Fowler A. Mathematical Geoscience

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830 D Melting, Dissolution, and Phase Changes

bands. In this section, we describe a model due to Keller and Rubinow (1981) which

aims to explain the phenomenon, based on the earlier ideas of Ostwald.

Keller and Rubinow consider the reaction scheme

A +B



−

→D, (D.56)

in which A would represent the silver nitrate seed crystal, B would be the dilute

dichromate solution, C is the reaction product silver dichromate, and D is the solid

precipitate. In one dimension, a suitable set of equations is

−r,

+r −p,

=p,

(D.57)

where r is the reaction rate and p is the precipitation rate, given respectively by

r =k

ab −k

−

c, (D.58)

and

p =



q(c −c

) if c ≥c

(> c

) or d>0,

0ifc<c

and d =0,

(D.59)

where c

is the saturation concentration of C and c

is the required supersaturation

for nucleation. Let us suppose that D

, and the reaction is very fast, so that

r ≈0. Then

c ≈Kab, (D.60)

where

K =

−

. (D.61)

Suitable initial conditions are

a =0,b=b

,c=d =0att =0, (D.62)

and suitable boundary conditions are

a =a

=0atx =0. (D.63)

Adding the equations for b and c, and deﬁning B =b +c, we obtain

−p, (D.64)

D.11 Liesegang Rings 831

and in addition (D.60) implies

c =AB, (D.65)

where

A =

1 +Ka

. (D.66)

Keller and Rubinow assume that the reaction term r can be neglected in the equation

for a, essentially on the basis that if b

 a

(the dichromate is very dilute), then

very little A is removed in forming the product. In this case A simply diffuses away

from the seed crystal, providing an expression for a as

a =a

erfc



√



. (D.67)

It is convenient to scale the equations, and we therefore choose the scales

d,c,B ∼b

,p∼qb

,t∼

,x∼



; (D.68)

then the dimensionless model is

−p,

=p,

(D.69)

where

p =



AB −A

if AB ≥A

(> A

) or d>0,

0ifAB < A

and d =0,

(D.70)

where we deﬁne

. (D.71)

The function A is given by

A =

κ erfcθ

1 +κ erfc θ

,θ=

βx

√

, (D.72)

in which

κ =Ka

,β=



. (D.73)

Note that A is a monotonically O(1) decreasing function of θ , which tends to zero

at inﬁnity. The initial and boundary conditions are

B =1,p=0att =0;

=0atx =0,

B →1asx →∞.

(D.74)

832 D Melting, Dissolution, and Phase Changes

It should be noted that since the time scale is that of precipitation (and thus quite

fast in the laboratory), we can expect the length and time scales to be small, so that

large space and time solutions of this model are of interest.

D.11.1 Central Precipitation

The maximum value of A =

1+κ

is at θ = 0, and thus precipitation will begin at

x = 0 providing

1+κ

; we presume this to be the case. Keller and Rubinow

give an ingenious (but heuristic) approximate solution for their model, which we

now emulate. Initially, there is a central precipitating region 0 <x<R(t), where

p>0, and p =0 outside this. First, they suppose that A is slowly varying in space,

and that R is slowly varying in time, so that a quasi-static solution is appropriate.

Since B is continuous at R, then AB =A

there, and this solution is

AB =A

−A

) cosh(

√

Ax)

cosh(

√

AR)

. (D.75)

For x>R, a stationary solution is not possible, but for slowly varying R,

B =1 +



−1



erfc



x −R

√



. (D.76)

Equating the derivatives B

at R±, we ﬁnd that R is determined by the relation

√

A −

√

=(A

−A

)

√

πt tanh



√



, (D.77)

in which A(θ) is given by (D.72), with

θ =

βR

√

. (D.78)

To solve this, we deﬁne

u =

√

AR, (D.79)

and then (D.77) can be written in the form

u tanh u =

2θ

√



A(θ) −A

−A



. (D.80)

The right hand side is a unimodal (one-humped) function of θ, while u tanh u is

an increasing function of u. Therefore u(θ) is a positive function in the range

0 <θ <θ

, where A(θ

) =A

. Consulting (D.77), we see that initially A =A

and

thereafter increases with t. Therefore, initially θ = θ

and decreases with increas-

ing t. Since A is increasing as is R, u must increase, but it cannot do so indeﬁnitely,

D.11 Liesegang Rings 833

because of the maximum value of u(θ). In consequence, there is a ﬁnite time t

∗

when R reaches a maximum R

∗

, and the solution cannot be continued beyond this

time.

Keller and Rubinow go on to suggest that a sequence of precipitation bands will

subsequently form, and they analyse these based on the same approximating solu-

tions. The question arises, whether there is any rational basis for supposing that their

approximation method is valid.

The two principal assumptions in the solution method are that A is slowly varying

in space for x<R, and that R is slowly varying in time. The ﬁrst of these requires

θ deﬁned by (D.78) to be small, and since A ranges from

1 +κ

(D.81)

to A

at x =R, this requires

δ =

1 +κ

−A

1. (D.82)

The assumption that R is slowly varying, i.e., that the time derivative in (D.69)

can be ignored, requires t  x

∼ R

, and thus, from (D.78), θ  β. Assuming

β ∼O(1), as seems likely, this condition is included by (D.82).

We write

θ =δΘ, (D.83)

and then (D.80) can be approximated by

≈

2δ

Θ(1 −a



Θ)

√

π(A

−A

)

, (D.84)

where



2κ

√

π(1 +κ)

. (D.85)

From (D.78) and (D.79), we then ﬁnd

R ≈

δκ

√

πt

{βA

+(A

−A

)πκt}

, (D.86)

and R reaches its maximum

∗



βA

−A

)



1/2

(D.87)

at time

∗

βA

πκ(A

−A

)

. (D.88)

These results provide a basis for a direct asymptotic approach, based, for example,

on the limit δ  1, with the other parameters being taken as O(1).

834 D Melting, Dissolution, and Phase Changes

D.12 Exercises

D.1 The density ρ, velocity u and internal energy e of a ﬂuid are given by the

conservation laws

∂ρ

∂t

+∇.(ρu) = 0,

∂ρu

∂t

+∇.(ρu

u) = ∇.σ

+ρf

∂

∂t



ρu

+ρe +ρχ



+∇.



ρu

+ρe +ρχ





= ∇.(σ

) −∇.q,

where σ

=σ

, q is the heat ﬂux, and the conservative body force f is deﬁned

f =−∇χ,

where χ is the potential.

Show that the momentum equation can be written in the form



∂u

∂t

+u.∇u



∂σ

∂x

+ρf

and that the energy equation can be reduced to

=σ

˙ε

−∇.q.

D.2 The perfect gas law may be written in the form

v =

where R is the gas constant, and M is the molecular weight. Show that β =

and deduce that for a perfect gas,

dh =c

dT,

where h is speciﬁc enthalpy.

Use the relation

de =Tds−pdv

and the deﬁnition of the speciﬁc heat at constant volume,



∂s

∂T



D.12 Exercises 835

to show that

de =c

dT −pdv+T



∂p

∂T



[hint: use the Maxwell relations]. Hence show that, for a perfect gas,

de =c

dT.

D.3 The functions g

g =Ac +B(1 −c),

for coefﬁcients A

and B

, and A

and B

, respectively, and these are deﬁned

A = a +RT ln c,

B =b +RT ln(1 −c),

with similar subscripting S,L of the coefﬁcients a and b.

Show that the conditions A

= A

and B

= B

are solved by values c

which satisfy



) =g



) =

g(c

) −g(c

)

−c

Appendix E

Averaged Equations in Two Phase Flow

E.1 Discontinuities and Jump Conditions

Suppose we have a conservation law of the form

∂φ

∂t

+∇.F =0, (E.1)

which is derived from the integral conservation law



φdV =−



∂V

F.ndS. (E.2)

From ﬁrst principles we can derive the jump condition across surfaces where φ and

F are discontinuous:

[φ]

−

]

−

, (E.3)

where +and −refer to the values either side of the surface of discontinuity, and n is

the unit normal at this surface (pointing either way); V

is the speed of the surface in

the direction of the normal, and F

=F.n. In the common case of a ﬂuid in motion,

where the conservation law takes the form

∂φ

∂t

+∇.(φu) =∇.J, (E.4)

the corresponding jump condition is



φ(u

−V

) −J



−

=0. (E.5)

The basic equations of conservation of mass, momentum and energy for a ﬂuid

with density ρ, velocity u and internal energy e weregivenin(D.12), and are re-

peated here:

∂ρ

∂t

+∇.(ρu) = 0,

A. Fowler, Mathematical Geoscience, Interdisciplinary Applied Mathematics 36,

837

838 E Averaged Equations in Two Phase Flow

∂ρu

∂t

+∇.(ρu

u) = ∇.σ

+ρf

, (E.6)

∂

∂t



ρu

+ρe +ρχ



+∇.



ρu

+ρe +ρχ





= ∇.(σ

) −∇.q;

in the last equation, χ is the potential energy. The corresponding jump conditions

are



ρ(u

−V

)



−

= 0,



ρu

−V

) −σ



−

= 0, (E.7)



ρu

+ρe +ρχ



−V

)



−

=[σ .u.n −q

]

−

Note that these jump conditions are implied automatically by the integral forms of

the conservation laws, assuming there is no production at the surface (e. g., of energy

by a surface reaction). Therefore the integral forms can be applied directly to ﬁnd

the total mass, momentum and energy conservation laws for a two phase ﬂow in

which the density and energy in particular may be discontinuous.

Let us deﬁne the interfacial source term

Γ =Γ

−

=−



ρ(u

−V

)



−

, (E.8)

where we deﬁne the unit normal n

−

here to be pointing from the − phase towards

the + phase. If we suppose that there is no slip across an interface, [u.t]

−

=0, where

t is any tangent vector at the interface, then the momentum jump condition (E.7)

implies

[σ

]

−

=0, [σ

]

−

=−Γ [u

]

−

, (E.9)

and the energy jump condition becomes





−

+[σ

−q

]

−

=0, (E.10)

since we take the potential energy χ to be continuous.

E.2 Averaging Methods

Next, we consider the derivation of averaged equations for two-phase ﬂows. This is

a subject which has been the subject of a number of different investigations, see for

example Ishii (1975) and Drew and Passman (1999), and also the thorough overview

by Drew and Wood (1985). Averaging proceeds as in the derivation of averaged

equations for turbulent ﬂows (see Sect. B.1), but the choice of average is not clear

E.2 Averaging Methods 839

cut. A local space average seems the most obvious choice, but only for homogeneous

ﬂows. A local time average is a better choice, but in fact preference is usually given

to the ensemble average over a number of realisations of the ﬂow. For stationary

ﬂows, this is likely to be equivalent to a local time average.

Further complication arises since often one is concerned with axial ﬂows in a

pipe (for example in a volcanic vent), where a cross-sectional average is appropriate

either as well as, or instead of a local time average. There seem to be few examples

where two-phase models in two or three dimensions are proposed.

There are various different ways to derive averaged equations. We follow Drew

and Wood (see also Fowler 1997) in using an indicator function X

which is equal to

one in phase k (k =1, 2) and zero otherwise. We denote averages by overbars, and

the averaged equations are obtained by multiplying the point forms of the governing

equations by X

and then averaging. This procedure introduces derivatives of the

piecewise continuous X

, and these must be interpreted using generalised functions.

To see how this works, consider a general conservation law of the form

∂

∂t

(ρψ) +∇.(ρψu) =−∇.J +ρf, (E.11)

where ψ is the conserved quantity (per unit mass), u is the ﬂuid velocity, J is the

ﬂux, and f is a volumetric source. Multiplying by X

and averaging yields the exact

equation

∂

∂t

(

ρψ) +∇.[X

ρψu]

=−∇.[

J]+X

ρf

ρψ



∂X

∂t

.∇X





ρψ(u −u

) +J



.∇X

, (E.12)

where u

is the velocity of the interface between the phases, and we assume that

∇f = ∇

f , ∂f /∂ t = ∂

f/∂t, which will be the case for sufﬁciently well-behaved

f . Derivatives of X

are interpreted as generalised functions. Thus, for example,

j.∇X

is deﬁned, for any smooth test function φ which vanishes at inﬁnity, through

the identity



φj.∇X

dV =−



∇.(φj)dV =−



∇.(φj)dV =−



φj

dS, (E.13)

where j

is the normal component of j at the interface, pointing away from phase k.

This suggests that

j.∇X

can be identiﬁed with the speciﬁc surface average of −j.n,

which is consistent with the fact that ∇X

is essentially a delta function centred on

the interface.