Numerical simulations of free convection about a … · In previous work [4], we investigated some mathematical questions, particularly ... In fact, for P−1/3, the physical solution

Computational Geosciences 7: 137–166, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.

Numerical simulations of free convection about a verticalflat plate embedded in a porous medium

Z. Belhachmi a, B. Brighi b, J.M. Sac-Epee a and K. Taous a

a Laboratoire Méthodes Mathématiques pour l’Analyse des Systèmes, Université de Metz, ISGMP,Batiment A, Ile du Saulcy, 57045 Metz-Cedex 01, France

E-mail: [email protected] Laboratoire de Mathématiques et Applications, Université de Haute-Alsace, 4 rue des Frères Lumière,

68093 Mulhouse, France

Received 21 November 2001; accepted 14 October 2002

We consider the free convection about a vertical flat plate embedded in a porous mediumwithin the framework of a boundary layer approximation. In some cases, similarity solutionsarise in the modelization of such phenomena, allowing a reduction of the dimension of theproblem. We suggest two complementary and rather simple numerical methods to computesuch solutions. When possible, the numerical analysis for the two discretizations is performedand convergence results are given. Numerical experiments with a physical model are presentedto confirm the efficiency of both approaches.

Keywords: boundary layer, boundary value problem, initial value problem, numerical simu-lations, porous medium, similarity solutions

AMS subject classification: 65L05, 80A20

1. Introduction

The free convection about a vertical impermeable surface embedded in a porousmedium belongs to a family of heat transfert phenomena which have a wide range of ap-plications in many geophysical and industrial fields (oil exploration, petroleum, chemicaland nuclear industries, . . .) [9,13,15,20]. The term “free convection” refers to motionsof fluids arising from the density differences due to temperature gradients. Various mod-elizations (system of partial differential equations) coupling the transport of heat throughporous media (thermal effects) and the motion of the fluid (based on Darcy’s laws) areavailable to describe such phenomena. Particularly interesting models lead to bound-ary layer approximation, and among them, those exhibiting some similarity solutionsfor which study and numerical computations are usually easier than the integration ofthe full system. This is actually the case of the model considered here, namely the freeconvection in a porous medium at high Rayleigh number. These problems are also par-ticularly interesting as far as pedagogy is concerned. In fact, different models and fields

138 Z. Belhachmi et al. / Numerical simulations of free convection

of research could lead to nearly similar governing equations: for example, the classi-cal boundary layer approximation of a steady two-dimensonal flow of slightly viscousincompressible fluid past a wedge at high Reynolds number [10,17,19].

The boundary layer approximation is relevant when the Rayleigh number Ra islarge, and it is well known that the thickness of the zone where this approximationtakes place is in order of R−1/2

a . Whence, solving the resulting system of PDEs in thisdomain (with high anisotropy) turns to be source of many numerical difficulties. Thesimilarity transformation reduces the integration of the system to that of an ordinarydifferential equation. The solutions obtained this way are called similarity solutions.The computation of such solutions is often simpler. Their accuracy is known to begood, far from the singularity generated by the transformation. Moreover, in engineeringproblems, they allow to obtain quickly, and with low cost, approximate results on somecrucial physical parameters (see section 5).

The determination of those similar solutions for the problem we consider amountsto solving the one-dimensional nonlinear boundary value problem Pα

f ′′′ + α + 1

2ff ′′ − αf ′2 = 0, on ]0,∞[, (1.1)

with the boundary conditions

f (0) = 0, f ′(0) = 1, (1.2)

f ′(∞) := limt→∞f ′(t) = 0, (1.3)

where α is a parameter. We will see in the next section how this problem is derived. Forphysical reasons, which will appear in the modelization, we also have to consider thefollowing constraint:

∀t ∈ [0,∞[ 0 � f ′(t) � 1. (1.4)

We will call “solution” any smooth function satisfying (1.1)–(1.3) and any solution sat-isfying the constraint (1.4) will be refered as a “physical” one.

There is an abundant engineering literature dealing with such models of freeconvection and, generally, heat transfer phenomena in porous media. However, mostcontributions consist in using ad-hoc methods to integrate numerically the equations[6–8,12,14,16]. Moreover, no systematic study is available, and only few special cases(few values of α) are considered.

In previous work [4], we investigated some mathematical questions, particularlyexistence and uniqueness of solutions to Pα . We summarize the main results of thismathematical analysis in the two following theorems:

Theorem 1.

• In the case α � − 12 , the problem (1.1)–(1.3) has no solution.

• For − 12 < α < − 1

3 , no solution is of physical type (i.e., do not satisfy (1.4)); more-over, such solution, if they exist, are unbounded.

Z. Belhachmi et al. / Numerical simulations of free convection 139

• For α � − 13 , there exist at least one physical solution. Moreover, this solution satis-

fies, for instance, the following estimates:

f (t) � 2√α + 1

. (1.5)

Note that the estimate (1.5) will be useful for computational purpose. With regardto uniqueness, a little is known.

Theorem 2.

• For α = − 13 there is an infinite number of solutions of the problem (1.1)–(1.3), and

among them, only one satisfies (1.4).

• For α � 0, uniqueness of physical solutions (problem (1.1)–(1.4)) holds.

• For α ∈ [0, 13 ], there exists one and only one solution. Moreover, this solution satis-

fies (1.4).

In addition, in the cases α = − 13 and α = 1, the physical solution is explicitly

known. In fact, for P−1/3, the physical solution is given by

f (t) = √6 tanh

(t√6

)(1.6)

and for P1, one has

f (t) = 1 − e−t . (1.7)

However, as far as we know, no other explicit solution is available except these twocases. Note that the case α = 0 is particularly interesting. It is known as Blasius equa-tion [5] and was abundantly studied in the field of Prandtl approximation to viscousincompressible fluid (with different boundary conditions than (1.2), (1.3) which con-sequently change the problem). In our case, the problem with α = 0 was essentiallyinvestigated from the mathematical point of view in [3].

The first numerical results about (1.1)–(1.4) for some values of α between − 13 and 1

can be found in [8]. Further numerical investigations for α close to − 12 and α > 1 are

in [1,2,12]. Other simulations for similar problem with f (0) �= 0 can be found in [14].We present two simple numerical methods to solve the problem Pα restricted to a

finite interval [0, T ].Our objective is twofold. First, the construction of an efficient tool which yields, for

any α, solutions which approximate the true ones on [0,+∞[ and which allow to obtainthe values of some physical parameters such as the thickness δ of the boundary layer,and f ′′(0) (note that f ′′(0) is linked to the convected energy for the model we consider,see section 5.3). This tool is a first step for computations in more realistic models (withmore physic) and complex situations requiring several parameters (see [13,20]). Notethat the comparison with earlier simulations and with exact solutions, when they areexplicitly known, is a major test for any numerical approach.


A second objective (more mathematic) is the aid to clear up some qualitative as-pects (such as uniqueness question, the role of the constraint (1.4), the existence ofsolutions for − 1

2 < α < − 13 , . . .) which are still obscure and essentially open questions.

Note that the parameter α comes from the distribution law of temperature on thewall, so that the range for which the model we study seems to be physically relevantis − 1

3 � α � 1 (for some thermicians only the range 0 � α � 1 is consistent). Oursimulations confirm this last opinion and give more details on the range α > 1. Conse-quently, the numerical experimentation presented here could help to test the consistencyof the model, leading to its improvment.

The first method is the well-known shooting method. It is easy to implement,and “works well” for all values of α, both to compute the solutions and the physicalones. However, the analysis of this method is, in general, not easy (i.e. convergenceresults, error analysis). Moreover, even if in our particular case, the situation seems quitesimple at first glance, (solving a one-dimensional problem), the results of the numericalexperiments are far from being well-understood and call for a careful interpretation.

The second method we present consists in a direct discretization of the problem Pα .This approach is constructed so as to compute only the physical solutions. The numer-ical analysis for this method is elementary but the main result of convergence is onlyavailable for α > 0. Nevertheless, computations are performed for more large intervalof α with a satisfactory accuracy.

An outline of the paper is as follows: in section 2, we will briefly recall the deriva-tion of the model. Section 3 will be devoted to the descripion of the shooting method:we will give the main algorithm and discuss the expected performances and limits ofthis approach. In section 4, we will introduce and analyze the second discretization,which consists in replacing the original nonlinear problem by a successive approxima-tions process, namely, in a linear system of second order (linear) boundary value prob-lems that we discretize by a standard finite differences method. Finally, we will assembleall numerical experiments in section 5 where we will discuss the results of the two meth-ods. We will also link the numerical experiments to a concrete geophysical model andgive some perspectives before concluding.

2. Derivation of the model

Let us consider the problem of steady free convection about a vertical impermeablesemi-infinite flat plate in a saturated porous medium, and consider a rectangular Carte-sian coordinate system with zero point fixed at the leading edge of the vertical surfacesuch that the x-axis is directed upwards along the wall and the y-axis is normal to it. Ifwe assume that, in particular, the convective fluid and the porous medium are everywherein the local thermodynamic equilibrium, the governing equations are given by

∂u

∂x+ ∂v

∂y= 0,


u= − k

µ

(∂p

∂x+ ρg

),

v = − k

µ

∂p

∂y,

u∂T

∂x+ v

∂T

∂y= λ

(∂2T

∂x2+ ∂2T

∂y2

),

ρ = ρ∞[1 − β(T − T∞)

],

where u and v are Darcy velocities in the x and y directions, ρ, µ and β are the density,viscosity and thermal expansion coefficient of the fluid, k is the permeability of thesaturated porous medium, and λ is the equivalent thermal diffusivity, p is the pressure,T – the temperature and g – the acceleration due to gravity. The subscript ∞ denotes avalue at infinity.

For our coordinate system, the boundary conditions at the wall are

v(x, 0) = 0, T (x, 0) = Tw(x) = T∞ + Axα, α ∈ R,

where A > 0 and the wall temperature is prescribed as a power function of distancefrom the zero point. The boundary conditions at a great distance from the wall are

u(x,∞) = 0 and T (x,∞) = T∞.

Introducing the stream function ψ as

u = ∂ψ

∂yand v = −∂ψ

∂x,

the previous equations are rewritten, after eliminating p, as

∂2ψ

∂x2+ ∂2ψ

∂y2= ρ∞βgk

µ

∂T

∂y,

λ

(∂2T

∂x2+ ∂2T

∂y2

)= ∂T

∂x

∂ψ

∂y− ∂T

∂y

∂ψ

∂x.

Hence, assuming that the convection takes place in a thin layer around the heating sur-face, we can use the boundary layer approximations similar to those of the classicaltheory of a boundary layer in a free viscous fluid [17]. So, with these simplificationswhich take into account the boundary layer thickness, we get

∂2ψ

∂y2= ρ∞βgk

µ

∂T

∂y, (2.1)

∂2T

∂y2= 1

λ

(∂T

∂x

∂ψ

∂y− ∂T

∂y

∂ψ

∂x

). (2.2)


The appropriate boundary conditions for the function ψ are given by

∂ψ

∂x(x, 0) = 0 and

∂ψ

∂y(x,∞) = 0.

We now define a local Rayleigh number as

Rax = ρ∞βgk(Tw(x) − T∞)x

µλ,

and introduce the following dimensionless similarity variables:

t = (Rax)1/2y

x, ψ(x, y) = λ(Rax)

1/2f (t), θ(t) = T (x, y) − T∞Tw(x) − T∞

.

Since the temperature is assumed to decrease from the wall, we have 0 � θ � 1. Interms of these variables, it is easy to see that the equations (2.1), (2.2) become

f ′′ − θ ′ = 0, (2.3)

θ ′′ + α + 1

2f θ ′ − αf ′θ = 0, (2.4)

with boundary conditions given by

f (0) = 0, θ(0) = 1,

and

f ′(∞) = 0, θ(∞) = 0, (2.5)

where the primes indicate differentiation with respect to η. Finally, integrating (2.3) andtaking into account the boundary conditions (2.5), we get

f ′ = θ. (2.6)

Using (2.4) and (2.6), we see that similarity solutions can be obtained by solving theboundary value problem (1.1)–(1.3) and that the constraint (1.4) derives from the rela-tion (2.6) and the definition of θ .

For more details on this model, see [9,13,15,20].

3. Shooting method

3.1. Description of the method

To apply this method, we will begin with considering a restriction of the originalproblem (1.1)–(1.3) to a finite interval [0, T ], where T is given. Condition (1.3) nowreads

f ′(T ) = 0. (3.1)


For this modified problem, the shooting method can be described in a standard way:it consists in the resolution of a Cauchy problem with initial data depending on someparameters, and in adjusting those parameters to achieve the prescribed boundary valuesof the original problem. More precisely, we consider the initial values problem Pα,(a,b,c)

f ′′′ + α + 1

2ff ′′ − αf ′2 = 0,

f (0) = a,

f ′(0) = b,

f ′′(0) = c,

where (a, b, c) are arbitrary real numbers. The aim is to seek values (a∗, b∗, c∗) so thatthe real function ϕ(a, b, c) = (f ′

(a,b,c))(T ) satisfies

ϕ(a∗, b∗, c∗) = (f ′(a∗,b∗,c∗))(T ) = 0, (3.2)

where f(a,b,c) is a solution of Pα,(a,b,c) which we will just denote by f . The original prob-lem corresponds to the particular case where a = 0, b = 1 so that only the parameter cis unknown.

Remark 3.1. Note that if f is a solution of the differential equation, then t �→ kf (kt)

is also a solution for any positive constant k. Therefore, Pα,(a,b,c) reduces to a two-parameter problem in the cases a = −1, a = 0, and a = 1. The case a > 0 correspondsto a suction and the case a < 0 to a lateral injection of the fluid through the wall assumedto be permeable [6,7,14]. In addition, taking b < 0 yields solutions which are far fromthe physical situation we intend to study here, and which arise in another model (Prandtlapproximation to incompressible Navier–Stokes equations [17]). To be close to the ini-tial model, we choose to consider Pα,(a,b,c) as a one-parameter problem (namely c) and,since for fixed a and b > 0, the structure of the set of solutions does not seem qualita-tively affected, we will stick to problem Pα,(0,1,c), with c � 0.

Remark 3.2. Observe also that we could choose ϕ as ϕ(a, b, c) = (f ′(a,b,c)(T ),

f ′′(a,b,c)(T )) and look for a zero of this function. It is easy to show that this definition

of ϕ takes into account the constraint

∀t ∈ [0, T [ 0 � f ′(t) � 1. (3.3)

We used this last definition numerically to improve the computation of the physical solu-tion, but the result is not convincing. In fact, we prefer to work without this constraint inorder to exhibit a general structure of solutions and to have a better understanding of therole played by (3.3) for uniqueness of solutions of the original problem (see section 5).

It is well known that when using the shooting method, one has to worry about twoquestions: is ϕ well defined as a function and does it satisfy (3.2)? Generally speaking,the answers to these questions are not obvious and depend on the particular problem onehas to solve.


The first difficulty in our case (with c < 0) is that the blow up of solutions to the(general Cauchy) problem Pα,(a,b,c) may occur. Indeed,

t �→ 6

t − τ,

is a solution of Pα,(−6/τ,−6/τ 2,−6/τ 3), which becomes infinite (with all its derivatives) at τ .As a result, the definition of ϕ is not always clear. However, given T0 > 0, and if we set

D = {c � 0; Tc � T0 and f ′(t) � 0, t ∈ [0, Tc[

},

where [0, Tc[ is the right maximal interval of existence of the solution to the problemPα,(0,1,c), we use the following process: when α = 0, we prove in [3] that D is an inter-val of the form [c∗, 0], with c∗ > −∞, which is due to the existence of a comparisonprinciple. When α �= 0, we do not know anything about the nature of set D. Never-theless, numerical simulations show that there is always an appropriate bounded intervalin D, where ϕ is well defined. Moreover, the smaller T0 is, the larger the interval.

The second question, which deals with the existence of solutions to equation (3.2)is usually less clear than the first one. In the (more difficult) case T = +∞, we give apositive answer to this question for α � − 1

3 . Note that, since the initial boundary valuesproblem has no solutions for the values α � − 1

2 , we are not interested in this interval.The case − 1

2 < α < − 13 is still an open problem. However, numerical evidences show

that the existence of solutions still holds.For both questions and for all α � − 1

2 , it follows from the computations that thefunction ϕ is well defined, smoother than expected by the general theory [11,18], andhas, at least, one zero in its domain of definition (see figures 1–5).

The practical algorithm for solving the original problem is classical. It can bedescribed as an alternate method which consists in generating a sequence (cn)n by aNewton-like method (secant method) where the values ϕ(cn) are updated by solving theproblem Pα,(0,1,cn). This is performed thanks to the fourth-order Runge–Kutta method.

Algorithm.

– initial step: take c0 and c1 arbitrarily in R

– step k: suppose known ck−1 and ck ,

• compute ϕ(ck−1) and ϕ(ck) by Runge-Kutta.

• compute ck+1:

ck+1 = ck − ck − ck−1

ϕ(ck) − ϕ(ck−1)ϕ(ck)

– repeat until convergence

The computations show that the generated sequence (ck)k is well defined at leastwhen the initial values c0, c1 are chosen not too far from the expected value c∗, thatit converges, and that convergence occurs quickly. Moreover, various numerical tests


are in accordance with a priori considerations and previuous results obtained in [8], forinstance. In fact, the algorithm allows us to compute with high accuracy the physicalyrelevant quantities: c∗, and the thickness δ of the boundary layer (see table 3).

4. Discretization by the finite differences method

The previous shooting method is very accurate, not costly, easy to implement andcertainly well adapted to our nonlinear problem. Moreover, it is well known that onecan improve it, for example, by using the multishoot variants. However, there are alsoknown drawbacks to this approach such as the dependency of the Newton-like part ofthe algorithm on the initial values, and the limitations in the choice of T imposed tohave a well-defined method. This last constraint is also of numerical nature, particularlywhen one uses an explicit method to integrate the differential equation. In addition,the numerical analysis is difficult (error estimates, convergence) since the function ϕ

is not known explicitly. Therefore, though the simulations give satisfactory results inaccordance with the physical interpretation and also with other numerical computations,some questions are not completely solved by this method. For instance, the convergenceof the algorithm to the “right” value c∗, when equation (3.2) has more than one solution,the dependency of the results on the choice of T or the role of the constraint (3.3) arenot well-understood in this framework.

More disconcerting is the response, after the convergence of the algorithm, to smallperturbations (see, for example, figure 4(a)). Indeed, solving the problem Pα,(0,1,c) withvalues c close to c∗ yields various solutions which are possible solutions of the originalboundary value problem, but which do not satisfy (3.3), or are unbounded. In the caseT = +∞, we call this type of solutions non physical ones and generally, we do notknow if they exist and, if so, for which values of α.

These observations lead us to introduce another approach which consists in a directmethod to approximate the solutions of the boundary value problem by a finite differ-ences scheme. The first important change in relation to the previous method is thatconstraint (3.3) is now included in the discretization. The idea is now to construct asequence of solutions to a linear system by a successive approximations process, whereinequalities in (3.3) are bounds acting as upper and subsolutions.

4.1. The successive approximations method

Given T > 0, let us denote by fT the solution of the restricted boundary valueproblem, where the condition at infinity is replaced by fT

′(T ) = 0 and the constraint(1.4) is replaced by (3.3). Then, we define, in a standard way, the so-called succes-sive approximations method which consists in replacing this nonlinear problem by asystem of linear boundary value problems with variable coefficients to which we apply


the discretization by the finite differences scheme. For this purpose, let us consider thefollowing system:

∀n � 0,(f(n+1)T

)′′′ + α + 1

2f(n)T

(f(n+1)T

)′′ − α(f(n)T

)′(f(n+1)T

)′ = 0, (4.1)

with the boundary conditions

f(n+1)T (0) = 0,

(f(n+1)T

)′(0) = 1, (4.2)

and (f(n+1)T

)′(T ) = 0, (4.3)

where the function f 0T is chosen so that 0 � (f 0

T )′ � 1, and that the boundary condi-

tions (4.2), (4.3) are respected. Our aim is to study the convergence of this system to thesolution fT of the problem (1.1)–(1.4) on [0, T ].

Notations. Let I be the interval ]0, T [. We denote by L2(I ) the Lebesgue space ofsquare integrable functions, equiped with the norm

∀ψ ∈ L2(I ), ‖ψ‖L2(I ) =(∫

I

∣∣ψ(x)∣∣2

dx

)1/2

.

We will also use the standard Sobolev spaces Hm(I), m � 1, provided with the norm

‖ψ‖Hm(I) =( ∑

0�k�m

∥∥ψ(k)∥∥2L2(I )

)1/2

,

and the semi-norm

|ψ |Hm(I) = (∥∥ψ(m)∥∥2L2(I )

)1/2,

where ψ(k) denotes the derivative of order k of ψ .

On C0(I ), we use the usual norm

‖ψ‖∞ = maxx∈I

∣∣ψ(x)∣∣.

Finally, we denote u(x) = (f(n+1)T (x))′, s(x) = (f

(n)T )′ and r(x) = f

(n)T (x) =∫ x

0 s(t) dt .The linear system (4.1)–(4.3) reduces to a sequence of second-order boundary

value problems in the form of

u′′ + α + 1

2r(x)u′ − αs(x)u(x) = 0, (4.4)

and

u(0) = 1, u(T ) = 0,


with

r(x) ∈ C1(I), r(x) � 0, r(0) = 0,

r ′(x) = s(x), 0 � s(x) � 1, s(T ) = 0.

We refer to this problem by (L). We have the following statement.

Proposition 4.1. Let α � − 13 . Then, the problem (L) has one and only one solution

u ∈ C2(I ). The solution is such that if s ∈ Ck(I ), then u ∈ C(k+2)(I ). Moreover, in thecase α � 0, we have 0 � u(x) � 1.

Proof. Let ub be a function of H 2(I ) consistent with the boundary conditions. Theproblem (L) can be written in a variational form: find u = u − ub ∈ H 1

0 (I ),

a(u, v) = -(v), ∀v ∈ H 10 (I ),

where

a(u, v)=∫ T

0u′v′ dx − α + 1

2

∫ T

0(ru′)v dx +

∫ T

0αs(x)uv dx,

-(v)= −a(ub, v).

It is clear that the bilinear form a(., .) is continuous with the constant

M = max

(1 + α + 1

2‖r‖∞, α‖s‖∞

).

Concerning the ellipticity, we obtain, after integration by parts and taking into accountthe relationship r ′ = s,

a(v, v) = |v|2H 1(I )

+ 5α + 1

4

∫ T

0s(x)v2 dx,

so we distinguish two cases: if α � − 15 , then we can directly check the ellipticity of

the bilinear form a(., .) with the constant m = 1. In the case − 13 � α < − 1

5 , as0 � s(x) � 1 and denoting by cp = T /π the constant of Poincaré, we get

a(v, v) �(

1 + 5α + 1

4c2p

)|v|H 1(I ).

Hence, we obtain the ellipticity property under the restriction T <√

6π , and with theconstant (in the worst case) m = (1 + ((5α + 1)/4)c2

p). In both cases, we immedi-ately get the well-possedness of the variational problem thanks to Lax–Milgram lemma.Moreover, we have the following stability estimate

‖u‖H 1(I ) � C(α, T ) (4.5)

where C = (M/m)‖ub‖H 1(I ), which can be easily bounded independently of r and s

thanks to the hypothesis on those functions.


Finally, let us note that the solution is at least in H 2(I ) ∩ H 10 (I ). Thanks to this

regularity, one can check, by going back to (4.4), that u is a classical solution. Applyingthe strong maximum principle leads to the bounds 0 � u(x) � 1, when α � 0. �

Remark 4.2. The restriction on T in the case α ∈ [− 13 ,− 1

5 [ is a limitation only due tothe variational approach. We know how to obtain the results for α � − 1

3 without anyrestriction on T . Anyhow, we will see that the analysis of the convergence towards thesolution u of the nonlinear problem limits us to α � 0. Besides, computations show thatthe method also applies for negative values of α until − 1

2 (see section 5.2).

Proposition 4.3. Concerning α � 0, the nonlinear problem (1.1)–(1.4), restricted to I ,has one and only one solution which is the limit of (4.1)–(4.3).

Proof. Uniqueness. Let f1 and f2 be two distinct solutions of the nonlinear problemwith f ′′

1 (0) > f ′′2 (0). Introducing h = f1 − f2, we have h(0) = 0, h′(0) = 0 and

h′′(0) > 0. Since h′(T ) = 0, let t0 > 0 be the smaller value such that

h′′(t0) = 0, h′ > 0 on ]0, t0], h(t0) > 0 and h′′′(t0) � 0. (4.6)

We have

h′′′(t0) = f1′′′(t0) − f2

′′′(t0) = αh′(t0)(f1

′(t0) + f2′(t0)

) − α + 1

2f ′′

1 (t0)h(t0).

Now, using the equality f ′′1 (t0) = f ′′

2 (t0), and by means of [4, lemma 3.1] (respec-tively [4, proposition 3.2]), we have f ′′

i (t0) < 0 (respectively f ′i (t0) > 0) for i = 1, 2,

from which we derive h′′′(t0) > 0, whence a contradiction with (4.6).Existence. From the energy estimate (4.4), we deduce that any sequence of solu-

tions (un) is bounded in H 1(I ). It follows the weak convergence of subsequences inH 1-norm and strong convergence in Hs , s < 1, to some function u. Moreover, the com-pactness of embedding of H 1 in C0(I ) will allow to pass to the limit in the variationalformulation: ∀v ∈ H 1

0 (I ),∫ T

0u′n+1v

′ dx − α + 1

2

∫ T

0

{∫ x

0un(t) dt

}u′n+1v dx +

∫ T

0αun(x)un+1v dx

= −an(ub, v),

where the bilinear form an(., .) is the left-hand side term with un+1 replaced by ub. Thelimit for the first and last terms, in the left-hand side expression, is just the strong L2

convergence. Concerning the second term, it is enough to observe that the dominatedconvergence applies for

∫ x

0 un dx. We proceed in the same way for the right-hand sideterm, and we get u as a solution of a variationnal nonlinear problem. Note that the Greenformula applied to the variationnal nonlinear problem shows that u ∈ H 2(I ) ∩ H 1

0 (I ),so that the initial function f is in C2 (recall that we have set f ′ = u), whence, thanksto (1.1), f is in C3, and even in C∞ by using again (1.1). Finally, using uniqueness, weget the convergence of the entire sequence. �


Remark 4.4. 1. The bounds 0 � u(x) � 1 are necessary to prove the convergence of thesequence. However, this introduces a restriction to α > 0. As we are interested in allα � − 1

3 , a classical way to avoid this restriction is to consider the linear problem (Lλ):

u′′ + α + 1

2r(x)u′ − (

αs(x) + λ)u(x) = −λs(x),

with the same boundary conditions. This “penalized” problem is obtained by adding theterm −λfT

′ on both sides of the original equation, so that the nonlinear problem remainsunchanged while the linearization leads to the last equation. Unfortunately, we cannotrecover the upper bound u(x) � 1. Moreover, numerical simulations show that there isno improvement by this formulation.

2. Observe that the linear problem is elliptic. Hence, it possesses a smoothingeffect which accelerates the convergence of the process.

4.2. Discretization by finite differences method

Notations. Let (xi)i=0,n+1 denotes a uniform mesh, namely, xi = i ∗ h, h = T /(n+ 1),and fi = f (xi), ui = u(xi). Also denote

(ui)′ = ui+1 − ui−1

2h, (ui)

′′ = ui+1 − 2ui + ui−1

h2.

In order to solve the linear problem L, we consider the following discretized linearsystem:

−ui+1 − 2ui + ui−1

h2− α + 1

2ri

(ui+1 − ui−1

2h

)+ αsiui = bi, (4.7)

where u0 = 1, un+1 = 0, b1 = −((α + 1)/4)(r1/h)+ 1/h2 and bi = 0, 2 � i � n.

The values ri are computed using the Simpson quadrature formula from si , 1 �i � n. The matrix Ah of the linear system is given by(

h2Ah

)ii

= 2 + αsih2 = δi,(

h2Ah

)i,i−1 = −1 + α + 1

4rih = γi,(

h2Ah

)i,i+1 = −1 − α + 1

4rih = βi.

We summarize the main properties of this discretization in the following proposi-tion:

Proposition 4.5.

(i) For α � 0, there exists h0 such that ∀h � h0, the matrix Ah is monotone (i.e.invertible and A−1

h is positive).

(ii) For α > 0, there exists h0 such that ∀h � h0, we have ‖A−1h ‖ � (α min si)−1.


(iii) For α > 0, the linear system (4.5) has one and only one solution uh. Furthermore,0 � uh � 1.

(iv) The solution u of the problem (L) is in C4([0, T ]), and

‖u − uh‖∞ � Ch2∥∥u(4)∥∥∞.

Proof. (i) We use the following equivalent characterization of the monotony: Ah ismonotone if

{v; Ahv � 0} ⊂ {v; v � 0}.There exists a subscript p ∈ {1, . . . , n} such that ∀i = 1, . . . , n, vp � vi .

(1) p = 1: h2(Ahv)1 � 0, so

−v2 + 2v1 − α + 1

4hr1v2 + αs1h

2v1 � 0

or similarly

(2 + αs1h2)v1 −

(1 + α + 1

4hr1

)v2 � 0.

We can choose a positive value h0 such that if h � h0, we have (1 − ((α + 1)/4)hr1 +αs1h

2)v1 � 0 and the result follows.(2) p = n: This case is similar to the previous one.(3) 1 < p < n: we obtain

−(

1 + α + 1

4hrp

)vp+1 −

(1 − α + 1

4hrp

)vp−1 + (

2 + αsph2)vp � 0

and we show in the same way that this implies αsph2vp � 0, which yields the result

when α > 0. In the case α = 0, we know that the property holds as a limit when α goesto zero.

(ii) Let w = (h2Ah)−1z, and denoting p ∈ {1, . . . , n} the subscript such that

|wp| = ‖w‖∞, we have

|zp| = |δpwp + βpwp+1 + γpwp−1|�

(|δp| − |βp| − |γp|)|wp| � α min

isih

2|wp|,hence the result.

(iii) is an immediate consequence of (i) and (ii).(iv) Since u = f ′ is C4([O,T ]), then the Taylor developpement for u and the

replacement of the exact derivatives by the corresponding finite differences yields

Ahu = fh + εh(u),

where εh is the rest in the Taylor formula, which is clearly bounded by the product|u(4)|∞h2. The estimate on u − uh then follows immediatly. �


The property (iii) is crucial to the convergence of the successive approximationsprocess and we retrieve at this discrete level the same (theoretical) limitation to α � 0.However, the numerical computations show that the method also works for negative α.

Remark 4.6. For α = 0, we have h2Ah = A0 + (h/4)C = A0(Id + (h/4)A0−1C), where

(A0)ii = 2, (A0)ij = −1 for j = i − 1 or j = i + 1, and

(A0)ij = 0 otherwise, and

Ci−1,i = ri, Ci,i+1 = −ri, and Cij = 0 otherwise.

Since ‖A0−1‖∞ � T /(8h2), we have∥∥∥∥h4A0

−1C

∥∥∥∥∞� h

4

∥∥A0−1

∥∥∞‖C‖∞ � T 2

16h.

So, under a very stringent restriction on T 2/h, one can easily bound the uniform normof A−1

h . In practice, this restriction on T 2/h is too strong to be used.

Remark 4.7. Obviously, the estimate in (ii) is not satisfactory. Let us bear in mind that sis such that s(T ) = 0. So, when h goes to zero (or when α is close but not equal to zero),the factor (α mini si)−1 in the first estimate becomes too large. For large T ’s, h must betoo small to satisfy the assumption. However, this estimate is used two times, first inthe error estimate of (iv), where the term h2 seems to be dominant, so that – even if theconstant is bad – the numerical results are not affected, and secondly to obtain uh � 1.In the worst cases, and also in the case α = 0, we observed that the upper-bound on uhis sometimes violated, but the convergence still holds.

5. Numerical results

5.1. The shooting method

Let us recall that this method is, for any fixed T , devoted to search for nonpositiveroots of function c �→ ϕ(c) = f ′

(0,1,c)(T ), where f is a solution of the Cauchy problemPα,(0,1,c) defined in section 3.

This method gives very accurate results in accordance with some previous experi-mentations (see [8]), and for special cases ((1.6) and (1.7)), where explicit solutions areknown. Moreover, this method is efficient to reach physical solutions whose existenceis established for α � − 1

3 (see [4]), but also for general solutions for α > − 12 (see

figure 1(c)).Figures 1(a)–(c) and 3(a)–(c) represent the graph of the function ϕ, obtained as fol-

lows: for given T > 0 and α > −1/2, we solve for each c the Cauchy problem Pα,(0,1,c)

by means of a Runge–Kutta method, improved by a standard prediction–correction al-gorithm. The step size is automatically updated during the computations, taking large


(a)

(b)

(c)

Figure 1. (a) The graph of the function ϕ(c) with T = 70 and α = −0.27; (b) zoom around the computedvalue c∗ in the case T = 20 and α = −0.27; (c) the graph of the function ϕ(c) with T = 40 and α = −0.42.


steps where the solution varies slowly, and small ones otherwise. Note that this approachis strongly recommended in our case, since for a given c, we have no guaranty to get avalue ϕ(c), because the blow up may occur.

Now, let us present in detail the numerical experimentation.The method presents two different behaviours depending on the values of α:

5.1.1. The case −1/2 < α � 0Figures 1(a)–(c) show that the function ϕ is smooth enough, concave, and possesses

one (unique) zero c which is quasi-independent of T (we tested for T varying from 5to 10000). This value c approximates the right value c∗ of the initial problem (1.1)–(1.3).Moreover, the zeros of ϕ, for various α, are obtained with a good accuracy, even withsmall T ’s (for instance, T = 20, figure 1(b)). Of course, the convergence of the sequence(ck)k depends on the choice of the initial values c0, c1. However, except when thoseinitial values are too far from c, where the convergence is very slow, one could usually“choose” a good interval where fast convergence occurs. Unfortunately, this interval willbe as small as T is large. The link between the length of the interval where ϕ is definedand how small is T is not easy to explicit analytically, nor to quantify numerically, sinceit depends in a nontrivial way on other parameters such as the choice of initial conditions,the values of α, . . . . In practice, we start with few iterations with large step-sizes forsolving the differential equation, which provides a reasonable interval.

It emerges from the simulations in the case − 12 < α � 0 that only one (computed)

solution to the boundary value problem exists. This solution is a physical one in therange − 1

3 � α � 0 and the computed value c = f ′′(0) is the expected one; the compar-ison is made with the values α = − 1

3 , 0, for which the solutions are known exactly orapproximately from previous simulations available in the literature.

Note that solutions computed on intervals [0, T [ are good approximations to thoseon the whole [0,+∞[. This is not always the case for boundary layer approximationsproblems. This is to be linked with the non-existence of oscillating solutions unlikesome initial value problems involving the Falkner–Skan equation [4,10].

Figure 2 points out an important feature of the shooting method. Indeed, we haveplotted the function f (and its derivatives f ′, f ′′) for three values c1 < c < c2, c beingthe value to which the method converges, while ci , i = 1, 2, are small perturbationsaround it. We observe (when varying T ) that fc exists globally, remains bounded withf ′c and f ′′

c vanishing quickly. Function fc2 also exists globally but seems unbounded. Itgoes slowly to infinity (so that f ′

c2 and f ′′c2

still remain small for large T ’s). Eventually,fc1 blows up near T = 85 and of course f ′

c1and f ′′

c1as well, all this functions going

to −∞.This observation induced us to take care in the interpretation of the numerical ex-

periments. In fact, the right value c corresponding to the problem on [0,+∞[ appearsto be the smallest one for which global solution to the Cauchy problem exists (this wasrigorously proved for α = 0, [3]), so that the Cauchy problem on [0, T [ (whence theshooting method) is always unstable (asymptotically).


Figure 2. Solutions f , f ′, f ′′ to P(α,c), with α = −0.27.

Remark 5.1. Finally, note that a good (and severe) test for the shooting method with neg-ative α is given by the solution (1.6), corresponding to α = − 1

3 . Recall that, in this case,we have an infinite number of solutions but one only is of physical type. What is some-what surprising is that the numerical method always yields the right c, namely c = 0,(independently of the initial values c0 and c1). However, small perturbations cδ around c

give the same results as those shown in figure 2.

The case α > 0 must be splitted into the two ranges 0 < α < 1 and α � 1.

5.1.2. The case 0 < α < 1Figures 3(a)–(c) show that the function ϕ is smooth, convex, and possesses two

distinct zeros c1 and c2, c1 < c2. The value c2 is near the expected one corresponding tothat of the problem on [0,+∞[ and is quasi-independent of T . The value c1 is dependenton the value of T , as follows: c1 is far from c2 for small T ’s but asymptotically, thedifference c2 − c1 (c2 is nearly constant) goes to zero when T goes (numerically) to +∞(see table 1).

The next two figures (4(a), (b)) reveal an important feature of the shooting methodapplied in this range of α; they precise the nature of the solutions related to these val-ues c1 and c2. In fact, figure 4(a) represents the two solutions fc1 and fc2 plotted on[0, T [, with c1 and c2 computed for this T . The function fc1 is solution of the problem(1.1)–(1.3), and fc2 is solution of the problem (1.1)–(1.4). Figure 4(b) represents thesame solution (with the same c1 and c2) plotted on [0, T [, T > T , and it shows thatonly fc2 is still solution of the problem (1.1)–(1.4), while fc1 is no more solution of(1.1)–(1.3). Our interpretation of this phenomenon is given at the end of this section.


(a)

(b)

(c)

Figure 3. The graph of the function ϕ(c) with (a) T = 10 and α = 0.4, (b) T = 100 and α = 0.4,(c) T = 500 and α = 0.4.


Table 1Effect of the variation of T on c1 and c2 for α = 0.4.

T c1 c2 c2 − c1

10 −0.78849 −0.71787 0.0706220 −0.74063 −0.71618 0.02445

100 −0.72171 −0.71609 0.00562500 −0.71727 −0.71615 0.00112

4000 −0.71688 −0.71615 0.00073

(a)

(b)

Figure 4. (a) The physical solution fc2 (top) and the general solution fc1 (bottom) (with their derivativesf ′, f ′′) for α = 0.4, T = 100. (b) The physical solution fc2 (bounded) and the general solution fc1 for

α = 0.4, T = 4000.


Table 2Effect of the variation of T on c1 and c2 for α = 1.13.

T c1 c2 c2 − c1

10 −1.09035 −1.05176 0.0385920 −1.06989 −1.05175 0.01814

100 −1.06437 −1.05174 0.01263500 −1.06437 −1.05174 0.01263

1000 −1.06437 −1.05174 0.01263

5.1.3. The case α � 1The function ϕ possesses also two distinct zeros c1 and c2, c1 < c2. The values

c1, c2 and also the distance c2 − c1, now remain quasi-constant and do not depend on T

(see table 2). However, on the contrary to the previous case, for large T ’s, not only c1

and c2 are zeros of ϕ, but each value on the interval [c1, c2] becomes a zero of ϕ (seefigure 5(c)). Moreover, each c ∈ [c1, c2] corresponds to a solution of the problem ((1.1)–(1.3)). Therefore, we have an infinite number of solutions but only one is of physicaltype, namely, that corresponding to c2.

Let us end this section with the following comments: The existence of more thanone solution in the case α > 0, the dependency on T which separates ranges α < 1and α � 1 are questions for which we have no definitive answers. However, in the caseα < 1, the solution corresponding to the value c1 does not seem to be just a numericalartefact. Indeed, there is no theoretical argument to exclude this kind of non-physicalsolutions on [0, T [. The only thing we are able to prove is the uniqueness of physicalsolutions on [0, T [ given in proposition 4.3 (this result holds in the case of the unboundedinterval [0,+∞[, see [4]).

In the case α � 1, the same observation holds, that is each value in the interval[c1, c2[ yields a solution whereas only c2 corresponds to the physical one.

Remark 5.2. If, for instance, we use an analogy with the bifurcation theory by consider-ing our problem under the form:

F(α, f ) = 0, (5.1)

it seems that for − 12 � α � 0, equation (5.1) possesses a unique solution that we denote

f (α), and the mapping α �→ f (α) is smooth. For 0 < α < 1, the former problempossesses two solutions (for fixed T ) but only one is of physical type. The mappingα �→ f (α), where f (α) is the physical solution, appears to be a branch of regularsolutions. Moreover, the two solutions coalesce asymptotically. When α � 1, it has aninfinite number of solutions but only one is of physical type. In this case, the notion ofregular solution to (5.1) is not clear.

The above results, enforced by the application presented in section 5.3, lead us toclaim that for T = +∞, we have the


(a)

(b)

(c)

Figure 5. The graph of the function ϕ(c) with (a) T = 10 and α = 1.13, (b) T = 100 and α = 1.13,(c) T = 500 and α = 1.13.


Conjecture.

• In the case − 13 � α < 0, uniqueness must occur for physical solutions (i.e., there is

a comparison principle of solutions).

• For 13 < α < 1, any solution on [0,∞] is a physical one. As consequence, no other

solution (than the physical ones) holds for 0 � α < 1.

• For 1 � α, even if we know that physical solutions are unique, there is no reason tonon-existence of other types of solutions (in fact an infinite number according to thesimulations!). This may imply that the model for this range of α is physically lessconsistent.

5.2. The direct simulation

The second numerical approach we have called the direct simulation method is apriori dedicated to the computation of the physical solutions only. It is also a very fastand accurate method (see figure 6). The solutions are computed for small T ’s, whichis important to have reasonable cost. In addition, unlike its mathematical analysis, thismethod gives good results for both positive and nonpositive α and also (to our surprise)for α in ]− 1

2 ,− 13 ] (non-physical solutions!). More precisely, we obtain the convergence

until α = −0.48 . . . . In fact, numerically, the bounds 0 � uh � 1 may be slightlyviolated without affecting the convergence of the scheme. This is confirmed by theconvergence curves (figure 7) which performs worst and worst as α decreases to − 1

2 .The following figures show f and f ′ for various values of α. Note that we retrievenearly the a priori bound (1.5) for solutions.

The comparison with the shooting method shows that if we are only interested bythe physical solution, this direct approach is a low cost and faster method. Indeed, thematrix of the linear system is a tri-diagonal one, so that classical algorithms allow tosolve the system in O(n2) operations (n ≡ 1/h). The nonlinear process also convergesquickly except when α tends to − 1

2 . The convergence of the iterative scheme is veryfast and behaves better as α increases as shown in figure 7, where we have plotted thelogarithm of the L2-norm of u−un, which we denote by en, as a function of the iterationnumber.

The values of c are computed using the formula

c = −2(u1 − 1) − h2α

2h+ O

(h2

)(where u1 denotes the first component of u(xi), i = 1, . . . , n) and are close to theexpected ones. Moreover, the comparison with [8] yields nearely the same thicknessand the same profile for the boundary layer (see table 3). The table shows the values ofc corresponding to different α while te are the abscissa from which the temperature isless than a given threshold e (here e = 10−2). It gives an idea over the thickness of theboundary layer.


(a)

(b)

(c)

Figure 6. Graph of the solution for (a) α = −0.47 (f top, f ′ bottom), (b) α = −0.1 (f top, f ′ bottom),(c) α = 3 (f top, f ′ bottom).


Figure 7. Convergence curves.

Table 3

α c∗ = f ′′(0) te

−0.33 0.00000002 7.33−0.25 −0.16203550 7.11

0.00 −0.44374828 6.371.00 −1.00000022 4.612.00 −1.34845469 3.783.00 −1.62435712 3.27

5.3. Application to a geophysical problem

Following [8], let us consider a geophysical problem concerning the free convec-tion about a dike. The dike consists in a molten magma intruded in an aquifer. Whenthe hot intruded magma is in contact with the cooler environment, its surface will bechilled to form a thin glassy selvage. The contact zone between the dike and the aquifercorresponds to our wall immersed in a porous medium. The complete description ofthis problem must take into account many physical phenomena and becomes quicklydifficult. However, a simplified model is given by (1.1)–(1.4) in the quite realistic situ-ation of non-uniform wall temperature of the dike. Constants chosen for the numericalsimulations stem from those taken in [8], and are precised below: the equivalent ther-mal diffusivity is λ = 6.3 × 10−7 m2/s, the permeability of the porous medium isk = 10−12 m2, the viscosity is µ = 0.27 g/s ·m, the thermal expansion coefficient of thefluid is β = 1.8×10−4/◦C, the density at infinity is ρ∞ = 106 g/m3 and the temperatureat infinity T∞ = 15◦C.

In [8], the authors only consider the case α = 0 (a uniform wall temperature),for which they give different numerical values related to the free convection and the


heat transfer. We are more interested in a qualitative description and, precisely, weintend to show how the thermal behaviour changes according to different values of α. Inaccordance with section 2, the local Rayleigh number Rax = Bxα+1, where B = 0.0105,the dimensionless similarity variable, and temperature are

t = √Byx(α−1)/2, θ(t) = T (x, y) − 15

Axα= f ′(t).

The stream function is

ψ(x, y) = λ√Bx(α+1)/2f (t).

The vertical component u in the x direction and the horizontal component v in the y di-rection of Darcy velocities are then given by

u(x, y) = λBxαf ′(t), v(x, y) = λ√B

2x(α−1)/2[(1 − α)tf ′(t) − (1 + α)f (t)

].

In figures 8(a) and (b), we plot, for various α and a given constant A = 1.0, the curvesIα(c) = {(x, y) | θ(t) = c}, c = 0.01, which are the dimensionless isotherms. The plotrepresents the height x of the wall as a function of the distance y from the wall (on theabscissa). These show the boundary layer profile.

We distinguish two cases:

• − 13 � α < 1: the isotherms are convex and have the following properties: if c < c′,

then Iα(c) < Iα(c′), and if α < α′ then Iα(c) < Iα′(c), which means that for a

given α, the boundary layer thickness increases when x increases, and so is α. Weretrieve in these computations that the hot water zone is very thin for large Rayleighnumbers.

• α � 1: the isotherms are still convex curves but now, for a given α, the boundarylayer thickness decreases when x increases and when α < α′, we have Iα(c) > Iα′(c).Unlike previous cases, these results are unusual in the classical boundary layer theory.Indeed, it is well known that the approximation in this theory becomes singular nearx = 0 and is more and more accurate away from the zero point. Unfortunately, forthis range of α, the zone where the approximation takes place is too small (as small asα is large). Therefore, the model in this interval may suffer from a lack of consistency.

Note that the value α = 1 is a limiting case where the isotherms are vertical linesparallels to the x axis.

To be exhaustive, we present in figure 9 the curves of constant values of ψ whichrepresent the flow pattern.

First, let us observe that the vertical velocity u is positive while the horizontal one,v, changes the sign as the function t �→ h(t) = [(1 − α)tf ′(t) − (1 + α)f (t)] does, sothat, for α � 0, v is positive and the fluid is attracted by the hot zone (the wall) in anascendant motion.

In the case − 13 � α < 0, the component v changes the sign for each α. Plotting

the isotherm where v vanishes, we see that this curve separates the plan in two regions


(a)

(b)

Figure 8. Boundary layer profile (a) for − 13 � α < 1: from the left (the wall) to the right α = 0.75, 0.,

−0.25 and −0.33; (b) for α � 1: from the left (the wall) to the right α = 3.0, 2.0, 1.5 and 1.25.

where the motion of the fluid is different. The first region, very thin, is delimited belowby the axis x = 0 and above by that isotherm, where the fluid is also attracted by thewall. The second region is above the isotherm where the fluid is pushed away from thewall.

It is not clear whether this motion is physically relevant. The last plot (figure 10)represents the true isotherms. In the case α � 0, we retrieve the classical behaviourof the boundary layer solutions. In the case α < 0, the significant zone (where thetemperature exceeds a threshold), is small and close to (or confined near) the zero point.


(a)

(b)

Figure 9. Flow pattern: (a) α < 0, (b) α � 0.

Figure 10. True isotherms.


6. Conclusion

We have suggested two simple numerical approaches to the computation of solu-tions to the problem Pα . Both of them are of course to be improve, even if they alreadygive satisfactory results. The shooting method, in addition to its efficiency, reveals therich structure of the set of solutions of the problem. We saw for example that at least twodifferent solutions could exist when α is positive, even if the uniqueness seems to occurfor the physical ones (in accordance with the mathematical results). We also observe, inthe case α < 0, that the monotony of ϕ and its smoothness suggest the existence of anunderlying comparison principle (available in the case α = 0) which would be the keyfor proving the uniqueness in this interval.

Of course, using this method requires a careful choice of T as the numerical exper-iments show, but this is not a major difficulty and there are several ways to remedy to it(adaptative refinement, the use of the second method to have a good initial start).

The second method is also efficient and complementary to the first one. In fact,both of them allow to obtain a very accurate approximation for physical parameters suchas the convected energy and the thin layer, for instance, and more generally, to validatethe model as in the last paragraph.

In another hand, from a mathematical point of view, the simulations have clearup some questions related to the problem and give us some impressions that we havesummarized in a conjecture. These questions, and other aspects emphasized such as thestability analysis and the bifurcation phenomenon, are actually under consideration.

The objective in a future work is to investigate mathematically and numerically thestability of the boundary layer.

Acknowledgements

The authors wish to thank anonymous referees for valuable hints and remarks.

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Numerical simulations of free convection about a … · In previous work [4], we investigated some mathematical questions, particularly ... In fact, for P−1/3, the physical solution