Self-similar solutions for classical heat-conduction...

Self-similar solutions forclassical heat-conduction

mechanisms

Imre Ferenc Barna & Robert Kersner

1) KFKI Atomic Energy Research Institute of the Hungarian Academy of Sciences

2) University of Pécs, PMMK Department of Mathematics and Informatics

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Outline

• Motivation (infinite propagation speed with the diffusion/heat equation)

• A way-out (Cattaneo equ. OR using a hyperbolic first order PDE system

• Derivation of a self-similar telegraph-type equation & analysing the properties

• Non-continuous solutions for thehyperbolic system for heat propagation

• Summary

Ordinary diffusion/heat conduction equation

U(x,t) temperature distribution

Fourier law + conservation law

• parabolic PDA, no time-reversal sym.

• strong maximum principle ~ solution is smeared out in time

• the fundamental solution:

• general solution is:

• kernel is non compact = inf. prop. speed

• Problem from a long time ��

• But have self-similar solution ☺☺☺☺

Important kind of PDA solutions

- Travelling waves: arbitrary wave fronts

u(x,t) ~ g(x-ct), g(x+ct)

- Self-similar solutions

Sedov, Barenblatt, Zeldovich

Cattaneo heat conductionequ.

Cattaneo heat conduction law, new term

Energy conservation law

T(x,t) temperature distributionq heat flux k effective heat conductivityheat capacity relaxation time

Telegraph equation(exists in Edyn., Hydrodyn.)

General properties of the telegraph eq. solution

decaying travelling waves

Bessel function

Problem: 1) no self-similar diffusivesolutions

2) oscillations, T<0 ? maybe not the best eq.

Our alternatives

• Way 1

• Def. new kind of Cattaneo law (with physical background)

new telegraph-type equation

with self-similar and compact solutions ☺

• Way 2

instead of a 2nd order parabolic(?) PDA

use a first order hyperbolic PDA system with 2 Eqs.

these are not equivalent!!!

non-continuous solutions and also self-similar

General derivation forheat conduction law

(Way 1)

the kernel can have microscopic interpretation

Cattaneo heat conduction law,there is a general way to derive

T(x,t) temperature distributionq heat flux

telegraph-type time dependent

eq. with self sim. solution

Solutions

There are two different solutions:

a total difference = conserved quantity

There are differential eqs. for only foror for

physically relevant solution, compact supportwith vanishing derivatives at the boardersI.F. Barna and R. Kersner, http://arxiv.org/abs/1002.099J. Phys. A: Math. Theor. 43, (2010) 375210

Not so nice ��

2 Important new feature: the solution is a product of 2 travelling wavefrontsno flux conservation problem

Solutions

• where F(a,b;c;z) is the hypergeometric function

• some elementary functions can be expressed via F

• In our case if is Integer or Half-Integer are important

the 4 basic cases:

Solutions

not-so-nice solutions, non-compact no-finite derivatives just have a rich mathematical structure

I.F. Barna and R. Kersner http://arxiv.org/abs/1009.6085

Adv. Studies Theor. Phys. 5, (2011) 193

Self-similar, non-continousshock-wave behaviour for

heat-propagation (Way 2)

general Cattaneo heat conduction law,+ cylindrically symmetric conservation law

heat conduction coefficient temperature dependent (e.q. plasmas)relaxation time also temperature dependent (e.q. plasmas)

using the first oder PDA system (not second order)looking for self-similar solutions in the form

Parameters are fixed, coupled sytem of ODEbut Eq. 2. can be integrated

only one ODE

Properties of the model

originaly there are 5 independent parameters, exponentsonly one remained independent , we fixed omega

the parameter dependence of the solutions is now dictated

heat conduction and relaxation time terms are also known

Properties of the solution

first order non-linear ODE (no analytic solution) BUT-Variable transformations, and considering the inverse of the first derivative linear inhomogeneous ODE

can be integrated

general solution of the homogeneous equation times the particular solution of theinhomogeneous one, there is only one parameter dependence

Properties of the inversesolution

it is not singular for so for

different means different kind of solutions

Non-continous solutions of PDE

applying the back-transformation (inversion + square root)

wewe maymay definedefine zerozero solutionssolutions outsideoutside thisthis etaeta domaindomain

principalprincipal valuevalue

havehave toto be fixedbe fixed!!!!

we presented the problem of the heat conduction eq. defined two possible way-outs

As a new feature we presented a new telegraph-typeequation with self-similar solutionsIt has both parabolic and hyperbolic properties

As a second point we use a hyperbolic system to investigate heat conduction, can have non-continuoussolutions

Summary and Outlook

Questions, Remarks, Comments?…

Self-similar solutions for classical heat-conduction...

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