## Abstract

The freezing of a supercooled droplet occurs in two steps: recalescence, that is, a rapid return to thermodynamic equilibrium at the freezing temperature leading to a liquid–solid mixture and a longer stage of complete freezing. The second freezing step can be modelled by the one-phase Stefan problem for an inward solidification of a sphere, assuming the droplet to be spherical. A convective heat transfer with the ambient immiscible fluid is modelled by a mixed boundary condition on the outer surface of the droplet. This condition depends on the Biot number (ratio of the heat transfer resistances inside the droplet and at its surface). A novel asymptotic solution is developed for a small Stefan number and an arbitrary Biot number. Applying the method of matched asymptotic expansions, uniformly valid solutions are obtained for the temperature profile and freezing front evolution in the whole stage of complete freezing. For an infinite Biot number, that is, for a fixed temperature at the droplet outer boundary, known solutions are recovered. In parallel, numerical results are obtained for an arbitrary Stefan number using a finite-difference scheme based on the enthalpy method. The asymptotic and numerical solutions are in good agreement.

## 1. Introduction

The freezing of supercooled liquids or melts has a broad spectrum of practical applications, namely in metal casting, ice accretion on power cables and on aircraft during flights, low-temperature biology and medicine. The freezing of supercooled droplets (being in a metastable thermodynamic equilibrium at a temperature below the freezing point) consists of two main stages: (i) recalescence, that is, an initial freezing stage with a rapid return to a stable thermodynamic equilibrium at the freezing temperature (as shown by Macklin & Payne (1967) and Feuillebois *et al.* (1995)) and (ii) complete freezing, when the whole droplet volume is transformed into solid.

During the first stage, a part of the droplet liquid becomes a solid. Feuillebois *et al.* (1995) discussed three possible locations for the formed solid: (i) at the droplet centre, (ii) spread uniformly in the droplet volume, forming a liquid–solid mixture, and (iii) at the outer droplet surface. The second scenario was considered to be the most relevant one. This was later confirmed experimentally for the case of water–ice phase change by Hindmarsh *et al.* (2005), who observed the water distribution in a freezing droplet by nuclear magnetic resonance. Note that this type of mushy solid–liquid mixture also occurs for other freezing conditions forming dendrites, e.g. Worster (1997). The physical properties of this mixture depend on the extent of the initial supercooling. For example, the mixture latent heat of fusion is different from the liquid latent heat of fusion. This mixture at the freezing temperature will be the initial condition for the complete freezing stage considered in this article.

The complete freezing stage of a free spherical droplet can be described as the one-phase Stefan problem of an inward solidification of a spherical ball initially at the freezing temperature. Note that formally it can be regarded as a single phase with some given latent heat of fusion. If the outer surface of the droplet is isothermal, below the freezing temperature, then freezing is governed by conduction. Although this one-phase Stefan problem with the first-order outer boundary condition is one of the simplest moving boundary problems, it is not solved exactly (up to our knowledge and as pointed out also by McCue *et al.* (2008)) and only approximate and numerical solutions exist in the literature. An extensive survey on this problem can be found, for example, in Crank (1984), Feuillebois *et al.* (1995) and McCue *et al.* (2008). The asymptotic power solution in a small Stefan number (defined as the ratio between the sensible heat and latent heat released during the phase change) becomes singular when the freezing front, *viz.* the liquid–solid interface, approaches the central zone of the spherical droplet. This is a boundary layer type problem with respect to a small time difference between the current time and the final freezing time. Different time scales have been introduced and the corresponding solutions for the interface position in time and the temperature distribution in space and time have been presented in a series of works: Pedroso & Domoto (1973*a*,*b*), Riley *et al.* (1974), Stewartson & Waechter (1976) and Soward (1980). The solution in the latter work elaborates the solutions of the previous ones. This solution has been used by Feuillebois *et al.* (1995) to construct a uniformly valid solution (using a low Stefan number based on the mixture, *S**t*_{m}≪1, in the perturbation method) for all time scales and to compare it with numerical solutions obtained for different models of solid nuclei distribution. Later, a rigorous construction of singular solutions for the interface shrinking to a point was performed by Herrero & Velázquez (1997). They proved a stronger estimate of the interface evolution in time close to the final freezing time than that of the previous authors (Stewartson & Waechter 1976; Soward 1980). However, the detailed study of the interface evolution in time close to the final freezing time, made by Stewartson & Waechter (1976) and Soward (1980) by introducing outer and inner solutions with respect to the small time difference to the complete freezing, is more appropriate for further applications and comparisons with numerical solutions and/or experimental results. The two–phase Stefan problem for the inward freezing of a sphere was considered by McCue *et al.* (2008) in the framework of a small Stefan number (note that McCue *et al.* (2008) used the reciprocal definition, i.e. their Stefan number is large) by a similar asymptotic analysis as for the one-phase problem. Their analysis is extended with the small-time perturbation solution and a comparison with a numerical solution.

When the droplet is placed in an ambient flow (for example, in an air flow as in the case of ice formation), the convective heat transfer is a significant cooling mechanism at the fixed outer surface. The Stefan problem (one or two phase) with convective (and possibly radiative) boundary conditions on the outer droplet boundary has received less attention in the literature devoted to approximate solutions for low Stefan numbers. The difficulty associated with the singular behaviour of the regular perturbation solutions in the power series of small Stefan numbers when approaching the sphere centre has been remedied with the aid of the method of strained coordinates by Milanez & Boldrini (1988) to obtain a uniformly valid solution (with only the first two terms of the asymptotic expansion) for the inward freezing of a sphere with convective surface cooling. However, this solution is correct only to the lowest order, like that of Pedroso & Domoto (1973*b*) for the case of a fixed temperature at the outer droplet surface. The same method was used by Parang *et al.* (1990) for both inward cylindrical and spherical solidification when the surface cooling occurs by simultaneous convection and radiation. The short-coming of the strained coordinates method is that it does not provide a physically meaningful and correct solution near the final freezing time, unlike the matched asymptotic expansion technique as seen above for the isothermal outer boundary case. A small-time perturbation solution and its comparison with the corresponding numerical solution in the case of radiation–convection type boundary conditions are proposed by Gupta (1987) and Gupta & Arora (1992).

All of the above papers concern a spherical droplet. However, a droplet in an ambient flow may distort by surface stresses. The droplet stays spherical if surface tension is high enough to overcome the distorting flow stresses. This may be estimated in terms of capillary and Weber numbers. Note that the freezing of a non-spherical droplet was considered by McCue *et al.* (2005), however, only for a fixed surface temperature.

In the present paper, the Stefan problem with a mixed boundary condition on the spherical droplet surface (a convective heat flux between the droplet and the ambient immiscible fluid) is studied analytically by means of a perturbation for the small Stefan number of the mixture, *S**t*_{m}≪1, and numerically for arbitrary *S**t*_{m}. The basic physical assumptions for modelling the mixture formation during the recalescence stage together with a dimensional analysis of the various time scales are given in §2. An asymptotic solution is constructed in §3. The analytical results are compared with the numerical ones found by the enthalpy method with a finite-difference implicit scheme in §4. Finally, a discussion on the obtained solutions is presented in §5. Some formulae used in the course of deriving the asymptotic solution are given in appendix A.

## 2. Formulation of the problem

### (a) Basic assumptions

The droplet is supposed to be spherical of radius *a* after the recalescence stage and to remain spherical during the subsequent phase change. Moreover, the liquid and solid densities are assumed to be equal: *ρ*_{l}=*ρ*_{s}. Thus, changes in volume during freezing are neglected.

At the end of the recalescence stage, the temperature is supposed to be uniform and equal to the equilibrium freezing temperature *θ*_{f}. The liquid mass fraction in the mixture is *ϕ*=(1−(*c*_{l}/*c*_{s})*S**t*), where *c*_{l} and *c*_{s} are the liquid and solid heat capacity, respectively, *S**t* is the Stefan number: *S**t*=*c*_{s}Δ*θ*/*L*; Δ*θ*=(*θ*_{f}−*θ*_{a}) is the supercooling and *L* is the liquid latent heat of freezing. The mixture may be regarded as a uniform phase with effective latent heat
2.1

### (b) Governing equations

The complete freezing stage of the liquid–solid mixture droplet can be regarded as a one-phase Stefan problem for an inward freezing of a sphere
2.2
where *κ*_{s} is the solid phase conductivity and *r*_{i} is the freezing interface position. The boundary conditions on the freezing interface *r*=*r*_{i} are a constant temperature equal to the equilibrium freezing temperature
2.3
and a heat flux jump
2.4
The heat flux boundary condition on the outer droplet surface is
2.5
where *α*_{a} is the heat transfer coefficient (reciprocal to thermal resistance) between the drop surface and ambient and *θ*_{a} is the ambient temperature. If the droplet surface is isothermal, condition (2.5) is replaced by *θ*=*θ*_{a}. The initial temperature condition states that at the first moment, the supercooled droplet has just returned to thermodynamic equilibrium at its freezing temperature
2.6
and the droplet surface is the initial position of the interface
2.7

### (c) Characteristic times

The stated problem (2.2)–(2.7) is in a dimensional form. In order to transform it into a dimensionless form, several characteristic time scales will be introduced. As the whole heat transfer process of freezing and cooling includes several mechanisms of heat transfer processes, *viz.* recalescence, conduction, phase change (freezing) and convection with the ambient, the following time scales can be generated: (i) one for the initial nucleation and freezing creating a liquid–solid mixture, *t*_{nuc}∼*f*(Δ*θ*^{−1}); (ii) a time for heat conduction inside the droplet, regardless of the phase change, *t*_{cond}=*a*^{2}*ρ*_{s}*c*_{s}/*κ*_{s}; (iii) a time scale for complete freezing, *t*_{fr}=*a*^{2}*ρ*_{s}*L*_{m}/*κ*_{s}Δ*θ*=*t*_{cond}/*S**t*_{m}, where *S**t*_{m}=*c*_{s}Δ*θ*/*L*_{m}; (iv) a convection time scale *t*_{conv} for cooling the outer boundary down to the ambient temperature, *t*_{conv}=*ρ*_{s}*c*_{s}*a*/*α*_{a}=*t*_{cond}/*B**i*, where *B**i* is the Biot number, *B**i*=*α*_{a}*a*/*κ*_{s}. The analysis presented below will be made in the complete freezing time scale, but the discussion will also refer to the other time scales.

## 3. Small Stefan numbers approximation

The mixture droplet radius *a*, the freezing time *t*_{fr} and the supercooling Δ*θ* are, respectively, chosen as the characteristic length, time and temperature. Then, the dimensionless analogue of the problem (2.2)–(2.7) is written as
3.1a
3.1b
3.1c
3.1dand
3.1e
where *x*=*r*/*a*, *τ*=*t*/*t*_{fr}, *u*(*x*,*τ*)=*x*(*θ*−*θ*_{f})/(*θ*_{a}−*θ*_{f}), *s*=*r*_{i}/*a*.

### a) Outer solution

The free boundary value problem (3.1a–*e*) can be written as a fixed boundary value problem, if the following variable is introduced:
3.2
where *ξ*∈[0,1] (*ξ*=0 represents the outer surface and *ξ*=1 represents the freezing front). Then, the variables (*x*,*τ*) are changed to (*ξ*,*s*) and the unknowns *u*=*u*(*ξ*,*s*), *τ*=*τ*(*s*) can be found as solutions of the transformed system
3.3a
3.3b
3.3c
3.3d
3.3e

In the small Stefan number approach, the solutions are sought as asymptotic expansions in *S**t*_{m}
3.4
and
3.5
The first three terms of equation (3.4), valid for arbitrary *B**i*>0, are as follows:
3.6a
3.6b
and
3.6c
where *A*=*s*+*B**i*(1−*s*).

The first three terms of equation (3.5) for the time function are also obtained for arbitrary *B**i*>0
3.7a
3.7b

3.7c

The zero-order solutions (3.6a) and (3.7a) have been obtained by Milanez & Boldrini (1988), but their first-order solutions are different, because the techniques of strained coordinates exploited by them do not give a correct solution of the physical problem of freezing. It must be noted that the dimensionless temperature profile can be easily found from the obtained solution (3.6) for *u*
3.8
where *T*(*x*,*τ*)=(*θ*_{a}−*θ*)/(*θ*_{a}−*θ*_{f}).

If the limit of equations (3.6) and (3.7) at is taken, then Pedroso & Domoto (1973*a*) solutions for an isothermal droplet surface are recovered. These expansions are regular when *s*=*O*(1), but become singular when , with singularities appearing in *u*_{1} and higher order terms for the temperature distribution and in *τ*_{2} for the time. The same type of singularity has been noticed by Riley *et al.* (1974) also for the isothermal case. Therefore, solutions (3.6) and (3.7) are valid for *τ*_{e}−*τ*≥*O*(1) and are referred to as outer solutions with respect to the time proximity to the dimensionless final freezing time *τ*_{e}. The dimensionless temperature *T*(*x*,*τ*) in the outer region will be denoted as *T*_{outer}(*x*,*τ*). In the next subsection, an inner solution will be constructed for *τ*_{e}−*τ*=*o*(1).

### (b) Inner solution

In the isothermal outer boundary case (), the singularity problem at the drop centre, close to the end of freezing, has been successively treated by different authors: Riley *et al.* (1974), Stewartson & Waechter (1976) and Soward (1980). In all works, an inner region of time near to the end of freezing has been introduced and a matching with the outer solution afterwards performed. However, the solution of Riley *et al.* (1974) is not valid near the end of freezing, and the solution of Stewartson & Waechter (1976) is the most complete one with some minor errors, but contains a complicated analysis that is difficult for further implementation. Soward (1980) obtained results similar to the latter using a simpler method, which follows the ideas of Riley *et al.* (1974). This method will be applied in the present study close to the final time of freezing.

As the standard method of matched asymptotic expansions will be used, an inner solution is constructed based on a stretched time
3.9
where *λ*=2/*S**t*_{m} is a large parameter.

In the outer region where solutions (3.7) hold, they can be rewritten in terms of the inner variable and parameter *λ* as
3.10
where
3.11

The unknown dimensionless final freezing time *τ*_{e} will be obtained after matching the outer expansion (3.10) with the inner solution. If equation (3.10) is solved with respect to *s* (as ), then is obtained, which will be used later when constructing the uniformly valid solution.

In the inner region, , the solution is sought following Soward (1980) by a method similar to the image method for potential problems. First, we return to the original problem (2.2)–(2.5) and write it down for the inner temperature *θ*_{inner}. Its dimensionless variant is given in appendix A by equations (A1)–(A6), for the sake of brevity. Then, the heat conduction problem (A3) with convection boundary condition (A4) is solved on a fixed domain (0<*r*≤1) using a fictitious heat source at the centre of the sphere with unknown intensity, which is determined afterwards so as to satisfy the boundary conditions (A5) and (A6) on the interface (). The matching of this solution with the outer solution (3.10), when , is treated as an initial condition for the parabolic equation (A3). This procedure will provide the unknown time *τ*_{e}. The region has no physical meaning; it is involved only to facilitate the mathematical model in the inner region.

A transient heat source of strength is imposed at *x*=0. Then, the inner temperature is sought close to *x*=0 in the form (Carslaw & Jaeger 1959)
3.12
where . Further, is chosen.

The unknown functions and will be determined during the construction of the inner solution. The solution method was used by Riley *et al.* (1974) and Soward (1980) for the isothermal outer boundary case, and its details will be omitted here.

Finally, the solutions for and are the following:
3.13
and
3.14
where , and *β*_{k} is the solution of the equation
3.15
where *β*_{k}∈[(*k*−1)*π*,*k**π*] and *k*=1,2,….

The forms of the inner solution of and the solution of are found when applying the boundary conditions (A5) and (A6) at the interface 3.16 and 3.17 where .

Then, the final form of *T*_{inner} becomes
3.18
where *D*=1−(1−1/*B**i*)*x*, as shown in equation (A7).

If in the upper formulae the limit at is performed and equations (A9)–(A12) are taken into account, the respective expressions (2.16*a*,*b*) and (2.18*a*) of Soward (1980) are recovered.

The obtained inner solutions (3.16) and (3.18) with (3.17) are valid for .

### (c) Uniformly valid solution

Up to now, the final freezing time *τ*_{e} remains unknown. It has to be determined when matching the solution of equation (3.10) with equation (3.16), as shown subsequently.

The matching is performed in the time overlap domain , where both the outer (3.10) and inner (3.16) solutions are valid and *s*≪1. The expressions containing *A* on the right-hand side of equation (3.10) are expanded in power series in *s* as given by equation (A15). Then, the inner solution (3.16) for *s* is substituted in equation (3.10) with (A15)
3.19

In this formula, using equation (A16), the function was replaced by a term containing .

The terms in front of equal powers of are successively equated in equation (3.19): the term in front of gives the unknown quantity *τ*_{s}
3.20
while the terms in , and are identically equal to zero, if the formula (A8) is applied. The term in will match with higher order terms, which are not given here.

Therefore, the final freezing time is explicitly determined by equations (3.11) and (3.20)
3.21
Finally, the freezing front evolution in time can be written by its uniformly valid expansion in *s*, following the matching procedure of van Dyke (1975):
3.22
where *s*_{outer} is the solution of equation (3.10), the inner solution *s*_{inner} is given by equation (3.16) and the common part of both the outer and inner solutions resulting from matching *s*_{match} is
3.23

The uniformly valid solution for the temperature *T* is constructed in a similar way. The outer solution *T*_{outer} given by equation (3.8) with (3.6a,*b*), is written in the inner variable *s*_{inner} from equation (3.16) and the terms of equal powers in are compared with their corresponding terms of *T*_{inner}. Finally, the uniformly valid solution in the whole domain 0<*r*≤*s* is obtained
3.24
where *T*_{outer} and *T*_{inner} are given, respectively, by equation (3.8) with (3.6a) and equation (3.18) with (3.17), while *T*_{match} is their common part
3.25

In the above expansions, the interface position in time *s*_{uni} has left-over terms of order *O*(*λ*^{−3/2}), whereas the temperature distribution in space and time *T*_{uni} has left-over terms of order *O*(*λ*^{−1}). However, when both *x*→0 and , there is another singularity named ‘inner core’, which has been studied by Soward (1980) and Herrero & Velázquez (1997). For the isothermal case (), the inner core time is very small: . The interface position is proved to be by Herrero & Velázquez (1997). As their analysis is independent of the far-field boundary conditions, we assume that this estimate holds also for our case of convective outer boundary conditions.

## 4. Asymptotic results for low *S**t*_{m} and comparison with numerical results for arbitrary *S**t*_{m}

The problem (2.2)–(2.7) at arbitrary Stefan numbers *S**t*_{m} will be solved numerically by means of the enthalpy method. Tabakova & Feuillebois (2004) have solved the isothermal problem () by the same method. Popov *et al.* (2005) have applied it for the convective freezing problem (arbitrary *B**i*>0) of a drop spread on a substrate. Here, we shall briefly summarize the enthalpy method. The heat conduction equation is rewritten in a form that is valid in both the solid and liquid–solid mixture. Then, an enthalpy function *H*(*θ*) is introduced (Voller & Cross 1981; Crank 1984) as
4.1
where *η* is the Heaviside step function
4.2
and *ρ*(*θ*) and *c*(*θ*) are the density and specific heat, expressed as functions of temperature: *ρ*(*θ*)=*ρ*_{m}=*ρ*_{l}*ϕ*+*ρ*_{s}(1−*ϕ*) and *c*(*θ*)=*c*_{m}=[*ρ*_{l}*c*_{l}*ϕ*+*ρ*_{s}*c*_{s}(1−*ϕ*)]/*ρ*_{m}, when *θ*=*θ*_{f} and *ρ*(*θ*)=*ρ*_{s} and *c*(*θ*)=*c*_{s}, when *θ*<*θ*_{f}. Then equation (2.2) with equation (2.4) is transformed into
4.3

The dimensionless form of equation (4.3) in terms of *T*(*x*,*τ*) becomes
4.4
where *δ*(*T*−1) is the Dirac delta function. As for the numerical calculation, a smoothing of this function is applied over a small temperature interval around *T*=1. An implicit finite-difference method is applied to provide a more flexible choice of the time steps.

Results for the interface evolution in time and temperature distribution inside the droplet for different times will now be presented for small Stefan numbers and various Biot numbers. Values obtained by the asymptotic solution of the preceding section will be compared with those of the numerical model.

The uniformly valid solution (3.22) for the interface is plotted in figure 1. As expected, a smaller Biot number increases the freezing time. In the limit , the outer droplet surface remains adiabatic and there is no heat flux entering through it, so that the freezing time is infinite.

The comparison between the outer, inner and uniformly valid solutions and the numerical solution for the freezing front evolution in time is presented in the case *B**i*=1 in figures 2 and 3, for *S**t*_{m}=0.01 and 0.1, respectively. The outer, inner and uniformly valid solutions are almost equal for *S**t*_{m}=0.01, while for *S**t*_{m}=0.1 there is a visible difference between them. The final freezing time is found from equation (3.21) to be *τ*_{e}=0.5043197 for *S**t*_{m}=0.01 and *τ*_{e}=0.5284889 for *S**t*_{m}=0.1. The circles show the points at which the uniformly valid expansion diverges approaching the inner core. The same behaviour of the uniformly valid expansion has been noticed by Feuillebois *et al.* (1995) for the isothermal case.

To illustrate the dependence of the freezing process on the convective boundary conditions, i.e. on the value of *B**i*, the temperature function is shown for *S**t*_{m}=0.01 and 0.1 at *B**i*=1. Several profiles are drawn: in figure 4 at *S**t*_{m}=0.01 for times *τ*=0.1, 0.2, 0.3, 0.4, 0.5 and in figure 5 at *S**t*_{m}=0.1 for times *τ*=0.1, 0.2, 0.3, 0.4, 0.5, 0.52. The comparison between the uniformly valid solution (3.24) (consisting of equations (3.8), (3.18) and (3.25)) and the numerical solution is very good for smaller times and smaller *S**t*_{m}, while for times approaching the final freezing time, the difference is significant because of the ‘inner core’.

The case is represented for comparison in figures 6 and 7, where several temperature profiles are provided for *S**t*_{m}=0.01 and 0.1 and times *τ*=0.05, 0.1, 0.15, 0.16, 0.168 and *τ*=0.05, 0.1, 0.15, 0.17, 0.179, respectively. Here, the comparison between the uniformly valid solution and the numerical solution is similar to the *B**i*=1 case. The final freezing time is calculated to be *τ*_{e}=0.1682362 and 0.1802603, respectively, for the cases *S**t*_{m}=0.01 and 0.1.

## 5. Discussion

A novel asymptotic solution has been presented for the Stefan problem of a freezing sphere in the case of a mixed boundary condition on the sphere surface. The convection with the ambient immiscible fluid is described by means of the Biot dimensionless number. It has a significant influence on the behaviour of the heat transfer inside the droplet. Results show that the behaviour of the uniformly valid solution of the mixed boundary condition problem near the complete freezing is similar to that for the first-kind boundary condition (a uniform temperature, i.e. infinite *B**i*). The validity of both solutions is limited to the region , which corresponds to *τ*∼*O*(*τ*_{e}−*S**t*_{m}). In parallel, a numerical solution has been presented using the enthalpy method. By comparison of these approaches, it appears that the practical range of application of the asymptotic analysis (without the inner core) is a little more limited for *B**i*=1 than for large *B**i*. The convection with the ambient immiscible fluid may be interpreted in terms of a convection time scale *t*_{conv}. As *t*_{conv}=*t*_{fr}*S**t*_{m}/*B**i*, then the convection with the outer ambient will depend on the ratio *S**t*_{m}/*B**i*. If *B**i*≥*O*(*S**t*_{m}), the freezing can be described by the asymptotic and/or numerical solution presented in this work. If *B**i*=*o*(*S**t*_{m}) at small Stefan numbers *S**t*_{m}, then the freezing will be very slow and practically the asymptotic and numerical solutions give an infinite time of complete freezing.

As the developed asymptotic solution covers the whole spectrum of the Biot number, *B**i*>0, then it can be applied to spherical droplets freezing at arbitrary heat transfer with the ambient fluid. If , we have recovered the analytical results of Soward (1980) and the uniformly valid solution of Feuillebois *et al.* (1995). Our previously obtained numerical results in Popov *et al.* (2005) for the case of a spherical drop are also confirmed by the present asymptotic solution. Note that in Popov *et al.* (2005), there is a typewriting error in table 1: *B**i*=100 must be read as *B**i*=32.5.

## Acknowledgements

The present study has been performed within the framework of the research project no. 21471 (2008–2009) between CNRS (France) and BAS (Bulgaria) entitled ‘Impact and freezing of supercooled droplets.’

## Appendix A

In the inner region, the stretched time (3.9) can be regarded as a dimensionless time in the conduction time scale
A1
where *t*_{e} is the dimensional final freezing time. The dimensionless inner temperature function is given as
A2

The dimensionless analogue of the problem (2.2)–(2.5) is as follows: A3 A4 A5

A6

The finite Fourier transforms used in the derivation of the inner solution exploited the expression . It has been proved using the computational algebra package MAPLE that this expression can be approximated as
A7
Similarly, the following formula has been proved:
A8
where *E*_{k} is given in §2*b*.

If , some of the expressions are simplified as A9 A10 A11

A12 To obtain the last two formulae, the infinite sums have been used (see Gradshteyn & Ryzhik 2007) A13 and A14

In order to perform the matching procedure between the inner and outer expansions, the expression 1/*A* is developed in power series in *s*≪1
A15

For large values of its argument *v*, the function can be approximated as (Gradshteyn & Ryzhik 2007)
A16

## Footnotes

- Received September 19, 2009.
- Accepted October 29, 2009.

- © 2009 The Royal Society