] He X-F. Anisotropy and isotropy: a model of fraction-dimensional space. Solid State Commun 1990;75(2):1114. [42] He X-F. Fractional dimensionality and fractional derivative spectra of interband optical transitions. Phys Rev B 1990;42(18): 117516. [43] He X-F. Excitons in anisotropic solids: the model of fractionaldimensional space. Phys Rev B 1991;43(3):20639. [44] Thilagam A. Excitonphonon interaction in fractional dimensional space. Phys Rev B 1997;56(15):9798804.

continuous phase. An emulsion that is a mixture of two immiscible liquids, one of which (the dispersed phase) is fractally dispersed in the other (the continuous phase). In this case the dispersed phase can be described as a fractal fluid by suggested continuum models with non-integer dimensional space. The proposed models can be used for a solution that is a homogeneous mixture composed of only one (liquid) phase, where one phase has a fractal dimension. We can consider a fractal distribution of a solute dissolved in a non-fractal solvent, then the solute is considered as a fractal fluid. The solvent that is fractally homogeneously mixed with solute can be considered as a fractal homogeneous fluid. The homogeneity property of the fractal fluid means that two regions W 1 and W 2 with the equal volumes V n ПW 1 ч ј V n ПW 2 ч have equal number of particles N D ПW 1 ч ј N D ПW 2 ч. In other words the fractal fluid is called homogeneous if the power law N D ПW ч $ RD (or M D ПW ч $ RD ) does not depend on the translation of the region W. We can consider a fractal distribution of small solid particles in the suspension. In this case, we have an internal phase (solid) that is fractally distributed through the external phase (fluid) by mechanical agitation. An object of investigations can be a liquid mixed with a solid particles, where the distribution of these particles in space can be characterized by non-integer mass dimensions, which can be caused by a power law distribution of particles by size or mass. As a complex medium which may exhibit fractal properties can be considered the blood that is composed of proteins, glucose, mineral ions, hormones, carbon dioxide, blood cells and other particles suspended in water. We assume that the blood as a multi-phase medium can have attributes of a fractal distribution for some blood components including bacteria, viruses and medicinal substances getting into the blood. We assume that the suggested approach to describe fractal fluids by continuum models with non-integer dimensional spaces may be important for fractal theory of blood flow in cardiovascular system, dynamics of fractal media in hydrologic modeling [6669] and it allows to develop the fractal dynamics of multi-phase media [70,71]. References
[1] Falconer KF. The geometry of fractal sets. Cambridge University Press; 1985. [2] Feder J. Fractals. New York: Plenum Press; 1988. [3] Carpinteri A, Mainardi F, editors. Fractals and fractional calculus in continuum mechanics. New York: Springer; 1997. [4] Tarasov VE. Fractional dynamics: applications of fractional calculus to dynamics of particles fields and media. New York: Springer; 2011. [5] Kugami J. Analysis on fractals. Cambridge University Press; 2001. [6] Strichartz RS. Differential equations on fractals. Princeton, Oxford: Princeton Univ. Press; 2006. 170p. [7] Strichartz RS. Analysis on fractals. Not AMS 1999;46(10):1199208. [8] Harrison J. Flux across nonsmooth boundaries and fractal Gauss/ Green/Stokes' theorems. J Phys A 1999;32(28):531728. [9] Kumagai T. Recent developments of analysis on fractals. Am Math Soc Transl 2008;223(202):8196. [10] Derfel G, Grabner P, Vogl F. Laplace operators on fractals and related functional equations. [Topical Review] J Phys A 2012;45(46):463001. 34p. Available from: . [11] Carpinteri A, Chiaia B, Cornetti P. Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput Methods Appl Mech Eng 2001;191(12):319.

V.E. Tarasov / Chaos, Solitons & Fractals 67 (2014) 2637 [45] Matos-Abiague A. Deformation of quantum mechanics in fractionaldimensional space. J Phys A 2001;34(49):1105968. Available from: . [46] Matos-Abiague A. Bose-like oscillator in fractional-dimensional space. J Phys A 2001;34(14):312538. [47] Lohe MA, Thilagam A. Quantum mechanical models in fractional dimensions. J Phys A 2004;37(23):618199. [48] Eid R, Muslih SI, Baleanu D, Rabei E. On fractional SchrЖdinger equation in a-dimensional fractional space. Nonlinear Anal Real World Appl 2009;10(3):1299304. [49] Muslih SI. Solutions of a particle with fractional d-potential in a fractional dimensional space. Int J Theor Phys 2010;49(9):2095104. Available from: . [50] Muslih SI, Agrawal OP. Schrodinger equation in fractional space. In: Fractional dynamics and control. New York: Springer; 2012. p. 20915 [chapter 17 ]. [51] Calcagni G, Nardelli G, Scalisi M. Quantum mechanics in fractional and other anomalous spacetimes. J Math Phys 2012;53(10):102110. Available from: . [52] Martins J, Ribeiro HV, Evangelista LR, da Silva LR, Lenzi EK. Fractional Schrodinger equation with noninteger dimensions. Appl Math Comput 2012;219(4):23139. [53] Lucena LS, da Silva LR, Tateishi AA, Lenzi MK, Ribeiro HV, Lenzi EK. Solutions for a fractional diffusion equation with noninteger dimensions. Nonlinear Anal Real World Appl 2012;13(4):195560. [54] Muslih SI, Baleanu D. Fractional multipoles in fractional space. Nonlinear Anal Real World Appl 2007;8(1):198203. [55] Baleanu D, Golmankhaneh AK, Golmankhaneh AK. On electromagnetic field in fractional space. Nonlinear Anal Real World Appl 2010;11(1):28892. [56] Muslih SI, Saddallah M, Baleanu D, Rabei E. Lagrangian formulation of Maxwell's field in fractional D dimensional space-time. Rom Rep Phys 2010;55(78):65963. [57] Zubair M, Mughal MJ, Naqvi QA. The wave equation and general plane wave solutions in fractional space. Prog Electromagn Res Lett 2010;19:13746.

37

[58] Zubair M, Mughal MJ, Naqvi QA. On electromagnetic wave propagation in fractional space. Nonlinear Anal Real World Appl 2011;12(5):284450. [59] Zubair M, Mughal MJ, Naqvi QA. An exact solution of the spherical wave equation in D-dimensional fractional space. J Electromagn Waves Appl 2011;25(10):148191. [60] Zubair M, Mughal MJ, Naqvi QA. An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space. Prog Electromagn Res 2011;114:44355. [61] Zubair M, Mughal MJ, Naqvi QA. Electromagnetic fields and waves in fractional dimensional space. Berlin: Springer; 2012. [62] Moon P, Spencer DE. The meaning of the vector Laplacian. J Franklin Inst 1953;256(6):5518. [63] Svozil K. Quantum field theory on fractal spacetime: a new regularization method. J Phys A 1987;20:386175. [64] Tarasov VE. Fractional systems and fractional Bogoliubov hierarchy equations. Phys Rev E 2005;71(1):011102. 12p. Available from: . [65] Tarasov VE. Fractional Liouville and BBGKI equations. J Phys Conf Ser 2005;7:1733. Available from: . [66] Zhu QY, Xie MH, Yang J, Li Y. A fractal model for the coupled heat and mass transfer in porous fibrous media. Int J Heat Mass Transfer 2011;54(78):14009. [67] Gabrys E, Rybaczuk M, Kedzia A. Blood flow simulation through fractal models of circulatory system. Chaos Solitons Fract 2006;27(1):17. [68] Jayalalitha G, Shanthoshini Deviha V, Uthayakumar R. Fractal model for blood flow in cardiovascular system. Comput Biol Med 2008;38(6):68493. [69] Benson DA, Meerschaert MM, Revielle J. Fractional calculus in hydrologic modeling: a numerical perspective. Adv Water Resour 2013;51:47997. [70] Soo S-L. Fluid dynamics of multiphase systems. Blaisdell Pub. Co.; 1967. [71] Nigmatulin RI. Dynamics of multiphase media, vols. 12. New York: Hemisphere; 1991.