ML

:i 7 SEP'BS

REFE i O l l IC/89/171

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

SELF SIMILAR SOLUTIONS OF THE SECOND KIND

OF NONLINEAR DIFFUSION-TYPE EQUATIONS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONS

EDUCATIONALSCIENTIFIC

AND CULTURALORGANIZATION

Javier Albert Diez

Julio Gratton

and

Fernando Minotti

1989 MIRAMARE- TRIESTE

1C/89/17I

International Atomic Energy Agency

and

United Nalions Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SELF SIMILAR SOLUTIONS OF THE SECOND KIND

OF NONLINEAR DIFFUSION-TYPE EQUATIONS -

Javier Albert Diez **

Departamento de Fisica, Facultad de Ciencias Exactas,

Universidad Nacional del Cemro de ta Provincia de Buenos Aires,

Pinto 390, 7000 Tandil, Argentina

Julio Gratton "*

International Centre for Theoretical Physics, Trieste, Italy

and

Fernando Minotti ™

LFP, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,

Pabellon I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.

MIRAMARE- TRIESTE

July 1989

* Submitted for publication."* Fellow of the Consejo Nacional de Investigaciones Cientificas y Tecnicas.'"* Researcher of the Consejo Nacional de Investigaciones Cientificas y Tecnicas.

Permanent address: LFP, Facultad de Ciencias Exactas y Naturales, Universidud dc Buenos Aires,

Pabellon I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.

AbslracL-

We study the self similar solutions of the problem of one-ditnensional nonlinear diffusion of apassive scalar u [ciiffusiviiy P — u m , m S 11 towards the centre of a cylindrical or spherical sym-metry. It is shown thiil this problem has a self similar solution of the second kind. The self simi-larity exponent S is found by solving a nonlinear eigenvalue problem arising from the requirementthat the integral curve that represents the solution must join the appropriate singular points in thephase plune of the diffusion equation. In this way the integral curves that describe the solution "be-fore, and after the diffusive current arrives at the centre of symmetry can be determined. Theeigenvalues for different values of the nonlinearity index m and for cylindrical and sphericalgeometry are computed. Numerical integration of the equations allows to determine the shape ofthe solution in terms of the physical variables. The application to the case m = 3, corresponding| for cylindrical symmetry] to the creeping gravity currents of a very viscous liquid is worked out indetail.

t-

1. Introduction

In this paper we shall be concerned with nonlinear diffusion equations of (he type1

(1)

in which the diffusivity D - u m \ m ^ 11. This equation is of considerable importance in malhe-malical physics, as it governs a variety of phenomena of different kind. We mention the followingexamples:(1) Flow in thin saturated regions in porous media in itie Dupuit-Forchheimer approximation [m =

1; see for example Polubarinova Kochina, 1962, Eagleson, 1970. Peletier, 19811.(2) Percolation of a compressible polylropic gas in a porous medium [ m = y- cpicv > 1; Muskat,

1937, Gilding and Peletier, 1976, Vasquez, 1983].(3) Heat conduction by electrons in a plasma [ m - 5/2, Zel'dovich and Raizer, 1966|.(4) Viscous gravity currents | m = 3, sec Buckmaster, 1977, Huppert 1982, Gralton and Minotli

19R91.(5) Heat conduction by radiation in a multiply ionized gas | m = 4.5-5.5, Zel'dovich and Raizer,

1966, Pert, 1977|.(6) Heat conduction by radiation in a fully ionized gas | Marshak waves, m = 13/2, see Marshak,

1958, Zel'dovich and Raizer, 1966, Larsen and Pomraning, 1980).

We shall consider only one dimensional problems, so that eq. (1) takes the form

du .„ d , 3H ,(2)

in which x denotes the spatial coordinate, and n = 0, 1, 2 for plane, cylindrical, and sphericalsymmetry, respectively.

As is well known, eq. (2) admits a self similar solution if the problem at hand involves no morethan one parameter, say b, with independent dimensions, [ b ] = L T - $. The numerical constant Sis the self similarity exponent. The similarity solutions of nonlinear diffusion-type equations havebeen studied by many authors, including Barenblatt [1954], Barenblatt and Zel'dovich [19591,Pattle [1959J, etc., and more recently by Pert 11977], Grundy [1979], Huppert [1982], andGratton and Minotti [19891, where more references can be found. With few exceptions all the so-lutions discussed in the literature belong to what is called self similarity of the first kind, in whichthe exponent & is determined by dimensional analysis in terms of the constant dimensional param-eters of the problem.

Among the various self similar solutions of eq. (2), we shall investigate here those corre-sponding to a particular class of problems that lead to self similar solutions of the second kind. Inthese cases the solution cannot be found by means of dimensional analysis, nor by the applicationof conservation laws [see Zel'dovich and Raizer, 1966, and Barenblatt, 1979, in which these mat-ters are discussed in detail[. The self similar solutions of the second kind can be obtained by any ofthe following procedures: (a) by starting from an adequate non self similar problem, and followingits solution (either numerically, or in an experimentl unlit the self similar asymptotic* is ap-proached, (b) by direct construction, requiring the existence at large of the desired self similar so-lution of equation (2). In the last case, one is led to a nonlinear eigenvalue problem that allows todetermine the solution t. This is the approach that shall be followed in the present paper.

A particular example of this type of self similar solutions of the second kind was first obtainedby Grallon and Minolti I 11J79] in the course of an investigation on viscous gravity currents, and

actually the solution is obtained only within some numerical factor that can be fixed only byexperiments, or by starting from the non self similar problem and following numerically theevolution of the solution until the self similar asymptotic;; is reached.

2

corresponds to m = 3, and n = 1 [axial symmetry] in (2). We shall take this example as the typecase, in order to keep the discussion as concrete as possible. I lowever, it must be kept in mind thatthe results of this paper are of a general nature, and can be applied, mutatis mutandis, to any of thephenomena described above, and also to spherical symmetry [that for viscous gravity currentscannot be realizedl. Accordingly, in what follows the term " current" must be understixxl in a gen-eralized sense, i. e., as meaning any of the phenomena described by cq. (1), if not otherwise indi-cated. To our best knowledge the family of self similar solutions of the second kind representingconverging diffusive currents had so far escaped detection |with the above mentioned exception].Clearly its investigation is of considerable theoretical interest in view of the wide range of applica-tions.

2. The collapse of a converging current : self similarity of the second kind

The type case we shall discuss is that of an axisymmetric gravity current of a very viscous liq-uid over a rigid horizontal surface, that converges towards the origin. This type of current can beproduced, for example, if there is initially a pool of liquid surrounding a circular wall; the liquidextends to infinity on the outside of the wall, while the circular inner pan of the supporting surfaceis dry. If at a certain moment the retaining wall is removed, a gravity current towards the centre[the origin] will ensue. This current will have a converging front, whose radius will decrease withthe passing of time, so that the front will ultimately collapse at the origin.

Using the lubrication approximation [for discussions of this model, in which it is assumed anearly horizontal motion governed by a balance between the forces due to gravity and those due toviscosity, and inertia is neglected, see Buckmaster, 1977, Huppert, 1982, also Gratton andMinotti. 1989[ the flow is described by eq. (2), in which u = (gl 3v )^H [ H (x,t) denotes thethickness of the current, g is the acceleration of gravity, v is the cinematic viscosity coefficient),and m = 3, n = 1 as already said.

We shall be interested in the behaviour of the current near the instant of collapse, which weshall assume occurs at ( = 0 [thus I < 0 correspond to times previous to the collapse, when theconverging front is approaching the origin, and t > 0 to times after the collapse, when the currenthas covered the initially dry region, but there is still flow tending to increase the thickness of theliquid in the central part. We shall only be interested in the properties of the current near the front,or equivalcntly, in its central part, i. e., for radii negligibly small as compared with any parameterthat ch;iracterb.es the initial conditions, such as the radius of the circular wall that contained the liq-uid. Clearly, in this region near the front, the thickness of the current will be also negligibly smallin comparison to the initial depth of the pool outside the circular wall.

In this situation we are left with no constant dimensional governing parameter: those arisingfrom the initial conditions are no longer adequate as a scale of the properties of the current in theregion of interest, and the characteristic parameters of the flow are functions of time. Therefore theflow is self similar, but the self similarity exponent cannot be determined by simple dimensionalanalysis jas it is not possible by these means to determine the parameter b ]. We are in the presenceof self similarity of the second kind.

The generalization of the present arguments to other types of converging diffusive currentscharacterized by different values of m and n can be easily obtained by analogy. Some exampleswill be discussed further on. It can be observed that these problems are strongly reminiscent of thatof a converging shock wave in gas dynamics [see Guderley, 1942, also Zel'dovich and Raizer,1966],which also yields a self similar asymptotics of the second kind under conditions of the sametype as those discussed above.

i. Basic c<|iiiitions and phase plane formulism

A convenient method to find the self similar solutions of eq. (2) is based on a phase plane for-malism, us discussed by Gratton and Minotti [1989]. In (his reference, only the case m = 3, n =

0,1 has been discussed in detail. In view of [he applications to the phenomena mentioned in theIntroduction, we shall extend the analysis to arbitrary values of m [ > 0 1 and to any symmetry.

We introduce a new dependent variable, v, as

v = - u "•-' =— (3)

I in the case of the viscous gravity currents, v represents the vertically averaged flow velocity |.With this definition the equation (2) is then equivalent to the following system of first order partialdifferential equations:

I™ +v = 0 . (^ + ^-{uv) + nm = oox ot ox x

This system admits self similar solutions of the form

(4)

(5)

where V, Z are usually called phase variables. Substituting (5) in (4) one obtains

f Z ' + 2 Z + m V = 0 , m(ZV-£Z'{S-V) + (pV - 1 ) 2 = 0

where the primes denote derivatives with respect to f, and

p = 2 + m ( n + 1 ) (7)

As the independent variable appears in (6) only through logarithmic derivatives, it is possible toreduce this system to an autonomous first order differential equation for V (Z)

+m(8 -V )V

dl m Z ( 2Z + m V )

Once (8) is solved, a simple quadrature allows to compute f:

JL in in = 12Z

(8)

(9)

Therefore, for any given 5, the solution of a self similar problem is essentially reduced to the inte-gration [usually numericall of the autonomous equation (8). Once V (Z) has been found one candetermine £(Z) using (9), and finally Z ( f ) and V(f) by inversion.

The plane ( Z, V ) is usually culled the phase plane. A solution of (8) is represented by an inte-gral curve in the phase plane. A single integral curve passes through any regular point of the phaseplane. Any integral curve represents a self similar solution of some kind. The solution of a givenself similar problem characterized by certain particular boundary conditions is represented in thephase plane by one or more pieces | adequately joined | of the appropriate integral curves, that sat-isfy at their ends the boundary conditions. Any piece represents the solution in a certain domain ofthe independent variable.

To ascertain which integral curve corresponds to the problem under study |i. e., to the giveninitial and boundary conditions] it is essential to know the behaviour of the solutions in the neigh-bourhood of the singular points of (8). A complete discussion of the singular points has been doneby Gratton and Minotti | I9K9| for m = 3, and is there presented in detail. We have here extendedthe analysis for arbitrary m , and the main results are summarized in the Appendix. There we givea list of the singular points of (8), their properties, the asymptotic behaviour of the solutions, and

the corresponding physical interpretation. A collection of examples, that show how to construct theexplicit self similar solutions for various problems^an be found in the above mentioned reference.

4. Construction of the self similar solutions for converging currents

It was shown in Section 2 that the solutions of (2) representing converging currents in cylindri-cal and spherical symmetry have a self similar intermediate asymptotics of the second kind. Weshall now determine this asymptotics by direct construction, i. e., starting from (R) and 0), andrequiring the existence of the solution at large.

Let us first consider the solution before collapse | f < 0 ]. For negative time, the integral curveof interest must lie in the Z < 0 half plane, since w must be a positive quantity (see the first of eqs.(5)|. Clearly, the solution we are seeking must have an moving front at a finite distance x/ of theorigin. Then, the corresponding integral curve must start at the singular point A \Z\ = 0, VA = S,see the Appendix] that represents a moving front. Since A is a saddle, there is a single integralcurve passing through it | beside the curve Z =0, which represents a trivial uninteresting solution].The trajectory leaving A must end at some other singular point, and for Z < 0, the only candidatesare B |2j, = - ml 20, VB = 1/0 ], O [Zo = 0, VO = 0 ], or C [Zc = 0, Vc = ± » ]; the properties ofthe solutions near these singular points are given in the Appendix.

We shall now show that only a trajectory joining A with O has the desired properties. In fact, itcan be verified that an integral curve going from A to B |or to C ) represents a current that for -( -»0, blows up | H, v -»<x> 1 or vanishes [ u, v -+ 0 ] at a finite distance from the origin. Then tra-jectories of these types cannot represent the required solution. On the other hand, an integral curvejoining A with O represents a current with the correct properties \u, v finite and non zero at anyfinite distance behind the front for -t -» 0|. Then our solution must be represented by a trajectorygoing from A to O .

Now, it must be observed (hat for curves coming from Z < 0, O is a saddle; in other words,there is a single integral curve in this half plane lhat arrives at O. All other curves in the neighbour-hood must either go to B, or to C . Then, for arbitrary S there is no integral curve joining A withO . Such a curve exists only if 5 has a particular value, 5 = SQ. This critical value must be foundnumerically |see below].

We then see that the requirement of the existence of a self similar solution leads to an eigenvalueproblem for the self similar exponent, as is characteristic of self similarity of the second kind.

After solving the eigenvalue problem it is possible to find the solution for f > 0, i. e., after thecollapse. It obviously corresponds to S = $ , it must represent a current having zero flow [flow -x»uv ) and a finite value of it for f = 0, and an incoming flow far from the origin of coordinates.Since t > 0, the integral curve must lie in the Z > 0 half plane. It can be verified that theappropriate integral curve starts at the singular point D [representing the origin of coordinates; Zp= », V'D = (I - 2S)lm (n + 1)] and ends at O [that represents f -+ » | ; D is a saddle, and there isa single integral curve leaving it that arrives at the Finite in the phase plane; we omit details forbrevity.

5 . Determination of the eigenvalue, integral curves, and properties of thesolutions

We shall show first that the eigenvalue lies in the interval 1/2 < 6c < 1. To this end we considerthe behaviour of the solution for r -> <*>. According to the results reported in the Appendix, near Oone has, approximately

- \)/mS - 1)13 (10)

Therefore, since v must vanish for x -> <*> | i . e. the liquid must be at rest, or the heat flow mustvanish, etc., according to the kind of phenomenon we are considering], we must have S^ < 1.Furthermore, u can noi decrease for x —> °=, as this would mean that we have there another front,

or a sink, [of mass, or of heat], so that 1/2 < ^ . This lower bound can also be demonstrated ob-serving that for f > 0, and near the origin |lhis corresponds to the neighbourhood of the singularpoint D, see the Appendix|, it must be an increasing function of time.

The eigenvalue can be found by a simple (rial and error method. A trial value S] is assumed,and eii. (8) is integrated numerically. Since il is not possible to integrate through the whole intervalof interest, the following technique is employed lo find 8^: the integration is started both from O,and from A, and continued on the two sides until the turves arrive, respectively, at the points Zo.iand Z^ i where they have a vertical slope. In general 7-o,\ and Z&\ will not coincide since ^ * fi[.Next, a second attempt is made, based on a new guess §2, and new values ZQ 2 ; ind f-A,2 are de-termined. Then, using Zo\ and 7-A \ | i = 1,2] one interpolates [or extrapolates | linearly to obtain anew guess £3 that will be a better approximation to S^. This procedure can be repeated as manytimes as necessary to obtain the value of ^ with the desired precision. The details of the computa-tions can be found in Diez et al. 11989]. In Table I and Fig. 1 we give the values of 5c for differentvalues of the geometrical index n, and the nonlinearity index m.

The asymptotic formula for the integral curve near the point A [the front], used to start the inlegration, is

V = S + ps-\ -Z + ...m(m+ 1)8

The corresponding solution near the front is given by

u=[( -mSxf/t )(1 - TJ)11/m

1m (m+ 1)5

(H)

(12)

and

v = S(xf It (13)

where r\ -xtxf and xf = K (l)S,K = const.. From these formulae it can be seen that as -t0 the front accelerates, us speed tending to infinity al the moment of collapse.

All the integral curves that arrive al O are given by

mS \ mS2!*l-3 + Az

f

for Z > 0 there is an infinity of integral curves arriving at O ; for Z < 0, as already said, there isonly one such line.

The integral curve that represents the solution after the collapse goes from D to O, and can beobtained [once <̂ has been found] by numerical integration starting from D. Near D this line isgiven approximately by

V =-• 1 -ps -1

m ( n + 1) [ m ( n + ! ) ( « +3 ) Z(15)

It can be noticed that the self similarity variable £ is determined within a constant numericalfactor, since the eigenvalue ^ fixes the dimensionality of b, but not its numerical value. This nu-merical value can only be obtained by actually performing an experiment, or by means of a nu-merical simulation of the complete non self similar problem, starting from adequate initial condi-tions, and following the calculations until the self similar intermediate asymptotics is attained [secBarcnhlatt, I'J79|.

The profiles of it, v before and after collapse are represented in the Figs. 2, 3, and 4, 5, respec-tively, lor HI =3, and n = 1,2, as functions of TJ = xlxf. In the problem of the circular wall \m =3,n = 1!, the solution represents a viscous current whose height and velocity are proportional to uand v , respectively. For oilier values of the nonlinearity index the shape of the profiles is similar,and only the numerical values are changed. In this way, the profiles of thickness and velocity of aliquid layer that flows in a saturated porous medium [m =1, n = 1| are similar to those representedin Pigs. 2-5.

The spherically symmetric case [n = 2) is of course meaningless in the above mentioned prob-lems that deal with essentially two-dimensional currents, but is of interest for three-dimensionalphenomena like the percolation of a polytropic gas in a porous medium \m = yi\\, and the non-linear heat conduction |either by radiation or by electrons!. In the first instance the physical prob-lem is that of an infinite porous medium in which there is initially a spherical |or a cylindrical! re-gion devoid of gas, which is being filled by the percolation of the gas from the surrounding re-gions. In the second case our solutions describe the asymptotics corresponding to an initial condi-tion in which there is a spherical [or cylindrical] cold region, surrounded by a hot [temperaturelarger than, say, 104 °K | region that extends to infinity; in this situation a converging thermal wavewill travel into the inner cold region. It can be observed that in this problem one can neglect themotion of the medium [partially ionized gas, or plasma] in which the heat propagates. In effect, letus consider, for simplicity, a gaseous medium: then a compression wave (or a shock wave] will beformed at the boundary between the hot and the cold regions, and will travel into the cold unper-turbed medium with a velocity of the order of the speed of sound in the hot region. The thermalwave has initially a velocity of the order of dxji dl - V(D/f), where D ~ iJi" | t9 = temperature].Then xf ~ \V"i 2 for f finite and non zero. On the other hand, the speed of sound in a high tem-perature gas is roughly proportional to i)"2. Consequently, if the temperature is very high, theinitial velocity of propagation of the thermal wave will be much larger than the speed of sound. Inaddition as the moment of collapse is approachedl -r —* 0], the velocity of the front of the thermalwave tends to infinity as (-J )o*l. Then, the propagation of the thermal front takes place in amedium ar rest.

Let us now discuss briefly the behaviour of the solution after the collapse, illustrated by theprofiles of Tigs. 4-5. It can be seen that near the centre of symmetry u - 1 (2<5- O/m ; of coursethis behaviour is limited to a short interval of time after the collapse; in fact, u cannot increase in-definitely, but for large t > 0 must eventually approach a constant value. It is also interesting loobserve that (a): as ( passes through zero, v [the flow velocity, in the case of the viscous current]which before collapse is very large near the centre, becomes very small later on [for reasons ofsymmetry v = 0 at the centre], and (b): for any fixed r, as x is varied from large values to the ori-gin, v increases at first, passes through a maximum at a certain locus S at a radius xs, and thendecreases to zero at the centre. The locus of maximum v moves outwards with time according toxs - 1 s . This is due to the nonlinearity of the diffusion equation (1), one of whose effects is thatperturbations are propagated with a finite speed. The locus 5 carries outwards the information thatthe current has reached the centre of symmetry, so that the rate of flow [proportional to v ] mustdiminish. The results obtained for different values of m and n show that the value of d dependson m but not on n .

6. Final remarks

We have studied one dimensional currents of the nonlinear diffusive type in the last stages ofconvergence towards the centre of [cylindrical and spherical| symmetry, and in the first stages after

the collupse of llie current from. General values of llic nonlincarily index are considered. For con-crclcness, the convergent viscous gravity current is taken as the type case and is discussed in de-tail. The analysis of the governing equations and parameters shows that the stages of interest fol-low a self similar intermediate asyniptotics of the second kind. This important type of self similar-ity is well known in other problems of mathematical physics, but until now the self similarasyniptotics of the second kind ot convergent nonlinear diffusive currents has not been discussedin the literature.Thc fact that the nonlinear diffusion equation describes a large number of interest-ing phenomena, as those mentioned in the Introduction, adds to the relevance of the results wehave derived.In Ihe present paper we have put a certain emphasis on the solutions corresponding to the viscousgravity currents \m =3, n = 11, since the theoretical results here obtained can be verified by meansof simple experiments, that can be carried out with modest technical resources. In fact, as acontinuation of this research, we are presently engaged in a series of experiments in which a con-vergent gravity current of a very viscous liquid is set up in a circular tank. Some very preliminaryresults of this work have already been obtained, that tend to validate the present theory, within ihelimitations of its approximations |Diez, 1989J.

Acknowledgements

One of the authors |J. G.| thanks Prof. Abdus Salam and the International Centre for Theoreti-cal Physics for their assistance. We also acknowledge grants of the Organization of AmericanSlates, the Consejo Nacional de Invesiigaciones Cieniificas y Tecnicas, the Universidad of BuenosAires and the Universidad del Centra de la Provincia de Buenos Aires at Tandil.

Appendix

We summarize here some relevant properties of the singular points of eq.(!i), (heir physical in-terpretation, and the behaviour of the solution in (heir vicinity. The present results are an extensionto arbitrary values of the nonlinearity index m of those reported in Gratton and Minotti [ 1989| forthe special case m =3. Other details of this particular case can be found in this reference.

The six singular points of (8) are denoted by O, A, B, C, D,E \K denotes a numerical con-stant. The flow is defined as F = nx"uv[n = \.2rr,4x for n = 0,1,2 respectively]. We also useihe following notations

6t = So±X . So = \ll + 2(m + l ) j / 2 / J (A l )

I = Ify2 - \(J) +2)(4m +p + 2) + 4/n (m + 2p )] /4/J z ) " 2 (A2)

We examine below Ihe different singular points:

I. O\Zo=0.Vo=0]

1.1. For 5=0,0 is a cusp; it represents a front at a fixed position x = xr. One has

u ~[-(x-x/)2/t]]'m , v~(x-xf)lt (A3)

1.2. For S * 0, O is a saddle-node [saddle for the curves arriving from Z < 0, node for ihecurves arriving from Z > 0]; it represents x = °°. Two cases must be distinguished:

1.2.1. 5*\/2:F # 0 atx = » , a n d :

M - x (25 - ])/ mS i v _ x (S - iy 5

notice that u and v are independent off,

1.2.2. S = \/2:F=0 a t* = » , a n d :

u ~const. , v ~(xl t ) exp ( - K fi)

Point A is a saddle, representing a moving front (xy. - f ^j with:

" m v ~ xlt

(A4)

(A5)

<A6)

3 . H\ZB = - m / 2 p , V B pThe nature of this singular point varies with S:(a) for 0 < 5 < S. is a node.(b) for £ < S< S+ is a focus.(c) for 5+ < S is a node.The physical interpretation also varies wilh 5:

3.1. l:or5<S(),B represents x = « , and F * 0 .

3.2. For S> S»,S represents x = 0. In both cases one has:

9

;» • •*« V * <*f-*me

u - ( - JC.2/ r ) " m , v ~ jf / /

, Point C is a node, representing a moving sink [Xf. - I * ] with:

*•') , v - t s -'

5. D \Z[) = °°,Vi} = (1 -25)!m (n + 1)|Point D is a saddle, it represents x - 0 with /•" = 0 . ; near D :

u _ , (25 - l)/m _ v xi t

notice that u is independent of x |for Jt near the origin)

(A7)

(AK)

(A9)

6. E \7.E = •», VE = « jThis singular point is a saddle-node, it represents x = 0. Two cases can be distinguished:

6.1. n =0|p!ane symmetry]; two types of behaviour are possible:

6.1.1, jr-dependent behaviour, only F > 0:

u - ^ 1/4 , 1(8 -m)5 -41/4m , v ~xml4- 1 ( (8-m).5/4m (A 10)

6.1.2. jt-independeni:

„ _ , (25 - l)/m , » ~ , M (Al l )

6.2. n *() ;only/• ' >0, and:

„ _ , ( 2 5 - l ) / m | l n C | 1/4 _ v „ , 2S-lx-l | l n f | - l /4 ( A 1 2 )

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Math. Gen. 10(4), 583.Pert, G. J., 1977, A class of similar solutions of the non-linear diffusion equation, J. Phys. A:

Math. Gen. 10, 583-593.Potubarinova-Kochina, P. Y., 1962, Theory of Ground Water Movement. Princeton Univ. Press.Vast|iie7. J. L., 1983, Free Boundary Problems: Theory and Applications, Vol I. Boston: Pitman

Adv. Pub. Prog.Zel'dovich Ya. B., and Yu. P. Raizer, 1966, Physics of Shock Waves and High Temperature

llydrodynamic Phenomena. New York: Academic Press.

1(1

Table I

Self similar exponents S^ for different values of the nonlinearily index m andgeometrical index n [ n = 1 for cylindrical, n = 2 for spherical geometry |.

ni

1

2

3

4

5

6

7

n = l

0.856554

0.796439

0.762116

0.739577

0.722874

0.710324

0.700152

n = 2

0.771452

0.698553

0.661124

0.637777

0.622229

0.609715

0.600678

12

1.0

0.8

0.6

0.4-

0.2-

0.0

n

n

= 1——.

= 2• — — _

" - —

— —

•

—

m

rigure 1: Self similar exponenis [eigenvalues] as function of the nonlinearityindex HI, for cylindrical [n = 11 and spherical [ n = 2] symmetry.

13

T

\(xlx f)2 ~ , 1/m

/ ]

1.5-

1.0-

0 .5 -

0.0-

n = 2

,

0 1 2 3 4

Xl Xf

Figure 2: Solutions before collapse \l < 0, m = 3 | . Profiles of H.

(x/xf)V

0.8

0 . 6 - - -

0.4-

0.2 -

0.0

\

n = 2

0 1 3 4

xl Xf

I-'igurc 3: Solutions before collapse [f < 0, m = 3j. Profiles of \v I.

1.5

1.4

1 . 3 - -

1.2-

1.1

1.0

0.9

A//2 4 6 8 10

c

Figure 4: Solutions after collapse [t < 0, m = 3|. Profiles of u ,

16

0.12

0.10

0.08 -

0.06 -

0.04

0.02 -

0.00

/

/

/ /

f—-̂.n = 2

^ - - — ."~——^

6 8 10

Figure 5: Solutions after collapse |/ <0, m = 3J. Profiles of Iv I.

17

T •

Stampato in proprio nella tipografia

del Centro Tnt.ernazionale di Fisica Teorica