- Email: [email protected]
Steady laminar flow of fractal fluids
Accepted Manuscript Steady laminar flow of fractal fluids
Alexander S. Balankin, Baltasar Mena, Orlando Susarrey, Didier Samayoa
PII: DOI: Reference:
S0375-9601(16)31699-1 http://dx.doi.org/10.1016/j.physleta.2016.12.007 PLA 24223
To appear in:
Physics Letters A
Received date: Revised date: Accepted date:
12 November 2016 4 December 2016 6 December 2016
Please cite this article in press as: A.S. Balankin et al., Steady laminar flow of fractal fluids, Phys. Lett. A (2017), http://dx.doi.org/10.1016/j.physleta.2016.12.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • • • •
Equations of Stokes flow of Newtonian fractal fluid are derived. Pressure distribution in the Newtonian fractal fluid is derived. Velocity distribution in Poiseuille flow of fractal fluid is found. Velocity distribution in a steady Couette flow is established.
Steady laminar flow of fractal fluids Alexander S. Balankin a)1), Baltasar Mena2), Orlando Susarrey1), Didier Samayoa1) 1)
Grupo Mecánica Fractal, ESIME, Instituto Politécnico Nacional, México D.F., México 07738
2)
Laboratorio de Ingeniería y Procesos Costeros, Instituto de Ingeniería, Universidad Nacional
Autónoma de México, Sisal, Yucatán, Mexico 97355
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived.
Keywords: Fractal; Fractal fluids; Laminar flow; Fluid velocity distribution
a)
Corresponding author. Adress: SEPI-ESIME-Zacatenco, U.P. Adolfo Lopez Mateos, Av. Politécnico Nacional s/n, México D.F. 07738, México. Tel.: 55-5729600 + Ext. 54589; E-mail: [email protected]
I. Introduction
A pressure driven laminar flow is ubiquitous in nature and widely used in scientific and engineering applications [1]. The Poiseuille law for the flow rate in a tube of circular cross section was apparently first derived from the Navier-Stokes equations by Stokes himself in 1845 (see Ref. [2]). Since, pressure driven laminar flows of viscous fluids remain an active topic of research (see Refs. [3-5] and references therein). Commonly, viscous fluids are viewed as homogeneous continua of an integer dimension. However, internal structures of many complex fluids exhibit scale-invariant features characterized by non-integer fractal dimensions [6,7]. Examples of fractal fluids include solutions with a fractal distribution of a solute dissolved in a nonfractal solvent [8], emulsions in which one phase is fractally dispersed in the other [9], suspensions in which solid particles are fractally distributed in a liquid [10,11], and classical fluids confined in a fractal configuration space [12].
Technological applications stimulate the studies of the hydrodynamics of fractal fluids [13-17] and fluids confined in fractal configuration space [5,18-21]. Functions defined on the fractal are not differentiable in the conventional sense [22,23]. Accordingly, several mathematical tools have been to deal with problems on fractals (see, for review, Refs. [24-27] and references therein). Specifically, it has been argued that the physical problems on fractals can be mapped onto boundary value problems either in a fractal continuum [14,19-21,28], or in a fractional dimensional space with an appropriate metric [29-31]. Axiomatic basis for spaces with non-integer dimension have been suggested by Stillinger [32]. Further, the theory of fractional dimensional space was developed in a number of works (see, for example, Refs. [27-37] and references therein). In this regard, it should be emphasized that the fractional dimensional space can be endowed with any
metric satisfying the conventional criteria required of metrics (see Refs. [38-40]). Accordingly, the mapping of physical problems on fractals to mathematical problems in the fractional dimensional space requires the definitions of the space dimension and the metric induced by the fractal topology [24].
In this way, the fractal features can be characterized by a set of dimension numbers [30]. Specifically, the fractal (Hausdorff or box-counting) measure is characterized by the mass fractal dimension D (see Table 1). The fractal topology is characterized by the connectivity dimension
d"
, also called the chemical, or spreading dimension (see, for
review, Ref. [25] and references therein). The ratio
D / d " is equal to the fractal (box-
counting) dimension of the minimum path ( d min ) between two points on the fractal1) [25]. The number of the effective dynamical degrees of freedom of the fractal is determined by the spectral dimension d s [41] (see Table 1), whereas the number of the effective spatial degrees of freedom2) ν (see Table 1) is equal to ν = 2d " − d s [30].
In this work, we study the laminar flow of fractal fluids possessing scale invariance along and across the flow direction. The fractal fluid flows are mapped into the flows in the
1
The connectivity dimension can be determined by the fractal covering by the
d " -dimensional cubes with
the geodesic (minimum path) metric [25,30]. Notice that for path-connected fractals
d" ≤ D
and
d min = D / d " ≥ 1 . For totally disconnected fractals (e.g. Cantor dusts) the connectivity dimension is equal d = n > D , while d min = D / d " < 1 to the dimension of the embedding Euclidean space n , such that " . 2)
The number of effective spatial degrees of freedom can be viewed as the number of directions that someone who lives on the fractal would experience [30]. Notice that for the Euclidean object of the
ν = d " = d s = D = d , whereas topological dimension d all dimension numbers are equal to d , that is
ν ≥ d" ≥ ds . for fractals D > d , while
fractional dimensional space with the metric induced by the fractal topology. This allows us to find the velocity and stress distributions in the steady Couette and Poiseuille flows of fractal fluids. The rest of the paper is organized as follows. In Sec. II the mapping of fractal fluid flow into the flow in a fractional dimensional space is discussed. The Stokes equations for incompressible flow of a viscous fractal fluid are derived. Some solutions of these equations are discussed in Sec. III. The main findings and conclusions are outlined in Sec. IV.
II. Map of fractal fluid flow into homogeneous flow in fractional dimensional space
The fractal attributes of a complex fluid can be typified by three independent dimension numbers (see Table 1). Specifically, the fractal mass distribution in the embedding Euclidean space E n is characterized by the mass fractal dimension ( D ) linked to the Hausdorff dimension [6-12]. The fractal topology is characterized by the number of effective spatial degrees of freedom of a walker on the fractal (ν ) [30]. The dynamic features are governed by the spectral dimension ( d s ) [41]. Accordingly, the power-law exponents describing particular properties of the fractal fluid can be calculated using the scaling relations between them and the dimension numbers. The values of dimension numbers for some classic fractals are listed in Table 2.
3 The physical problems on the fractal F ⊂ E with ν effective spatial degrees of freedom
ν can be mapped into the ν -dimensional space F with the metric induced by the fractal ν 3 topology as follows [30]. The distance " between two points A and B in F ⊂ E is
related to the Euclidean distance between these points
l = ( x1A − x1B ) 2 + ( x2A − x2B ) 2 + ( x3A − x3B )2
(1a)
as
" = al α ,
(1b)
1−α where a = κr0 , r0 is the characteristic size of elementary components which constitute
the fractal structure, κ is a geometric constant, and
α=
D
ν ,
(1c)
is the fractal metric exponent [30]. The fractal mass density is generally scale and space dependent [8-12]. Specifically, the mass of the fractal fluid in a sphere of radius l scales as M ∝ l D , while the sphere volume is V ∝ l 3 . Accordingly, the density of the fractal ν fluid scales as ρ ∝ l D − 3 . The mapping (1) implies that M ∝ " , while the volume of the ν ν -dimensional sphere in Fν is Vν ∝ " . So, the mapping of fractal fluid flow into the
ν flow in F implies
ρ ∝ l D −3 → ρν = M / Vν = const ,
(2)
where the flow density ρν can be obtained from the requirement of mass conservation (see, for example, Ref. [20]).
If the number of effective spatial degrees of freedom in the fractal fluid is ν < 3 , the ν 3 number of independent coordinates which can be defined in F ⊂ E is less than 3 [25].
Nonetheless, one can define three fractional coordinates " i = ai xiα
(3a)
associated with the Cartesian coordinates in the embedding Euclidean space (see [4245]), such that
ª 3 § " · 2ν / D º " = a «¦i ¨¨ i ¸¸ » «¬ © ai ¹ »¼
D / 2ν
.
(3b)
where ai are the constants accounting for the fractal anisotropy [30].
ν Furthermore, in Ref. [30] it has been argued that the intrinsic time τ in F is related to 3 the time t in the embedding space E as 1− β τ = bt β , where b = t0 ,
(4a)
t0 is the intrinsic time scale and
β = d s /ν ≤ 1
(4b)
is the time metric exponent (see, also Ref. [46]). Consequently, the velocity of material ν point in F is defined using the time metric derivative as 3 ∂" & & u = ¦i i ei ∂τ i ,
(5)
ν while the limiting distribution of random walker displacements in F is a Gaussian
distribution with a variance proportional to the first power of the intrinsic time of the walk [30].
The introduction of spatial coordinates (3) allows to define (see Ref. [38]) the partial metric derivatives3):
∇νi f = lim Δ" i
f ( " i ) − f ( " i + Δ" i ) , Δ" i
(6a)
3 which are related to the conventional partial derivatives in the embedding space E as
3)
Notice that in Refs [19-21,25-27,44,46] the metric derivative defined by Eq. (6a) along with Eq. (3a) is termed as the Hausdorff derivative.
§ x 1−α ∇νi f = ¨¨ i © αai
· ∂ ¸¸ f . ¹ ∂x i
(6b)
Furthermore, Palmer and Stavrinou [33] have argued that the Laplace operator in the fractional dimensional space has the form 2 3§ ∂ f ν / 3 − 1 ∂f · ¸, Δν f = ¦ i ¨¨ 2 + " i ∂" i ¸¹ © ∂" i
(7)
where (" 1 , " 2 , " 3 ) are the mutually orthogonal coordinates in Fν ⊂ E 2 . In the fractional cylindrical coordinates in F ν ⊂ E 2 :
[
" r = a (" 1 / a1 )
2ν / D
]
2ν / D D / 2ν
+ (" 2 / a2 )
D /ν ( ) , φν = arctan " 2 / " r , " z = a 3 " 3 ,
(8)
where 2 < D ≤ ν ≤ 3 , Eq. (7) takes the form Δν f = "1r− 2ν / 3
∂ § 2ν / 3 −1 ∂ · 1 ∂ 2 ∂ § 1−ν / 3 ∂ · ¨¨ " r ¨"z + "1z−ν / 3 f ¸¸ + 2 f ¸. 2 ∂" r © ∂" r ¹ " r ∂φν ∂" z ¨© ∂" z ¸¹
(9a)
Taking into account that Δν f = divν gradν f ,
(9b)
the divergence and gradient operators in F ν ⊂ E 2 can be defined as
& ∂ 2ν / 3 −1 ∂ ∂ divν f = "1r− 2ν / 3 "r f r + " −r1 fφ + "νz / 3−1 fz ∂" r ∂φν ∂" z
(
)
(10)
and
gradν f =
respectively, where
∂f & ∂f & ∂f & er + " −r1 eφ + "νz / 3−1 ez , ∂" r ∂φν ∂" z
& f = ( f r , fφ , f z )
(11)
& e& & ν 3 is a vector in F ⊂ E and er , φ and ez are the unit
3 vectors. Notice that Eqs. (6) – (11) convert in the conventional forms in E when
D =ν = 3.
In continuum mechanics, a Newtonian fluid is defined as a fluid in which the stress is linearly proportional to the strain rate. For an isotropic incompressible Newtonian fluid, the shear stress
τ ij
is related to the strain rate as
§ ∂υ ∂υ · τ ij = μ ¨¨ i + j ¸¸ © ∂x j ∂xi ¹
(12)
x where μ the dynamic viscosity of the fluid, j is the spatial coordinate in j direction,
and υi is the fluid velocity in the direction of i axis [1-3].
The mass density and viscosity of fractal fluids are generally scale and space dependent. ν The mapping defined by Eqs. (3) - (5) implies that the mass flow in the space F is
homogeneous. Hence, we can define the Newtonian fractal fluid as a fluid obeying the following constitutive equation
§ ∂ui
τ ijν = μν ¨¨
© ∂" j
where
τ ijν
+
∂u j · ¸ ∂" i ¸¹ ,
(13)
ν is the shear stress arising from the homogeneous flow in F and μν = const is
ν μ ( xi ) in E 3 is space dependent. the flow viscosity in F , while the fluid viscosity
Accordingly, the incompressible Stokes flow of the Newtonian fractal fluid obeys the momentum balance equation &
μν Δν u& = gradν pν + fν
(14a)
and the continuity equation
& divν u = 0 ,
(14b)
where of
pν
& f is the pressure and ν is an applied body force. Notice that in the limiting case
ds = D = ν
equations (14) convert into the classical equations of the incompressible
Stokes flow of homogeneous Newtonian fluids. III. Laminar flows of Newtonian fractal fluid in circular tubes
Let us first consider an incompressible flow of the Newtonian fractal fluid in a straight tube with a circular cross-section of radius R . In the case of steady laminar flow, the velocity field is a function only of the radial coordinate r , whereas the pressure gradient depends only on the coordinate z
in the flow direction [3]. The mapping
( r ,θ , z ) → (" r ,θν , " z ) implies that the flow velocity in Fν is related to the fluid flow
velocity υ z ( r ) as uz =
∂" z αa z = βb ∂τ
§ z α −1 · ∂z α § " z t · ¨¨ β −1 ¸¸ = ¨ ¸υ z © t ¹ ∂t β © zτ ¹ .
(15)
Taking into account that in the case of steady laminar flow u z (" r ) should be independent of " z and τ , from Eq. (15) it follows that "z
τ
=c
z t ,
(16)
ν where c is a geometrical constant, while β ≤ α < 1 . Therefore, the flow velocity in F
is related to the fluid velocity along the tube as
u z (" r ) = c
whereas
ur = uθ = υ r = υθ = 0
D υ z (r ) ds ,
. Accordingly, Eq. (14a) gets the form
(17)
"νz / 3 −1
§ ∂ 2u ∂p 2ν / 3 − 1 ∂u z · ¸ = − μν ¨¨ 2z + ∂" z ∂" r ¸¹ r © ∂" r
,
(18)
while the continuity equation (14b) holds trivially.
In the case of ν = 3 , the general solution of Eq. (18) is ur = C1" 2r + C2 ln " r + C3
,
(19a)
whereas if 2 < ν < 3 , the general solution of Eq. (18) has the form ur = C2 " 2r + C3 + C4 " 2r − 2ν / 3 ,
(19b)
where 0 < 2 − 2ν / 3 < 2 / 3
(19c)
and Ci are constants which can be determined using appropriate boundary conditions.
3.1. Couette flow of a fractal fluid
Now let us consider the flow between axially moving concentric cylinders with radii R1 ν and R2 > R1 . If there is no applied pressure gradient, the flow in F obeys the following
Poisson equation with no source § ∂ 2u z 2ν / 3 − 1 ∂u z · ¨¨ 2 + ¸=0 r ∂" r ¸¹ © ∂" r .
(20)
The mapping of the Dirichlet boundary conditions
υ z ( R1 ) = υ1 and υ z ( R2 ) = υ2
(21a)
ν into the boundary conditions in F reads as
u z (" 1 ) = u1 and u z (" 2 ) = u2 .
(21b)
α α where " 1 = ar R1 and " 2 = ar R2 .
In the case of ν = 3 the solution of Eq. (20) with the boundary conditions (21b) is
u z = u2
ln(" r / " 1 ) ln(" r / " 2 ) − u1 ln(" 2 / " 1 ) ln(" 2 / " 1 )
(22a)
and so, the distribution of the fractal fluid velocity between axially moving concentric cylinders
υ z = υ2
ln( r / R1 ) ln(r / R2 ) − υ1 ln( R2 / R1 ) ln( R2 / R1 ) .
(22b)
is the same as in the case of homogeneous fluids (see, for example, Ref. [1]), even when D <ν = 3 .
However, if 2 < ν < 3 , the solution of Eq. (20) with the boundary conditions (21) has the form ur = C4" 2r − 2ν / 3 + C3 ,
(23a)
where
C3 =
u −u u1 + u2 u1 − u2 " 2R−12ν / 3 + " 2R−22ν / 3 C4 = 2 − 2ν / 13 22 − 2ν / 3 − 2 − 2ν / 3 2 − 2ν / 3 " R1 − " R2 2 2 " R1 − " R2 and .
(23b)
Accordingly, the distribution of fractal fluid velocity between axially moving concentric cylinders takes the form
υr =
υ2 − υ1 ζ
ζ
2( R2 − R1
(2r )
ζ
)
− R1ζ − R2ζ +
υ1 + υ2 2
,
(24a)
where the scaling exponent
§2
2·
2
ζ = D¨ − ¸ = 2α − D 3 ©ν 3 ¹
(24b)
is in the range of 0 < ζ < 2 / 3 , while 2 < ν < 3 and D ≤ ν . The values of ζ for some fractals are given in Table 2. The graphs of velocity distributions in the Couette flows of fractal fluids characterized by different sets of dimension numbers are shown in Fig. 1.
The fluid velocity distribution (24) depends on the fractal metric of flow (characterized by α ), as well as on the fluid mass distribution (characterized by D ). Notice that the averaged velocity of fractal fluids having the same metric ( α = const ) decreases as the mass fractal dimension D increases. Conversely, in the case of fluids having the same number of effective degrees of freedom (ν = const ) the averaged fluid velocity increases as D increases. On the other hand, the averaged velocity of fluids of the same dimension ( D = const ) decreases as the number of effective spatial degrees of freedom decrease.
3.2. Poiseuille flow of a fractal fluid
A Poiseuille flow is driven by a pump that forces fluid to flow by modifying the pressure along a pipe of length L , while the pressure gradient remains constant. Fully developed Poiseuille flow exists only far from the entrance and exit of the pipe, where the flow is ν aligned parallel to the walls [1]. The steady Poiseuille flow in F obeys Eq. (18), while
∂p / ∂" r = 0 . Applying the no-slip boundary condition together with the condition that the fluid velocity is zero if the pressure gradient is zero, we find that the velocity distribution in steady Poiseuille flow for a Newtonian fractal fluid in a cylindrical tube of radius R has the form
ª § r · 2α º υ z = υ max «1 − ¨ ¸ » ¬« © R ¹ ¼» ,
(25)
while the pressure gradient along the flow direction is equal to
4μ υ ∂p = − eff2 Dmax zγ /ν ∂z ar R ,
(26)
where
μ eff =
· ¸¸ μν ¹
cD ( D + ν )(ν − 1) § a z ¨¨ 4νd s © ar
(27a)
is the effective viscosity of the Newtonian fractal fluid obeying the constitutive equation (13),
υ max = −
a r R 2α Δ P 4 μ eff L1+ γ
(27b)
is the maximum velocity imposed by the pressure drop ΔP along the tube of length L , and
ν
D ν − 3
(28a)
1 1 <γ < 3 3 ,
(28b)
γ= is the scaling exponent. Notice that −
while 2 < ν ≤ 3 and D ≤ ν .
From Eq. (26) with the boundary conditions p ( z = 0) = p0
and p( z = L) = pL ,
(29)
it follows that the pressure distribution along the steady laminar flow of the Newtonian fractal fluid behaves as 1+ γ
§z· p( z ) = p0 − ( p0 − pL )¨ ¸ ©L¹
,
(30)
where γ is defined by Eq. (28). In this regard, it is pertinent to point out that the velocity distribution (25) is governed by the fractal metric (1), whereas the pressure distribution (30) depends on the fractal metric and the fractal topology of the fractal fluid. The values of γ for some fractals a given in Table 2. Velocity and pressure distributions in steady Poiseuille flows of fractal fluids characterized by different sets of dimension numbers are shown in Fig. 2. Notice that if 2 < D =ν < 3 ,
(31a)
then the fluid velocity distribution (25) is parabolic (as this is in the case of homogeneous fluids), whereas the pressure distribution (30) is no-linear. On the other hand, if 2 < ν = 3D < 3 ,
(31b)
the pressure distribution along the flow direction is lineal, but the velocity distribution (25) is not parabolic.
In this regard, it is pertinent to note that the laminar flow of fractal fluid in the cylindrical tube was studied in Refs. [16,17]. However, the radial distribution of fluid velocity distribution was derived only for a special case when the fluid density is homogeneous along the flow, but it is fractal across the tube. Furthermore, in Refs. [16,17] it was implicitly assumed that the number of effective degrees of freedom across the tube is equal to ν = 2 . Consequently, the pressure gradient remains constant along the flow, while the radial velocity distribution is not parabolic [17]. It is a straightforward matter to see that in the case of ν = 2 Eq. (25) converts in the Eq. (20) of Ref. [17].
Using Eq. (25) we find that the average velocity of the fractal fluid is equal to R
υ avg =
ν 2 υ (r )rdr = υ max 2 ³ z R 0 D +ν
.
(32)
Accordingly, the modified Poiseuille law relating the flow rate
Q = πR 2υ avg
to the
pressure drop Δp along the tube of length L reads as Q=−
ΔP πar R 2( D +ν ) /ν 8μeff L1+ D / ν −ν / 3
.
(33)
So, the fractal nature of the complex fluid can either assist or hinder the pressure driven flow in a pipe (see also Fig. 2). Notice that if ν = D = d s = 3 , Eqs. (25), (26), (30), (32), and (33) convert into the classical equations describing the Poiseuille flow of homogeneous Newtonian fluids.
VI. Conclusions
The mapping of laminar flows of a fractal fluid into the flow in the fractional dimension space with metric induced by the fractal topology is discussed. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. Next, we obtain expressions for the radial distributions of flow velocities in steady Couette and Poiseuille flows of a fractal fluid. The pressure distribution along the steady Poiseuille flow of the fractal fluid is also found explicitly.
We state that the radial distribution of the fluid velocity in the Couette flow depends upon the fractal metric of flow, as well as on the fluid mass distribution. However, if the number of effective spatial degrees of freedom of a random walker in the fractal fluid (ν ) is equal to 3, the radial distribution of the fractal fluid velocity is the same as in the Couette flow of the homogeneous fluid, even when the mass density of the fractal fluid is scale and space dependent. Conversely, the radial distribution of the fluid velocity in the steady Poiseuille flow of the fractal fluid is governed only by the fractal metric. In particular, the
radial distribution of the fluid velocity is parabolic when the mass fractal dimension of the fluid is equal to the number of effective spatial degrees of freedom. The pressure distribution along the steady Poiseuille flow of a fractal fluid depends on the fractal topology of flow, as well as on the fractal metric. In particular, if ν = 3D , the pressure distribution is linear, but the radial distribution of the fluid velocity is not parabolic.
In this regard, it is pertinent to point out that the approach this work is applicable to any type of Newtonian fractal fluids obeying the constitutive law (13). For this purpose, the real fractal fluid (as for example, studied in Refs. [6-10]) should be characterized not only by the mass fractal dimension, but the spectral and connectivity dimension should be also determined from independent experiments. So, we expect that our findings will stimulate further experimental and theoretical studies of the fractal and hydrodynamic properties of fractal fluids.
Acknowledgments
The authors wish to thank Armando Garcia Jaramillo and Michael Shapiro for fruitful discussions. This work was supported by PEMEX under the SENER-CONACYT research Grant No. 143927.
Table 1. Dimension numbers characterizing a fractal [30]. Dimension number
Symbol
Definition
Mass fractal dimension
D
Defined via the scaling behavior of the fractal mass
M ∝ l D , where l is the characteristic size of the fractal Number of effective spatial degrees of freedom of a walker on the fractal Spectral dimension
ν
ds
in the embedding Euclidean space The number of independent directions on the fractal in which a walker on the fractal can move without violating any constraint imposed on it. Defined via the scaling behavior of the probability of a random walker the fractal to return to its starting point
P (0, t ) ∝ t d s .
3
Table 2. Dimension numbers of some classic fractals in E [30] and the values of scaling
D
ds
ν
2α
Eq. (1b)
Menger sponge 2.727 2.51 2.94 1.85 Siepinski pyramid 2 1.55 2.45 1.63 Infinite random 2 4/3 8/3 3/2 graphs [47] 2D / 3 3 3 Cantor dusts in E 3 : D 3 1.65 3 3 1.1 Cantor dust in E 3 2.1 3 3 1.4 Cantor dust in E 3 2.9 3 3 1.93 Cantor dust in E Diamond fractals 2 D D D Diamond fractal 2.1 2.1 2.1 2 Diamond fractal 2.5 2.5 2.5 2 Diamond fractal 2.9 2.9 2.9 2 exponents characterizing steady laminar flows of fractal fluids
ζ Eq. (24b)
0.03 0.3 1/6
1+ γ Eq. (28a)
0.94 0.9997 31/36
D/3 -1) 1) 0.55 1) 0.7 1) 0.97 2 − 2D / 3 2 − D / 3 0.6 1.3 1/3 7/6 0.07 1.03 with mass distributions
modeled by these fractals.
1)
In the Couette flow of suspensions with the Cantor dust-like distributions of solid particles, the radial distribution of fluid velocity is logarithmic (see Eq. (22b)).
Figure captions
Figure 1. Velocity distributions for Couette flow between concentric cylinders (
R1 / R2 = 0.1 , υ1 = 0 ) of homogeneous Newtonian fluid (1) and for fractal fluids having the structure of the infinite random graph (2), Siepinski pyramid (3), and diamond fractal with D = 2.1 (4). See Table 2.
Figure 2. (a) Velocity distributions for steady laminar flow in a pipe of circular cross
section for a homogeneous Newtonian fluid (1) and for fractal fluids having the structure of the Menger sponge (2), Siepinski pyramid (3), infinite random graph (4), Cantor dust with D = 2.1 (5), and Cantor dust with D = 1.65 (6). (b) Pressure distribution along the laminar flow of a homogeneous Newtonian fluid (1) and for fractal fluids having the structure of the diamond fractal with D = 2.1 (2), diamond fractal with D = 2.5 (3), Menger sponge (4), infinite random graph (5), Cantor dust with D = 2.1 (6), and Cantor dust with D = 1.65 (7), while pL = 0 . See Table 2.
References [1] O. Darrigol, Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, Oxford, 2005. [2] S. P. Sutera, R. Skalak, The history of Poiseuille’s law, Annu. Rev. Fluid Mech. 25 (1993) 119; DOI: 10.1146/annurev.fl.25.010193.000245. [3] M. Z. Bazant, Exact solutions and physical analogies for unidirectional flows, Phys. Rev. Fluids 1 (2016) 024001; DOI: 10.1103/PhysRevFluids.1.024001. [4] J. Cai, E. Perfect, C.-L. Cheng, X. Hu, Generalized modeling of spontaneous imbibition based on HageníPoiseuille flow in tortuous capillaries with variably shaped apertures, Langmur 30 (2014) 5142-5151; DOI: 10.1021/la5007204. [5] A. S. Balankin, J. C. Valdivia, J. Marquez, O. Susarrey, M. A. Solorio-Avila, Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity, Phys. Lett. A 380 (2016) 2767-2773; DOI: 10.1016/j.physleta.2016.06.032. [6] W. Wei, J. Cai, X. Hu, Q. Han, S. Liu, Y. Zhou, Fractal analysis of the effect of particle aggregation distribution on thermal conductivity of nanofluids, Phys. Lett. A 380 (2016) 2953–2956; DOI: 10.1016/j.physleta.2016.07.005. [7] J. Cai, X. Hu, B. Xiao, Y. Zhou, W. Wei, Recent developments on fractal-based approaches to nanofluids and nanoparticle aggregation, Int. J. Heat Mass Transf. 105 (2017) 623–637; DOI: 10.1016/j.ijheatmasstransfer.2016.10.011 [8] H. Shui and H. Zhou, Viscosity and fractal dimension of coal soluble constituents in solution, Fuel Process. Tech. 85 (2004) 1529-1538; DOI: 10.1016/j.fuproc.2003.10.024. [9] B. P. Hills, P. Manoj, C. Destruel, NMR Q-space microscopy of concentrated oil-in-water emulsions, Magn, Res. 18 (2000) 319-333; DOI: 10.1016/S0730-725X(99)00143-5. [10] A. A. Shiyan, Viscosity for fractal suspensions: dependence on fractal dimensionality, Phys. Lett. A 220 (1996) 117-119; DOI: 10.1016/0375-9601(96)00479-3. [11] L. Haider, P. Snabre, M. Boynard, Rheology and ultrasound scattering from aggregated red cell suspensions in shear flow, Biophys. J. 87 (2004) 2322-2334; DOI: 10.1529/biophysj.104.041665. [12] M. Heinen, S. K. Schnyder, J. F. Brady, H. Löwen, Classical liquids in fractal dimension, Phys. Rev. Lett. 115 (2015) 097801; DOI: 10.1103/PhysRevLett.115.097801. [13] V. I. Lesina, Y. A. Koksharov, G. B. Khomutov, Viscosity of liquid suspensions with fractal aggregates: Magnetic nanoparticles in petroleum colloidal structures, Coll. Surf. A: Physicochem. Eng. Aspects 392 (2011) 88-94; DOI: 10.1016/j.colsurfa.2011.09.038.
[14] V. E. Tarasov, Fractional hydrodynamic equations for fractal media, Ann. Phys. 318 (2005) 286-307; DOI: 10.1016/j.aop.2005.01.004. [15] X.-J. Yang, D. Baleanu, J. A. Tenreiro-Machado, Systems of Navier-Stokes equations on Cantor sets, Math. Probl. Eng. (2013) 769724. DOI: 10.1155/2013/769724. [16] V. E. Tarasov, Flow of fractal fluid in pipes: Non-integer dimensional space approach, Chaos Solitons Fractals 67 (2014) 26-37; DOI: 10.1016/j.chaos.2014.06.008. [17] V. E. Tarasov, Poiseuille equation for steady flow of fractal fluid, Int. J. Mod. Phys. 30 (2016) 1650128; DOI: 10.1142/S0217979216501289. [18] M. Sahimi, Flow and Transport in Porous Media and Fractured Rocks, VCH, New York, 1995. [19] A. S. Balankin, B. Espinoza-Elizarraraz, Hydrodynamics of fractal continuum flow, Phys. Rev. E 85 (2012) 025302(R); DOI: 10.1103/PhysRevE.85.025302. [20] A. S. Balankin, B. E. Elizarraraz, Map of fluid flow in fractal porous medium into fractal continuum flow, Phys. Rev. E 85 (2012) 056314; DOI: 10.1103/PhysRevE.85.056314. [21] J. Cai, L. Luo, R. Ye, X. Zeng, X. Hu, Recent advances on fractal modeling of permeability for fibrous porous media, Fractals 23 (2015) 1540006; DOI: 10.1142/S0218348X1540006X. [22] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Willey, New York, 1990. [23] C. Tricot, Curves and Fractal Dimension, Springer, New York, 1993. [24] A. S. Balankin, Physics of fracture and mechanics of self-affine cracks, Eng. Fract. Mech. 57 (1997) 135-203; DOI: 10.1016/S0013-7944(97)00007-6. [25] A. S. Balankin, A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric, Eur. Phys. J. B 88 (2015) 90; DOI: 10.1140/epjb/e2015-60189-y. [26] A. S. Balankin, A continuum framework for mechanics of fractal materials II: elastic stress fields ahead of cracks in a fractal medium, Eur. Phys. J. B 88 (2015) 91; DOI: 10.1140/epjb/e2015-50703-8. [27] A. S. Balankin, Toward the mechanics of fractal materials: mechanics of continuum with fractal metric, arXiv:1409.5829 (2014) 1-80. [28] M. Ostoja-Starzewski, Continuum mechanics models of fractal porous media: Integral relations and extremum principles, J. Mech. Mat. Struct. 4 (2009) 901-912; DOI: 10.2140/jomms.2009.4.901. [29] A. S. Balankin, B. Mena, J. Patiño, D. Morales-Matamoros, Electromagnetic fields in fractal continua, Phys. Lett. A 377 (2013) 783-788; DOI: 10.1016/j.physleta.2013.01.030. [30] A. S. Balankin, Effective degrees of freedom of a random walk on a fractal, Phys. Rev. E 92 (2015) 2322; DOI: 10.1103/PhysRevE.92.062146.
[31] M. Zubair, L. K Ang, Fractional-dimensional Child-Langmuir law for a rough cathode, Phys. Plasm. 23 (2016) 072118; DOI: 10.1063/1.4958944. [32] F. H. Stillinger, Axiomatic basis for spaces with non integer dimension, J. Math. Phys. 18 (1977) 1224-1234; DOI: 10.1063/1.523395. [33] C. Palmer, P. N. Stavrinou, Equations of motion in a non-integer-dimensional space, J. Phys. A: Math. Gen. 37 (2004) 6987-7003; DOI: 10.1088/0305-4470/37/27/009. [34] S. I. Muslih, O. P. Agrawal, A scaling method and its applications to problems in fractional dimensional space, J. Math. Phys. 50 (2009) 123501; DOI: 10.1063/1.3263940. [35] M. Zubair, M. J. Mughal, Q. A. Naqvi, Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer, New York, 2012. [36] A. S. Balankin, Physics in space–time with scale-dependent metrics, Phys. Lett. A 377 (2013) 1606-1610; DOI: 10.1016/j.physleta.2013.04.040. [37] W.-P. Zhong, M. R. Belic, B. A. Malomed, Y. Zhang, T. Huang, Spatiotemporal accessible solitons in fractional dimensions, Phys. Rev. 94 (2016) 012216; DOI: 10.1103/PhysRevE.94.012216. [38] A. S. Balankin, J. Bory-Reyes, M. Shapiro, Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric, Physica A 444 (2016) 345-359; DOI: 10.1016/j.physa.2015.10.035. [39] J. Weberszpil, M. J. Lazo, J.A. Helayël-Neto, On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric, Physica A 436 (2015) 399-404; DOI: 10.1016/j.physa.2015.05.063. [40] J. Weberszpil, J. A. Helayël-Neto, Variational approach and deformed derivatives, Physica A 450 (2016) 217–227; DOI: 10.1016/j.physa.2015.12.145. [41] U. Mosco, Invariant Field Metrics and Dynamical Scalings on Fractals, Phys. Rev. Lett. 79 (1997) 4067-4070; DOI: 10.1103/PhysRevLett.79.4067. [42] A. S. Balankin, B. Mena, C. L. Martinez-Gonzalez, D. Morales-Matamoros, Random walk in chemical space of Cantor dust as a paradigm of superdiffusion, Phys. Rev. E (2012) 86052101; DOI: 10.1103/PhysRevE.86.052101. [43] X.-J. Yang, H. M. Srivastava, J.-H. He, D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Phys. Lett. A 377 (2013) 1696-1700; DOI: 10.1016/j.physleta.2013.04.012. [44] A. S. Balankin, Stresses and strains in a deformable fractal medium and in its fractal continuum model, Phys. Lett. A 377 (2013) 2535-2541; DOI: 10.1016/j.physleta.2013.07.029. [45] X.-J. Yang, D. Baleanu, M. C. Baleanu, Observing diffusion problems defined on Cantor sets in different co-ordinate systems, Therm. Sci. 19 (2015) S151-S156; DOI: 10.2298/TSCI141126065Y.
[46] W. Chen, Time–space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006) 923-929; DOI: 10.1016/j.chaos.2005.08.199. [47] B. Durhuus, Hausdorff and spectral dimension of infinite random graphs, Acta Phys. Pol. 40 (2009) 3509-3532; http://www.forskningsdatabasen.dk/en/catalog/2193011880.
Figure 1-rev
Figure 2
Alexander S. Balankin, Baltasar Mena, Orlando Susarrey, Didier Samayoa
PII: DOI: Reference:
S0375-9601(16)31699-1 http://dx.doi.org/10.1016/j.physleta.2016.12.007 PLA 24223
To appear in:
Physics Letters A
Received date: Revised date: Accepted date:
12 November 2016 4 December 2016 6 December 2016
Please cite this article in press as: A.S. Balankin et al., Steady laminar flow of fractal fluids, Phys. Lett. A (2017), http://dx.doi.org/10.1016/j.physleta.2016.12.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights • • • •
Equations of Stokes flow of Newtonian fractal fluid are derived. Pressure distribution in the Newtonian fractal fluid is derived. Velocity distribution in Poiseuille flow of fractal fluid is found. Velocity distribution in a steady Couette flow is established.
Steady laminar flow of fractal fluids Alexander S. Balankin a)1), Baltasar Mena2), Orlando Susarrey1), Didier Samayoa1) 1)
Grupo Mecánica Fractal, ESIME, Instituto Politécnico Nacional, México D.F., México 07738
2)
Laboratorio de Ingeniería y Procesos Costeros, Instituto de Ingeniería, Universidad Nacional
Autónoma de México, Sisal, Yucatán, Mexico 97355
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived.
Keywords: Fractal; Fractal fluids; Laminar flow; Fluid velocity distribution
a)
Corresponding author. Adress: SEPI-ESIME-Zacatenco, U.P. Adolfo Lopez Mateos, Av. Politécnico Nacional s/n, México D.F. 07738, México. Tel.: 55-5729600 + Ext. 54589; E-mail: [email protected]
I. Introduction
A pressure driven laminar flow is ubiquitous in nature and widely used in scientific and engineering applications [1]. The Poiseuille law for the flow rate in a tube of circular cross section was apparently first derived from the Navier-Stokes equations by Stokes himself in 1845 (see Ref. [2]). Since, pressure driven laminar flows of viscous fluids remain an active topic of research (see Refs. [3-5] and references therein). Commonly, viscous fluids are viewed as homogeneous continua of an integer dimension. However, internal structures of many complex fluids exhibit scale-invariant features characterized by non-integer fractal dimensions [6,7]. Examples of fractal fluids include solutions with a fractal distribution of a solute dissolved in a nonfractal solvent [8], emulsions in which one phase is fractally dispersed in the other [9], suspensions in which solid particles are fractally distributed in a liquid [10,11], and classical fluids confined in a fractal configuration space [12].
Technological applications stimulate the studies of the hydrodynamics of fractal fluids [13-17] and fluids confined in fractal configuration space [5,18-21]. Functions defined on the fractal are not differentiable in the conventional sense [22,23]. Accordingly, several mathematical tools have been to deal with problems on fractals (see, for review, Refs. [24-27] and references therein). Specifically, it has been argued that the physical problems on fractals can be mapped onto boundary value problems either in a fractal continuum [14,19-21,28], or in a fractional dimensional space with an appropriate metric [29-31]. Axiomatic basis for spaces with non-integer dimension have been suggested by Stillinger [32]. Further, the theory of fractional dimensional space was developed in a number of works (see, for example, Refs. [27-37] and references therein). In this regard, it should be emphasized that the fractional dimensional space can be endowed with any
metric satisfying the conventional criteria required of metrics (see Refs. [38-40]). Accordingly, the mapping of physical problems on fractals to mathematical problems in the fractional dimensional space requires the definitions of the space dimension and the metric induced by the fractal topology [24].
In this way, the fractal features can be characterized by a set of dimension numbers [30]. Specifically, the fractal (Hausdorff or box-counting) measure is characterized by the mass fractal dimension D (see Table 1). The fractal topology is characterized by the connectivity dimension
d"
, also called the chemical, or spreading dimension (see, for
review, Ref. [25] and references therein). The ratio
D / d " is equal to the fractal (box-
counting) dimension of the minimum path ( d min ) between two points on the fractal1) [25]. The number of the effective dynamical degrees of freedom of the fractal is determined by the spectral dimension d s [41] (see Table 1), whereas the number of the effective spatial degrees of freedom2) ν (see Table 1) is equal to ν = 2d " − d s [30].
In this work, we study the laminar flow of fractal fluids possessing scale invariance along and across the flow direction. The fractal fluid flows are mapped into the flows in the
1
The connectivity dimension can be determined by the fractal covering by the
d " -dimensional cubes with
the geodesic (minimum path) metric [25,30]. Notice that for path-connected fractals
d" ≤ D
and
d min = D / d " ≥ 1 . For totally disconnected fractals (e.g. Cantor dusts) the connectivity dimension is equal d = n > D , while d min = D / d " < 1 to the dimension of the embedding Euclidean space n , such that " . 2)
The number of effective spatial degrees of freedom can be viewed as the number of directions that someone who lives on the fractal would experience [30]. Notice that for the Euclidean object of the
ν = d " = d s = D = d , whereas topological dimension d all dimension numbers are equal to d , that is
ν ≥ d" ≥ ds . for fractals D > d , while
fractional dimensional space with the metric induced by the fractal topology. This allows us to find the velocity and stress distributions in the steady Couette and Poiseuille flows of fractal fluids. The rest of the paper is organized as follows. In Sec. II the mapping of fractal fluid flow into the flow in a fractional dimensional space is discussed. The Stokes equations for incompressible flow of a viscous fractal fluid are derived. Some solutions of these equations are discussed in Sec. III. The main findings and conclusions are outlined in Sec. IV.
II. Map of fractal fluid flow into homogeneous flow in fractional dimensional space
The fractal attributes of a complex fluid can be typified by three independent dimension numbers (see Table 1). Specifically, the fractal mass distribution in the embedding Euclidean space E n is characterized by the mass fractal dimension ( D ) linked to the Hausdorff dimension [6-12]. The fractal topology is characterized by the number of effective spatial degrees of freedom of a walker on the fractal (ν ) [30]. The dynamic features are governed by the spectral dimension ( d s ) [41]. Accordingly, the power-law exponents describing particular properties of the fractal fluid can be calculated using the scaling relations between them and the dimension numbers. The values of dimension numbers for some classic fractals are listed in Table 2.
3 The physical problems on the fractal F ⊂ E with ν effective spatial degrees of freedom
ν can be mapped into the ν -dimensional space F with the metric induced by the fractal ν 3 topology as follows [30]. The distance " between two points A and B in F ⊂ E is
related to the Euclidean distance between these points
l = ( x1A − x1B ) 2 + ( x2A − x2B ) 2 + ( x3A − x3B )2
(1a)
as
" = al α ,
(1b)
1−α where a = κr0 , r0 is the characteristic size of elementary components which constitute
the fractal structure, κ is a geometric constant, and
α=
D
ν ,
(1c)
is the fractal metric exponent [30]. The fractal mass density is generally scale and space dependent [8-12]. Specifically, the mass of the fractal fluid in a sphere of radius l scales as M ∝ l D , while the sphere volume is V ∝ l 3 . Accordingly, the density of the fractal ν fluid scales as ρ ∝ l D − 3 . The mapping (1) implies that M ∝ " , while the volume of the ν ν -dimensional sphere in Fν is Vν ∝ " . So, the mapping of fractal fluid flow into the
ν flow in F implies
ρ ∝ l D −3 → ρν = M / Vν = const ,
(2)
where the flow density ρν can be obtained from the requirement of mass conservation (see, for example, Ref. [20]).
If the number of effective spatial degrees of freedom in the fractal fluid is ν < 3 , the ν 3 number of independent coordinates which can be defined in F ⊂ E is less than 3 [25].
Nonetheless, one can define three fractional coordinates " i = ai xiα
(3a)
associated with the Cartesian coordinates in the embedding Euclidean space (see [4245]), such that
ª 3 § " · 2ν / D º " = a «¦i ¨¨ i ¸¸ » «¬ © ai ¹ »¼
D / 2ν
.
(3b)
where ai are the constants accounting for the fractal anisotropy [30].
ν Furthermore, in Ref. [30] it has been argued that the intrinsic time τ in F is related to 3 the time t in the embedding space E as 1− β τ = bt β , where b = t0 ,
(4a)
t0 is the intrinsic time scale and
β = d s /ν ≤ 1
(4b)
is the time metric exponent (see, also Ref. [46]). Consequently, the velocity of material ν point in F is defined using the time metric derivative as 3 ∂" & & u = ¦i i ei ∂τ i ,
(5)
ν while the limiting distribution of random walker displacements in F is a Gaussian
distribution with a variance proportional to the first power of the intrinsic time of the walk [30].
The introduction of spatial coordinates (3) allows to define (see Ref. [38]) the partial metric derivatives3):
∇νi f = lim Δ" i
f ( " i ) − f ( " i + Δ" i ) , Δ" i
(6a)
3 which are related to the conventional partial derivatives in the embedding space E as
3)
Notice that in Refs [19-21,25-27,44,46] the metric derivative defined by Eq. (6a) along with Eq. (3a) is termed as the Hausdorff derivative.
§ x 1−α ∇νi f = ¨¨ i © αai
· ∂ ¸¸ f . ¹ ∂x i
(6b)
Furthermore, Palmer and Stavrinou [33] have argued that the Laplace operator in the fractional dimensional space has the form 2 3§ ∂ f ν / 3 − 1 ∂f · ¸, Δν f = ¦ i ¨¨ 2 + " i ∂" i ¸¹ © ∂" i
(7)
where (" 1 , " 2 , " 3 ) are the mutually orthogonal coordinates in Fν ⊂ E 2 . In the fractional cylindrical coordinates in F ν ⊂ E 2 :
[
" r = a (" 1 / a1 )
2ν / D
]
2ν / D D / 2ν
+ (" 2 / a2 )
D /ν ( ) , φν = arctan " 2 / " r , " z = a 3 " 3 ,
(8)
where 2 < D ≤ ν ≤ 3 , Eq. (7) takes the form Δν f = "1r− 2ν / 3
∂ § 2ν / 3 −1 ∂ · 1 ∂ 2 ∂ § 1−ν / 3 ∂ · ¨¨ " r ¨"z + "1z−ν / 3 f ¸¸ + 2 f ¸. 2 ∂" r © ∂" r ¹ " r ∂φν ∂" z ¨© ∂" z ¸¹
(9a)
Taking into account that Δν f = divν gradν f ,
(9b)
the divergence and gradient operators in F ν ⊂ E 2 can be defined as
& ∂ 2ν / 3 −1 ∂ ∂ divν f = "1r− 2ν / 3 "r f r + " −r1 fφ + "νz / 3−1 fz ∂" r ∂φν ∂" z
(
)
(10)
and
gradν f =
respectively, where
∂f & ∂f & ∂f & er + " −r1 eφ + "νz / 3−1 ez , ∂" r ∂φν ∂" z
& f = ( f r , fφ , f z )
(11)
& e& & ν 3 is a vector in F ⊂ E and er , φ and ez are the unit
3 vectors. Notice that Eqs. (6) – (11) convert in the conventional forms in E when
D =ν = 3.
In continuum mechanics, a Newtonian fluid is defined as a fluid in which the stress is linearly proportional to the strain rate. For an isotropic incompressible Newtonian fluid, the shear stress
τ ij
is related to the strain rate as
§ ∂υ ∂υ · τ ij = μ ¨¨ i + j ¸¸ © ∂x j ∂xi ¹
(12)
x where μ the dynamic viscosity of the fluid, j is the spatial coordinate in j direction,
and υi is the fluid velocity in the direction of i axis [1-3].
The mass density and viscosity of fractal fluids are generally scale and space dependent. ν The mapping defined by Eqs. (3) - (5) implies that the mass flow in the space F is
homogeneous. Hence, we can define the Newtonian fractal fluid as a fluid obeying the following constitutive equation
§ ∂ui
τ ijν = μν ¨¨
© ∂" j
where
τ ijν
+
∂u j · ¸ ∂" i ¸¹ ,
(13)
ν is the shear stress arising from the homogeneous flow in F and μν = const is
ν μ ( xi ) in E 3 is space dependent. the flow viscosity in F , while the fluid viscosity
Accordingly, the incompressible Stokes flow of the Newtonian fractal fluid obeys the momentum balance equation &
μν Δν u& = gradν pν + fν
(14a)
and the continuity equation
& divν u = 0 ,
(14b)
where of
pν
& f is the pressure and ν is an applied body force. Notice that in the limiting case
ds = D = ν
equations (14) convert into the classical equations of the incompressible
Stokes flow of homogeneous Newtonian fluids. III. Laminar flows of Newtonian fractal fluid in circular tubes
Let us first consider an incompressible flow of the Newtonian fractal fluid in a straight tube with a circular cross-section of radius R . In the case of steady laminar flow, the velocity field is a function only of the radial coordinate r , whereas the pressure gradient depends only on the coordinate z
in the flow direction [3]. The mapping
( r ,θ , z ) → (" r ,θν , " z ) implies that the flow velocity in Fν is related to the fluid flow
velocity υ z ( r ) as uz =
∂" z αa z = βb ∂τ
§ z α −1 · ∂z α § " z t · ¨¨ β −1 ¸¸ = ¨ ¸υ z © t ¹ ∂t β © zτ ¹ .
(15)
Taking into account that in the case of steady laminar flow u z (" r ) should be independent of " z and τ , from Eq. (15) it follows that "z
τ
=c
z t ,
(16)
ν where c is a geometrical constant, while β ≤ α < 1 . Therefore, the flow velocity in F
is related to the fluid velocity along the tube as
u z (" r ) = c
whereas
ur = uθ = υ r = υθ = 0
D υ z (r ) ds ,
. Accordingly, Eq. (14a) gets the form
(17)
"νz / 3 −1
§ ∂ 2u ∂p 2ν / 3 − 1 ∂u z · ¸ = − μν ¨¨ 2z + ∂" z ∂" r ¸¹ r © ∂" r
,
(18)
while the continuity equation (14b) holds trivially.
In the case of ν = 3 , the general solution of Eq. (18) is ur = C1" 2r + C2 ln " r + C3
,
(19a)
whereas if 2 < ν < 3 , the general solution of Eq. (18) has the form ur = C2 " 2r + C3 + C4 " 2r − 2ν / 3 ,
(19b)
where 0 < 2 − 2ν / 3 < 2 / 3
(19c)
and Ci are constants which can be determined using appropriate boundary conditions.
3.1. Couette flow of a fractal fluid
Now let us consider the flow between axially moving concentric cylinders with radii R1 ν and R2 > R1 . If there is no applied pressure gradient, the flow in F obeys the following
Poisson equation with no source § ∂ 2u z 2ν / 3 − 1 ∂u z · ¨¨ 2 + ¸=0 r ∂" r ¸¹ © ∂" r .
(20)
The mapping of the Dirichlet boundary conditions
υ z ( R1 ) = υ1 and υ z ( R2 ) = υ2
(21a)
ν into the boundary conditions in F reads as
u z (" 1 ) = u1 and u z (" 2 ) = u2 .
(21b)
α α where " 1 = ar R1 and " 2 = ar R2 .
In the case of ν = 3 the solution of Eq. (20) with the boundary conditions (21b) is
u z = u2
ln(" r / " 1 ) ln(" r / " 2 ) − u1 ln(" 2 / " 1 ) ln(" 2 / " 1 )
(22a)
and so, the distribution of the fractal fluid velocity between axially moving concentric cylinders
υ z = υ2
ln( r / R1 ) ln(r / R2 ) − υ1 ln( R2 / R1 ) ln( R2 / R1 ) .
(22b)
is the same as in the case of homogeneous fluids (see, for example, Ref. [1]), even when D <ν = 3 .
However, if 2 < ν < 3 , the solution of Eq. (20) with the boundary conditions (21) has the form ur = C4" 2r − 2ν / 3 + C3 ,
(23a)
where
C3 =
u −u u1 + u2 u1 − u2 " 2R−12ν / 3 + " 2R−22ν / 3 C4 = 2 − 2ν / 13 22 − 2ν / 3 − 2 − 2ν / 3 2 − 2ν / 3 " R1 − " R2 2 2 " R1 − " R2 and .
(23b)
Accordingly, the distribution of fractal fluid velocity between axially moving concentric cylinders takes the form
υr =
υ2 − υ1 ζ
ζ
2( R2 − R1
(2r )
ζ
)
− R1ζ − R2ζ +
υ1 + υ2 2
,
(24a)
where the scaling exponent
§2
2·
2
ζ = D¨ − ¸ = 2α − D 3 ©ν 3 ¹
(24b)
is in the range of 0 < ζ < 2 / 3 , while 2 < ν < 3 and D ≤ ν . The values of ζ for some fractals are given in Table 2. The graphs of velocity distributions in the Couette flows of fractal fluids characterized by different sets of dimension numbers are shown in Fig. 1.
The fluid velocity distribution (24) depends on the fractal metric of flow (characterized by α ), as well as on the fluid mass distribution (characterized by D ). Notice that the averaged velocity of fractal fluids having the same metric ( α = const ) decreases as the mass fractal dimension D increases. Conversely, in the case of fluids having the same number of effective degrees of freedom (ν = const ) the averaged fluid velocity increases as D increases. On the other hand, the averaged velocity of fluids of the same dimension ( D = const ) decreases as the number of effective spatial degrees of freedom decrease.
3.2. Poiseuille flow of a fractal fluid
A Poiseuille flow is driven by a pump that forces fluid to flow by modifying the pressure along a pipe of length L , while the pressure gradient remains constant. Fully developed Poiseuille flow exists only far from the entrance and exit of the pipe, where the flow is ν aligned parallel to the walls [1]. The steady Poiseuille flow in F obeys Eq. (18), while
∂p / ∂" r = 0 . Applying the no-slip boundary condition together with the condition that the fluid velocity is zero if the pressure gradient is zero, we find that the velocity distribution in steady Poiseuille flow for a Newtonian fractal fluid in a cylindrical tube of radius R has the form
ª § r · 2α º υ z = υ max «1 − ¨ ¸ » ¬« © R ¹ ¼» ,
(25)
while the pressure gradient along the flow direction is equal to
4μ υ ∂p = − eff2 Dmax zγ /ν ∂z ar R ,
(26)
where
μ eff =
· ¸¸ μν ¹
cD ( D + ν )(ν − 1) § a z ¨¨ 4νd s © ar
(27a)
is the effective viscosity of the Newtonian fractal fluid obeying the constitutive equation (13),
υ max = −
a r R 2α Δ P 4 μ eff L1+ γ
(27b)
is the maximum velocity imposed by the pressure drop ΔP along the tube of length L , and
ν
D ν − 3
(28a)
1 1 <γ < 3 3 ,
(28b)
γ= is the scaling exponent. Notice that −
while 2 < ν ≤ 3 and D ≤ ν .
From Eq. (26) with the boundary conditions p ( z = 0) = p0
and p( z = L) = pL ,
(29)
it follows that the pressure distribution along the steady laminar flow of the Newtonian fractal fluid behaves as 1+ γ
§z· p( z ) = p0 − ( p0 − pL )¨ ¸ ©L¹
,
(30)
where γ is defined by Eq. (28). In this regard, it is pertinent to point out that the velocity distribution (25) is governed by the fractal metric (1), whereas the pressure distribution (30) depends on the fractal metric and the fractal topology of the fractal fluid. The values of γ for some fractals a given in Table 2. Velocity and pressure distributions in steady Poiseuille flows of fractal fluids characterized by different sets of dimension numbers are shown in Fig. 2. Notice that if 2 < D =ν < 3 ,
(31a)
then the fluid velocity distribution (25) is parabolic (as this is in the case of homogeneous fluids), whereas the pressure distribution (30) is no-linear. On the other hand, if 2 < ν = 3D < 3 ,
(31b)
the pressure distribution along the flow direction is lineal, but the velocity distribution (25) is not parabolic.
In this regard, it is pertinent to note that the laminar flow of fractal fluid in the cylindrical tube was studied in Refs. [16,17]. However, the radial distribution of fluid velocity distribution was derived only for a special case when the fluid density is homogeneous along the flow, but it is fractal across the tube. Furthermore, in Refs. [16,17] it was implicitly assumed that the number of effective degrees of freedom across the tube is equal to ν = 2 . Consequently, the pressure gradient remains constant along the flow, while the radial velocity distribution is not parabolic [17]. It is a straightforward matter to see that in the case of ν = 2 Eq. (25) converts in the Eq. (20) of Ref. [17].
Using Eq. (25) we find that the average velocity of the fractal fluid is equal to R
υ avg =
ν 2 υ (r )rdr = υ max 2 ³ z R 0 D +ν
.
(32)
Accordingly, the modified Poiseuille law relating the flow rate
Q = πR 2υ avg
to the
pressure drop Δp along the tube of length L reads as Q=−
ΔP πar R 2( D +ν ) /ν 8μeff L1+ D / ν −ν / 3
.
(33)
So, the fractal nature of the complex fluid can either assist or hinder the pressure driven flow in a pipe (see also Fig. 2). Notice that if ν = D = d s = 3 , Eqs. (25), (26), (30), (32), and (33) convert into the classical equations describing the Poiseuille flow of homogeneous Newtonian fluids.
VI. Conclusions
The mapping of laminar flows of a fractal fluid into the flow in the fractional dimension space with metric induced by the fractal topology is discussed. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. Next, we obtain expressions for the radial distributions of flow velocities in steady Couette and Poiseuille flows of a fractal fluid. The pressure distribution along the steady Poiseuille flow of the fractal fluid is also found explicitly.
We state that the radial distribution of the fluid velocity in the Couette flow depends upon the fractal metric of flow, as well as on the fluid mass distribution. However, if the number of effective spatial degrees of freedom of a random walker in the fractal fluid (ν ) is equal to 3, the radial distribution of the fractal fluid velocity is the same as in the Couette flow of the homogeneous fluid, even when the mass density of the fractal fluid is scale and space dependent. Conversely, the radial distribution of the fluid velocity in the steady Poiseuille flow of the fractal fluid is governed only by the fractal metric. In particular, the
radial distribution of the fluid velocity is parabolic when the mass fractal dimension of the fluid is equal to the number of effective spatial degrees of freedom. The pressure distribution along the steady Poiseuille flow of a fractal fluid depends on the fractal topology of flow, as well as on the fractal metric. In particular, if ν = 3D , the pressure distribution is linear, but the radial distribution of the fluid velocity is not parabolic.
In this regard, it is pertinent to point out that the approach this work is applicable to any type of Newtonian fractal fluids obeying the constitutive law (13). For this purpose, the real fractal fluid (as for example, studied in Refs. [6-10]) should be characterized not only by the mass fractal dimension, but the spectral and connectivity dimension should be also determined from independent experiments. So, we expect that our findings will stimulate further experimental and theoretical studies of the fractal and hydrodynamic properties of fractal fluids.
Acknowledgments
The authors wish to thank Armando Garcia Jaramillo and Michael Shapiro for fruitful discussions. This work was supported by PEMEX under the SENER-CONACYT research Grant No. 143927.
Table 1. Dimension numbers characterizing a fractal [30]. Dimension number
Symbol
Definition
Mass fractal dimension
D
Defined via the scaling behavior of the fractal mass
M ∝ l D , where l is the characteristic size of the fractal Number of effective spatial degrees of freedom of a walker on the fractal Spectral dimension
ν
ds
in the embedding Euclidean space The number of independent directions on the fractal in which a walker on the fractal can move without violating any constraint imposed on it. Defined via the scaling behavior of the probability of a random walker the fractal to return to its starting point
P (0, t ) ∝ t d s .
3
Table 2. Dimension numbers of some classic fractals in E [30] and the values of scaling
D
ds
ν
2α
Eq. (1b)
Menger sponge 2.727 2.51 2.94 1.85 Siepinski pyramid 2 1.55 2.45 1.63 Infinite random 2 4/3 8/3 3/2 graphs [47] 2D / 3 3 3 Cantor dusts in E 3 : D 3 1.65 3 3 1.1 Cantor dust in E 3 2.1 3 3 1.4 Cantor dust in E 3 2.9 3 3 1.93 Cantor dust in E Diamond fractals 2 D D D Diamond fractal 2.1 2.1 2.1 2 Diamond fractal 2.5 2.5 2.5 2 Diamond fractal 2.9 2.9 2.9 2 exponents characterizing steady laminar flows of fractal fluids
ζ Eq. (24b)
0.03 0.3 1/6
1+ γ Eq. (28a)
0.94 0.9997 31/36
D/3 -1) 1) 0.55 1) 0.7 1) 0.97 2 − 2D / 3 2 − D / 3 0.6 1.3 1/3 7/6 0.07 1.03 with mass distributions
modeled by these fractals.
1)
In the Couette flow of suspensions with the Cantor dust-like distributions of solid particles, the radial distribution of fluid velocity is logarithmic (see Eq. (22b)).
Figure captions
Figure 1. Velocity distributions for Couette flow between concentric cylinders (
R1 / R2 = 0.1 , υ1 = 0 ) of homogeneous Newtonian fluid (1) and for fractal fluids having the structure of the infinite random graph (2), Siepinski pyramid (3), and diamond fractal with D = 2.1 (4). See Table 2.
Figure 2. (a) Velocity distributions for steady laminar flow in a pipe of circular cross
section for a homogeneous Newtonian fluid (1) and for fractal fluids having the structure of the Menger sponge (2), Siepinski pyramid (3), infinite random graph (4), Cantor dust with D = 2.1 (5), and Cantor dust with D = 1.65 (6). (b) Pressure distribution along the laminar flow of a homogeneous Newtonian fluid (1) and for fractal fluids having the structure of the diamond fractal with D = 2.1 (2), diamond fractal with D = 2.5 (3), Menger sponge (4), infinite random graph (5), Cantor dust with D = 2.1 (6), and Cantor dust with D = 1.65 (7), while pL = 0 . See Table 2.
References [1] O. Darrigol, Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, Oxford, 2005. [2] S. P. Sutera, R. Skalak, The history of Poiseuille’s law, Annu. Rev. Fluid Mech. 25 (1993) 119; DOI: 10.1146/annurev.fl.25.010193.000245. [3] M. Z. Bazant, Exact solutions and physical analogies for unidirectional flows, Phys. Rev. Fluids 1 (2016) 024001; DOI: 10.1103/PhysRevFluids.1.024001. [4] J. Cai, E. Perfect, C.-L. Cheng, X. Hu, Generalized modeling of spontaneous imbibition based on HageníPoiseuille flow in tortuous capillaries with variably shaped apertures, Langmur 30 (2014) 5142-5151; DOI: 10.1021/la5007204. [5] A. S. Balankin, J. C. Valdivia, J. Marquez, O. Susarrey, M. A. Solorio-Avila, Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity, Phys. Lett. A 380 (2016) 2767-2773; DOI: 10.1016/j.physleta.2016.06.032. [6] W. Wei, J. Cai, X. Hu, Q. Han, S. Liu, Y. Zhou, Fractal analysis of the effect of particle aggregation distribution on thermal conductivity of nanofluids, Phys. Lett. A 380 (2016) 2953–2956; DOI: 10.1016/j.physleta.2016.07.005. [7] J. Cai, X. Hu, B. Xiao, Y. Zhou, W. Wei, Recent developments on fractal-based approaches to nanofluids and nanoparticle aggregation, Int. J. Heat Mass Transf. 105 (2017) 623–637; DOI: 10.1016/j.ijheatmasstransfer.2016.10.011 [8] H. Shui and H. Zhou, Viscosity and fractal dimension of coal soluble constituents in solution, Fuel Process. Tech. 85 (2004) 1529-1538; DOI: 10.1016/j.fuproc.2003.10.024. [9] B. P. Hills, P. Manoj, C. Destruel, NMR Q-space microscopy of concentrated oil-in-water emulsions, Magn, Res. 18 (2000) 319-333; DOI: 10.1016/S0730-725X(99)00143-5. [10] A. A. Shiyan, Viscosity for fractal suspensions: dependence on fractal dimensionality, Phys. Lett. A 220 (1996) 117-119; DOI: 10.1016/0375-9601(96)00479-3. [11] L. Haider, P. Snabre, M. Boynard, Rheology and ultrasound scattering from aggregated red cell suspensions in shear flow, Biophys. J. 87 (2004) 2322-2334; DOI: 10.1529/biophysj.104.041665. [12] M. Heinen, S. K. Schnyder, J. F. Brady, H. Löwen, Classical liquids in fractal dimension, Phys. Rev. Lett. 115 (2015) 097801; DOI: 10.1103/PhysRevLett.115.097801. [13] V. I. Lesina, Y. A. Koksharov, G. B. Khomutov, Viscosity of liquid suspensions with fractal aggregates: Magnetic nanoparticles in petroleum colloidal structures, Coll. Surf. A: Physicochem. Eng. Aspects 392 (2011) 88-94; DOI: 10.1016/j.colsurfa.2011.09.038.
[14] V. E. Tarasov, Fractional hydrodynamic equations for fractal media, Ann. Phys. 318 (2005) 286-307; DOI: 10.1016/j.aop.2005.01.004. [15] X.-J. Yang, D. Baleanu, J. A. Tenreiro-Machado, Systems of Navier-Stokes equations on Cantor sets, Math. Probl. Eng. (2013) 769724. DOI: 10.1155/2013/769724. [16] V. E. Tarasov, Flow of fractal fluid in pipes: Non-integer dimensional space approach, Chaos Solitons Fractals 67 (2014) 26-37; DOI: 10.1016/j.chaos.2014.06.008. [17] V. E. Tarasov, Poiseuille equation for steady flow of fractal fluid, Int. J. Mod. Phys. 30 (2016) 1650128; DOI: 10.1142/S0217979216501289. [18] M. Sahimi, Flow and Transport in Porous Media and Fractured Rocks, VCH, New York, 1995. [19] A. S. Balankin, B. Espinoza-Elizarraraz, Hydrodynamics of fractal continuum flow, Phys. Rev. E 85 (2012) 025302(R); DOI: 10.1103/PhysRevE.85.025302. [20] A. S. Balankin, B. E. Elizarraraz, Map of fluid flow in fractal porous medium into fractal continuum flow, Phys. Rev. E 85 (2012) 056314; DOI: 10.1103/PhysRevE.85.056314. [21] J. Cai, L. Luo, R. Ye, X. Zeng, X. Hu, Recent advances on fractal modeling of permeability for fibrous porous media, Fractals 23 (2015) 1540006; DOI: 10.1142/S0218348X1540006X. [22] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Willey, New York, 1990. [23] C. Tricot, Curves and Fractal Dimension, Springer, New York, 1993. [24] A. S. Balankin, Physics of fracture and mechanics of self-affine cracks, Eng. Fract. Mech. 57 (1997) 135-203; DOI: 10.1016/S0013-7944(97)00007-6. [25] A. S. Balankin, A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric, Eur. Phys. J. B 88 (2015) 90; DOI: 10.1140/epjb/e2015-60189-y. [26] A. S. Balankin, A continuum framework for mechanics of fractal materials II: elastic stress fields ahead of cracks in a fractal medium, Eur. Phys. J. B 88 (2015) 91; DOI: 10.1140/epjb/e2015-50703-8. [27] A. S. Balankin, Toward the mechanics of fractal materials: mechanics of continuum with fractal metric, arXiv:1409.5829 (2014) 1-80. [28] M. Ostoja-Starzewski, Continuum mechanics models of fractal porous media: Integral relations and extremum principles, J. Mech. Mat. Struct. 4 (2009) 901-912; DOI: 10.2140/jomms.2009.4.901. [29] A. S. Balankin, B. Mena, J. Patiño, D. Morales-Matamoros, Electromagnetic fields in fractal continua, Phys. Lett. A 377 (2013) 783-788; DOI: 10.1016/j.physleta.2013.01.030. [30] A. S. Balankin, Effective degrees of freedom of a random walk on a fractal, Phys. Rev. E 92 (2015) 2322; DOI: 10.1103/PhysRevE.92.062146.
[31] M. Zubair, L. K Ang, Fractional-dimensional Child-Langmuir law for a rough cathode, Phys. Plasm. 23 (2016) 072118; DOI: 10.1063/1.4958944. [32] F. H. Stillinger, Axiomatic basis for spaces with non integer dimension, J. Math. Phys. 18 (1977) 1224-1234; DOI: 10.1063/1.523395. [33] C. Palmer, P. N. Stavrinou, Equations of motion in a non-integer-dimensional space, J. Phys. A: Math. Gen. 37 (2004) 6987-7003; DOI: 10.1088/0305-4470/37/27/009. [34] S. I. Muslih, O. P. Agrawal, A scaling method and its applications to problems in fractional dimensional space, J. Math. Phys. 50 (2009) 123501; DOI: 10.1063/1.3263940. [35] M. Zubair, M. J. Mughal, Q. A. Naqvi, Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer, New York, 2012. [36] A. S. Balankin, Physics in space–time with scale-dependent metrics, Phys. Lett. A 377 (2013) 1606-1610; DOI: 10.1016/j.physleta.2013.04.040. [37] W.-P. Zhong, M. R. Belic, B. A. Malomed, Y. Zhang, T. Huang, Spatiotemporal accessible solitons in fractional dimensions, Phys. Rev. 94 (2016) 012216; DOI: 10.1103/PhysRevE.94.012216. [38] A. S. Balankin, J. Bory-Reyes, M. Shapiro, Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric, Physica A 444 (2016) 345-359; DOI: 10.1016/j.physa.2015.10.035. [39] J. Weberszpil, M. J. Lazo, J.A. Helayël-Neto, On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric, Physica A 436 (2015) 399-404; DOI: 10.1016/j.physa.2015.05.063. [40] J. Weberszpil, J. A. Helayël-Neto, Variational approach and deformed derivatives, Physica A 450 (2016) 217–227; DOI: 10.1016/j.physa.2015.12.145. [41] U. Mosco, Invariant Field Metrics and Dynamical Scalings on Fractals, Phys. Rev. Lett. 79 (1997) 4067-4070; DOI: 10.1103/PhysRevLett.79.4067. [42] A. S. Balankin, B. Mena, C. L. Martinez-Gonzalez, D. Morales-Matamoros, Random walk in chemical space of Cantor dust as a paradigm of superdiffusion, Phys. Rev. E (2012) 86052101; DOI: 10.1103/PhysRevE.86.052101. [43] X.-J. Yang, H. M. Srivastava, J.-H. He, D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Phys. Lett. A 377 (2013) 1696-1700; DOI: 10.1016/j.physleta.2013.04.012. [44] A. S. Balankin, Stresses and strains in a deformable fractal medium and in its fractal continuum model, Phys. Lett. A 377 (2013) 2535-2541; DOI: 10.1016/j.physleta.2013.07.029. [45] X.-J. Yang, D. Baleanu, M. C. Baleanu, Observing diffusion problems defined on Cantor sets in different co-ordinate systems, Therm. Sci. 19 (2015) S151-S156; DOI: 10.2298/TSCI141126065Y.
[46] W. Chen, Time–space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006) 923-929; DOI: 10.1016/j.chaos.2005.08.199. [47] B. Durhuus, Hausdorff and spectral dimension of infinite random graphs, Acta Phys. Pol. 40 (2009) 3509-3532; http://www.forskningsdatabasen.dk/en/catalog/2193011880.
Figure 1-rev
Figure 2