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Nanofluids for Heat Transfer – Potential and Engineering Strategies

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Heat Transfer in Nanostructures Using the

Fractal Approximation of Motion

Maricel Agop

1,2

, Irinel Casian Botez

Luciu Razvan Silviu

and Manuela Girtu

Laboratoire de Physique des Lasers, Atomes et Molécules (UMR 8523),

Université des Sciences et Technologies de Lille,

Physics Department, “Ghe. Asachi” Technical University, Iasi,

Department of Electronics, Telecommunication and Information Technology,

“Gh. Asachi” Technical University, Iasi,

Faculty of Civil Engineering, "Ghe. Asachi” Technical University, Iasi,

”Vasile Alecsandri” University of Bacau, Department of Mathematics, Bacau

France

2,3,4,5

Romania

1. Introduction

If at a macroscopic scale the heat transfer mechanism implies either diffusion type

conduction or phononic type conduction (Zhang, 2007; Rohsenow et al., 1998), at a

microscopic scale the situation is completely different. This happens because the

macroscopic familiar concepts cannot be applied at a microscopic scale, e.g. the concept of a

distribution function of both coordinates and momentum used in the Boltzmann equation

(Wang et al., 2008). Moreover, fundamental concepts such a temperature cannot be defined

in the conventional sense, i.e. as a measure of thermodynamic equilibrium (Chen, 2000).

Thus anomalies might occur: the thermal anomaly of the nanofluids (Wang&Xu, 1999;

Keblinski et al., 2002; Patel et al., 2003), etc.

According to our opinion, anomalies become normalities if their specific measures depend

on scales: heat conduction in nanostructures differs significantly from that in

macrostructures because the characteristic length scales associated with heat carriers, i.e. the

mean free path and the wavelength, are comparable to the characteristic length of

nanostructures (Chen, 2000). Therefore, we expect to replace the usual mechanisms

(ballistic thermal transport, etc.) by something more fundamental: a unique mechanism

in which the physical measures should depend not only on spatial coordinates and

time, but also on scales. This new way will be possible through the Scale Relativity (SR)

theory (Notalle, 1992, 2008a, 2008b, 2007). Some applications of the SR theory at the

nanoscale was given in (Casian Botez et al., 2010; Agop et al. 2008). In the present

paper, a new model of the heat transfer on nanostructures, considering that the heat

flow paths take place on continuous but non-differentiable curves, i.e. an fractals, is

established.

Two Phase Flow, Phase Change and Numerical Modeling

452

2. Consequences of non-differentiability in the heat transfer processes

Let us suppose that the heat flow take place on continuous but non-differentiable curves

(fractal curves). The non-differentiability implies the followings (Notalle, 1992, 2008a, 2008b,

2007):

i. A continuous and a non-differentiable curve (or almost nowhere differentiable) is

explicitly scale dependent, and its length tends to infinity, when the scale interval tends

to zero. In other words, a continuous and non-differentiable space is fractal, in the

general meaning given by Mandelbrot to this concept (Mandelbrot, 1982);

ii. There is an infinity of fractals curves (geodesics) relating any couple of its points (or

starting from any point), and this is valid for all scales;

iii. The breaking of local differential time reflection invariance. The time-derivative of the

temperature field

T can be written two-fold:

dT T(t dt) - T(t)

lim

dt dt

dt 0

dT T(t)- T(t -dt)

lim

dt dt

dt 0

→

(1a,b)

Both definitions are equivalent in the differentiable case. In the non-differentiable

situation these definitions fail, since the limits are no longer defined. In the framework

of fractal theory, the physics is related to the behavior of the function during the

“zoom” operation on the time resolution

tδ , here identified with the differential

element

dt (“substitution principle”), which is considered as an independent variable.

The standard temperature field

T(t) is therefore replaced by a fractal temperature field

T(t,dt) , explicitly dependent on the time resolution interval, whose derivative is

undefined only at the unobservable limit

dt 0→ . As a consequence, this lead us to

define the two derivatives of the fractal temperature field as explicit functions of the

two variables

t and dt ,

T(t + dt,dt) - T(t,dt)

= lim

dt 0

dt dt

d T T(t,dt) - T(t - dt,dt)

=lim

dt 0

dt dt

→

(2a,b)

The sign, +, corresponds to the forward process and, -, to the backward process;

iv. the differential of the fractal coordinates,

dX(t,dt)

, can be decomposed as follows:

d X(t,dt) d x(t) d (t,dt)

±±±

(3a,b)

where

dx(t)

is the “classical part” and d(t,dt)

is the “fractal part”.

v. the differential of the “fractal part” of

satisfies the relation (the fractal equation)

()

ddt

±±

ξ=λ

(4a,b)

where

λ are some constant coefficients, and

D is a constant fractal dimension. We

note that for the fractal dimension we can use any definition (Kolmogorov, Hausdorff

Heat Transfer in Nanostructures Using the Fractal Approximation of Motion

453

(Notalle, 1992, 2008a, 2008b, 2007; Casian Botez et al., 2010; Agop et al. 2008;

Mandelbrot, 1982), etc.);

vi. the local differential time reflection invariance is recovered by combining the two

derivatives,

ddt

and

ddt, in the complex operator:

dd d-d

d1 i

dt 2 dt 2 dt







(5)

Applying this operator to the “position vector” yields a complex speed

dddidd

Vii

dt dt dt

22 22

+− +− +−+−

+−+−



== − = − =−





XXX XXVVVV

VU (6)

with

+−

−

(7a,b)

The real part,

V , of the complex speed,

, represents the standard classical speed, which

is differentiable and independent of resolution, while the imaginary part,

U , is a new

quantity arising from fractality, which is non-differentiable and resolution-dependent;

vii. the average values of the quantities must be considered in the sense of a generalized

statistical fluid like description. Particularly, the average of

X is

±±

=Xx

(8a,b)

with

d ξ =0

(9a,b)

viii. in such an interpretation, the “particles” are indentified with the geodesics themselves.

As a consequence, any measurement is interpreted as a sorting out (or selection) of the

geodesics by the measuring devices.

3. Covariant total derivative in the heat transfer processes

Let us now assume that the curves describing the heat flow (continuous but non-

differentiable) is immersed in a 3-dimensional space, and that

X of components

()

Xi=1,3

is the position vector of a point on the curve. Let us also consider the fractal temperature

fluid

T( ,t)X , and expand its total differential up to the third order:

iik

dT dt T d dXdX dXdXdX

XX XXX

±±±± ±±±

∂

∂∂

=+∇⋅

∂∂ ∂∂∂

∂

++X

(10a,b)

where only the first three terms were used in the Nottale’s theory (

i.e. second order terms in

the equation of motion). The relations (10a,b) are valid in any point of the space manifold

Two Phase Flow, Phase Change and Numerical Modeling

454

and also for the points X on the fractal curve which we have selected in relations (10a,b).

From here, the forward and backward average values of this relation take the form:

dT dt T d dXdX

tXX

dXdXdX

XXX

±±±±

±±±

∂∂

=+∇⋅+ +

∂∂∂

∂

∂∂∂

(11a,b)

We make the following stipulation: the mean value of the function

and its derivatives

coincide with themselves, and the differentials

and dt are independent, therefore the

average of their products coincide with the product of averages. Thus, the equations (11a,b)

become:

iik

TTT

dT dt T d dXdX dXdXdX

tXXXXX

± ± ±± ±± ±

∂∂ ∂

=+∇⋅+ +

∂∂∂∂∂∂

(12a,b)

or more, using equations (3a,b) with the property (9a,b),

(

)

(

)

ikik

dT dt T d dxdx d d

tXX

dxdxdx d d d

XXX

±±±±±±

±±± ±±±

∂∂

=+∇⋅+

ξξ

∂∂∂

∂

+ξξξ

∂∂∂

(13a,b)

Even the average value of the fractal coordinate,

is null (see (9a,b)), for the higher order

of the fractal coordinate average the situation can be different. First, let us focus on the

mean

ξξ

±±

. If ij≠ , this average is zero due the independence of

and

. So,

using (4a,b), we can write:

()

d d dt dt

ξξ λλ

−

±± ±±

= (14a,b)

Then, let us consider the mean

ddd

ξξξ

±±±

. If ijk≠≠ , this average is zero due the

independence of

and

. Now, using equations (4a,b), we can write:

()

ikik

ddd dt dt

ξξξ λλλ

−

±±± ±±±

= (15a,b)

Then, equations (13a,b) may be written under the form:

()

dT dt d T dxdx

tXX

dt dt

dxdxdx

XXX

dt dt

XXX

λλ

λλλ

±± ±±

−

±±

±±±

−

±±±

∂∂

=+⋅∇+ +

∂∂∂

∂

∂∂

∂

∂∂∂

∂

∂∂∂

(16a,b)

Heat Transfer in Nanostructures Using the Fractal Approximation of Motion

455

If we divide by dt and neglect the terms which contain differential factors (for details on the

method see (Casian Botez et al., 2010; Agop et al. 2008)), the equations (16a,b) are reduced

to:

()

dT T T

Tdt

dt t X X

XXX

()

λλ

λλλ

−

±±±

−

±±±

∂∂

=+⋅∇+ +

∂∂∂

∂

∂∂∂

(17a,b)

These relations also allows us to define the operator:

()

dt X X

XXX

()

λλ

λλλ

−

±±±

−

±±±

∂

=+⋅∇+ +

∂∂

∂

∂∂∂

(18a,b)

Under these circumstances, let us calculate

()

∂∂. Taking into account equations (18a,b),

(5) and (6) we obtain:

()

iik

TdTdTdTdT T

tdtdtdtdt t

dt dt

XX XXX

Tdt

tXX

TiTi

XXX t

21 31

111

222

412

11 1

22 4

12 2

λλ λλλ

λλ

λλλ

+− +−

−−

++ +++

−

−−−

−

−−−



∂∂



=+−−=+⋅∇+





∂∂





∂∂

++ +

∂∂ ∂∂∂

∂∂

++⋅∇+ +

∂∂∂

∂∂

+−−

∂∂∂ ∂

()

iik

jj j

ii i

iTi T

dt dt

XX XXX

iT i i T

Tdt

tXX

XXX

21 31

212

22 2

λλ λλλ

λλ

λλλ

λλ λλ λλ

−−

++ +++

−

−−−

−

−−−

+− +−

−

++ −− ++

⋅∇ −

∂∂

−− +

∂∂ ∂∂∂

∂∂

++⋅∇+ +

∂∂∂

∂

∂∂∂

∂+ +



=+ − ⋅∇+



∂



++−−

VV VV

()

()()

()

()()

()

jj jj

ikik ikik

jj jj

ii ii

jj jj

ikik iki

XXX

tXX

λλ

λλλ λλλ λλλ λλλ

λλ λλ λλ λλ

λλλ λλλ λλλ λλλ

−−

−

+++ −−− +++ −−−

−

++ −− ++ −−

−

+++ −−− +++ −−−

∂





∂∂

∂



++−− =



∂∂∂

∂∂



=+⋅∇+ + − − +



∂∂∂



++−−



()

XXX

∂





∂∂∂

(19a,b)

Two Phase Flow, Phase Change and Numerical Modeling

456

This relation also allows us to define the fractal operator:

()

()()

jjj

ii ii

jj jj

ikik ikik

tt XX

XXX

λλ λλ λλ λλ

λλλ λλλ λλλ λλλ

−

+−− ++−−

−

+++ −−− +++ −−−

∂∂ ∂



=+⋅∇+ + − − +



∂∂ ∂∂

∂



++−−



∂∂∂

(20)

Particularly, by choosing:

jjij

jj ijk

ik ik

λλ λλ δ

λλλ λλλ δ

++ −−

+++ −−−

=− =

(21a,b)

the fractal operator (20) takes the usual form:

()

idt dt

21 31

−−

∂∂

=+⋅∇− Δ+ ∇

∂∂

V DD

(22)

We now apply the principle of scale covariance, and postulate that the passage from

classical (differentiable) mechanics to the “fractal” mechanics can be implemented by

replacing the standard time derivative operator,

ddt, by the complex operator t

∂∂ (this

results in a generalization of the principle of scale covariance given by Nottale in (Nottale,

1992)). As a consequence, we are now able to write the equation of the heat flow in its

covariant form:

()

Ti dt T dt T

21 31

−−

∂∂

=+⋅∇− Δ+ ∇=

∂∂

V DD

(23)

This means that at any point of a fractal heat flow path, the local temporal term,

T∂ , the

non-linearly (convective) term,

()

⋅∇V , the dissipative term, TΔ and the dispersive one,

∇ , make their balance. Moreover, the behavior of a fractal fluid is of viscoelastic or of

hysteretic type,

i.e. the fractal fluid has memory. Such a result is in agreement with the

opinion given in (Ferry& Goodnick, 1997; Chiroiu et al., 2005): the fractal fluid can be

described by Kelvin-Voight or Maxwell rheological model with the aid of complex

quantities e.g. the complex speed field, the complex structure coefficients.

4. The dissipative approximation in the heat transfer processes

4.1 Standard thermal transport

In the dissipative approximation of the fractal heat transfer, the relation (23) becomes a

Navier-Stokes type equation for the temperature field:

()

Ti dt T

−

∂∂

=+⋅∇− Δ=

∂∂

V D (24)

with an imaginary viscosity coefficient:

()

idt

−

= D (25)

Heat Transfer in Nanostructures Using the Fractal Approximation of Motion

457

Separating the real and imaginary parts in (24), i.e.

()

Tdt T

−

∂

+⋅∇=

∂

−⋅∇= Δ

U D

(26a,b)

We can add these two equations and obtain the thermal transfer equation in the form:

() ()

()

Tdt T

21−

∂

+−⋅∇= Δ

∂

VU D

(27)

The standard equation for the thermal transport,

i.e.:

()

Tdt

αα

−

∂

=Δ

∂

= D

(28a,b)

results from (27) on the following assertions

i. the fractal heat flow are of Peano’s type (Nottale, 1992), i.e. for

D 2= ;

ii. the movements at differentiable and non-differentiable scales are synchronous, i.e.

V=U;

iii. the structure coefficient

D , proper to the fractal-nonfractal transition, is identified with

the diffusivity coefficient α, i.e.

≡ D .

In the form

txy







∂∂∂

(29)

where we used the normalized quantities

ttxkxykyT

,,,

====

(30a-d)

and the restriction

(31)

the equation (29) can by solved with the following initial and boundaries conditions:

()

Txy x y

0, , ,for 0 1and 0 1

=≤≤≤≤

(32a,b)

() ()

Tt y Tt y

Ttx Ttx

,0,14,,1,14

12 12

,,0 exp exp , ,,1 14

12 12















−−

=− − =

(33a-d)

In the Figures (1a-j) we present the solutions obtained with the finite differences method

(Zienkievicz &Taylor, 1991). Furthermore, using tha same method from (Zienkievicz

Two Phase Flow, Phase Change and Numerical Modeling

458

&Taylor, 1991), if the thermal transport occurs in the presence of a “wall”, condition which

involves substituting (33d) with

()

,,1 0

∂

(34)

then obtain the numerical solutions from the figures (2a-j). It results that the perturbation of

the thermal field, either disappear because of the rheological properties of the fractal

environment (Figures 1a-j), or it regenerates (Figures 2a-j).

Fig. 1. a-j. Numerical 2D contour curves of the normalized temperature field in the absence

of a “wall”. Thermal field perturbation disappears due to the rheological properties of the

fractal environment