- •LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
- •CONTENTS
- •FOREWORD
- •PREFACE
- •Abstract
- •1. Introduction
- •2. A Nice Equation for an Heuristic Power
- •3. SWOT Method, Non Integer Diff-Integral and Co-Dimension
- •4. The Generalization of the Exponential Concept
- •5. Diffusion Under Field
- •6. Riemann Zeta Function and Non-Integer Differentiation
- •7. Auto Organization and Emergence
- •Conclusion
- •Acknowledgment
- •References
- •Abstract
- •1. Introduction
- •2. Preliminaries
- •3. The Model
- •4. Numerical Simulations
- •5. Synchronization
- •6. Conclusion
- •Acknowledgments
- •References
- •Abstract
- •1. Introduction: A Short Literature Review
- •2. The Injection System
- •3. The Control Strategy: Switching of Fractional Order Controllers by Gain Scheduling
- •4. Fractional Order Control Design
- •5. Simulation Results
- •6. Conclusion
- •Acknowledgment
- •References
- •Abstract
- •Introduction
- •1. Basic Definitions and Preliminaries
- •Conclusion
- •Acknowledgments
- •References
- •Abstract
- •1. Context and Problematic
- •2. Parameters and Definitions
- •3. Semi-Infinite Plane
- •4. Responses in the Semi-Infinite Plane
- •5. Finite Plane
- •6. Responses in Finite Plane
- •7. Simulink Responses
- •Conclusion
- •References
- •Abstract
- •1. Introduction
- •2. Modelling
- •3. Temperature Control
- •4. Conclusion
- •References
- •Abstract
- •1. Introduction
- •2. Preliminaries
- •3. Second Order Sliding Mode Control Strategy
- •4. Adaptation Law Synthesis
- •5. Numerical Studies
- •Conclusion
- •References
- •Abstract
- •1. Introduction
- •2. Rabotnov’s Fractional Operators and Main Formulas of Algebra of Fractional Operators
- •4. Calculation of the Main Viscoelastic Operators
- •5. Relationship of Rabotnov Fractional Operators with Other Fractional Operators
- •8. Application of Rabotnov’s Operators in Problems of Impact Response of Thin Structures
- •9. Conclusion
- •Acknowledgments
- •References
- •Abstract
- •1. Introduction
- •3. Theory of Diffusive Stresses
- •4. Diffusive Stresses
- •5. Conclusion
- •References
- •Abstract
- •Introduction
- •Methods
- •Conclusion
- •Acknowledgment
- •Abstract
- •1. Introduction
- •2. Basics of Fractional PID Controllers
- •3. Tuning Methodology for Fuzzy Fractional PID Controllers
- •4. Optimal Fuzzy Fractional PID Controllers
- •5. Conclusion
- •References
- •INDEX
In: Fractional Calculus: Applications |
ISBN: 978-1-63463-221-8 |
Editors: Roy Abi Zeid Daou and Xavier Moreau |
c 2015 Nova Science Publishers, Inc. |
Chapter 9
TH EO RY O F DIFFUSIVE STRESSES BASED
O N TH E FRACTIO NAL ADVECTIO N-DIFFUSION
EQUATIO N
Yuriy Povstenko
Institute of Mathematics and Computer Science
Jan Długosz University in Cze¸stochowa, Poland
Abstract
Conventional theory of diffusive stresses is based on the principles of the classical theory of diffusion, specifically on the classical Fick law, which relates the matter flux to the concentration gradient. In combination with the balance equation for mass, this law leads to the classical diffusion equation. We study time-nonlocal generalizations of the diffusive flux governed by the Fick law and of the advection flux associated with the velocity field. The nonlocal constitutive equations with the long-tail power memory kernel result in the time-fractional advection-diffusion equation. The nonlocal constitutive equations with the middle-tail memory kernel expressed in terms of the Mittag-Leffler function lead to the fractional advection-diffusion equation of the Cattaneo type. The theory of diffusive stresses based on the fractional advection-diffusion equation is formulated.
Keywords: Fractional calculus, advection-diffusion, diffusive stresses, Mittag-Leffler function
1.Introduction
The classical Fourier law constitutes the relation between the heat flux q and the temperature gradient
q = −k grad T . |
(1) |
In combination with a law of conservation of energy, equation (1) leads to the parabolic heat conduction equation
∂T |
= aT |
T |
(2) |
∂t |
E-mail address: j.povstenko@ajd.czest.pl
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Yuriy Povstenko |
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with aT being the heat diffusivity coefficient. |
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From the mathematical point of view, the Fourier law in the theory of heat conduction |
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corresponds to the Fick law in the theory of diffusion |
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j = −a grad c, |
(3) |
where j is the matter flux, c is the concentration. In combination with the balance equation for mass, equation (3) results in the classical diffusion equation:
∂c |
= a c |
(4) |
∂t |
with a being the diffusivity coefficient.
The classical heat conduction and diffusion equations based on the Fourier and Fick laws, respectively, are quite acceptable for different physical situations. However, many theoretical and experimental studies testify that in media with complex internal structure the standard parabolic equations are no longer accurate enough. In nonclassical theories, the Fourier and Fick laws as well as the parabolic heat conduction and diffusion equations are replaced by more general equations. Some of these theories were formulated in terms of the theory of heat conduction, other in terms of diffusion. For an extensive bibliography on this subject and further discussion see Chandrasekharaiah (1986, 1998), Ignaczak and Ostoja-Starzewski (2010), Joseph and Preziosi (1989), Metzler and Klafter (2000, 2004), Zaslavsky (2002) and references therein.
Gurtin and Pipkin (1968) proposed the general time-nonlocal dependence between the heat flux vector and the temperature gradient. Nigmatullin (1984a, 1984b) considered the
following general form of such an equation: |
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q(t) = −k Z0 t K (t − τ ) grad T (τ ) dτ |
(5) |
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resulting in the heat conduction equation with memory |
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∂t = aT Z0 |
K (t − τ ) T (τ ) dτ. |
(6) |
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Nigmatullin (1984a) and Green and Naghdi (1993) proposed the constitutive equation of heat conduction in the case of constant kernel (full memory with no memory decay)
q (t) = −k |
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grad T (τ )dτ . |
(7) |
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The wave equation for temperature
∂2T |
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(8) |
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∂t |
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obtained from Eq. (7) is a constituent part of thermoelasticity without energy dissipation developed by Green and Naghdi (1993).
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Cattaneo (1948) introduced the generalized constitutive equation for the heat flux which can be rewritten in a non-local form with the “short-tail” exponential time-nonlocal kernel
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−ζ Z0 |
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q (t) = |
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exp |
t − τ |
grad T (τ ) dτ, |
(9) |
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where ζ is a nonnegative constant. This equations leads to the telegraph equation for tem-
perature |
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∂T |
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∂2T |
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(10) |
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Based on this equation, Lord and Shulman (1967) proposed the generalized theory of thermoelasticity.
The time-nonlocal dependences between the heat flux vector and the temperature gradient with the “long-tail” power kernel (see Povstenko (2005, 2009a, 2013))
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q (t) = − |
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Z0 |
(t − τ )α |
1 grad T (τ ) dτ, |
0 < α ≤ 1, |
(11) |
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q (t) = − |
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Z0 t (t − τ )α−2 grad T (τ ) dτ, |
1 < α ≤ 2, |
(12) |
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can be interpreted in terms of fractional calculus |
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q (t) = −k D1−α grad T (t), |
0 < α ≤ 1, |
(13) |
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q (t) = −k I α−1 grad T (t), |
1 < α ≤ 2, |
(14) |
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and result in the time-fractional heat conduction equation |
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∂αT |
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T , |
0 < α ≤ 2, |
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∂tα |
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or in terms of diffusion |
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∂αc |
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c, |
0 < α ≤ 2, |
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∂tα |
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with the particular cases corresponding to subdiffusion (weak diffusion) (0 < α < 1); normal diffusion (α = 1); superdiffusion (strong diffusion) (1 < α < 2), and ballistic diffusion (α = 2).
In Eqs. (13)–(15) I αf (t), Dαf (t) and |
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are the Riemann-Liouville fractional |
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integral, Riemann-Liouville fractional derivative and the Caputo fractional derivative, respectively (Kilbas et al. (2006), Podlubny (1999)):
Dαf (t) =
d αf (t)
dtα
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I αf (t) = (α) |
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(t − τ )α−1f (τ ) dτ, |
α > 0, |
(17) |
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dtm |
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(t − τ )m−α−1f (τ ) dτ , |
m − 1 < α < m, |
(18) |
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= (m − α) |
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dτ m dτ, |
m − 1 < α < m . |
(19) |
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dmf (τ ) |
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The general fractional Cattaneo-type equations were considered by Povstenko (2012, 2013):
q (t) = −ζ |
Z0 |
(t − τ )β |
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2 Eβ−α, β−1 |
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(t − ζ |
grad T (τ ) dτ, |
(20) |
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τ )β−α |
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where Eα,β (z) is the two-parameter Mittag-Leffler function (see Kilbas et al. (2006), Podlubny (1999))
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Eα,β (z) = |
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, α > 0, β > 0, |
(21) |
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being the generalization of the exponential function. The constitutive equation (20) leads to the fractional telegraph equation for temperature
∂αT |
+ ζ |
∂β T |
= aT T . |
(22) |
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Several particular casers of Eqs (20) and (22) corresponding to the different choise of α and β were analyzed by Povstenko (2011) (see also Compte and Metzler (1997)) and were used in construction of the corresponding theories of thermoelasticity (Yussef (2010), Sherief et al. (2010), Povstenko (2011)). Analysis of space-fractional generalizations of the Fourier law and the heat conduction equation as well as of the thermoelasticity theories based on these equations can be found in Povstenko (2009a, 2009b, 2012, 2013).
2.Time-fractional Advection-diffusion Equation
Consider the following constitutive equation for the matter flux (see, for example, Kaviany (1995)):
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j = −a grad c + vc, |
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where v is the velocity vector. |
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In combination with the balance equation for mass we get |
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∂c |
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(24) |
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∂c |
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c − (div v) c − v · grad c. |
(25) |
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Supposing v = const (or div v = 0 for generality), we obtain the standard advectiondiffusion equation
∂c
∂t = a c − v · grad c (26) which has several physical interpretations in terms of Brownian motion, diffusion or heat transport with external force or with additional velocity field, diffusion of charge in the electrical field on comb structure, transport processes in porous media, groundwater hydrology, etc. (Bird et al. (2002), Feller (1971), Kaviany (1995), Nield and Bejan (2006), Risken (1989), Scheidegger (1974), Van Kampen (1981)).
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In the case of one spatial coordinate x, equation (26) has the following form |
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∂c |
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∂2c |
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∂c |
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= a |
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− v |
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(27) |
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∂t |
∂x2 |
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The constitutive equation (23) as well as the advection-diffusion equation (26) can be generalized like the Fick law (3) and the standard diffusion equation (4) similarly to equations considered in the previous section. We will restrict our consideration to the case
v = const. For example, the constitutive equation |
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Z t |
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j(t) = |
K (t − τ ) [−a grad c(τ ) + vc(τ )] dτ |
(28) |
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leads to the advection-diffusion equation with the general memory kernel:
Z
∂c t
∂t = 0 K (t − τ ) [a c(τ ) − v · grad c(τ )] dτ. (29)
Equations describing the ballistic advection-diffusion have the following form:
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j (t) = |
[−a grad c(τ ) + vc(τ )] dτ , |
(30) |
0
and
∂2c
∂t2 = a c − v · grad c. (31) The time-nonlocal constitutive equation with the “short-tail” exponential kernel
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Z0 |
− ζ |
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t − τ |
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j (t) = |
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exp |
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[ a grad c(τ ) + vc(τ )] dτ |
(32) |
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results in the telegraph advection-diffusion equation
∂c |
+ ζ |
∂2c |
= a c − v · grad c. |
(33) |
∂t |
∂t2 |
The time-nonlocal constitutive equations for the matter flux with the “long-tail” power kernel
j (t) = D1−α [−agrad c(t) + vc(t)] , |
0 < α ≤ 1, |
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j (t) = I α−1 [−agrad c(t) + vc(t)] , |
1 < α ≤ 2, |
(35) |
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give the time-fractional advection-diffusion equation |
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∂αc |
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= a c − v · grad c, |
0 < α ≤ 2. |
(36) |
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∂tα |
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Time-fractional generalizations of the advection-diffusion equation were considered by many authors (Barkai (2001), Barkai et al. (2000), Huang and Cao (2013), Huang and Liu (2005), Jumarie (1992), Karatay and Bayramoglu (2012), Liu et al. (2003), Metzler et al. (1999), Zheng and Wei (2010)). In majority of the abovementioned papers, the
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