On the non-linear eqlations of the selfsimilar motion of a viscous fluid

194 REFERENCES

1. VOSKRESENSKII G.P. and CHUSHKIN P.I., Numerical methods of solving problems of supersonic flow round bodies, Itogi Nauki i Tekhn. VINITI. Mekhan Zhidkosti i Gaza, 11, 5-65, 1978. 2. TOLSTYKH A.I., On implicit schemes of higher accuracy for systems of Navier-Stokes equations, Zh. vychisl. Mat. mat. Fiz., 21, 2, 339-354, 1981. 3. TUSHEVA L.A., On animplicit fourth-order scheme for the system of gas-dynamic equations, Chisl. Metody Mekhan. Sploshnoi Sredy. Novosibirsk: VTs SO Akad. Nauk SSSR, 8, 5, 120131, 1977. 4. KRAIKO A.N., Some problems of constructing numerical algorithms for calculating flows of an ideal gas. The construction of algorithms and the solution of problems of mathematical physics, Moscow, 1987. 5. MORROW R. and CRAM L.E., Flux-corrected transport on a non-uniform mesh in plasma boundary problems in computational techniques and applications, Eds. J. Noye and C.A. Fletcher, CTAC-83, North Holland, 1984. 6. BORIS J.P. and BOOK D.L., Methods in Computational Physcis, Ed. J. N. Killen, Academic Press, New York, 1976. 7. COLELLA P.A., Direct eulerian MUSCL scheme for gas dynamics: Report LBL-14104, Lawrence Berkeley Lab., 1982. 8. KOLGAN V.P., Application of the minimum derivative r ilue principle to the construction of finite-difference schemes for calculating discontinuous solutions of gas dynamics, Uch. Zap. TsAGI, 3, 6, 67-77, 1972. 9. CHAKRAVARTHYS.R. and OSHER S., A new class of high-accuracy TVD schemes for hyperbolic conservation laws: AIAA Paper 85-0363, 1985. 10. ORAN E.S.and BORIS J.P., Numerical simulation of reactive flow, Nav. Res. Lab., Washington, D-C., 1987. Method for Solving Multidimensional Problems of 11. YANENKO N.N., The Fractional-Step Mathematical Physics, Nauka, Novosibirsk, 1967. NASA 12. COAKLEY T.J., A compressible Navier-Stokes code for turbulent flow mdelling, TM 85-899, 1984. method in schemes of the predictor13. KOVENYA V.M. and LEBEDEV A-S., Use of thesplitting corrector type for solving problems of gas dynamics, Chisl. Metody Mekhan. Sploshnoi Sredy. Novosibirsk: ITPM SO Akad. Nauk SSSR, 15, 2, 49-60, 1984. 14. KOVENYA V.M. andCHERNY1 S.G., On iterative algorithms for solving stationary problems, Modelling in Mechanics, Novosibirsk, 1 (18), 1, 59-74, 1987.

Translated

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

Vo1.28,No.6,pp.194-203,1988

by H.T.

0041-5553/88 $10.00+0.00 01990 Pergamon Press plc

GN THE NON-LIf'iEAF. EQUTIONS OF TtiE SELFSIriILAli F;OTIONOF E. VISC(iliS FLLJIT;*

V.I. NAIDBNOV

A mathematical theory of the convective viscous fluids in tubes is constructed.

instability

of the motion of a

Introduction. A non-isothermal instability of the forced motion of a viscous fluid in a cooled tube, due to the strong temperature dependence of the viscosity was predicted in /l-4/. As a result of this dependence, the thermal and hydrodynamic effects are interconnected, and at critical pressure drops they increase spontaneously, which leads to a break-down of the steady-state conditions and to a jumpwise increase (decrease) in the rate of flow of the fluid for a small change in the pressure drop. This physical effect was analysed using an elementary thermal balance model, without taking account of the radial temperature and velocity distributions in the tube, where the heat transfer coefficient used in /l, 2/ was a quite undetermined quantity that depended on the nature of the viscous fluid flow. In /S-11/ an independent investigation was made of convective instability, which was based on the analysis of the exact (selfsimilar) solutions of the Navier-Stokes equations and on convective heat exchange, and was free from this defect. *Zh.vychisl.Mat.mat.Fiz.,28,12,1884-1896,1988

195 This instability has not been investigated sufficiently comparedwith, for instance, the classical hydrodynamic instabilities of isothermal flow, the Rayleigh and Marangoni instabilities, etc. There are experimental investigations of the isothermal motion of very viscous fluids (melts of polymers and glycerine), which support the theory /12/. The practical importance of investigations of convective instability is connected first and foremost with the search for the optimal processes for the thermal processing of polymers when they are being extruded, with the realization of regimes of fluid slag and ash removal in highly-forced powerplants, with thecomputation of lines for viscous petroleum, and also with the prediction of some natural phenomena, for example, palaeomagmatogenic processes in the depths of the Earth. 1. Convective analogue of the phenomenon of a thermal explosion. !Ihe mathematical apparatus that is used to analyse convective instability is closely connected with the methods of investigating non-linear boundary-value problems that contain positive linear differential equations and monotonic functions of the dependent variable /13/. Henceforth, we will make extensive use of the ideas and methods of the mathematical theory of thermal explosions /14-19/. One of the physical set-ups that leads to "position" problems is connected with the temperature distribution in a fluid filtering through a prismatic tube which is filled with a porous medium. If the viscosity of the fluid depends exponentially on temperature, and a constant negative temperature gradient is specified along the walls, then the corresponding problem leads to the equation /9/ AH=-.@, Ols=O, (1.1) which

is considered in some bounded region GclR' with a boundary S. Eq.(l.l) describes the stationary temperature distriburion as a function of the applied pressure drop A. The non-linear term in (1.1) is proportional to the rate of convective heat supply to the cooled volume, and the linear term is proportional to the rate of return of this amount of heat to the walls due to molecular thermal conduction. Problem (1.1) is well-known in stationary thermal explosion theory /16/; there are general methods for investigating it /14, 15/. It has been shown that (1.1) can only be solved for fairly small positive values of the parameter A, and that for A>.\. it cannot be solved. The basis for this assertion in a fairly general form and methods of computing A. are given in /14, 15, 18, 19/. AS it applies to the problem of isothermal filtering, this means that at supercritical pressure drops A>,\, , the convective heat supply begins progressively to outstrip the heat return to the walls due to thermal conduction, which leads to the development of instability. Many other problems of mathematical physics with distributed heat sources lead to (1.1) /20-24/, in particular, the problem of the dissipative heating of a viscous liquid of variable viscosity /21, 22/. 2. Selfsimilar motion of a viscous liquid in a tube. For an exponentialdependence of the viscosity on temperature, the Navier-Stokes equations of thermal convection have a selfsimilar solution, which corresponds to the rectilinear motion of a viscous liquid with a constant temperature gradient along the walls. In this problem with respect to the excess temperature of the liquid has the case, the boundary-value form

e(i)=e’(o)=o.

e”(1)=-(nS-1)9’(1),

(2.lb)

n=-1 corresponds to the current in Here n=O corresponds to the flow in a circular pipe, for cooling a plane channel, the parameter A is proportional to the pressure drop, and .I>0 and ,\
Green's

function of the operator

K(x,t)=

and we write

I3"'=0

t-0.5 (t*+X*))

1

t--rt,

(2.2) in the form of a non-linear

e(x)= we

(2.2)

w(i)=e(i)=e’(o)=o.

e”‘=.\d, We construct

integral

ASKC+, t)te

with boundary

conditions

(2.2):

za. rat. equation

of the Hammerstein

type:

(2.3)

dt.

show that .i<.i. where a solution of (2.2) exists. We assume that a solution of (2.2) exists for any .1)0.

will

Since K(s, t)>O,

dK/dx
d’K/

196

axzO. will be used in future to analyse stability. We function 0(x) linear problem for the eigenvalues: 11, z =-_clxz, z(0)=2'(1)=2"(0)=0. This equation

is equivalent

to the linear integral

z=p

These properties of the will consider the auxiliary (2.4)

equation

jc(x,t)tP(t)dt,

(2.5)

0

where

( x(1-t)7

G(x.t)=

Z-0.5(z~+tZ),

xe, xx.

The linear problems for the eigenvalues (2.41, (2.5) are non-trivial, since the non-selfadjointness of the operator z"'= 0 with boundary conditions (2.4) and the non-symmetry of Green's function do not allow us to use the normal Stunn-Liouville theorem, which is widely employed in the analysis of selfadjoint equations, for example (1.1). TO solve (2.4) and (2.5)we have to UseKrein's oscillation theorems /25-27/, which are applied to an extensive class of boundary value problems irrespective of whether these Krein's theory of oscillatory kernels was first used to problems are selfadjoint or not. There are numerous examples of investigate problems of hydrodynamic stability in /28, 29/. its application in /30/. is written in an iterated form with positive weights, and Since the operator L(z)=z"' The boundary-value problem Green's function is positive, the kernel of (2.5) is oscillatory. to which such a kernel corresponds has a discrete spectrum of real eigenvalues, and the eigenfunction belonging to the smallest eigenvalue does not change sign in a closed interval. Consequently, there are eigenvalues (2.41, (2.5) that satisfy the inequalities O
I”d,=$Y’ .

1

I -_)

z,(t)t8(t)dt

1j

tzo(t) eecrl dt.

0

holds, whence it follows that .l+Oe-'. the inequality e'>ee Since 0(x) 20, Thus, in the region where the solutions (2.2), (2.3) exist, all the A have an upper limit A.dp,e-'. We will point out some useful properties of (2.3). It is not difficult to then it exists for all values of h from the show that if (2.3) exists for some .A=&, interval [0,&J.. As a consequence, we obtain the assertion that if the solution does not Using the contraction principle, exist when i\-.\,. then it also does not exist when .!>.iO. it is not difficult to show that for sufficiently small A the non-linear problems (2.21, (2.3) are always solvable. is the absence of solutions for Thus, the most important property of (2.2), (2.3) A.Guoe-' holds. sufficiently large positive A. For A., the estimate We will investigate the properties of (2.2), (2.3) when ii<.\,. We introduce the sC0 implies heating) auxiliary parameter @'(1)=-s (s>O implies cooling of the liquid, and and consider the Cauchy problem f3"'=,\see, where

the dependence

e(i)=o,

e’(i)=-+,

.4=.\(s) will be determined

(2.6)

e”(i)=o,

from the fourth boundary

condition

e’to,

A(s))=O. Theorem. For any s>O there is a unique ,\ for which the solution of (2.6) satisfies the equality e'(O)=& As an example, we will consider the case of liquid heating: s
y"'=-_TreH!', We will show that for each Lemma.

Suppose

y(l)=O, y'(l)=l, y"(1)=0. ff>I there is a unique .\ that satisfied

the function

y(S)

(2.7) equality

*obeys the inequality

y"'Gf(& .2), where /(rl, 5) is monotonically decreasing in y, continuous and continuously with respect to y. And let z(z) be a solution of the Cauchy problem Z""/(Z,

Then the inequalities

.r),

Y'(O,,~(H))=O.

z(i)=P(~),

y"'Cz'", y"&z", y'z

z'(~)=y'(l), hold.

z~‘w=errw.

differentiable

197 This lemma is a direct consequence of theorems concerning differential inequalities for systems of equations with off-diagonal monotonic right-hand sides /31, 32/, and we will not dwell on its proof. We will now prove that (2.7) is solvable. To start with, we will establish the 1 in the theorem when

required

the equality

Y'(O,x)=O

H>O.

We denote by

Y(x,.x)

the solution

when iT>O.

can only be satisfied

of (2.7) and show that

In fact, by virtue of the constant

we have Y"'(r,x) 20 (O<~dl), whence by sign of the right-hand side of (2.7) with .\l. Thus, we will consider with ;1>0. In this case, Y(.r,n) satisfies the conditions

y"'(r,X)60, Y'(1,X)=1, y”(1, Y(l, ;i)=O, (by virtue of the lemma) the following inequalities

and therefore

y’(x,

Y" (5, X) 30, We note

that Y(s,X)=s--l, into the interval [O, 11.

11)<1,

y(s,

(2.7)

1X)=0

hold:

X)2s-I.

(2.8)

if n=O, that is, with 1\=0 the function Y(z, X) is continuable Consequently, from the theorem and the continuous dependence of the

solutions on the paramete_r, ficiently small positive 12. We consider the maximum

is also continuable

Y(? n)

connected

into the interval

set that contains

zero of the 2

[0,1] for

suf-

at which the function

is continuable into the interval It is clear that this is an interval of the form ]8, 11. (0, E). Formally speaking, there are two possibilities: either the inequality Y(O,&)>O is satisfied at some A, , or the inequality is satisfied at all y(0, ;c)
;i=](JE) .

We

~(0, 11)<0

and

(2.9) r-lGy(r, X)ao. The uniform boundedness of y(.r,i) and the form of (2.7) show that the functions Y"'(S),Y"!Z). But then, by virtue of the theorem concerning the y'(r) will also be uniformly bounded. continuous dependence of the solutions on the parameter with ;\=E (if E is finite), the is also continuable into the interval the ]O, 11 r which contradicts This shows that E=m, that is, when the second possibility is satisfied

solution y(z, 3) definition of E. for each

>%&O,

the solution y(O,r\)

we will estimate -.Lexp

y(x,T\)

for large

is satisfied,

[H(z-I)]

is continuable

;i.

whence,

into the interval

By virtue of (2.9), for any

[O, II.

In this case,

.I the inequality

y"'G

by virtue of the lemma,

y(s.,\)br-1+0.5,1 -s (2-t)ZtezP]H(t-l)ldt r

and, consequently, Y(O..W-I++

For sufficiently Thus, y”20,

for some y’(l,

&)=I,

large Ax wehave &[O,E) y(1,

As was mentioned follows that an

with

prove

respect

to X.

Y(O,;L)aO.

of

1..

follows the inequality

of the lemma. function

So

But, then, by virtue of the relationships ~'(0..%)CO.

Consequently,

The function

z=8yiar\. To determine Z(~~=_reXL;\H&Ur

Hence

y(O,x)
~'(0,O)=l.

Hence

and from (2.8) it

such that ~'(0, .x)=0.

the uniqueness We set

t which contradicts the assumption

have the inequality

earlier, Y(r, 0)=x-l.

x. exists

We will

we

0

Y(o,:)>o

we will have X0)=0

-r tJexp[H(t--l)]dt,

z'(z, I\)<0

1

y(s,2)

is obviously

differentiable

r(z,-\), we have the equation

2(1)=2’(1)=2”(1)=0.

;"'<-.~lIe"Q, which proves

and, in particular,

z'(O,x)
that

r(r)

obeys the conditions

But this signifies

that the

y'(O,i\) decreases monotonically with respect to x whence the uniqueness of the solution of Y'(0,:I)=0 also follows. Above, we considered the case of liquid heating. and the solvability For cooling ;DO, of (2.6) is proved in ananalogous fashion. O"(s, s)0(x)>8(0) (l-s) holds by virtue of convexity. problems (2.2), (2.6) in an equivalent form:

198

It is easy to verify that the equality 8'(i)=-s is automatically satisfied. Using the expression for Green's function K(0, t) it is not difficult to establish the inequalities We finally obtain an upper and 0.5S<:8(O)~S, from which it follows that s>13(x)~OSs(l-x). lower estimate for the function

A(s)=s

(we recall that, according

[J

to the preceding

theory,

such a function

exists)

:

1 -_(

3sP
It follows

S.

tz

(2.10)

A(s) have an upper limit of

from (2.10) that all the

A.-

max A(s), 0<*<*

with which, strictly, the non-solvability of (2.2) for ABA. is also connected. It also follows from (2.10) that when AO when s
3. Stability. We will investigate the stability of the solution of (2.2), (2.3). Searching for a solution of the non-stationary Navier-Stokes equations and of the heat-exchange equations under corresponding boundary conditions in the form 9(x, z)=O(x)+u(x, z), where u(x, z)=V(x)e"' is a small quantity, and linearizing these equations in the neighbourhood of their stationary solutions, we obtain the spectral problem V

"'-ou'=Axe@v,

u(l)=u"(1)=u'(0)=0,

(3.la)

(3.lb)

For

0>0, the solution O(x) is unstable, and for 0<0 it is stable. In corresponding to (3.1), we formulate the auxiliary spectral problem 2"'-ox'=-AxeOx,

2(0)=2'(1)=2"(0)=0.

(3.2)

Unlike (3.1), Eq.(3.2) does not have any direct physical meaning, and it is introduced to overcome the difficulties connected with the non-selfadjoint nature of (3.1). We multiply (3.1) by z,(x) and integrate by parts, taking into account the boundary conditions, and we have no difficulty in ascertaining that the smallest eigenvalues of both the boundary-value problems are identical. Differentiating (2.2) with respect to s and writing X=&I/&, we obtain the equation

dil

W"'=A(s)x@~+-xee,

w(l)=w"(l)=w'(O)=O.

ds

Multiplying (3.3) by the constant-sign we arrive at the relationship

Oa

eigenfunction

Since

dw/dx
dA/ds>O

and

the

do>0

by parts,

jzogclx=% j z,xe'(")dx. 0

for

z,(x) of (3.2) and integrating

(3.3)

0

sign of a0 is the same as the sign of -d.\lds. Consequently, oo
199 drop decreases when the rate of flow increases correspond at least two stationary temperature e,(s,)cf3,(s,)

if

are unstable. distributions:

there To each AE(O,:?.) 81(S,) and 0,(s:) where

~6s~.

follows from the analysiswe have giventhat the non-linear operators (2.2), (2.3) have a bifurcation in the simpleeigenvalue at the point A=n.. Thus, the most dangerous perturbations are symmetric perturbations with an infinitely large wavelength which, evolving monotonically in time, lead to a breakdown in the regime when critical pressure drops are reached. We also note that isothermal Poiseuille motion in pipes is stable with respect to this class of perturbations /30/, since there is no dynamical source to maintain their growth in rectilinear shear flows. In the non-isothermal motion under consideration, the perturbation, perwhich does not depend on the axial coordinate, is effective for transporting thermal turbations since it is maintained bytheconvective heat supply. It

4.

The non-linear problem of the notion of a liquid in a circular pipe. For the motion of a liquid in a circularpipe (n=(l) , the boundary-valueproblem (2.1) takes the form ~"'+2-~e"_~-Ze'=,~~~e,e”(i)=-e’(i), e(i)=e’(o)=o. (4.1) for sufIt has been established that in this case problem (4.1) does not have a solution the boundary-value problem (4.1) has ficiently large positive A.We can prove that forA<& A=A. I and when ii-&k one solution corat least two solutions, which merge into one when Investigation of the stability responds to a higher temperature and the other to a lower one. shows that the smallest positive solution is positive and the remaining solutions are unstable. For This can be proved as follows. We make :1so, the solution of (4.1) is unique. the substitution

e=-g,,\--.x

and write

(4.1) in the form

e=i jR(Z,t)texp--IB(t)ldt=x(B). (I

Since

K(r, t)>O

For sufficiently

(Green's function of a linear operator small

x

the operator

X(6)

is positive),

is a contracting

operator,

then

B>O,x(B)>O.

since the inequality

iexp(-B,)-exp(-02)1~18,-621 holds, from which follow the existence and uniqueness of the solution of (4.1) when .140 (we recall that for cooling of the liquid the solution is nonunique even for small A). Critical phenomena in liquid heating are not observed, and the processes of motion and heat-exchange proceed smoothly with some degree of non-isothermality.

5. Non-Newtonian

uledia.

The theory given above can be generalized to the case of the motion of a liquid with a non-linear fluidity law /33/. The curve of rheologically-complicated media in a definite temperature range may be approximated by the power function

(5.1) Here duldn is the velocity gradient, n is the normal to the streamline, T is the temperature of the medium, m is the index of non-Newtonian behaviour of the medium, m>Z corresponds to a dilatant liquid, and f?l
8 I,, T Y$!

e/r _ I$!

e”(i)=-(n+i)e’(i),

e’EA\zf/,,~ cxp (e/m), e(i)=e’(o)=o.

(5.2a) (5.2b)

The form of (5.2a) shows that the introduction of a new positive parameter m does not change the basic properties of this equation (positive Green function, monotonic increase of the dependent-variable function, etc.). Using the results of Sects.2, 3, and 4 we can show that for .\>;I., (5.2) does not have a solution, and for .\<.\. there is non-uniqueness. Naturally, the critical value .\. will be a function of m. 'Ihe flow of viscoplastic Shvedov-Binqham media with the following rheologicalequation of state can be investigated similarly:

200

(5.3)

is the velocity gradient, T. is the limit shear stress, u is the plastic Here dwldy viscosity. FOX rGz. the medium behaves like a solid, and viscous flow only begins at t>z.. In the case of an exponential dependence of the limit shear stress and the plastic viscosity on temperature in the motion of the medium (5.3) in a plane channel on whose walls a constant temperature gradient is maintained, we can have a structural regine with a viscous flow zone adjacent to the walls and with a solid zone on the axis of the channel. The flow is rectilinear in the viscous zone. The corresponding boundary-value problem has the form e’(u)=aW’(a),

e(l)=e”(l)=O,

e,,,=~bee_n,

Aaeecal=II.

(5.4)

Here n is the so-called plasticity parameter. We will show that when liquid is cooled (A>O) (5.41 does not have a solution, and when Asn. we have non-uniqueness. We make the substitution f3=u+a is a solution where u(f) of the boundary-value problem u"'=-n

,

u(l)=u"(l)=o, u'(a)=au"(a).

u = $ (-%“+3%*+3a*%-2-3~~). In this case, we obtain

the equation

f(%,rr)=Q+',

u"'=Af(f,n)e~, v(l)=v"(l)=O,

u'(a)=av"(u),

(%a) Aue”‘“‘e”‘~“=lI.

(5.5b)

we note that f(%,n)<% since u(t)~O. v"'=O We introduce Green's function of the operator with boundary-value conditions (5.5b) and reduce (5.5a) to an integral equation with a positive, non-symmetric, oscillatory kernel /25-27/:

u(%)=A 5 K(%,t)f(t,II)e”(lldt, 0

K(%,t) =

( t-o.s(t’+%‘), t-St,

%a %a.

for 1430. Since K(%, t)>O, then u(e)>0 We introduce the auxiliary linear equation ~"'=-sf(e,

rI)2,

2’(1)=d’(u)=O, z(u)=ua’(a).

(5.6)

According to Krein's theory of oscillatory kernels /25-27/, (5.6) has a discrete spectrum of real eigenvalues, and the eigenfunction that corresponds to the smallest eigenvalue 60 does Further, multiplying both sides of (5.5a) by ZO not change sign in the interval [a, I]. and integrating by parts, we obtain the relationship

-=80

2’ Since u(t)>O, as .\+w.

we have

[

Sf(t,n)z,(t)e”‘)dt][ J f(t,n)z,(t)u(t)dt]-'. ,I

”

It is obvious e‘>,eu. Hence follows the inequality .1<&(A)e-'. 6, is the smallest eigenvalue of the problem

that

6,(‘1)+50 where

z"'=-S"f(E, rI)z.

z(O)=z'( l)=z"(O)=O.

The right-hand side of the inequality i\<&(A)e-' has a limit as _\+-, and the lefthand side increases without limit, and so the contradiction that arises invalidates the ,1>0 _ Consequently, there is a A. such assumption of the solubility of (5.4) for large It is interesting to note that f(%, n)<% .I=-.\., (5.4) does not have a solution. that with for n>O, a>0 and the critical value .I. will be higher than in a Newtonian liquid - that is, the non-isothermal instability effect we described will be realized more easily in a Newtonian liquid than in a Shvedov-Bingham medium.

6.

Computational of the critical values.

(5.2) represents quite a simple problem for the From the calculation point of view, Eq.(5.2) admits of a decrease eigenvalues, and their values can be determined as follows. in order, as a result of which it becomes possible to determine the critical value S. exactly. Eq.(5.2) is invariant under the group of transformations

201

where k is an arbitrary positive number. We introduce the new variables u=-&l/d~,~=lnt. have the relation

g4inl=P-

3mfl-u

dq

m

du ’

Let t=-U"-(n-2)u'+

t=A exp

e+(3n+l)‘1 [

m

2nu.

From

(5.2) we

1

from this relationship, we can obtain a second-order differential equation Eliminating du/dq Since that connects t and u. Analysis of this equation shows that dt/du=O when u=3m+l. then dA/ds=O when s.=3m+l. u-es,t--A as 2~1, It is interesting to can be calculated exactly. Thus, the critical value of s.=3m+l note that it does not depend on the flow geometry and is the same for both a plane channel and a circular pipe. Further, for a specified S. the following Cauchy problem is solved numerically:

e(t)=o,

e’(l)=-s.,

er’(l)=(n+i)s.

and the critical 21. is determined from the condition lY(O,A.(s,))=O. Using the Runge-Kutta method, the dependences n(s) were also calculated for both Newtonian and non-Newtonian liquid (Figs.1 and 2 respectively; the dashed lines are the unstable regimes). The characteristic feature of the non-linear boundary-value problem that describes liquid flow in a circular pipe (Fig.1) is the absence of solutions not only for but also for s>s... For a plane channel (Fig.21 as follows from the theorem A>A., in sect.2, a solution exists for all s>O. As the index of non-Newtonian behaviour of the A., e.(O) increase in a roughly straight-line fashion liquid increases, the critical values in(m increases from 0.5 to 5 in steps of 0.5). The data given show that the convective stability effect arises in an essentially non-linear region of viscosity variation, where the viscosity changes by a factor of 10. 7.

Non-linear partial differential equations.

The selfsimilar problem of the motion of a viscous liquid in a prismatic pipe of noncircular section leads to a non-linear biharmonic equation with Dirichlet boundary conditions /9/:

e1,=Ael,=o.

div[e-egrad(Ae)]=&

(7.1)

One-dimensional analogues of (7.1) were considered in the preceding sections: here we will demonstrate some other approachesin finding the critical .I.. We will consider the most interesting case A>O. Let G be a connected open set of a Euclidean space withapiecewise-smooth boundary S, GER’. Integrating (7.1) over G, applying the well-known Green's formulae and taking account of the boundary conditions, we obtain the relationship

A=[_f(es-l)dG]-'j(Ae)rdG. G

(7.2)

0

Here .\(a) is a non-linear functional defined on the set of functions 8. We will formulate the problem of determining the relative extremum of this functional on the set of superharmonic functions TV>O, AW
WI,=AWls=O.

it is realized

(7.3)

on the solutions

I

25s Fig.1

Fig.2

of (7.3).

202 Using the technique developed in the preceding not have a solution for farily large values of A.& A.. is estimated by the eigenvalue of the Laplace .I..<2po*,

A9=-+,9,

sections, we can prove that (7.3) does The super limit of the critical value operator: .Pjs=o.

Thus, if the functional ,4(e) has a relative extremum (this functional does not have an absolute extremum, since there are derivatives in the numerator of 81, then the value of A* has an upper limit set either by the critical value of (7.3) or by the smallest eigenvalue of the Laplace operator. Consequently, we have the chain of inequalities A.
REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29.

PEARSON J.R.A., SHAN Y.T. and VIERA A.S., Stability of non-isothermal flow in channels. 1. Temperature-dependent Newtonian fluid without heat generation, Chem. Engng. Sci. 28, 11, 2079-2088, 1973. SHAN Y.T. and PEARSON J.R.A., Stability of non-isothermal flow in channels, Chem. Engng Sci., 29, 6, 1485-1493, 1974. PEARSON J.R.A., Heat-transfer effects in flowing polymers, Progress Heat and Mass Transfer, Pergamon, Oxford - NY, 5, 73-81, 1972. PEARSON J.R.A. and SHAN Y.T., On the stability of non-isothermal flow in channels, Rheological Acta Internat. J. Rheology, 12, 2, 240-244, 1973. AIEKSAPOL'SKII N.B.and NAIDENOV V.I., Critical phenomena in the non-isothermal flow of a viscous liquid along pipes, Teplofiz. Vys. Temp, 4, 783-790, 1979. NAIDENOV V.I., On integral equations that describe the temperature distribution in the plane flow of non-Newtonian media, Prikl. Mekh. i Tekhn. Fis. 5, 103-109, 1983. NAIDENOV V.I., On viscous explosion in the non-isothermal flow of an incompressible liquid, Teplofiz. Vys. Temp. 22, 3, 501-506, 1984. NAIDENOV V.I., Stationary theory of viscous explosion in non-Newtonian media, Inzh.-Fiz. zh. 46, 5, 837-845, 1984. NAIDENOV V.I. and POLYANIN A.D., On some non-linear thermal-convective effects in the theory of filtration and hydrodynamics, Dokl. Akad. Nauk SSSR, 279, 3, 575-579, 1984. NAIDENOV V.I., Thermal-convective instability of the motion of a viscous liquid in a circular pipe, Teplofiz. Vys. Temp. 24, 1, 82-87, 1986. motion of a viscous liquid, NAIDENOV V.I., Bifurcation in the selfsimilar non-isothermal Teor. Osnovy Khim. Tekhnol. 21, 2, 215-221, 1987. SKUL'SKII 0.1. and SLAVNOV B.V., Non-uniqueness of the forced-flow characteristic in the Proceedings of the 2nd All-Union non-isothermal flow of a power liquid in a pipe. Conference on the Mechanics of Anomalous Systems, Baku, 1977 theory, KELLER G.V., Some non-linear problems brought out by non-linear heat-generation In: Branching Iheory and Non-linear Eigenvalue Problems, Mir, Moscow, 1974. VOL'PERT A.I. and KHUDYAYEV S.I., Analysis in Classes of Discontinuous Functions and the Equations of Mathematical Physics, Nauka, Moscow, 1975. KHUDYAYEV S.I., Mathematical Theory of Thermal Explosion: Preprint, Chernogolovka: OIKhF, Akad. Nauk SSSR, 1984, FRANK-KAMENETSKII D.A., Diffusion and Heat Transport in Chemical Kinetics, Nauka, Moscow, 1967. GEL'FAND I.M., Some problems of the theory of quasilinear equations, Uspekhi. Matem. Nauk, 14, 2, 1 87-158, 1959. KHUDYAYEV S.I., Criteria for the solvability of the Dirichlet problem for elliptic equations, Dokl. Akad. Nauk SSSR, 148, 1, 44-46, 1963. KHUDYAYEV S.I., Boundary-value problems for some elliptic equations, Dokl. Akad. Nauk SSSR, 154, 4, 787-790, 1964. GRUNTFEST I.J., YOUNG J.P. and JOHNSON M.L., Temperature generated by the flow of liquid in pipes, J. Appl. Phys., 35, 18-22, 1964. liquid in a plane KAGANOV S.A., On the steady-state laminar flow of an incompressible channel and a circular cylindrical pipe taking account of frictional heat and the dependence of viscosity on temperature, Prikl. Mekh. i Tekh..Fis., 3, 96-99, 1962. thermal explosion, BOSTANDZHIYAN S.A., MERZHANOV A.G. and KHUDYAYEV S.I., On hydrodynamic Dokl. Akad. Nauk SSSR, 163, 1, 133-136, 1965. RATN'ER S.B. and KOROBOV V.I., Selfheating of polymers under multiple deformation, Dokl. Akad. Nauk SSSR, 161, 4, 824-827, 1965. MALININ N.I. et al., On the vibrational creep of polymer BARENBLATT G.I., KOZYREVYU.I., materials, Dokl. Akad. Nauk SSSR, 166, 4, 813-816, 1966. KREIN M.G., On one-symmetric oscillatory Green's functions of ordinary differential operators, Dokl. Akad. Nauk SSSR, 25, 8, 643-646, 1939. KREIN M.G. and RUTMAN M.A., Linear operators that leave a cone in a Banach space invariant, Uspekhi. Mat. Nauk, 3, 1(23), 3-96, 1948. KPEIN M.G., Oscillatory theorems for normal linear operators of an arbitrary order, Dokl. Akad. Nauk SSSR, 25, 9, 717-720, 1939. YUDOVICH V.I., Secondary flows and instability of the liquid flow between rotating cylinders, Prikl. Mat. Mekh. 30, 4, 688-698, 1966. BABSKII V.G. and MYSHKIS A.D., On the montonicity of the change in the first eigenvalue for a class of non-selfadjoint problems of hydrodynamic instability, Zadachi Mekhan. Mat.

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Translated

U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain

.,Vo1.28,No.6,pp.203-207,1988

by H.T.

0041-5553/88 $10.00+0.o0 01990 Pergamon Press plc

SHORT COMMUNICATIONS UNDECOMPOSABLEHAMILTONIAN MATRICES WITH PURE IMAGINARY SPECTRUM*

KH.D.

IKRAMOV

A subclass of Hamiltonian matrices with pure imaginary spectrum, reducible to Hamiltonian block-triangular form, is described. 1.

Let J be a square matrix of order 2x1 with the following

not

structure:

J=II“I. Y where I, is the (axn) identity matrix. We recall that a real (2nxZn) matrix N is said to be Jsymmetric if I'.YJ=N'.If we divide a J-symmetric matrix into four minors of order n:

then the minors NIZ,KZI will be skew-symmetric matrices and .\',I. Y22 will be transposes another. As shown in /l, 2/. any J-symmetric matrix may be reduced to the form

of one

Reduction to the form of (1) is achieved by a chain of similitude transformations; the Jsymmetry of the matrix is preserved at each step of the process, since the transforming matrices are symplectic (i.e., such that SIS=J). these matrices are all orthogonal, Moreover, so the process is numerically stable. Conservation of J-symmetry is importantin that it halvesthe storage space necessary for the matrix N and also reduces the number of arithmetic operations performed at each step of the reduction process. By applying this process one in fact halves the order of the spectral problem. It would be of practical value to have an analogous result for Hamiltonian matrices, i.e., real matrices of order 2n such that I'MI= -MT. The block structure of a Hamiltonian matrix M may be described by the formula

equations can As we know /3/, the numerical solution of algebraic matrix Riccati reduced to spectral problems for matrices of this type. We shall say that a Hamiltonian matrix M is decomposable if it can be reduced by similarity transformation to the form *Zh.vychisl.Mat.mat.Fiz.,28,12,1897-1902,1988

be a