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Annals of Physics 318 (2005) 286-307 www.elsevier.com/locate/aop

Fractional hydrodynamic equations for fractal media
Vasily E. Tarasov
*
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia Received 4 November 2004; accepted 17 January 2005 Available online 5 March 2005

Abstract We use the fractional integrals in order to describe dynamical processes in the fractal medium. We consider the ``fractional'' continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media. The sound waves in the continuous medium model for fractional media are considered. Ó 2005 Elsevier Inc. All rights reserved.
PACS: 03.40.Gc; 47.10.+g; 47.53.+n Keywords: Hydrodynamic equations; Fractal media; Fractional integral

1. Introduction The pore space of real media is characterized by an extremely complex and irregular geometry [1-3]. Because the methods of Euclidean geometry, which ordinarily deals with regular sets, are purely suited for describing objects such as in nature, stochastic models are taken into account [4]. Another possible way of describing a
*

Fax: +7095 9390397. E-mail address: tarasov@theory.sinp.msu.ru

0003-4916/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2005.01.004


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complex structure of the pore space is to use fractal theory of sets of fractional dimensionality [5-8]. Although, the fractal dimensionality does not reflect completely the geometric and dynamic properties of the fractal, it nevertheless permits a number of important conclusions about the behavior of fractal structures. For example, if it is assumed that matter with a constant density is distributed over the fractal, then the mass of the fractal enclosed in a volume of characteristic dimension R satisfies the scaling law M (R) $ RD , whereas for a regular n-dimensional Euclidean object M (R) $ Rn. Let us assume that a network of pore channels can be treated on a scale R as a stochastic fractal of dimensionality D < 3 embedded in a Euclidean space of dimensionality n = 3. Naturally, in real objects the fractal structure cannot be observed on all scales but only those for which Rp < R < R0, where Rp is the characteristic dimensionality of the pore channel, and R0 is the macroscopic scale for uniformity of the investigated structure and processes. For example, Katz and Thompson [10] presented experimental evidence indicating that the pore spaces of a set of sandstone samples are fractals in length extending from œ 10 A to 100 lm. The natural question arises: what happens to the ``laws of fluid flow'' when the medium through which the fluid actually flows is fractional? The scientific reason for our interest in this topic is immediately obvious: the laws associated with fluid flow through fractional media are only beginning to be understood [5-9]. In the general case, the fractal media cannot be considered as continuous media. There are points and domains that are not filled of the medium particles. These domains are the porous. We suggest to consider the fractal media as special (fractional) continuous media. We use the procedure of replacement of the fractal medium with fractal mass dimension by some continuous medium that is described by fractional integrals. This procedure is a fractional generalization of Christensen approach [11]. Suggested procedure leads to the fractional integration and differentiation to describe fractal media. The fractional integrals allow us to take into account the fractality of the media. To describe the fractal medium by continuous medium model we must use the fractional integrals. In many problems the real fractal structure of matter can be disregarded and the medium can be replaced by some ``fractional'' continuous mathematical model. To describe the medium with non-integer mass dimension, we must use the fractional calculus. Smoothing of the microscopic characteristics over the physically infinitesimal volume, we transform the initial fractal medium into ``fractional'' continuous model that uses the fractional integrals. The order of fractional integral is equal to the fractal mass dimension of the medium. More consistent approach to describe the fractal media is connected with the mathematical definition the integrals on fractals. In [12], was proved that integrals on net of fractals can be approximated by fractional integrals. In [13], we proved that fractional integrals can be considered as integrals over the space with fractional dimension up to numerical factor. To prove we use the well-known formulas of dimensional regularizations [14]. The fractional continuous models of fractal media can have a wide application. This is due in part to the relatively small numbers of parameters that define a ran-


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dom fractal medium of great complexity and rich structure. The fractional continuous model allows us to describe dynamics of wide class of fractal media. Fractional integrals can be used to derive the fractional generalization of the equations of balance for the fractal media. In this paper, we use the fractional integrals in order to describe dynamical processes in the fractal medium. In Section 2, we consider the ``fractional'' continuous medium model for the fractal medium. In Sections 3-5, we derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. In Section 6, the fractional generalization of Navier-Stokes and Euler equations are considered. In Section 7, we derive the equilibrium equation for fractal media. In Section 8, we consider the fractional generalization of Bernoulli integral. In Section 9, the sound waves in the continuous medium model for fractional media are considered. Finally, a short conclusion is given in Section 10.

2. Fractal media and fractional integrals The cornerstone of fractals is the meaning of dimension, specifically the fractal dimension. Fractal dimension can be best calculated by box counting method which means drawing a box of size R and counting the mass inside. The mass fractal dimension [21,22] can be easy measured for fractal media. The properties of the fractal media like mass obeys a power law relation M ðRÞ ? kR
D

ð D < 3Þ ;

ð 1Þ

where M is the mass of fractal medium, R is a box size (or a sphere radius), and D is a mass fractal dimension. Amount of mass of a medium inside a box of size R has a power law relation (1). The power law relation (1) can be naturally derived by using the fractional integral. In this section, we prove that the mass fractal dimension is connected with the order of fractional integrals. Therefore, the fractional integrals can be used to describe fractal media with non-integer mass dimensions. Let us consider the region WA in three-dimensional Euclidean space E3, where A is the midpoint of this region. The volume of the region WA is denoted by V (WA). If the region WA is a ball with the radius RA, then the midpoint A is a center of the ball, and the volume V ðW A Þ ? ð4=3ÞpR3 . The mass of the region WA in the fractal medA ium is denoted by MD (WA), where D is a mass dimension of the medium. The fractality of medium means that the mass of this medium in any region WA of Euclidean space E3 increase more slowly that the volume of this region. For the ball region of the fractal media, this property can be described by the power law (1) where R is the radius of the ball WA that is much more than the mean radius Rp of the porous sphere. Fractal media are called homogeneous fractal media if the power law (1) does not depends on the translation and rotation of the region. The homogeneity property of the media can be formulated in the form: For all regions WA and WB of the homogeneous fractal media such that the volumes are equal V (WA) = V (WB), we have


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that the mass of these regions are equal MD (WA) = MD (WB). Note that class of the fractal media satisfies the homogeneous property. In many can consider the porous media [15,16], polymers [17], colloid aggregates aerogels [19] as homogeneous fractal media. To describe the fractal media, we must use the continuous media model the fractality and homogeneity properties can be realized in the form:

the wide cases, we [18], and such that

(1) Fractality: The mass of the ball region W of fractal medium obeys a power law relation D R M D ðW Þ ? M 0 ; ð 2Þ Rp where D < 3 and R is the radius of the ball. In the general case , we have the scaling law relation dM D ðkW Þ ? kD dM D ðW Þ; where kW = {kx, x 2 W}. (2) Homogeneity: The local density of homogeneous fractal medium is translation and rotation invariant value that have the form q (r) = q0 = const. We can realize these requirements by the fractional generalization of the equation Z M 3 ðW Þ ? qðrÞ d3 r: ð 3Þ
W

Let us define the fractional integral in Euclidean space E3 in the Riesz form [20] by the equation Z D ðI qÞðr0 Þ ? qðrÞ dV D ; ð 4Þ
W

where dVD = c3 (D, r, r0)d3r, and c3 ðD; r; r0 Þ ? 2
3ÀD

C ð 3=2Þ j r À r0 j CðD=2Þ

DÀ3

;

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 uX jr À r0 j ? t ðxk À xk0 Þ2 :
k ?1

The point r0 2 W is the initial point of the fractional integral. We will use the initial points in the integrals are set to zero (r0 = 0). The numerical factor in Eq. (4) has this form in order to derive usual integral in the limit D fi (3À0). Note that the usual numerical factor cÀ1 ðDÞ ? Cð1=2Þ=2D p3=2 CðD=2Þ, which is used in [20], leads to 3 cÀ1 ð3 À 0Þ ? Cð1=2Þ=23 p3=2 Cð3=2Þ in the limit D fi (3À0). 3 Using notations (4), we can rewrite Eq. (3) in the form M3 (W) = (I3q)(r0). Therefore, the fractional generalization of this equation can be defined in the form Z 23ÀD Cð3=2Þ DÀ3 D M D ðW Þ ? ðI qÞðr0 Þ ? qðrÞjr À r0 j d3 r: ð 5Þ CðD=2Þ W


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If we consider the homogeneous fractal medium (q (r) = q0 = const) and the ball region W, then we have Z 23ÀD Cð3=2Þ DÀ3 M D ð W Þ ? q0 jRj d3 R; CðD=2Þ W where R = r À r0. Using the spherical coordinates, we get Z p25ÀD Cð3=2Þ 25ÀD pCð3=2Þ q0 q RD ; RDÀ1 dR ? M D ðW Þ ? CðD=2Þ DCðD=2Þ 0 W where R = |R|. As the result, we have M (W) $ RD, i.e., we derive Eq. (2) up to the numerical factor. Therefore, the fractal medium with non-integer mass dimension D can be described by fractional integral of order D. Note that the interpretation of the fractional integration is connected with fractional dimension [13]. This interpretation follows from the well-known formulas for dimensional regularizations [14]: Z Z1 2pD=2 f ðxÞ dD x ? f ðxÞxDÀ1 dx: ð 6Þ CðD=2Þ 0 Using Eq. (6), we get that the fractional integral Z f ðxÞ dV D ;
W

can be considered as a integral in the fractional dimension space Z CðD=2Þ f ðxÞ dD x 2pD=2 CðDÞ up to the numerical factor C (D/2)/(2p
D/2

ð 7Þ

C (D)).

3. Equation of balance of mass density The fractional integrals can also be used to calculate the mass dimensions of fractal media. Fractional integration can be used to describe the dynamical processes in the fractal media. Using fractional integrals, we can derive the fractional generalization of dynamical equations [13,23]. In this section, we derive the fractional analog of the equation of continuity for the fractal medium. Let us consider the region W of the medium. The boundary of this region is denoted by oW. Suppose the medium in the region W has the mass dimension D. In general, the medium on the boundary oW has the dimension d. In the general case, the dimension d is not equal to 2 and is not equal to (D À 1). The balance of the mass density is described by the equation Z d qðR; tÞ dV D ? 0; ð 8Þ dt W where we use dMD(W)/dt = 0 and the following notations:


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dV

D

?

2

3 ÀD

Cð3=2Þ D jR j CðD=2Þ

À3

dV 3 ;

dV 3 ? d3 R: The field inte-

Here, and late we use the initial points in the integrals are set to zero (r0 = 0). integral (8) is considered for the region W which moves with the medium. The of the velocity is denoted by u = u (R, t). The total time derivative of the volume gral is defined by the equation Z Z Z d oA dV D þ A dV D ? Aun dS d : dt W W ot oW

ð 9Þ

Here un is defined by un =(u, n)= uknk, the vector u = ukek is a velocity field, and n = nkek is a vector of normal. The surface integral for the boundary oW can be represented as a volume integral for the region W. To realize the representation, we derive the fractional generalization of the Gauss theorem (see Appendix A). As the result, we have the following equation for the total time derivative (9) of the volume integral: Z Z d oA þ cÀ1 ðD; RÞdivðc2 ðd ; RÞAuÞ dV D : A dV D ? ð10Þ 3 dt W ot W For the integer dimensions d = 2 and D = 3, we have the usual equation. Let us introduce notations that simplify the form of equations. We use the following generalization of the total time derivative: d o o ? þ cðD; d ; RÞuk ; ð11Þ dt D ot ox k where the coefficient c (D, d, R) is defined by cðD; d ; RÞ ? cÀ1 ðD; RÞc2 ðd ; RÞ ? 3 2DÀd À1 CðD=2Þ jRj Cð3=2ÞCðd =2Þ
d þ1ÀD

:

Note that the media with integer dimensions (D =3, d = 2) have c (D, d, R) = 1. We use the following generalization of the divergence: DivD ðuÞ ? cÀ1 ðD; RÞ 3 o 2DÀd À1 CðD=2Þ jRj ð c 2 ð d ; RÞ u Þ ? oR Cð3=2ÞCðd =2Þ
3ÀD

divðjRj

d À2

uÞ ;

ð12Þ

and the derivative with respect to the coordinates r D A ? c À1 ð D ; R Þ k 3 oc2 ðd ; RÞA 2DÀd À1 CðD=2Þ jRj ? ox k Cð3=2ÞCðd =2Þ
3ÀD

o jRj ox k

d À2

A:

ð13Þ

Here DivD ? ek rD and DivD ðAuÞ ? rD ðAuk Þ. Note that the rule of term-by-term difk k ferentiation for the operator rD is not satisfied k rD ðABÞ 6? ArD ðBÞþ BrD ðAÞ: k k k This operator satisfies the following rule: oA rD ðABÞ ? ArD ðBÞþ cðD; d ; RÞB : k k ox k

ð14Þ


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Note that rD ð1Þ 6? 0, and we have k rD ð1Þ ? cðD; d ; RÞðd À 2Þxk =R2 : k Using these notations, we rewrite Eq. (10) for the total time derivative of the integral in an equivalent form Z Z d d A dV D ? A þ A DivD ðuÞ dV D : ð15Þ dt W dt D W To derive the fractional generalization of the A = q (R, t) in Eq. (15) and the equation A = q (R, t) in Eq. (15), we get Z Z d d q dV D ? q þ qDivD ðu dt W dt D W equation of continuity [13], we consider of balance of mass (8). Substituting Þ dV D :

ð16Þ

Therefore, equation of balance of mass (8) has the form Z d q þ qDivD ðuÞ dV D ? 0: dt D W This equation is satisfied for all regions W. Therefore, we have the fractional equation of continuity d q þ qDivD ðuÞ ? 0: ð17Þ dt D The fractional generalization of equation of continuity has the form d q ? ÀqrD uk : k dt D Using Eqs. (69) and (70) of Appendix, the equation of continuity can rewritten in an equivalent form oq 2DÀd À1 CðD=2Þ þ jRj ot Cð3=2ÞCðd =2Þ Using ojRj d À3 o j R j ? ðd À 2ÞjRj ? ðd À 2ÞjRj oR oR we have the equation of continuity in the form gradjRj
d À2 d À2 3ÀD

divðjRj

d À2

quÞ ? 0:

ð18Þ

?

d À4

R;

oq oq ðu; RÞ þ cðD; d ; RÞu þ cðD; d ; RÞq divðuÞþðd À 2Þ 2 ot oR jRj

! ? 0; ð19Þ

where we use the following notation (u, R) = ukxk. For the homogeneous media, we have q = const and the equation of continuity leads us to the equation ðd À 2ÞðR; uÞ divðuÞþ ? 0: jRj2


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Therefore, we get the non-solenoidal field of the velocity (div u = ou/oR ,, 0). In addition to mass density, q, the continuity equation includes the density of momentum qu. To obtain the equation for the density of momentum, we consider the mass force and surface force. 4. Equation of balance of momentum density Let the force f = fkek be a function of the space-time point (R, t). The force FM, that acts on the mass MD (W) of the medium region W, is defined by Z FM ? qðR; tÞf ðR; tÞ dV D : ð20Þ
W

The force FS, that acts on the surface of the boundary oW of continuous medium region W, is defined by Z FS ? pn ðR; tÞ dS d ; ð21Þ
oW

where p = p (R, t) is a density of the surface force, and pn = pklnkek. Here, n = nkek is the vector of normal. Let P be a momentum of the medium mass that is situated in the region W. If the mass dMD (W)= q (R, t)dVD moves with the velocity u, than the momentum of this mass is dP = dMD (W)u = qu dVD. The momentum P of the mass of the region W is defined by the equation Z qðR; tÞuðR; tÞ dV D : ð22Þ P?
W

The equation of balance of density of momentum dP ? FM þ FS : dt ð23Þ

Substituting Eqs. (20)-(22) into Eq. (23), we get the balance equation in the form Z Z Z d qðR; tÞuðR; tÞ dV D ? qðR; tÞf ðR; tÞ dV D þ pn ðR; tÞ dS d : ð24Þ dt W W oW Using Eq. (71) of Appendix, the surface integral can be represented as the following volume integral: Z Z Z o ð c 2 ð d ; RÞ pl Þ À 1 pn dS d ? c2 ðd ; RÞpn dS 2 ? c3 ðD; RÞ dV D ox l oW oW W Z ? rD pl dV D : l
W

The balance equation of vectors u = ukek, f = fkek, Z Z d quk dV D ? dt W W

momentum density can be written for the components of and pl = pklek, in the form À Á qfk þrD pkl dV D : l


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The relation for the derivative of the volume integral with respect to time has the form (15). Using this relation for A = quk, we get the equation for the total time derivative of the integral Z Z d d q uk d V D ? ðquk Þþðquk Þ DivD ðuÞ dV D : ð25Þ dt W dt D W Therefore, we can rewrite the balance momentum density (24) in the form Z d ðquk Þþðquk Þ DivD ðuÞÀ qfk ÀrD pkl dV D ? 0: l dt D W This equation is satisfied for all regions W. Therefore, d ðquk Þþðquk Þ DivD ðuÞÀ qfk ÀrD pkl ? 0: l dt D Using the rule of the term-by-term d d ðquk Þ ? q u k þ uk dt D dt D differentiation with respect to time d q; dt D

we get the following form of the equations: d d q uk þ uk q þ q DivD ðuÞ À qfk ÀrD pkl ? 0: l dt D dt D Using the continuity equation, we reduce the fractional equation of balance of density of momentum to the form d q uk ? qfk þrD pkl : ð26Þ l dt D These equations can be called the equation of balance of momentum of fractal medium.

5. Equation of balance of energy density In the general case, the internal energy for the inhomogeneous medium is a function of the space-time point (R, t): e = e (R, t). The internal energy dE of the mass dMD (W) is equal to dE = e (R, t)q (R, t)dVD. The internal energy of the mass of the region W is defined by the equation Z E? qðR; tÞeðR; tÞ dV D :
W

The kinetic energy dT of the mass dMD (W) = q dVD, which moves with the velocity u = u (R, t), is equal to dT ? dM
D

u2 qu2 ? dV D : 2 2


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The kinetic energy of the mass of the region W is Z qu2 dV D : T? 2 W The total energy is a sum of the kinetic and internal energies Z 2 u U ?T þE ? þ e dV D : q 2 W The change of the total energy is defined by U ðt2 ÞÀ U ðt1 Þ ? AM þ AS þ QS ; where AM is the work of mass forces; AM is the work of surface forces; QS is the heat that are influx into the region. The mass dMD (W) = q dVD is subjected to force fq dVD. The work of this force is (u, f)q dVD dt, where (u, f) = ukfk. The work of the mass forces for the region W and time interval [t1;t2] is defined by the following equation: Z t2 Z dt ðu; f Þq dV D : AM ?
t
1

W

The surface (pn, u)dSd dt. [t1;t2] is defin Z AS ?
t

element dSd is subjected to force pn dSd. The work of this force is The work of the surface forces for the region W and time interval ed by the following equation: Z t2 dt ðu; pn Þ dS d :
oW

1

The heat that are influx into the region W through the surface oW is defined by Z t2 Z QS ? dt qn dS d ;
t
1

oW

where qn = (n, q) = nkqk is the density of heat flow. Here n is the vector of normal. The velocity of the total energy change is equal to the sum of power of mass force and the power of surface forces, and the energy flow from through the surface: Z 2 Z Z Z d u þ e dV D ? q ð u; f Þ q d V D þ ðu; pn Þ dS d þ qn dS d : ð27Þ dt W 2 W oW oW Using Eq. (15) for A = q (u2/2 + e), we can rewrite left-hand side of Eq. (27) in the form Z 2 d u þ e dV D q dt W 2 2 Z 2 d u u þe þq þ e DivD u dV D ? q dt D 2 2 W 2 Z 2 d u d u þe þ þe dV D : ? q q þ q DivD u dt D 2 dt D 2 W


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Using the equation of continuity, we get Z 2 Z 2 d u d u þ e dV D ? þe dV D q q dt W dt D 2 2 W Z d d ? qu uþq eðR; tÞ dV D : dt D dt D W

ð28Þ

The surface integrals in the right-hand side of Eq. (27) can be represented as volume integrals Z Z ðu; pn Þ dS d ? rD ðpl ; uÞ dV D ; ð29Þ l
oW W

and

Z
oW

qn dS d ?

Z
W

rD qk dV D : k

ð30Þ

Substituting Eqs. (28)-(30) in Eq. (27), we get Z Z À Á d d qu uþq eðR; tÞ dV D ? ðu; f Þq þrD ðpl ; uÞþrD qk dV D : l k dt D dt D W W ð31Þ These equations can be rewritten in an equivalent form Z Z À Á d d quk uk þ q e dV D ? quk fk þrD ðpkl uk ÞþrD qk dV D : l k dt D dt D W W ð32Þ Let us use equations of balance of momentum (26). Multiplying both sides of these equations on the components uk of vector u and summing with respect to k from 1 to 3, we get the equation d quk uk ? quk fk þ uk rD pkl : ð33Þ l dt D Substituting Eq. (33) in Eq. (32), we get Z d D quk fk þ uk rl pkl þ q e dV dt D W Using Eq. (14) in the form rD ðpkl uk Þ ? uk rD pkl þ cðD; d ; RÞp l l
kl

D

?

Z
W

À

Á quk fk þrD ðpkl uk Þþ rD qk dV D : l k ð34Þ

ouk ; ox l

we get Z d ouk D q e À cðD; d ; RÞpkl Àrk qk dV dt D ox l W

D

? 0:

ð35Þ


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This equation is satisfied for all regions W. Therefore, we get the fractional equation of balance of density of energy in the form d ouk q e ? cðD; d ; RÞpkl þrD qk : ð36Þ k dt D ox l

6. Fractional Navier-Stokes and Euler equations In Sections 3-5, we derive the fractional generalizations of the balance equations for fractal media. As the result we have the fractional hydrodynamic equations in the form: (1) The fractional equation of continuit d q ? ÀqrD uk : k dt D (2) The fractional equation of balance of density of momentum d uk ? qfk þrD pkl : q l dt D (3) The fractional equation of balance of density of energy d ouk q e ? cðD; d ; RÞpkl þrD qk : k dt D ox l

ð37Þ

ð38Þ

ð39Þ

Here, we mean the sum on the repeated index k and l from 1 to 3. We use the following notations: rD A ? aðD; d ÞR k
3ÀD

oÀ R oxk

d À2

Á A:
d þ1ÀD

ð40Þ o : oxl vffiffiffiffiffiffiffiffiffiffiffiffi u3 uX R?t x2 : k
k ?1

d o o o ? þ cðD; d ; RÞul ? þ að D ; d Þ R dt D ot ox l ot

u

l

ð41Þ

where a (D, d), and c (D, d, R) are defined by the equations cðD; d ; RÞ ? aðD; d ÞR
d þ1ÀD

;

að D ; d Þ ?

2 CðD=2Þ ; Cð3=2ÞCðd =2Þ

DÀd À1

The equations of balance of density of mass, density of momentum and density of internal energy makes up a set of five equations, which are not closed. These equations, in addition to the hydrodynamic fields q (R, t), u (R, t), and e (R, t), include also the tensor of viscous stress pkl (R, t) and the vector of thermal flux qk (R, t). Let us start with the definition of tensor pkl = pkl (R, t). According to NewtonÕs law, the force of viscous friction is proportional to the relative velocity of motion


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of medium layers (that is to the gradient of the relevant component of velocity). We further assume that tensor pkl (R, t) is symmetrical, and characterizes the dissipation due to viscous friction. The most general form of tensor of viscous stress, which satisfies the above requirements, is determined by two constants (l and n) which can be chosen so that ouk oul 2 oum ou m pkl ? Àpdkl þ l þ À dkl : ð42Þ þ ndkl ox l ox k 3 ox m ox m This expression contains a second coefficient of viscosity n, called the coefficient of internal viscosity because it reflects the existence of internal structure of particles. In case of structureless particles n = 0. The definition of the vector of heat flux qk = qk (R, t), is based on the empirical Fourier law qk ? À k oT ; ox k ð43Þ

where T = T(R, t) is the field of temperature. The value of heat conductivity k can be found experimentally. Now we have a closed set of Eqs. (37)-(39), (42), and (43) for fields q (R, t), uk (R, t), T (R, t)--a set of fractional equations of hydrodynamics. The fractional generalization of the equations of theory of elasticity for solids can be obtained in a similar way. Let us consider the special cases of the set of Eqs. (37)-(39), (42), and (43). 1. Let us consider the fluids that are defined by pkl ? Àpdkl ; qk ? 0;

where p = p (R, t) is the pressure. The fractional hydrodynamic equations for these fluids have the following form: d q ? ÀqrD uk ; ð44Þ k dt D d 1 uk ? fk À rD p; dt D qk d p o uk e ? ÀcðD; d ; RÞ : dt D q ox k ð45Þ

ð46Þ

These equations are the fractional generalization of the Euler equations. 2. Let the coefficients l, n, and k are constants. If we consider homogeneous viscous fluid, then we have d q ? 0: dt D


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299

For the fractal media, we have the non-solenoidal field of the velocity (div (u) ,, 0), that satisfies the relation divðuÞ ? oum ð 2 À d Þ x k uk ? P3 : 2 ox m l ?1 x l ð47Þ

Using qk = 0 and Eq. (42), we get the fractional generalization of Navier-Stokes equations in the form ouk ouk ouk oul q þ cðD; d ; RÞul þ l rD ? qfk ÀrD p þ l rD k l l ot ox l ox l ox k 2 oul þðn À lÞrD : ð48Þ k 3 ox l Equations (47) and (48) form the system of 4 equations for 4 fields u1 (R, t), u2 (R, t), u3 (R, t), and p (R, t). Note that Eq. (48) can be rewritten in an equivalent form ouk ouk op o2 uk þ cðD; d ; RÞul þ l c ðD ; d ; R Þ q ? qfk À cðD; d ; RÞ ox k ot ox l ox l ox l 2 l o ul þ n þ cðD; d ; RÞ þ Lk ðD; d ; R; uÞ: ð49Þ 3 ox k ox l Here we use the following notations: ouk o ul 2 oul D D D Lk ðD; d ; R; uÞ ? l þ r ð 1Þ ; rl ð1ÞÀ prk ð1Þþ n À l 3 oxl oxk ox l k where rD ð1Þ ? cÀ1 ðD; RÞoc2 ðd ; RÞ=oxk . If c (D, d, R) = 1 and Lk (D, d, R, u) = 0, then k 3 Eq. (49) has the usual form of the Navier-Stokes equations. 7. Fractional equilibrium equation The equilibrium state of media means that we have the conditions oA ? 0; ot oA ? 0; ox k

for the hydrodynamic fields A = {q, uk, e}. In this case, the fractional hydrodynamic equations have the form qfk þrD pkl ? 0; l r D qk ? 0: k

Using oul/oxk = 0, we get that the tensor pkl has the form pkl = Àpdkl. Using the Fourier law, we have the following system of equations: 1 fk ? rD p; qk oT D rk k ? 0; ox k ð50Þ

ð51Þ


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which are the fractional generalization of the equilibrium equations for the media. Eq. (50) can be rewritten in an equivalent form o ð c 2 ð d ; RÞ p Þ ? qc3 ðD; RÞfk : oxk Let us consider the homogeneous medium with the density q0 ? cÀ1 ðD; RÞq0 ? 3 q0 CðD=2Þ R 23ÀD Cð3=2Þ
3ÀD

:

In this case, we have the equation c3 ðD; RÞfk ? oðc2 ðd ; RÞp=q0 Þ : ox k

If the force c3 ðD; RÞfk is a non-potential force such that fk = ÀoU/oxk, then we get the fractional generalization of equilibrium equation in the form c2 ðd ; RÞp þ q0 U ? const: ð52Þ

As the result, we have the power law relation for the pressure in the external gravity field. If the force fk is non-potential force that is defined by the equation fk ? ÀcðD; d ; RÞ oU ; ox k

then we have the condition o ð c 2 ð d ; RÞ p Þ oU ? qc2 ðd ; RÞ : oxk oxk If the density q is defined by the relation q ? q0 cÀ1 ðd ; RÞ, we have the following equi2 librium equation (52). 8. Fractional Bernoulli integral Let us consider the equation of balance of momentum density with the tensor pkl = Àpdkl. Using the relation 2 d u d ? uk uk ; dt D 2 dt D and Eq. (38), we get 2 d u 1 ? uk fk À uk rD p: k dt D 2 q

ð53Þ

If the potential energy U and pressure p is time-independent fields (oU/ot = 0, op/ot = 0), then we can use the following relation:


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301

d d ? cðD; d ; RÞ : dt D dt Let us consider the non-potential force that is described by the equation fk ? ÀcðD; d ; RÞoU =oxk :

ð54Þ

ð55Þ

If D = 3 and d = 2, then this force is potential. Using Eqs. (54) and (55), we can rewrite Eq. (53) in the form d u2 þ U ðD; d Þþ P ðd Þ ? 0; dt 2 where the function P is defined by the usual relation Zp d ð c 2 ð d ; RÞ p Þ P ðd Þ ? : c 2 ð d ; RÞ q p0 As the result we have that the integral
3 X u2 k þ U þ P ðd Þ ? const: 2 k ?1

that can be considered as a fractional generalization of Bernoulli integral for fractal media. If the forces fk are potential, then the fractional analog of the Bernoulli integral does not exists. If the density is described by q ? q0 cÀ1 ðd ; RÞ ? q0 2 C ð d =2Þ R 22Àd
2Àd

;

ð56Þ

then we have the fractional generalization of Bernoulli integral in the following form: q0 u2 þ q0 U ðD; d Þþ c2 ðd ; RÞp ? const: 2 If uk = 0, then we get equilibrium Eq. (52) for non-potential force (55) and the density (56).

9. Sound waves in fractal media Let us consider the motion of the medium with small perturbations. Fractional equations of motion (44) and (45) have the form oq oq þ cðD; d ; RÞul ? ÀqrD uk ; k ot ox l ouk ouk 1 þ c ð D ; d ; R Þ ul ? fk À rD p: qk ot ox l Let us consider the small perturbation q ? q0 þ q0 ; p ? p0 þ p0 ; uk ? u0k ; ð59Þ ð57Þ ð58Þ


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where q 0 ( q0, and p 0 ( p0. Here p0 and q0 describe the steady state that is defined by the conditions oq0 ? 0; ot oq 0 ? 0; ox k op 0 ? 0; ot op 0 ? 0: ox k

Suppose that fk = 0. Substituting (59) in Eqs. (57) and (58), we derive the following equations for the first order of the perturbation: oq0 ? Àq0 rD u0k ; k ot ou0k 1 ? À rD p 0 : q0 k ot ð60Þ ð61Þ

These equations are equations of motion for the small perturbations. To derive the independent equations for perturbations, we consider the partial derivative of Eq. (60) with respect to time: o2 q0 ou0 ? À qr D k : k 2 ot ot Substituting Eq. (61) in this equation, we get o2 q0 ? rD rD p 0 : k k ot 2 ð63Þ ð62Þ

If we consider the adiabatic processes, we can use equation p = p (q, s). For the first order of perturbation we have the equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffi op 0 20 p ?v q; v? : oq s As the result, we get the following fractional generalization of the wave equations: o2 q0 À v2 rD rD q0 ? 0; k k ot 2 o2 p 0 À v2 rD rD p0 ? 0: k k ot 2 ð64Þ ð65Þ

Let us consider the simple example of the fractional wave equations. If we consider one dimensional case (n = 1), where D < 1 and c2 = 1, then we get the following equation for pressure: o2 p 0 o op 0 c1 ðD; xÞ c1 ðD; xÞ 2 À v2 ? 0; ð66Þ ox ot ox where the coefficient c1 (D, x) is defined by the relation c1 ðD; xÞ ? jxjDÀ1 : CðDÞ


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303

10. Solution of the fractional wave equation The fractional generalization o2 u ou 2 v c1 ðD; x c1 ðD; xÞ 2 ? ot ox of wave equation has the following form: ou Þ ; ox

ð67Þ

where c1 (D, x) P 0. Let us consider the region 0 6 x 6 l and the following conditions: uð x ; 0 Þ ? f ð x Þ ; uð 0; t Þ ? 0; ðot uÞðx; 0Þ ? gðxÞ;

u ð l; t Þ ? 0 :

The solution of Eq. (67) has the form 1 X gn fn cosðkn tÞþ pffiffiffiffiffi sinðkn tÞ y n ðxÞ: uð x ; t Þ ? kn n?1 Here fn and gn are the Fourier coefficients for the functions f (x) and g (x) that are defined by the equations Zl Zl fn ? ky n kÀ2 f ðxÞy n ðxÞ dlD ? ky n kÀ2 c1 ðD; xÞf ðxÞy n ðxÞ dx;
0 0 l À2

g n ? ky n k
2

Z
0

f ðxÞy n ðxÞdlD ? ky n k Z
0 l

À2

Z
0

l

c1 ðD; xÞf ðxÞy n ðxÞ dx:

ky n k ?

Z
0

l

y ðxÞ dlD ?

2 n

c1 ðD; xÞy 2 ðxÞ dx;

where dlD = c1 (D, x)dl1, and dl1 = dx. Note that the eigenfunctions yn (x) satisfy the following condition: Zl y n ðxÞy m ðxÞ dlD ? dnm :
0

The eigenvalues kn and the eigenfunctions yn (x) are defined as solutions of the equation v2 ?c1 ðD; xÞy 0x x þ k2 c1 ðD; xÞy ? 0;
0

y ð 0Þ ? 0;

y ð lÞ ? 0 :

This equation can be rewritten in an equivalent form v2 xy 00 ðxÞþðD À 1Þy 0x ðxÞþ k2 xy ðxÞ ? 0: xx The solution of this equation has the form y ðx Þ ? C 1 x
1ÀD=2

J m ðkx=vÞþ C 2 x

1ÀD=2

Y m ðkx=vÞ;

where m = |1 À D/2|. Here Jm (x) are the Bessel functions of the first kind, and Ym (x) are the Bessel functions of the second kind.


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As an example, we consider the case that is defined by l ? 1; v ? 1; 0 6 x 6 1; f ðxÞ ? xð1 À xÞ; g ð x Þ ? 0: The usual wave has D = 1 and the solution 1 X 4ð1 ÀðÀ1Þn Þ sinðpnxÞ cosðpntÞ : uð x ; t Þ ? p3 n3 n?1 The approximate solution for the usual wave with D = 1 that has the form uð x ; t Þ '
10 X 4ð1 ÀðÀ1Þn Þ sinðpnxÞ cosðpntÞ p3 n3 n?1

is shown in Fig. 1 for 0 6 t 6 3 and velocity v = 1. If D = 1/2, then we have the fractal medium wave with pffiffiffi 1 x3=4 J 3=4 ð 2kn x=2Þ: y n ðxÞ ? C ð 1=2Þ The eigenvalues kn are the zeros of the Bessel function pffiffiffi kn : J 3=4 ð 2kn =2Þ ? 0: For example, k1 ' 4:937; k2 ' 9:482; k3 ' 13:862; k4 ' 18:310; k5 ' 22:756: The approximate values of the eigenfunctions À2 Z l pffiffiffi ky n k fn ? x5=4 ð1 À xÞJ 3=4 ð 2kn =2Þ dx; CðDÞ 0

Fig. 1. Usual wave (D = 1) with the velocity v =1.


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305

Fig. 2. Fractal media wave (D = 1/2) with the velocity v =1.

are following f1 ' 1:376; f 2 ' À0:451; f 3 ' 0:416; f 4 ' À0:248; f 5 ' 0:243: The solution of the fractional equation 1 X pffiffiffi fn cosðkn tÞJ 3=4 ð 2kn x=2Þ: uð x ; t Þ ?
n?1

The approximate solution for the fractal media wave with D = 1/2 that has the form uð x ; t Þ '
10 X n?1

fn cosðkn tÞJ

3=4

pffiffiffi ð 2kn x =2Þ

is shown in Fig. 2 for the velocity v = 1. 11. Conclusion The fractional continuous models of fractal media can have a wide application. This is due in part to the relatively small numbers of parameters that define a random fractal medium of great complexity and rich structure. The fractional continuous model allows us to describe dynamics for wide class fractal media. In many problems the real fractal structure of matter can be disregarded and the medium can be replaced by some ``fractional'' continuous mathematical model. To describe the medium with non-integer mass dimension, we must use the fractional calculus. Smoothing of the microscopic characteristics over the physically infinitesimal volume transform the initial fractal medium into ``fractional'' continuous model that uses the fractional integrals [24]. The order of fractional integral is equal to the fractal mass dimension of the medium.


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The experimental research of the hydrodynamics of fractal media can be realized by introducing a neutral indicator, its distribution. It allows one to obtain much information on the motion and mixing of fluid. The fractal nature of damage and porosity has been experimentally detected over a wide range of scales. Note that the fractional hydrodynamic equations for fractal media can be derived from the fractional generalization of the Bogoliubov equations that are suggested in [23].

Acknowledgment I would like to thank Prof. G.M. Zaslavsky for very useful discussions.

Appendix A. Fractional Gauss theorem To realize the representation, we derive the fractional generalization of the Gauss theorem Z Z Aun dS 2 ? divðAuÞ dV 3 ; ð68Þ
oW W

where un is defined by un =(u,n)= uknk, the vector u = ukek is a velocity field, and n = nkek is a vector of normal. Here divðAuÞ ? oðAuÞ oðAuk Þ ? : oR ox k

Here and later we mean the sum on the repeated index k and l from 1 to 3. Using the relation dS d ? c2 ðd ; RÞ dS 2 ; we get Z
oW

c 2 ð d ; RÞ ?

22Àd jRj C ð d =2Þ

d À2

;

ð69Þ

Aun dS d ?

Z
oW

c2 ðd ; RÞAun dS 2 :

Note that we have c2 (2, R) = 1 for the d = 2. Using the usual Gauss theorem (68),we get Z Z c2 ðd ; RÞAun dS 2 ? divðc2 ðd ; RÞAuÞ dV 3 :
oW W

The relation dV
D

? c3 ðD; RÞ dV 3 ;

c3 ðD ; R Þ ?

2

3ÀD

C ð 3=2Þ jRj CðD=2Þ

DÀ3

ð70Þ


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307

in the form dV 3 ? cÀ1 ðD; RÞ dV D allows us to derive the fractional generalization of 3 the Gauss theorem: Z Z Aun dS d ? cÀ1 ðD; RÞdivðc2 ðd ; RÞAuÞ dV D : ð71Þ 3
oW W

References
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