Crank J. Free and Moving Boundary Problems

32

Free-boundary problems: formulation

enough for

the

flow

to

be considered as two-dimensional in the

(x,

y)

plane.

The

flow

is

taken to

be

laminar and

to

be

governed by Darcy's law

(1856) expressed in the form

-K

-k

q1

=-grad

h

=-grad{p+pgy}

pg I.t

(2.1)

where

q1

is

the velocity vector, p

is

the pressure in the fluid, h

is

the

hydraulic head,

p

is

density, I.t

is

the viscosity of the fluid, the scalar

constant K

is

called

the

hydraulic conductivity, and k = l.tK/pg is

the

permeability of

the

medium.

The

vertical coordinate y is positive up-

wards.

For

the

assumptions already made about

the

fluid k

is

constant

and so the function

<1>1

= (k/ I.t )(p + pgy)

(2.2)

is a velocity potential and

q1

=

-grad

<1>1.

But

the equation of continuity

divq1

=0

(2.3)

must also

be

satisfied by the incompressible

flow

and hence from (2.1-3)

div(grad

<1>1)

=

'\7

2

<1>1

= 0, (2.4)

i.e.

<1>1

satisfies Laplace's equation in the seepage region n in Fig. 2.1.

It

is

convenient

to

introduce a modified velocity potential

so that

(x,

y)

1.t<l>1

(x,

y)

kpg

q=-grad

and

where q =

I.tq1/kpg, a modified

flow

rate.

p

+y

pg

(2.5)

(2.6)

The

seepage region

is

bounded by parts of the walls of the dam

AF

and

BD

and its base

AB

but also by the free surface

FD

whose shape

and

position are

to

be

determined, including the location

of

the 'point

of

detachment' D on the wall BE. The

part

of the boundary CD, known as

the

'seepage surface', must exist for physical reasons.

The

conditions

to

be

satisfied by

(x,

y)

on

the

different parts of the

boundary of the region

n are derived as follows. Since there can

be

no

flow

across an impervious surface the normal derivative

a<l>/an

must

be

zero on any such sUrface, e.g. on the base AB,

a<l>/ay

=

O.

Since the free

surface

FD

is

the

interface between

the

water in

flow

region n and the air

above, into which no water penetrates, the condition

a<l>/an

holds on

FD

also. The second condition on the free boundary

is

that pressure must

be

continuous across it. But outside the flow region and above the two

reservoirs

the

pressure

is

constant and may

be

taken

to

be

zero. Putting

Simple rectangular dam

33

p = 0 in (2.5) gives

= Y

on

PD.

Similarly

on

the

seepage surface

CD

=

y,

since p = 0, because again

the

water there is in contact with air.

The

condition along a porous boundary in contact with a reservoir

depends

on

the

fact

that

the

fluid velocity

in

a reservoir is negligible

compared

to

that

in

the

porous medium. A zero velocity means

the

velocity potential

= y +

pI

(gp) is constant and since p = 0 at a reservoir

surface we have

=

YI

or

=

Y2

along

AF

or

BC

respectively (Fig. 2.1).

The

model problem of

the

simple rectangular dam

is

therefore defined

with reference

to

Fig. 2.1 by

the

following equation and conditions:

~=O

in

0,

O<x<x

io

O<y<[(x),

(2.7)

=

YI

on

AF,

=

Y2

on

BC, (2.8)

=

y,

a<l>lan

= 0 on

PD,

(2.9)

= Y

on

CD,

a<l>lay

= 0

on

AB,

a<l>lax..;;o

on

CD.

(2.10)

(2.11)

(2.12)

The

condition (2.12) expresses

the

fact

that

some fluid leaves

the

porous

medium

on

the

seepage face CD,

but

none can enter

the

medium there.

The

complete solution includes

the

determination

of

the

free boundary

Y =

[(x)

such

that

[(0)

=

Yio

d

dx

[(0)=0,

The

final condition in (2.13) is discussed

more

fully

in

§8.3.

In

this

example, orthogonal

to

the

equipotential curves

= constant

there

will

be

streamlines along which

a<l>lan

is zero.

It

follows that

the

impervious

base

and

the

free surface

are

both streamlines. Fluid particles move along

a streamline always in

the

same direction from

the

upper

to

the

lower

reservoir which is

the

direction

of

x increasing in Fig. 2.1. Thus

the

fluid

velocity

is

positive

in

the

flow region

and

must decrease along any

streamline.

In

particular, along

the

streamline free surface

is

a decreas-

ing function

of

X,

and

since

on

PD

= Y =

[(x),

it follows

that

[(x)

is

monotone decreasing.

Mathematical properties

of

the

free boundary Y =

[(x)

have been

examined for

the

simple

dam

and

other

problems by several authors

including Friedman and Jensen (1977), Jensen (1977, 1980),

and

Boieri

and

Gastaldi (1980). Jensen (1980) gives

rderences

to

a series

of

earlier

papers

on

which his own is based.

In

many cases

of

steady flow it has

been

proved analytically

that

[ is a continuous function strictly decreas-

ing, with [(0) =

YI

and [(Xl) =

Y2,

and

that

[ is concave. Different results

have been proved in some evolution dam problems (see §2.9). Collatz

34

Free-boundary problems: formulation

(1980) used monotonicity

to

establish upper and lower bounds for some

free-boundary problems.

Finally, because the base and the free surface are streamlines they

define a stream tube through any section of which the fluid

flow

is

the

same. In particular through any vertical section of the dam we have a

constant

flow

rate given by

I

f

(X) a

- -

</>(x,

y)

dy = constant = q

ax

(2.14)

and sometimes referred

to

as

the

'discharge'.

2.2.2. Stream function

The

components of the velocity q in (2.6) are given by

u =

-a</>/ax,

v =

-a</>/ay.

(2.15a)

There also exists a stream function

I{I

which satisfies Laplace's equation

V21{1

= 0, and for which

u =

-al{llay,

v

=al{llax.

(2.15b)

The Cauchy-Riemann equations

</>x

=

«/Iy

and

-</>y

=!/Ix

follow from

(2.15a) and (2.15b) as usual.

The simple dam problem defined by equations (2.7-12) can also

be

expressed in terms of the two functions

</>

and

I{I

as

a</>I

ax

-

al{ll

ay = 0,

a</>/ay

+al{llax

= 0

in

n,

</>

=

Yl

on AF,

</>

=

Y2

on BC,

</>

=

y,

I{I=O on PD, (2.16)

</>=y

on CD,

l{I=q

on AB,

where

q is

the

flow

rate

through the dam defined in (2.14).

It

is

important to note that, in this example, q can be explicitly

determined from

the

dimensions of the dam and

the

water levels. In fact,

Charni (1951) showed that

(2.17)

with reference

to

Fig. 2.1.

2.2.3. Formulation on fixed domain: Baiocchi transformation

In

1971 Baiocchi proposed a new way of formulating

flow

problems

through porous media which has proved highly effective in both theoreti-

cal and numerical respects.

It

allows existence and uniqueness theorems

to

be proved rigorously; it also yields new numerical algorithms which are

Simple rectangular dam

35

simple

and

efficient

to

use

and

which compete well with older schemes

based

on

the

classical formulations of equations (2.7-12)

or

(2.16).

Baiocchi (1972) developed

the

theoretical side

of

his

idea

for

the

simple

rectangular

dam

problem

and

the

numerical application is

due

to

Comin-

cioli

et

al.

(1971). Extensive systematic developments

and

applications

to

more

general problems were subsequently pursued

by

Baiocchi

and

his

colleagues

at

the

Laboratorio

di Analisi Numerica del Consiglio

Nazionale della Richerche, Pavia, Italy.

The

early theoretical results were

reported

by

Baiocchi, Comincioli, Magenes,

and

Pozzi (1973b). Baiocchi,

Comincioli,

Guerri,

and

Volpi (1973a) developed numerical

methods

and

presented

solutions for a

number

of

problems

of

practical importance,

some giving rise

to

mathematical

and

numerical difficulties which

do

not

arise

in

the

simple

dam

problem.

These

authors also surveyed

the

theoretical results

of

the

companion

paper

by

Baiocchi, Comincioli,

Magenes,

and

Pozzi (1973b). A most helpful survey

of

more

recent

work

by

Bruch

(1980) outlines

the

basic ideas

of

the

Baiocchi

method

and

its

application

to

a wide

range

of

important

practical problems. Solutions

are

presented

graphically

and

in tables

and

SOme

are

compared with results

of

earlier workers.

Chapter

8 is concerned with numerical methods

of

solution including some based

on

Baiocchi's ideas.

The

Baiocchi approach is introduced

here

with reference

to

the

simple

rectangular

dam

(Fig. 2.1).

The

unknown region

n,

in which

the

problem

has

been

formulated

by

equations (2.7-12), is extended

into

a known

region with fixed boundaries,

in

this case

ABEF.

The

dependent

variable

is

extended

continuously across

the

free surface

and

thus defined

over

the

whole

of

the

known

extended

region.

The

domain

n in Fig. 2.1 is extended

to

the

known region

ABEF

denoted

by

D(O<x<Xl>O<Y<Yl)

so

that

D=n+next+r,

where

r is

the

free

boundary

FD including

the

points F

and

D,

and

by

O<x

<Xl>

f(x)

< Y

<Yl'

The

dependent

variable

cf>

is extended continu-

ously

and

defined

on

D,

i.e.

the

domain D plus its boundaries,

by

putting

(x,

y) =

cf>(x,

y)

in

n,

(x,

y)=y

inD-n.

(2.18)

A weak solution

of

equations (2.7-12) is a triplet {t,

n,

cf>}

such that: f(x) is

a continuous strictly decreasing function in

O::S;;X::S;;Xl

with

f(O)

=

Yl>

f(Xl);;:'

Y2;

n is

the

open

domain already indicated;

cf>(x,

y) is

square

summable

together

with its first derivatives in

n,

continuous in

n,

and

in

this

sense

satisfies (2.8), (2.10),

and

the

first of (2.9);

cf>(x,

y) satIsfies

f f

(~x~

+

</>y~)

dx dy = 0

(2.19)

.0

36

Free-boundary problems: formulation

for all ;j,(x, y) defined

on

[0,

Xl]

x [0,

Yl]

with continuous first derivatives

and with

;j,

= 0 on a neighbourhood of

AF

and

BE.

As

usual, the weak

solution (2.19) implies that

V'l<t>

= 0

on

0 and

<t>

satisfies (2.11) and the

second of (2.9).

The

Baiocchi transfonnation is now used

to

define a new dependent

variable

w which is

(2.20)

First, certain properties of

w(x, t) are investigated.

On

the free surface

r and in

15

-

n,

W = 0, and so w is continuous

on

15

because (2.20)

implies

(2.20a)

Furthennore, substitution of

t$

= Y +

pi

pg in (2.20) gives

w(x,

y)=

--d1},

1

f

("')

p(x,

1})

pg

and hence, since p and

pg

are positive quantities in

0,

w(x,

y»O

in

O.

Differentiation of (2.20a) with respect

to

X yields

J

f(X)

wx(x, y) = f",(x){t$(x,

f(x))

-

f(x)}

+

t$",(x,

1})

d1}

y

= t$Ax,

1})

d1},

1

f

("')

since, from

the

first of (2.9),

t$(x,

f(x))

=

f(x).

Similarly

Wy

=-{t$(x,

y)-y}.

Thus

W'" = Wy

=0

on y =

f(x)

and

in

15-n,

and

the

first derivatives of W are continuous throughout D.

(2.21)

(2.22)

A convenient starting point for

the

derivation of

the

differential equa-

tion satisfied by

w is

the

Green's theorem

f f

V;j,

.

Vw

dx dy =

io

;j,

::

dS - f f

;j,V

2

w

dx

dy

o D

= - f f ;j,V'lw dx

dy

(2.23)

D

Simple rectangular dam

37

by virtue of the boundary conditions for

aw/an

and

~.

Introduction of a

function

(x,

y) which is given by

(2.24)

i.e.

~(x,

y) =

<l>y

(x, y),

(2.25)

leads to a seCond expression for

JSV~

.

Vw

dx dy

as

follows:

D

{}

after integration by parts and noting that

vanishes together with all its

derivatives

on

FA,

AB,

and

Be

and that w vanishes on FD. Now

substitution from

(2.22), (2.18), and (2.25) into (2.26) gives

f

Jv~.

Vw

dx

dy

=-

f

f[(<f>-Y)""

+(<f>-y)yy]dx

dy

D

{}

= f

f-~

dx

dy = f

J-~X{}dX

dy (2.27)

{}

D

by a further use

of

Green's theorem, inserting

the

boundary conditions

and V

2

<f>

= 0 in

fl

and where

X{}

= 1 in

fl

and zero outside fl. Finally,

(2.23) and (2.27) together yield

V

2

w = 1 in fl,

w>o,}

V

2

w=0

inD-fl,

w=O.

(2.28)

To

complete

the

formulation of the simple dam problem we need

to

establish

the

conditions satisfied by w(x, t) on

the

boundaries of D. Thus

onAF,

~(O'Y)=Yb

w(O,

y)=f'(Yl-1J)d1J

=!(YCy)2,

(2.29)

(2.30)

38

Free-boundary problems:

formulation

on

CD,

on

DE,

on

FE,

(Xb

y) =

y,

(x

b

y) =

y,

(x,

YI)

=

y,

Since AB is a streamline

W(x

b

y)=O,

w(x

b

y)=O,

w(x,

YI)

=0.

f

Y

'

wx(x,

0)

=.10

x(x,

1])

d1]

=

-q,

(2.31)

(2.32)

(2.33)

where

q is

the

flow

rate

through

the

dam

and

is independent of

x.

Using

(2.29) and (2.30) with

Y = 0 shows that

Wx

=

-!(yi-

y~)fXb

and on integration with respect

to

x,

remembering that

at

x = 0,

w(O,

0)

=

hi.

we have finally

onAB,

y(x,

0) = 0,

(2.34)

The

properties

of

W and

the

equations in (2.28) can

be

expressed by

the

following differential equation and inequalities in

the

domain

D:

w(1-V

2

w)=0,

1-V

2

w(x,

y);;=oO,

W(x,

y);;=oo,}

(2.35)

A numerical algorithm for finding

w(x, y) satisfying (2.35)

and

the

boundary conditions (2.29-33) with (2.34) is given by

the

relations (8.86)

in Chapter

8.

We

remark only at this stage that once such a solution

is

found, the domain n can

be

identified as

the

region in which w

>0,

i.e.

n={(x,y)l(x,y)

in

D;

w(x,y»O};

also

the

free boundary

y=f(x)

is

given by

f(x)

= sup{y I (x, y) in n},

with

f(O) =

lim

f(x)

and

f(x

l

)

=

lim

f(x).

x-+o+

X-7Xi'

2.2.4. Variational inequality formulation

It

is well established (Duvaut and Lions, 1976; Mosco, 1973) that

if

w(x, y) satisfies (2.35) and

the

conditions (2.29-33) with (2.34) it also

satisfies

the

variational inequality

a(w,

v-w);;=o(v-w)

(2.36)

for

all functions v which, together with their first derivatives, exist and are

square summable

on

D and which agree with w on

the

boundary of D.

The

function

wand

its first derivatives are likewise square summable

on

Simple rectangular dam

39

D.

The

nomenclature in (2.36)

is

that

a(w,v-u)=

JJVW.V(V-U)dxdY

D

and

= J J{Wx(Vx-w:J+Wy(Vy-Wy)}dXdY (2.37)

D

(V-W)

=-

J

J(v-w)

dx dy.

(2.38)

D

A proof outlined

by

Bruch (1980) runs as follows. Introduce the

function

(3

= v -

w,

which is thus zero

on

the

boundary of

D;

then

J

J{V(3'

Vw}dx

dy+

J

J(3

dx

dy

D

=JJ{V(3'VW}dxd

y

+JJ(3Xn

dxd

Y+

JJ

vdxdy

(2.39)

D

D-n

since W = 0 in 15-.0. Defining

(x,

y) =

Po

(3(x,

'1})

d'1}

and hence

(3(x,

y)

==

y(x,

y)

and noting

that

vanishes together with all its derivatives

on

FA,

AB,

and

Be

while W vanishes

on

FE,

the

argument leading

to

equation (2.26) above gives again

J

J{V(3

.

Vw}

dx

dy = J

J(Wy=

+wyyy)

dx

dy. (2.40)

D D

Substituting from (2.18) and (2.22) in (2.40) and repeating

the

derivation

of (2.27) leads

to

J

J{V(3'

Vw}dxdy

=-1

(3Xn

dx

dy.

(2.41)

D

The

proof of (2.36) follows immediately by using (2.41) in (2.39) and

remembering

that

v

~

0 in

15

-.0.

Reverting

to

the

formulation of

the

problem, because of the symmetry

in

the

inequality (2.36),

the

required solution w(x,

y)

is also a solution

of

the

minimization problem

J(W).::;J(V),

(2.42)

40

Free-boundary problems: formulation

where

the

functions v(x,

y)

are as in (2.36)

and

J(v)=~ff(V;+V;)dXdY+

ffVdXdY.

(2.43)

D

This follows by writing (2.36)

in

the

fonn

- f

f(w;+w;)

dx

dy-

f f w dx

dy

D

~

- f f(W

x

Vx

+WyVy)

dx dy - f f v dx dy (2.44)

D D

using (2.37).

The

general inequality

(a-b)2~0,

i.e. 2ab':;;;a

2

+b

2

,

gives

2w

x

Vx':;;;w;+v;

and also

2wyvy

':;;;w;+v;

and

hence (2.44) implies (2.42).

Proofs

of

the

existence

and

uniqueness

of

a solution

of

the

inequality

(2.36)

and

hence of

the

minimization problem (2.42) are given by

Baiocchi et

al.

(1973b, 1976), Lions

and

Stampacchia (1967), and

Glowinski (1976).

Baiocchi (1980a) extended

the

treatment

of

the

simple rectangular

dam

to

include a derivative condition

orJJ/ox

=

A.

(rJJ

-

Yl),

A.

> 0,

on

the

inlet face

AF

(Fig. 2.1) instead

of

the

condition (2.8),

rJJ

=

Yl'

Apparently

the

influence of impurities

in

the

reservoir when carried

to

the

face

AF

is

better

expressed by

the

derivative condition. Baiocchi derives

the

varia-

tional inequality

f

Jvw

.

V(v-

w)

dx dy

+A.

f

1

[w(0,

y){v(O,

y)-w(O,

y)}]

dy

D

~

f

f(w

-v)

dx dy -A. fl[(Y1Y -y2/2){v(O,

y)-

w(O,

y)}]dy,

D

where D is

the

fixed domain

of

the

whole dam. Baiocchi (1980a) finds

this problem

to

be

well posed,

that

existence

and

uniqueness can

be

proved, and that numerical solution is straightforward.

Baiocchi's extensive list

of

references gives a good impression

of

the

wide application

to

which his transfonnation

and

method

have

been

put.

2.2.5.

In

terms

of

stream junctions

An

alternative definition

of

the

Baiocchi transfonnation (2.20) incorpo-

rates

the

stream function

I{I

introduced in §2.2.2

and

extended

to

the

fixed

Other two-dimensional dams

41

domain D

by

setting

~(x,

Y)

=

.p(x,

Y)

_

in_n,}

.p(x,

Y)

= 0 in

D-O

(2.45)

analogous

to

the

extension

of

</>(x,

y) in (2.18).

The

alternative form of

(2.20) is

(2.46)

where, with reference

to

Fig. 2.1, P is a point in D and

FP

is a smooth

path

joining F

to

P.

The

integration in (2.46) is independent of

the

path

because

x

-

.jJy

= 0 from

the

Cauchy-Riemann equation.

The

first derivatives

of

(2.46), namely

WX

(P) = -.jJ(x, y),

are

seen

to

be

continuous through

jj

and

equal

to

zero

on

the

free

surface

and

throughout

jj

-

n.

The

differential inequalities (2.35) together with

the

conditions

(2.29-

33)

and

(2.34) can

be

derived also from

the

form (2.46)

of

the

Baiocchi

transformation.

In

addition

.p(x,

y)=-wx(x,

y)

in

n

could

be

deduced from

the

solution for w(x, y).

2.3.

Other

two-dimeusional

dams

(2.47)

Free-boundary problems presented by porous flow through

other

types

of

dam,

more

complicated

than

the

simple rectangular dam, will now

be

formulated.

They

provide examples

of

other

boundary conditions which

in some cases call for modifications in

the

methods of solution described

later

in

Chapter

8.

2.3.1. Rectangular dam with

toe

drain

In

Fig. 2.2

the

portion

AB

of

the

base is impervious

but

BC

is

the

toe

drain

through which fluid can escape.

The

free surface does

not

meet

the

vertical wall

and

so

there

is

no

separation point

to

be

considered. Instead,

the

free surface meets

the

toe

drain

at

a point C which is unknown a

priori. Accordingly,

the

solution domain is extended

to

a point D along

the

drain, as well as

up

to

y =

Yl

in

the

y direction.

The

problem can

be

formulated in terms

of

the

modified potential

</>

and

stream function

.p

as for

the

simple dam. Also,

the

Baiocchi-

transformation is still defined by (2.20)

or

(2.46) where now D denotes

Crank J. Free and Moving Boundary Problems

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