Similarity solutions for partial differential equations generated by finite and infinitesimal groups

SIMILARITY SOLUTIONS FOR PARTIAL

DIFFERENTIAL EQUATIONS GENERATED BY

FINITE AND INFINITESIMAL GROUPS

by

Henry S. Woodard and William F. Ames

Sponsored by Office of Naval Research Contract No. N00014-68-A-0196-0004

and

Project Themis

Contract No. DAAFO3-69-C-0014

uHR Report No. 132

Iowa Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa

July 1971

Approved for Public Release; Distribution Unlimited

a

ABSTRACT

The problem of developing systematic methods for obtaining

similarity variables is considered for partial differential equations. Similarity variables are a set of transformations which reduce a

partial differential equation to an ordinary differential equation. This paper considers two methods of generating similarity vari-ables. The first method uses a group of finite transformations and

the second uses a group of infinitesimal transformations. The

math-ematical theory for both techniques is described and illustrated. The two methods of obtaining similarity variables are applied to the Burgers' equation u + uu u and to the laminar boundary

y X XX

layer equations with a pressure gradient. In all cases considered, new types of similarity variables are found. In addition, the

aux-iliary conditions are discussed in the light of the new similarity

Chapter Page LIST OF FIGURES

LIST OF SYMBOLS

vi

I INTRODUCTION a.

II

SIMILARITY BY FINITE GROUP METHODS 9

Continuous Groups g

Invariant Transformations u.

Absolute Invariants

Similarity 16

Boundary Conditions . 20

III

SIMILARITY BY INFINITESIMAL GROUPS . 22

Infinitesimal Transformations ...

22

Invariants 25

Canonical Variables . 26

Reduction of the Number of Independent Variables

in a Partial Differential Equation . . . .. 28

Similarity Solutions in Practice 31

IV APPLICATION OF THE FINITE GROUP METHOD 35

Burgers' Equation . 35

Laminar Boundary Layer

TABLE OF CONTENTS CONT'D.

Chapter Page

V APPLICATION OF THE INFINITESIMAL GROUP METHOD. . . 61

Burgers' Equation 61

Boundary Layer Equations 7

VI

CONCLUSION 88

BIBLIOGRAPHY 91

LIST OF FIGURES

Figure Page

1.1 Heat Conduction in Circular Cylinders 2

1.2 Prcfile for Velocity in Boundary Layer _8S

£ = infinitesimal parameter

a1,a = finite continuous parameter = free stream velocity

= kinematic viscosity

u = characteristic velcitv

C

-u' component of velocity in the x-direction v* _{= component of velocity in the y-direction}

r, 2, , X., g. = absolute

invariants

f. = functions which define a finite transformation

x, Y, U, V, X functior which define an

infinitesimal

t ran s forma t ion.

= a differential form

7, Q differential operators

I = invariant solutions

i = Bessel function of the first kind

/3

1

= Bessel function of the second kind

/3

P unknown function in the boundary layer

equations

J = Jacobian

a0 = identity element

CHAPTER I

INTRODUCTION

This manuscript concerns itself with the problem of developing systematic methods for finding new similarity variables for partial

differential equations in engineering. Similarity variables are a set

of transformations which, when applied, reduce the number of

indepen-dent variables in a partial differential equation. The advantage of

such a transformation is the simplification of the problem at hand.

Similarity is often associated with the physical nature of the

problem. For example, steady state heat conduction in a hollow

dr-cular cylinder is described by Laplace's equation Txx + Tyy =

where T is the temperature and x and y are cartesian coordinates. If

the temperature is constant on the inner and outer surface (see Fig. 1.1).

the physical symmetry suggests that the isotherms are concentric circu-lar cylinders and the temperature distribution could be described by a single variable whose constants corresponded to the radii of the

circular isotherms. Hence, we may postulate from a physical standpoint that a reduction in the number of independent variables is plausible

and a transformation such as

T(x, y) - T(R)

2

2-R Cx

_{+ y}

)2

variables for Laplace's equation.

Boltzmann[1]* considered the problem of nonlinear diffusion which is expressed by the equation

3

=

_[(T)

and found the similarity variables

T(x, y) -T(n).

Later Blasius [2] postulated the form of similarity variables by

reasoningabout the physical nature of the problem. _{Although this} method worked adequately for some problems, many other problems _were too complicated to attack without a more systematic approach. _One answer to this shortcoming was a method which will be referred to _as

the separation of variables. _{As an illustration, suppose we have a}

partial differential equation with one dependent variable T and two

independent variables x and y. _{Separation of variables entails} search-ing for a transformation of the form

n = y g (x)

(1.4)

T = h(x, y) F(n)

such that the resulting application yields an ordinary differential

equation in F and n. _{In practice, severe restrictions of the functional}

form of g(x) and h(x, y) are often necessitated and many writers refer (1.2)

(1.3)

similarity variables.

Though restrictive, the method of separation of variables proved to be suessful for many problems in boundary layer theory. Herzig

and Hansen [3] considered three dimensional flows over a flat plate and assumed that the leading edge was under main flow streamlines which are translates and representable by polynomial expressions. In a later paper 1-lerzig and Hansen [1+] used separation of variables to attack

three -dimensional boundary layer flows in polar coordinates and found that similarity could be achieved by restricting the functional form

of the free stream velocity. Cohen and Reshotko [5], when considering compressible boundary layer flow with heat transfer and pressure grad-ients, found that the necessary condition for the existence of similar-ity placed restrictions on the combination of Prandtl number and

external Mach number together with requirements on the free stream

velocity. A more general attack on the boundary layer problem in curvilinear coordinates given by Hansen [6] revealed that again simil-arity using separation of variables placed restrictions on the free

stream velocity for various curvilinear systems. These and other examples show that the method of separation of variables is effective; but because of the nature of applying the method, definite restrictions on the generality of the similarity variables occur.

Chapter II of this report describes a group theoretic technique

for generating similarity variables. The groups which will be referred to here are continuous transformations with one or more parameters and have the following form:

f1 (x, y, a)

(1.5)

= f2 (x y, a)

In order that the above transformation be a group, three properties

must be satisfied. First, if composition' is the operator of the group and distinct but arbitrary values of the parameter a determine the

nbers of the group, then the composite is also a member of the group. This requirement is called the closure property. Second, the inverse

of each transformation must exist and be a member of the group. The

existence of an inverse (see [7]) is guaranteed if

f1(x, y, a) and f 2(x, y, a) are continuously differentiable at every point.

the Jacobian J )/3(x, y) is nonzero at every point.

Third, there must exist an identity element a0 such that f1(x,y,a°) = x

and f2(x,y,a°) y.

The general theory of group techniques is placed on firm mathema-tical ground by Morgan [83 and Michal [9]. Morgan [lo] gives a

dis-cussion of the application of his theory to the problem considered by Hansen [6] and finds that a simple groupt essentially produces the same results as the method of separation of variables. For problems in

engineering, detailed descriptions of the application of group techniques

are given by Birkhof f [il], Hansen [12], and Ames [13]. Application of

'The composition of two transformations is the successive perfor-mance of them.

tA _{group of continuous transformations with one parameter will be}

the group method for non-Newtonian boundary layer flows may be found in [iLe] and [15].

Perhaps the first effort to find and use a more general group was

given by Krzywoblocki and Roth [16], [17] and [18]. For Laplace's

equation in two dimensions they found the rotation group was applicable:

= x cosa - y sina

= x sina + y cosa (1.6)

= u.

Because Laplace's equation has the property that V2 ü = w(x,y,a) V2 u,

it will be called conf ormally invariant. As will be seen later, con-formal invariance is an essential property of the group technique.

Krzywoblocki and Roth considered the Laplace's equation in higher dim-ensions and some problems in viscous fluid mechanics, but only found

simple transformation groups for the latter. Their failure to find new similarity variables for fluid problems was probably due to the restrictive form

of

their transformation group.

The works of Morgan and Michal lacked a formal mathematical proce-dure to find the invariants of a group. A function is an invariant of a group if it has the same form in the transformed variables as in

the untransformed variables. For example, in the rotation group

x2 + y2 is an invariant of the group because 2 + 2 = x2 + y2.

Gaggioli and Moran ([19] through [23]) found a theorem in Cohen [2LeJ

for the necessary and sufficient conditions for the existence of

in-variants of a group. Combining this theorem with the theory of Morgan and Michal, Gaggioli and Moran considered the problem of finding

similarity variables with a general class of finite groups, but in

their illustrations only simple groups were used. In addition these

authors considered two parameter groups of the type

X. X.

1

x. = C (a, b) x.+ k 1 (a, b)

1 1

and applied the result to a three-dimensional boundary layer problem. Finally, Gaggioli and Moran developed a method for handling auxiliary conditions and demonstrated it with an illustration.

Chapter III of this report describes a relatively new method which will be referred to as the infinitesimal group method. An infinitesimal group of transformations assznes the form

i. = x. + c X.

_(x

...

x )

1 1 1

l

'

fl

where c is an infinitesimal parameter. Infinitesimal groups were

originated by Lie [25] in order to unify the methods of solution for

ordinary differential equations. In addition, Lie developed a theory

of one parameter groups including both the finite and infinitesimal

cases. Some authors refer to one parameter groups of transformations

as Lie groups. A discussion of the application of Lie groups to ordinary differential equations and partial differential equations of the first order is given by Cohen _[2L1]. Ovsjannikov [26] was perhaps the first one to use infinitesimal groups to generate similarity variables of partial differential equations; he considered the non-linear diffusion problem. Later Cole and Bluman [27] published a

des-cription of the infinitesima], group method and an application to the linear heat conduction equation. Bluman [28] gives a more complete discussion of the infinitesimal method and several examples.

Chapters IV and V of this report are devoted to expansion of both the theoretic group technique and the infinitesimal group method and to the application of these developments to a number of cases.

The main contribution of this work is threefold. First, extanding

upon the ideas of Krzywoblocki and oth, a practical method for obtaining generalized groups is presented together with the development of

appropriate similarity variables. In addition a systematic method for carrying an unknown function through the extended analysis is presented together with a procedure for determining restrictions on the nature of that function so that similarity is preserved. Boundary conditions are examined in the light of a set of necessary and suff

i-cient conditions to determine whether they are compatible with the

generalized group. Second, following the ideas suggested at the end of Cole and Bluman [27], this report uses the infinitesimal group to

generate nonclassical similarity variables and extends the method to handle simultaneous partial differential equations. The treatment of the boundary conditions and unknown functions is illustrated. A physical interpretation of some new similarity variables is presented and some numerical results for problems using such variables are given. Third, a possible pitfall in the method of infinitesimal groups is noted with an explanation of its occurrence and supporting examples are

CHAPTER II

SII4ILARITY BY FINITE GROUP METHODS

Continuous Groups

In Chapter I continuous transformation groups have been defined. Because the properties of such groups are extremely important for the discussion which follows, they will be reviewed here in greater

detail.

An r parameter continuous group of transformations has the form

= f(x1,..., x; a1,..., ar), i

It will be assumed that the f. are continuously differentiable func-tions of the variables xi, i l,...,n, and of the parameters

j = l,...,r. Further, the parameters a. must be essential in the

sense that a small variation in each of them produces a definite change in the functions f. For example, suppose that each parameter

a is incremented by an arbitrary small quantity .; then the

para-meters are not essential if

(x1,..., x; ai,...ar) f (x1,..., x; a1 + c1,..., _{ar + tr)} An example of a group with a non-essential parameter is

= x + (a1 + a2)

y + a1.

unchanged. A necessary and sufficient condition (see Eisenhart [29]) for r parameters to be essential is that the functions f. do not satisfy an equation of the form

af. i

a ) - = O.

kL.

Xk (a1,..., r

In addition we require that the inverse set of transformations exist and that we can find parameters a., which are functions of the

parameters a., such that

xi = f1 (x1,..., a1,..., a)

and that

= fX1,...,

; ä1,, ¡) = f1(f1,..., f; ä,.., ä) = x.

The inverse property is guaranteed if the Jacobian J f )

n

J=

is not zero at any point.

The closure property as stated in the introduction requires that

f1

1,

,xn;1,s, a

ii = f(x1,..., "n' b1,..., br) That is 3(x1,...,

x)

= f.(x1,...

X;

c1,...

c)

where the parameters c. are functions of the parameters a. and

b1.

if

ci = i1(a1,..., ari b1,..., b).

Next, we require that there exists a set of parameters a called the identity element such that

i. f.(x1,..., x; a10,..., ar°) = x.

Finally, the operation of composition must be associative.

Continuous r parameter groups of transformations which satisfy the inverse property, the closure property, and have an identity element are called r parameter Lie groups of transformations.

As an example consider the group al

xa x

a2 y a y.

The identity element is a = 1. The closure property implies that if

al ci1

b

=(ab) x

al

then there exists a _Cc) = (a b) such that

a =

X =

(c) X

belongs to the group. The inverse transformation is

i

xa

5

-a2

y=a

y

Invariant Trans formations

is denoted by

A differential form of the kth order and m independent variables

11

y

i n

(x1,..., xm; yl,...,

()k '''

A continuous group of transformations G1 represented by

= f.(x1,..., xm; a)

- G1 (2.2)

y1 v1(y1,..., y; a)

may be enlarged by appending the partial derivatives of y. with

respect to X. up to and including the kth derivative. The resulting

set of transformations forms a group Gk. _{Such a group will be} called the kth enlargement of the group G1. Denoting the arguments

of by z1,..., z, we say that the differential form _{is conformally} invariant under the enlargement Gk if

= E(z1,..., z; a) (z1,..., z). (2.3)

If E is a function of a only, is said to be constant conforinaily

invariant; and if E is a constant, then is said to be absolutely

invariant.

Now consider a path curve h(x1, x2) = O and a one parameter

group

= f1(x1, x2, a)

g2 = f2(x1, x2, a)

with the requirement that the identity element is a° O. The curve h(x1, x2) = O is an invariant curve if h(1, _whenever

h(x1, x2) O. _{Expanding h(1, x2) in a Taylor series we obtain}

h(1, X2) h(x1, x2) + Qha + Q2ha2/2 +

where

II

I ( )

a0o

+2j

_;;

ao

and Q2 QQ. A necessary condition (see Cohen [2+]) for h(x1, x2) to be an invariant curve is that Qh O whenever h(x1, x2) O; that is

Qh = w(x1, x2) h(x1, x2). In that case we find that Q2h = QQh = Qwh + wQh = (Qw + w2)h and similarly if Q% = 8(x1, x2)h then QQh = (Qe + 8w)h. Hence the vanishing of Qh, whenever h(x1, x2) does, is both a necessary and sufficient condition for h(x1, x2) O to be an invariant curve. This argument extends to the differential

form and leads to the following theorem (see Morgan [8]):

* ₍₁₎

Theorem 2.1. If is at least in the class C with respect to each of its arguments, then a necessary and sufficient condition for

to be conformally invariant under a one-parameter group of transfor-mations is that V = w1(z1,..., z) 41(z1,..., z ) p where V

z)

_{( )} i-

...

+ (z1,..., z )

ì_(

) p z p 3f. and =

-i.

3a

z;

'). (2.k) (2.5)

To give an example: the group given by equations (1.6) may be applied to Laplace's equation with the result that V2ü = V2u and we conclude that Laplace's equation is absolutely invariant with respect

to that group. _{Suppose we consider the Poisson equation of the form}

*

The class C1 is the class of continuous functions with _one

2 2

3u

;:_

;--H(x, y, u)

y

and inquire as to the necessary conditions for this equation to be

absolutely invariant under the group (1.6). In that case we have

2

au

au

+ - H(, , )

=

4

+

-

H(x, y, u)

a ay ay ax

which requires that H(, , ü) = H(x, y, u). Now differentiating with respect to a and setting a e0 we see that with group (1.6)

ah

- .y + .-37x O.

(2.6)

(2.7)

(2.8) In this case the w1 in (2.5) is zero and the above equation is the necessary and sufficient condition for H(x, y, u) to be absolutely

conformally invariant. The solution of (2.8) is

1-i

H(

2

= x

i-y,u)

where H1 is an arbitrary function. Of course, H1 can be just a

con-stant.

Absolute Invariants

Let us inquire under what condition a function D(x, y) is an absolute invariant of the one parameter group

f1(x, y, a)

= y, a).

Without loss of generality, we may assume that a0= O is the identity

element. In order for D(x, y) to be an absolute invariant we require

D(,

) = D(x, y).

Expanding D(, ) in a Taylor Series we have

D(,

) = D(x, y) + QDa + Q2D a2/2

j

(29)

Now the oper:tor Q has the property that Q2D = QQD, Q3D = QQ2D,...,

Qfl

= QQD, and if QD = O, then au. the higher order terms in the

series given by (2.9) are zero. As a result we have the theorem given by Cohen [21e]:

Theorem 2.2. The necessary and sufficient condition for

D(,

) D(x, y) is that QD = O.

The above result extends to two parameter groups. For the group = x C.(a, a2) + K(a1, a2), we have the following theorem (see

Gaggioli and Moran [23]):

Theorem 2.3. The necessary and sufficient conditions for to be an invariant are =j aa1 _ax1 a1 = a1 o a2 = a2

!.

.L._

_(n)

= O 1:l 3a2 i i a1 = a1° o a2 = a2 (2.10) (2 11)

A group with m transformations and r parameters has only the limited number of functionally independent invariants expressed by the

following theorem (see Gaggioli and Moran [23]):

where Q

-

a aa a

-ox

ai

+ -i-aa a i

-ay

Theorem 2»+. Let a group of transformations be expressed by i f1(x1..., _Xm a1,..., ar) and i 1,...,m. Also define

3f. = o (2 12) j o a -r r

The number of

functionally

independent invariants is equal to m -where o is the rank of the matrix whose elements are given by (2.12).

As an example consider the group given by (1.6). According to the theory presented above, the invariant n must satisfy the equation

-y-an an

(2.13) A general solution to (2.13) is n h1(x2 s y2) where h1 is an

arbi-trary function.

Similarity

When a differential equation is conformally

invariant

under a

group, then the solutions of the transformed differential equation will have precisely the same form as that of the original equation. For example, if u = u(x, y) is a solution to Laplace's equation, then

= ) is a solution of that equation under group (1.6) and

is the same

function

of and as u(x, y) is of x and y. A solution which has this property is called an invariant solution with the understanding that a particular group of transformations is

associated with it.

which may be expressed by

tn

(2.1L)

g1(Y1,. .., y; x1,. .., g(y1,. ..,

y.

X1. .. X ) (2.15) where the x's may be thought of as independent variables and the y's may be thought of as dependent variables. According to Morgan [8] the invariants have the property that the Jacobian J

g1,..., g) J

= _{y1,..., y)}

is not zero and the rank of the Jacobian matrix

is equal to m - 1.

For Laplace's equation under the group (1.6) the invariants

are-2 2

n = h1(x + y ) g = u.

Suppose that an invariant solution of Laplace's equation is known arid is denoted by u = I(x, y). By definition of invariant solutions

I(,

). Substitution of the invariant solution into the invariant

g = u implies that = I(, ) = u I(x, y) and that I is an absolute

invariant. The necessary and sufficient condition for I to be an

invariant is 3x 3a

aa

31

aa°

3 -(2.16) (2.17) (2.18)

The solution to (2.19) is

I h (x2 + y ) =2

where h2 is an arbitrary function. Hence, the similarity variables

for the problem are

u h2(n)

2 2

n = h1(x +y ).

Letting h1(x2 + y2) = X2 + y2 and transforming Laplace's equation with the similarity variables, we find

ah 2 an (2.19) (2.20) (2.21) (2.22) (2.23)

Equation (2.23) can be integrated by elementary methods and has the

solution

2 2

h1=b1ln(x +y)+b2

(2 2)

where b1 and b2 are constants.

Before stating the general theory of similarity variables, let's summarize the procedure illustrated for Laplace's equation in more

general form. First, referring to group (2.2) take an invariant

g1 g.(y1,..., y;

X1,..., x)

and substitute in it an invariant

solution y. I.(x ,...,

x )

and observe that g. becomes an invariant

2.

ii

_m i

of the subgroup i = 1,..., m. Second, apply the necessary and sufficient condition expressed by theorem 2.2 and obtain the result

where the functions g. may be determined by substitution into the differential equation under consideration,

The above procedure is a general one and is placed on firm math-ematical grounds by the following two theorems due to Morgan [8].

Theorem 2.5. Suppose we consider y6 and to be implicitly defined as functions of the x. and iL by the equations

z6(x1,..., x) g6(y1,..., Y; x1,...,

x )

in

6l'"'

=

.''

where g6 are absolute invariants of a group. A necessary and

suffi-cient condition for y6 to be exactly the saine

functions

of x1,.,.,

x

as the are of the iZ is that

and

where n1,..., _{nm_i} are the invariants of the subgroup:

X),

i = 1,..., m.

Theorem 2.6. If each of the differential forms . (see (2.1)) in i

a system of partial differential equations is conformally invariant

under the kth enlargement of a group, then the

invariant

_{solution of} that system can be expressed in terms of the solutions of a system of

assume the form

n. = n.(x1,...,

_i

_i

x)

g

g(n1,...,

(2.25) (2.26)

the form akF n A6(n1,..., _cm-1 F1,..., F;

''

_k an1 Boundary Conditions

As pointed out by Gaggioli and Moran, the auxiliary conditions must be compatible with invariant solutions. For an illustration of the concept consider Laplace's equation and denote the invariant solutions by u = I(x, y) and ü = I(, ). Suppose that the auxiliary

conditions are

A(u, x, y) = O when r(x, y) = 0. (2.27)

Transforming (2.27) under the group (1.6), we have

A(ü, E, ) = 0 when r(, ) o. (2.28)

Because in the group (1.6) = u, the above expression can be written

as

A(I(x, y), x cosa - y sina, x sina + y cosa) 0 (2.29)

when

r(x cosa - y sina, x sina + y cosa) 0. (2.30)

The auxiliary conditions are said to be compatible if (2.27) agrees with (2.29) and (2.30) for all values of the parameter a.

As is easily seen by comparing (2.27)with (2.28), the boundary conditions are compatible under a group of transformations if they

are invariant curves. Consequently, we may require that A(u, x, y) O

and F(x, y) = 0 are invariant curves and according to theorem 2.1 we

and I ai'

.2J

ai' aa

j

y) r(x, y) I aA a aA aü

i-

o + o

V +

= -a a=a a=a (x, _y, (2.31) u)A (2.32)

CHAPTER III

SIMILARITY BY INFINITESIMAL GROUPS

Infinitesimal Transformations

A set of infinitesimal transformations can be generated from a

set of continuous transformations. For example, consider the con-t inuous con-transformacon-tion

(x, y, a)

(3.1) = (x, y, a)

and let a0 be the identity element. Replacing the parameter a with

a° plus an infinitesimal óa and expanding in a Taylor series gives g (x, y, a0

t

óa)

x + -

óa + O(6a2)

a°

(3.2) (x, y, a° + 6a) = y + - óa t O(6a2).

a0

The system (3.2) can be thought of as an infinitesimal transformation.

By again using Taylor series, the effect of an infinitesimal

transfor-mation on a function (x, y) is seen to be

- - aw w(x, y) = w(x,

y) t

a° + a0 ]ôa t... Let X = and Y -a°

-.-

; then the second term in the above series

can be written as Qwóa where Q + ,.

Once the infinitesimal ax

transformation is known the term Qw can be written and for this reason

an infinitesimal transformation with the understanding that it represents one only in the above sense.

Using the notation introduced above and letting óa = e the system

(3.2) becomes

= x + Xc + O(e2)

(3.3) = y + ye + O(e2).

Let us investigate whether the transformation (3.3) is a group to

order 2. The identity element is e O and the Jacobian of the

trans-formation is 3x ay =

:i-i

3x 3y ax ay 2 = i + c(X + Y ) + O(e ) O. X _y

Hence the inverse transformation always exists. Letting i = x + Xe1 and = y + Ye2 and substituting into x + Xe2, we obtain

X2(X1 y1) x + Xe1 X(x + e1X, y c1Y)e2 x + Xe1 + Xe2 + O(c1 e2).

Because e and e are arbitrarily small infinitesimals, their sum is an infinitesimal and successive transformations lead to a member of

the group. All properties are satisfied and the system (3.3) is

a group.

As has

been

noted above an infinitesimal transformation can be

continuous group of transformations. For example, consider the group - a a a x = e X + e (1 - e )y

-y=e

y.

The corresponding infinitesimal transformation is

This question arises: Once the system (3.5) is

known,

can the system

(3.4) be found? There are two procedures illustrated by Cohen [24]

and one will be demonstrated here. To find the system (3L4) from

(3.5) solve the following set of equations:

(3.6) 2 i

As can be easily checked by the method presented in Chapter II, the

constant n of

integration

for the first pair of equations is an

invar-iant of the systems (3.4) and (3.5) and is found to be

or

x+v x+

n = ---t-

-

--y y

Integrating the second pair of equations (3.6), we have

in 2a

(3.7)

(3.8)

where c2 is an integration constant. Because at a = O = y, then = x + (x - y)c

(3.5)

= y. Equations (3.7) and (3.8) are simultaneous equations representing the equivalent finite continuous transformation which

is generated by the infinitesimal group and their solution is easily found to be the system (3M).

Invariants

The necessary and sufficient condition for a function

n(x,

y) to

be invariant under an infinitesimal transformation is the same as for continuous transformations and is given by

Qn = 0. (3.9)

Also, as in the case of continuous transformations, the necessary and sufficient condition for a curve H(x, y) = O to be an invariant curve under an infinitesimal transformation is

QH w(x, y)H (3.10)

where w(x, y) is an unspecified function. Considering x and y as independent variables and u as a dependent variable, an infinitesimal transformation can be written as

= x + Xe + O(e2)

y + Ye + O(e2) (3.11)

ü u + Uc + O(e2).

If we append to (3.11) the partial derivatives of ü with respect to

and up to and including the kth derivative, then we obtain another

th

k

3u 3u (x, y,

_;,

ax1ayi

=

where i + j k. Letting z1 through z denote the arguments of (3.12)

th

and z1 through Zn be the corresponding coefficients of £ in the k enlargement of the group (3.11), then (3.10) can be written as

a a

- +

. . . + z - = w (z , . . . , z )4 .

Z1 1 Z2

ai n 3z 1 .i. n

1 2 n

Equation (3.13) is then a necessary and sufficient condition for the

to be an invariant curve.

Canonical Variables

As was mentioned earlier an infinitesimal transformation may be represented as Qw and referring to the system (3.3)

a a

Q = X ( ) + Y ( )

With (3.l) we see that

Qx = X

Qy = Y

Consider a change of variable which is denoted by

X1 = x1(x, y) y1 = y1(x, y). Then (3.12) (3.13) (3.15) (3.16) sufficient enlargement of the group (3.11) the conditions expressed by (3.9) and (3.10) can be written for a differential equation of the

ax1 ax1

Qx X +Y-X

_{1 3x ay}

i

ay1 _ay1

Q y1 X - + Y._.... - Y1

and the new infinitesimal transformation under the change of variables

becomes

= + 1 _(3.18)

where is a function of X1 and y1. This procedure allows us to make

a change of variables and find the representation of the new

infini-tesimal transformation by solving the system of equations (3.17). As an illustration consider the rotation group (1.6) and the change of variables defined by

/2

2 p

=\jX

+y

(3.19) -1 8 = tan (y/x). a ( ) a , we find that

Since for the rotation group Q - -

y

ay

Qp = - y + x

ax ay

2

2-½

2

2t½

=-yx(x +y)

+yx(x +y)

=0

and ae ae

Qe = - y

_;.

+ x 2 2 -

y

X 2 2 2 2

X-y

x-y

hence the new infinitesimal transformation is represented by

Qw(p,

e)

=

and the new infinitesimal transformation is

U= e + £

(3.20)

It is interesting to note that the invariant of the group (3.20) is p. The variables which reduce an infinitesimal transformation to the

3w1

form Qw1 =

_r-

or

-r-

are called canonical variables and the

Y.L xl

resulting infinitesimal group is said to be in canonical form.

Following this notation the transformation (3.19) is a set of

canon-leal variables for the rotation group.

Reduction of the Number of Independent Variables

in a Partial Differential Ecuation.

Z.Z.

+Z.c i1,...,n

i i i

The infinitesimal representation of (3.22) is Q = Z1 .- + ... + Z

n

Letting z1 through z denote the arguments of the differential equation (3.12) and rewriting, we have

4(z1, Z2I

z)

= O. (3.21)

The infinitesimal transformations for the z' s may be written symbol-ically as

(3.22)

(3.23)

As shown in the previous section of this chapter we may find a set of canonical variables which will reduce the transformation (3.22) to

canonical form. Let V. i = 1,..., n denote the appropriate canonical variables and let the

canonical

form of the group (3.22) be denoted by

= V1 +

(3, 21+) = y. i = 2,..., n.

i i

If the differential equation (3.21) is conformaily invariant (see Chapter II) to the order when transformed by the group (3.22), then according to theorem (2.1) of chapter II we have

z

±+

...

+ z

i z1 n az w(z1,..., z) 4(z1,...,

za).

n

Introducing the canonical variables y1 we find that (3.25) becomes

a

-w

-

2(v1,..., y) (y1,..., v)

where is the differential equation when transformed by the

canon-ical variables. Also since v, i 2,..., n are absolute invariants

of the group (3.21+), we may apply the necessary and sufficient

condi-ditions given by theorem (2.2) to find that

3v. i

= O i = 2,..., n. (3.27)

Vi

If

l is absolutely invariant wider the group (3.21+), then (3.26) become s 1 = o. 3v1 (3.25) (3.26) (3.28)

Equations (3.27) and (3.28) guarantee that the variable y1 does not appear either implicitly or explicitly in

.

Consequently, the number of variables has been reduced by one. If z1 were an

This result is precisely the one needed to obtain similarity solu-tions.

Example

The infinitesimal transformations corresponding to the rotation

group (1.6) are

= X - yE

= y + x (3.29)

Laplace's equation in two dimensions is absolutely invariant under the group (3.29). We seek canonical variables s1, s2, and q1

such that

) = s1(s, y) + c

S2( ) = s2(x, y) (3.30)

q1,

) = q1(x, y).

An appropriate choice for

l and s2 is given by letting s1 e and s2 p in (3.19). Inspection of (3.29) shows that u is an appropriate

choice for q1. With these choices (3.30) becomes

= etc

p=p

(3.31)

In group (3.31) u is an invariant and accordingly = o. Treating 6 and p as new independent variables, Laplace's equation

transforms to

i

The similarity variables generated by the group

(3.29)

are

/2

2 p

x +y

u = u(p)

As is seen from this example, the process of

_introducing

_polar coordinates and requiring that = U is the same as searching for similarity variables under the group of rotations.

Similarity Solutions in Practice

In practice we may construct similarity solutions by a procedure

described by Bluman and Cole [27J. _{The method will be described for}

the case of two independent variables and two dependent variables; however, the ideas are applicable to other cases.

The infinitesimal transformation is assumed to have the form

= x + X(x, y)c + 0(c2)

= y + Y(x, y)c

0(c2)

ü = u + U(u,

V, X,

y)c +

(2)

= y -

V(u, v

x

y)

*0(c2)

and the independent variables are x and y while the dependent variables

are u and y. _{The differential equation may be represented by}

k k 3u av au

au

_av

il'

nl

axy

axay

(3.33) (3 34) (3.35)

where i + j k and n s i = k. We may append to the group (3.3Le) _the

transformations of the derivatives of _{u and y which appear in $. As} the determination of the transform of a derivative requires some careful

au

manipulations, the transform for

wil1 be presented here. By the

chain rule

ìL

ax ax a

aya

To apply (3.36) we must determine and remenbering that terms

of order

e2 and higher are dropped.

Differentiating the first of

equations

(331) with respect to Z ,

we bave

Differentiating the second of (3.3L1) with respect to and using

(3.38), we have O = Z. + (Y) +

O(e2)

ax 3 or ax a i = . + - (X)e +

O(e2).

dX

ax

From (3.36) we see that

a

()

a

ax

a

ax

= -

(Y 1- Y

*

+

O(e2).

a2

x

The above equation tells us that is

at most of order e

and e

is at most of order (e2). Hence (3.37) becomes ax O(e2) i =

=-+ eX

ax X

or

ax

ax

_(e2).

-

i - cX + 0

a x

The above eçpression tells us that .j. i + O(e) and that we may

further simplify to obtain

(3.36)

(3.37)

(3.38)

Similarly

- c + 0(c2). (3.42)

Substituting (3.141) and (3.42) into (3.36) gives

and = 1 - c(X ) + 0(c2). ax x au (1 - £ X ) + ( - c Y ) + 0(c2). -ax ax x ay x From (3.34)

au_au

-+ Uc-+ O(c)

aü

au

Cc2).

=---+cU +0

ay ay y

Substituting (3.44) and (3.45) into (3.43) gives

3ü

_{= - + cL -}

au

X

xx

u + U - Y u J + 0(c2). x X y (3,143) (3.45) (3.46)

Having calculated ail derivatives in the barred variables and

letting denote the differential equation (3.35) in the barred

var-i ables, we requvar-ire that X, U, Y, and V be chosen such that

= (i + £ (x, y, u, v)). (3,47)

Equation (3.47) is the necessary

condition

that be conformally

in-variant under the grout (3.314). If u = 01(x, y) and y = 82(x, y) is a solution to , then ) and 82(, ) is a solution to by

virtue of (3.47). Using (3.34) we see that

Expanding (3.48) in a Taylor series we have

e1(x, y) + (X u + Y u) 61(x, y) + tic + 0(c2)

and we conclude that

Xu +Yu

U. (3.'ig)

X y

Similarly we find that

X

sYv =V

(3.50)

X y

Once U, V, X, and Y have been determined to satisfy (3.47), then (3.49) and (3.50) define the similarity variables for the differential

equation 4. Actually (3.49) and (3.50) can be used when satisfying

the invariance condition expressed by (3.47). According to the term-inology of Bluman and Cole [27] the similarity variables obtained are cafled classical if (3.49) and (3.50) are not used in (3.47) and

non-classical if they are used. All similarity variables generated in this thesis by infinitesimal methods will be the non-classical type.

CHAPTER IV

APPLICATION OF THE FINITE GROUP METHOD

Burgerst Equation

The Burgers' equation u + uu = u has been used as a

rnathe-y

X XX

matical model of turbulence [30] and in the approximate theory for

weak nonstationary shock waves in a rea]. fluid [31]. We seek groups

of transformations of the form

= f1(x, y, a)

y, a)

_(L1.l)

= fu(a) u + f5(x, y, u, a)

such that the Burgers' equation is conformally invariant. _To accom-plish this task, we append to (4.1) the relevant partial derivatives

which are 3f 3f 5 5 3u 3x +

)

-= (f + 3x

43x

3x (14.2) 3u 5 3u.. 3v

- + - +

)

-Lê3y 3y u 3y 3x 3f 3f 3u 5 5 3u 3x - = (f ay

14 -j +

+

-

)

.-3f 3f 3u 5

53u

+ (f14

-s

+ .-

.._.) (14.3)

2_ 2 _3f 3f

3u

3x

5

)

53u

3u 3u

f -+- +

₄₃

ay au 35E y 2 32f 2 3f

3x2

3u

5

+--+ 2

3u

5 5 3u + (-w)

(f

-+ 2 2 au ia 3x 4 2 2 32f 3f 2 32f

3u

+ .

!.

(2 f4

axay

+ 2 3x3y + 2 3 u + 2 5 3u 3x 3x _3u

_axay

_{au3y 3x}

+2

5 3u 3u3x _3y 2 2 32f

5__ 3u2

3f 2 5 3u

()

+ 2

au

3u3y 14) a ay

_ay

32f5

In (i.L) it must be assumed that O; otherwise 2

the square of the derivatives of u would appear on the right-hand side of (4.4) and would force this assumption in the present example. In problems where the squares of derivatives appear in the differen-tial equation, such terms should be retained. Using (4.l),(4.2), (4.3) and (4.4) we form the Burgers' equation in the barred variables and

equate it to zero. Since we cannot place any restrictions on u or

its derivatives, the coefficients of such terms must vanish if _they do not appear in the differential equation. _{Inspection of (4.4)}

2

reveals that the coefficient of is equal to

3x3 y

-2f

-2--.-

ìL

43x 3x

3u 35E 35E

3f

and for this to vanish, either

_{O or -. =f}

_{The latter}

37

alternative is rejected since

it

would cause ü to vanish identically.

The coefficient of u, which arises from the term ü

_{ti_,}

is

3f 5 3x

I- - -

_43x

_3x

3f

5

which forces -

_dx = O. With the above observation a search for terms

which do not have u or its derivatives as coefficients reveals that

3f

ay 3y

which forces f5 to be a constant. The coefficient of is

a2

+ f4 f5

* -

4

-

o.

3x

The final, requirement is that the coefficients of the terms which

appear in the differential equation be equated; so that

2

3X_f

,=

3x 3 143y From (4.6) we have 3x 3

f4

(.7)

2 ay 4

Substituting the results of (4.7) into (4.5) gives

3x

5 4 (4.8)

The inverse transformation of (4.9) is

i

9' -

y

f4

The transformed Burgers1 equation is

+uu -u

)

y x xx 4 y x xx

which satisfies the conformal invariance requirement. For convenience

let f4 and i-i(a) = f5(a) f2(a). Using (4.10) and (4.1) the transformation group becomes

= f x + Hy - 2

y=fy

- i H u = - U + -s-f'

in Chapter II. Individual transformations can be denoted by

f(a.)x I- H(a.)y i i

- _2

y1 = f (a1)y H(a.)

-w

_u+

2 - f(a.,) f2(a ) i i (4.10) (4.11)

The transformation (4.11) must satisfy the closure property defined

(4.12)

A constant multiplied by a function of the parameter a could

be added to obtain a more general group.

x f4 - 5

(4.9)

where a. is any real number.

The closure

property

requires

that if

and x2 are transformations then there exists an x3 such that

x2(x1, a) = E3(x, y, a3).

S irnilarly,

'2"l' S1 a2) = 3(x, y, a3)

U2(U1 a2) = ü3(u, a3).

Substituting (4.12) into (4.13), (4.14) and (4.15)

X3 f(a2) [f(a1) x + Fi(a1) y] * Fi(a2) f2(a1)y

(4.16)

= f(a3) x + H(a3)y

- f(a2) (f2(a1) x) f2(a3)y (4.17)

H(a)

H(a2) = f(a) f(a1) + f2 +

f2()

u Fi(a3) + = f(a3) f2(a3)

The above three equations require that

H(a3) f(a2) Fi(a1) + Fi(a2) f2(a1)

f(a3) = f(a1) f(a2)

(4.13)

(4.18)

Equation (4.19b) implies that interchanging a1 and a2 does not change

a3 because the real numbers are commutative. Using this result in

(4.19a) gives

Fi(a3) = f(a2) Fi(a1) + Fi(a2) f2(a1) f(a1) Fi(a2) + Fi(a1) f2(a2)

which may be rearranged as

(L ₁₄₎ (4.15)

The subgroup iZ and possesses an

invariant.

separation of variables) which forms the subsystem of equations

H(a1) f(a2) [1 - f2(a2)] H(a2)[f(a1) (1 - f2(a)]. (4.20)

Equation (4.20) is satisfied if H(a) f(a) (1 - f(a)) and (4.11)

becomes

= f(a) x + f(a) (1 - f(a))y

= f2(a)y _(4.21)

= f(a) + (1 - f(a)) / f(a).

Satisfying the closure property has produced a result which immediately satisfies the existence of an identity element a if f(a ) 1. Since

o o

equation (4.21) is seen to possess an inverse, it satisfies all the

requirements of a group. The invariants solutions of of (4.12) ax. -3v +

-

-'-are described ax . -i + au by

functionally

independent ax.

io.

(4.22) 3x 3a

aa

o a =a ay 3a o a=ao

Substituting (4.21) into (4.22) gives

ax. ax. ax.

f'(a ) (x-y)

o -.- + 2 fT(a )yax o 3y - f'(ao) (u+l)

-.

0. (4.23)

Assuming that f'(a) does not vanish and that A1 = A1(x,

y)*

ax ax

we have

(x - y) ._! + 2y = 0. (4.24)

or

ax2 ax2 ax

(x - y) - + 2

ax

y_-(u+i)i-o.

au

Again using the method of Lagrange the solution of (4.27) is found to be

(X+Y

A2 , (u +

l)V½)

Equations

(4.26) and (4.28) implicitly define the sought

similar-ity variables. A special case is

= n = _(4.29)

= (u + l)Y½ F(n). (4.30)

Substituting (4.29) and ('+.30) into the ßurgers' equation gives

F"-FF'+½nF'-s-½F=O

(4.31)

[F' - ½ F2 + ½ nF] O.

One integration results in

F' - ½ F2 -s- ½ nF const. (4. 32) (4.27) (4.28) x-y 2y O (4.25) The solution to (4.25) is x+ (4.26)

y-A second invariant of (4.21) is found by letting y-A2 = y-A2(x, y, u) and (4.23) becomes

A =

r b(b+l) ... (bi-r-1)r

Laminar Boundary Layer

The boundary layer equations for steady, incompressible two-dimensional flow (see Schlichting [35]) are

* _{dU 2}

*3u

*au

au

U --

t V = U

* 2

ax

ay

which is a form of the Riccati equation (see ref. [33]) and it may be transformed into a linear second order differential equation by the

trans format ion

F - 2 Pfl

(L33)

Applying (L.33) to (.32) results in 2

+ n

+ C 4 O. (43l) fin n

i

The solution to (L3 J4) _{is found by standard methods (see [3k]) and is}

C C = C10 1Fj ( ; ½; z1) C11 (½ + ; /2 z1)(Ll.35) where 4 (ri) =

(zr)

2

z1 = - ¼ n

1F1 (c2, b, z1) =

A (z )"

o r i ('4. 37) c2(c2+1) .., (ctx-1) ('4.38a)

au

av

r

+ r =

O.

3x _3y

A set of dimensionless coordinates is defined by

y L

lu

L

VC

V * * x =

Lx

u U u C

where U _{is a characteristic velocity, L is a characteristic length,} and y is the kinematic viscosity. Introducing equation (4.39) into

(4.38) we have U dU

au

au_

32u

uf V

_{3x +}

ay

C 3u

av_

axtay

O

y

v=

U C

lu

L C

'4V

(4.38b) V (L1.39) ('+.40a) (4.4Gb)

We want to determine continuous one parameter groups of tran8-formations which will keep (4.40a) and (4.4Gb) conformally invariant.

U _dU

For reasons which will become clear later, the term will be assumed to be a function of x and y denoted by P(x y). A general

group

= f(a)u + f5(u, y, x, y, a)

f6(a)v + f7(u, y, x, y, a)

= f1(x, y, a)

is assumed. The relevant derivatives are

3f

3f

3f

3u 5 5 au 5

3v3x

--

¼f +t

_{4 a,}

t -e-w

_ax

_{3v 3x 3x}

3f

3f

3u 5 3u 5

t

(f

₄

tt

_{3y 3u}

s-

_{3y 3v 3y 3x}

3f

3f

3f

3v_

_!!-+...2+

7 3u 7 3v 3x a y

-

3x 3x

5t

(4.43)

+ (f

3f7

3f7

3f

7 3v

6 3y

T

3y

57

ay

3f

3f

3f

5 5 3u 5 3v 3x

s

s

+ ciy

_{4 3x}

T

(Li.

44)

3u

3f

5 3v

(f4

3y 3u 3y

V

2-

3f

3f

3f

2 3 _{U_ 3u 5} 5 3u 5 3v

3 x

7

3x 3x 3u

3x +

'

7

ay

ay

3f

3t

3f

2 5 5 3u 5 3v 3

(f

_{4 +}

+ .

+

.-2

32f

2

3f

a2f

3f

2

32f

₂

3u

5

3u

5 3u 5 5

3v

53v

3x

+ s +2 + +2

)(--)

4 _3x2 _3x2 _3x2 3u 3x

3u3x

3v _3x2

3v3x 3x

3y 2

32f

32f

2

f

+(2f

3u

+2

+2

3u

3u

35

2 5 3v

4 3x3y

3x3y

3u3y

+ 2

3x3y

3u

3vay

3f

2

32f5

3u 3v

_{(3x) (!L)}

3v

₊₂

₊₂

) 3v 3x3y

3u3x

3v3x

3y 3y 3 + 2

32f

2

3f5

32f

_53u

3f

_53v

2

32f

5 + 2

4

+ 2

3u3_{y . +} 2

+ 2

3v3 y ay

ay

_ay

_ay

(4.42)

3v)()2

(4.45)

Using (Li.1U) through

(L45),

the system of equations

(L.iO)

is written in the barred variables. The first step is to search for terms with coefficients which do not appear in the original system.

2 Since the coefficient of

2f

39 39 39 39

i'

3x

must be equal to zero, we choose - = O. Actually we could, at this

ay

point, choose O. However, such an assumption uld cause the

Y2

coefficient of --to be zero. To achieve conformal

invariance

all ay

other coefficients of terms in the differential equation uld be

re-quired to be identically zero, but this requirement uld cause

2 3f

2

unnecessary restriction. Since the coefficient of 4

,

(p.)

,

3f5 _ay Y

must be zero, we choose = O. With these assumptions the coefficient of in the momentum equation becomes

ay

3f

f

5 _ay

3f

which

requires that O. _{Finally, observing that the terms without}

any coefficient reduce to

3f

r 3f

5 + u f

-e- is taken to be zero to avoid f

_5-

ecualing - u f

If f5 were equal to a function of the parameter a multiplied by

u, nothing would be added to the group (L1.tll). Consequently, f5 is

taken to be zero. With 'the above assumptions the transformed

rtmentun1

equation becomes au 2 ax

-{uf2

t+f

f f

u {f --}+

_ax

14 3x ay 14 3x 14 n 3 6 14

ay

V 2 2 )} - PCx, ) = O.

f

}aU{f (!Z

1 ..2 y ay

ay

Before proceeding further with the nmentum equation, the continuity equation will be considered to see if there are any additional

res-trictions. The transform of the continuity equation is

au

ax

au

_.L}

. f14 - } .$. ax ay

4ax

au ay

af

av

+ {

f '. + 7

ay

=

ay6

The last term requires = O and the second term requires

af

-+--2.

z.=

. 14 ax au ay ax a f

-IIl-=f

14 ax 6 ay Then (11.147) becomes au av C 6

i

ay (4.146) (14.148)

To satisfy the condition of conformal invariance for the continuity equation, it is necessary to require

(11.119)

Because 0,

cannot

be a function of if

(4.49)

is to be

y y

a2

satisfied. Hence, we assume that --= 0.

Returning to the momentum equation (4.46), we assume that

and

uf2

f

¿4 ax 4 7

ay

in order to eliminate terms which do not appear in the original momen-tum equation. To satisfy (4.51) we set f7 p1(a) h1(x) u.

Substitu-ting into (4.51) furnishes

a p1(a) h1(x)

O. _{(4. 52)}

Finally, to satisfy the

condition

of conformal invariance we let

2 ax f f4 = 4 a a = f9(a) (4.51) (4.53) P(g, ) f9(a) P(x, y). _(4.54)

Thus carrying an unknown function can be accomplished by simply forcing it to satisfy the condition of conformal invariance for the equation

as a whole and accepting whatever functional form arises.

To satisfy the

condition

_{of conformal invariance for the system}

of equations (4.40), we have the following restrictions on (4.41): ax av f5 0 (4.55a) (4.55b) 47

3f 7 ay f f h1(x)

f= o

+ ax 6 Case I. 0.

From (Lf.55e) 0, whereupon the group (L14l) becomes

= f1(a) u

may be found by forming simple ratios and are

(4.55c)

(L.55d)

(1.55e)

(L455f)

The system of (L55) does not entirely determine the transformation (LLLl). Equation (L 55f) implies that = f6 and (L.55d) implies that

-. Several cases must be examined. 3x f

As long as there exists an a° such that f(a°) = f 5(a°) = 1, then (L.56) meets the requirements of a group. To generate the

Al=

Upon integrating we find that f6 y = f - p1(a) .-. x.

-Since x = --x, equation (4.59) is easily inverted and we have

f6

y=

i

p1(a)

y.'.

X.

f6(a) f62(a)

To satisfy the closure property there must exist an a3 such that

a2) = y, a3). With (4.60) we require that

p1(a1) p1(a2) f4(a1)

f5(a2) f6(a1) + 2

X] +

f5 (a1) f62(a2)

[2]x

p1(a3) (4.61) + X. f6(a3) f62(a3) (4.59) (4.60) A2 u (4.57) A3

_{= y V}

If in = O the invariants reduce to the classical ones used for the case of a flat plate.

Case IL h1 = 1.

Equation (4.55)e is solved to give

As in the case of the Burgers' equation we notice that

f6(a3) = f6(a2) f6(a1) and that a3 is not affected by interchanging

a1 and a2. Hence, we write

p1(a3) p1(a1)f6(a2) p1(a2)f(a1)

f62(a3) f62(a1)f62(a2) f62(a2)f62(a1)

p1(a2)f5(a1) p1(a1)f(a2)

f62(a2)f6 (a2) f62(a2)f6 (a1)

or

p1(a1) [f6(a2) - f4(a2)] = p1(a2) [f6(a1) - f(a1)]. The above relation implies that p1(a) = [f6(a) - f4(a)] and the group of continuous transformations becomes

X -

x

6

_1

X y

= çy +

(f6 -u = f u = f6 y + (f6 - f14)u (L,62) o

Letting a denote the identity element for the group, the

invar-iants are found by solving

(4,63) Since there exists one invariant of the subgroup and , we choose

= X1(x, y) and obtain ax.

-+

ax.

_iau

- ax. i

.-

-av

+ -

_aa

ax.

i

= 0.

0ax

aa

ax ax

(fk'(a°) - 2f6t(a0))x._. + [-f6a°)y + (ft(aO)_ft(aO))xj_ o.

(L

₆₄₎

The solution to (14.64) can be found by the method of Lagrange and is

f(a°)

f1(a°)-f(a°)

(f1(a°)-2f(a°))

=

(4.65)

To check whether X1 is an invariant, form (4.65) in the barred

varia-bles and use the group (4.62) to obtain

r_

LJ

+Lxj

f'

₆

f'-f'

₄₆

(f1-2f)

f-2f

i

'

11

4

ri

L>'i +__TLXJ

where f4' and f6' are evaluated at a

a0.

Expanding (4.66) we find

f'

₆

f'

₆

f

f

+1

(f-2f)

f1-2f

_(f'-2f6t)

f-2f

[f4 i

f6-f4

[f4

1

y[x]

Li

[xl

f'-f'

₄₆

ft-f'

₄₆

f-2f

f,-2f

[xJ (14.66) 51

(4.67)

Requiring equality between (4.67) and (4.65), forces

f,

6 i and (f'-2f6t)

=1

f t_f 6 4 6 'LI-(f '-2f') f,'-2f6' f6-fI+ f 62 L .1: - - r-c ta I IiL+

-.I

+

I2t

l.

f6J

_L6J

(14.68) (4.69)

Equations (4.68) and (4.69) are satisfied if f4 (f6)m where m is an

arbitrary constant. Our earlier choice is thus justified.

A second invariant may be found by choosing A2 X2(x, y, u) and solving equation (4.63). The result is

m

-7

u x ) = O

in i m-i

--

-

-ii--H2(A1,ux

,vu

+u

)=O

(4.70)

where H1 is an arbitrary function.

The third and last invariant of the group (4.62) is found by

cosing A3

A3(u, y, x, y) and again solving (4.63). The result is

(4.71)

where h2 is an arbitrary function. The functions H1 and H2 implicitly define a set of similarity variables for the boundary layer equations. A special choice of these functions is

i in-1 m-2

A1yx

(L1.72) In m-2 A2

x2(x1)

= u x (L473) In In A3 = X3(A1) = y u + u . ('1.74)

In addition to the above transformation, a transformation consistent with the condition of conformal invariance must be

devel-oped for F(x, y). Differentiating (4.54) with respect to the parameter

a produces

a a a a a

- + -.

=

-ax aa ay aa 3a p(x, y).

m+2

Since f (a) = 2 = , the above equation becomes

9

46

a a a

Z-=

(in + 2) f m+3. p(x, y)

-._.+-aa ax a aa

and setting a = a we have o aa 3V

-

+ -'-ax 3a a=a o a

-

(in + 2) f '(a ) P(x, y) (4.75) ay 6 o a=a o

According to threm (2.1) of Chapter II, (4.75) represents a necessary and sufficient condition for P(x, y) to be conformafly

in-variant under the group (4.62). Substituting the appropriate deriva-tives from (4.62), (4.75) yields

a

f '(a ) (in - 2) x + f '(a ) [-y+(l-rn)xJ (rn+2) f '(a ) P(x, y)

or

(in- 2)x+ [ y + (l-m) x]

(in + 2) P(x, y)

ax ay

The solution of ('4.76) is found by the method of Lagrange to be

m+2

H3(X1, P x ) = O (.77)

where FI3 is an arbitrary function. Equation ('4.77) implicitly

speci-fies the functional form of P(x, y) which can be retained in the momen-tum equation without destroying the similarity generated by group

(L4.61).

A special choice of ('4.77) is

m+2

P = xH'4(A1)

('4.78)

If it is desired that P be a function of x only, we just need to

choose H'4 to be constant.

substituting (L1.72), (i473), (4.7') and ('4.73) into (4.'4.Oa)

and (L4.L40b) gives

AA'

L

22

in 2 H (A ) s X " ______ (X ) + A + A ' X A 4 1 2 = in-2 i m-2 2 2 3 2 i A 'X 2 1 in

in[1A x_ix

f +A3]-O

in-2 in 3 2 2 (1475) (4.79) (4.80)

Equations (4.79) and (4.80) are ordinary differential equations with

as the independent variable and A2 and X3 as the dependent varia-bles.

Case III. h1(x) x.

Substituting h1(x) into (11.55e), we obtain

..=-xp1

pl

r:

-xv-

f6. 3 f6

()2

y = f6 y - p1 4

Inverting (4.82), the set of transformations gives

-X =

x

f6 = f u f6 V 1- X U _p1 - i l 2

f62f2

X

The procedure for obtaining p1, such that (4.83) satisfies the closure property, is the same as in Case I, whence p1 is found to be

f 2(a)

(a) = f (a) (4.84)

f6 (a)

Substituting (11.811) into (4.83) yields

(4.81)

(4.82)

(4.83) f

As in Cases I and II, - -x, and (11.81) may be integrated to

obtain

or - f4

X = i- X

f6 2 2

)x

S:;

y+(_

2f64 = f4 u.

The invariants of (4.85) must satisfy r6 12 f4 -f6 y + x u [-f6 - J

ax.

-1 au

aa

T

+

ax.

_i

_av

. +

-ou aa

aa

o ax

_i

+ 6' y + x u(3f6' - 2f4' )] ₌

_o.

Letting A A1(x, y) we find that

f6' 2f4t-3f6' ) (f '-2f6') f -2f

A1_y[x]

+½[x] 4 6

w-a w-a o (4.85)

ax.

1=0

ax. ax.

ax.

1

(f4t

_{2fb') x -+ [-f6ty + ½}

_{x2 (3ft} 2f4t)]

u-

1

ax 4 3u

(4.86)

(4.87)

where f4' and f6' are evaluated at the identity element a. As in Case II the requirement that A1 be absolutely invariant under (4.85)

forces f4(a) [f6(a)]m and ('+.87) becomes

i 2m-3 m-2

X1yx

+½x

(4.88) ax

ax.

-3a

ax

aa

aa0

Following the same procedures as in Case II, two other

partic-ular invariants are found to be

_l

i

-

1-.-m In A3

X(A1)

y u + x u In A2 = A2(A1) = u X

Again as in Case II the functional foru of P(x, y) is determined by solUtionS of

ap 2

(m - 2) x - t

L

y + ½ x (3 - 2m)] - (m + 2) P(x, y)

ax

ay

and a special form of P(x, y) is mi-2

H5(A),

where H5 is an arbitrary function of A1. If P is required to be a

function of x only, then H5 may be taken to be a constant. Substitution of (4.89), (4.90), and (4.92) into equations (L#.1+Oa) and (4.40b) furnishes

1

AA'

22

m 2

5 1 2 m-2 i m-2 2 2' A3

rn

H(A)+A"

A i--A

i-A

1 A2tA1 A3 A -1

i-A L

m-2 m-2 2 2 m 2 2

+A3]=0

(4.89) (4.90) (4.91) (4.92) (4.93) (4.94)

It is interesting to note that equations (4.93) and (4.94) are

identi-cal to (4.79) and (4.80). Also interesting is that A1 = O in Case II

Boundary Conditions

The boundary conditions for (14.L1Oa) and (L1.4Ob) for the case

of a zero pressure gradient are

u y O on the surface (4.95)

and

liTn U 1 (4.96)

Is the group (4.62) consistent with the above conditions? The necessary and sufficient conditions for u = O and y O to be invar-iant curves are

where 1(x, y) and 2(x, y) are arbitrary functions of their argu-ments. Choosing w1 = mf6'(a°) and recalling that f4 = f, (i.i.97)

integrates to

In

-H6(u x , À1) O (4.99)

where H6 is an arbitrary function and À1 is defined by (4.72). A particular form of (i99) is

in

in-2

u x H7 (À1). (4.100)

If the surface is described by À1 0, then choosing H7(0) = O we conclude that the condition u 0 on the surfaces is consistent with

3a 3a a 3a

aa

a=a 3u 3v -w(x, y) u

w(x,

y) V

(4.97) (4.98)

03x

aa

av

+

03x

a =a

the group (4.62). A similar argument holds for the condition y = o

on the surface.

The condition (4.96) requires that u(x, y) approach a constant

for large y. We address our attention to finding the nature of the

family of invariant curves u(x, y) = 1. According to Cohen [24] the necessary and sufficient condition for u = i to be an invariant curve is ax 3a

cx

a

aa

au

-= F (u)

aa0

3y (4.101)

where F1(u) is an arbitrary function. Following the method of Lagrange the solution to (4.101) is

1(u) = x H8(À1) (4.102)

where 1(u)

ÍU(U)

and H8 is an arbitrary function. If ü is to

'1

be constant when u = 1, then the right hand side of (4.102) must be a

constant under the group (4.62). Unless x is taken to be constant this condition is not satisfied and we are forced to search for a

restriction. If F1(u) is taken to be zero, the preceding analysis is

invalid. In that case a particular solution to (4.101) becomes

u = _(4.103)

where H9 is an arbitrary function. _{Equation (4.103) implies that}

u and that = i when u 1; hence, the boundary condition (4.96) may be satisfied if u in the group (4.62). _{Consequently, we}

may take f4 1 and derive a new set of invariants which will be com-patible with the boundary conditions (4.95) and (4.96).

On the other hand if the boundary condition (.96) is in m-2

limux

=1,

U dU m-2 u 2 dx C

=x

in in

m-2.

(L.lOO) confirms that is constant whenever u x is and

no modification of the group is necessary. Equation (L1.lOLi) implies

that the free stream velocity is

Equation (Le.106) meets the requirement set forth in (14.78).

(L4.104)

(.lO6)

m

U =U

X m-2 (.1o5)

C

condition

CFLAPTER V

APPLICATION OF THE INFINITESIMAL GROUP METHOD

Burgers Equation

For the Burgers' equation u + uu = u we shall seek an

t X XX

infinitesimal transformation of the form

+ X(x, t, u) + 0(c2) = t s c

Q(2)

= u + U(x, t, u) + 0(c2) such that - 2- ₂ au

-au

_au

_)[3U

au

au

u _{- --- w3(x, t,} at ax u ax _ax

The relevant derivatives required are

--= u

_{ax x} + c[U_x + (u - X )u - x u 2 + 0(c2) u X x u x

2-au

= u_xx t c[U_xx + (2U_xu - X_xx)u + (U - 2X )u x uu xu X dX

x

uu

u3+(U

x u

-2X)u

x xx

-3X u

u xx

u]+O(e2)

x

)u x u x u u]+O(c2)

ut

t X U

Xt

2 (5.1) (5.2) (5.3) (5.5)

Xu

fu

=U

X t

u

=u +uu

U+(u-X)u.

XX t X X

Using (5.3) through (5.7), we write

3ü

au_

- _{- c{U s UU + u (-XU - X - X u) + (u} )2 _{(X X)) + 0(c2)} t t u x t u X U

(5.8)

u -

u U + u (U + uU - UX _{) + (u )2(-X u)} t 0(c2)} 3x x x u X x u (5.9) 2_ 2 - - ì.) _{c { - U - U(u - 2X ) t u [2U} _2 2 xx u x x xu x x - X - (u - X) (u - 2X ) + 3X U] xx u X U

+ (u)2 [-U t 2x t 3X(u - X)]

+

(u)3 [X] }

+ 0(c2). (5.10)

To satisfy (5.2), the sum of the right-hand sides of (5.8) (5.9),

and (5.10) must vanish. As or powers of u are not identically

0 2 3

zero, we require that the coefficients of (u) , u, (u) , and (u)

in the sum mentioned above vanish.

Setting the sum of the coefficients of (u)3 equal to zero gives

X 0. (5 li)

uu

Similarly, from the coefficients of (u)2, (us), and (u)° we find

t hat

(5.6)

determine the similarity variables and also use it in satisfying (5.2). Using (5.6) and the Burgers' equation we see that