58 ZAMM . Z. angew. Math. Mech. 73 (1993) 1

t Gi

Fig4. Part of the bifurcation tree of the H~NON map with control parameter u = [ ( l + /3)/(l - p)] At. Solid (dashed) lines present stable (unstable) fixed points for /3 = 0.3, z2 = 1.453356

of the quadratic map [7], [S], [9]. Negative values of the parameter b have been considered in [9].

When map (31) is considered as a formula for the determination of approximations uk of solution (2), then the periodic behaviour of the solutions of equation (31) for

initial conditions u(tJ = uo > - 1 for A t > 0 @(ti) = uo < 1 for A t < 0), and the value u , equal to u(ti + A t ) or according to (lo), can be considered as periodic properties of the errors &k = 1 - u k for A t > 0 ( E ; = -1 - uk for A t < 0), which tend to and oscillate between .C3 = 1 - ti, and .C4 = 1 - ti4 for A t > 0 (if$ = -1 - ti; and tTi = -1 - C i for A t < 0).

Similarly, periodic and irregular behaviour of the solutions of map (31) for integration steps A t 2 > a + $[(1 - b)/(l + B)]’ are properties of the errors gk, too.

The “escape” to infinity for At2 E (0, $ + $ [(l - B)/(1 + b)]’), initial conditions u(ti) = u,, < -1 for A t > 0 (u(t i ) = uo > 1 for A t > 0) and u1 = u(ti + At) becomes evident. This behaviour is very sensitive to starting errors [7].

6. Conclusions

The quadratic map and the H ~ N O N map can be considered as formulae for the determination of approximations of the solutions u = u(t) of differential equation du/dt = - (u2 - 1). When small integration steps are used no erratic behaviour occurs and the solutions tend to 1, -1, co or -m. For integration steps At2 > 1 in case of the quadratic map, and At’ > [(l - @/(I + 8)12, IBI < 1 in case of the HBNON map bifurcations of the fixed points occur. The periodic or irregular behaviour of the successive values uk of these maps appear as periodic and irregular properties of the numerical errors &k = u(t,) - uk.

References 1 GUCKENHEIMER, J.; HOLMES, PH : Nonlinear oscillations, dynamical systems, and bi-

2 COLLATZ, L. : The numerical treatment of differential equations. Springer Verlag, Berlin

3 LICHTENBERG, A. J.; LIEBERMAN. M. A,: Regular and stochastic motion. Springer Ver-

4 HENON, M.: A two-dimensional mapping with a strange attractor. Commun. math.

5 HENRICI, P.: Discrete variable methods in ordinary differential equations. John Wi-

6 KLwp, H. J.: Approximate solutions, numerical errors and chaos of the logistic

7 THOMPSON, J. M. T.; STEWART, H. B.: Nonlinear dynamics and chaos. John Wiley &

8 COLLET, P.; ECKMANN, J.-P.: iterated maps on the interval as dynamical systems.

9 GRUBOGI, C.; O n , E.; YnRK, J. A,: Crises, sudden changes in chaotic attractors, and

furcations of vector fields. Springer Verlag, New York 1983, pp. 227, 268.269.

1966, pp. 52-61.

lag, New York 1983, pp. 389-391, 397-408, 422-426.

Phys. 50 (1976) 50, 69-77.

ley 8 Sons, New York 1962, pp. 201, 204,242.

equation. ZAMM 71 (1991) 7/6, 289-291.

Sons, Chichester 1986, pp. 162, 163, 174, 177-183.

Progress in Physics, Vol. 1. Birkhauser Verlag, Boston 1980, pp. 25-62.

transient chaos. Physica 7D (1083), 181 -200.

Received October I . 1990

Address: Privat-Dozent Dr. rer. nat. HORST KLEPP, Ruhr- Universitiit Bochum, Institut fur Mechanik, Universitats- straDe 150, W-4630 Bochum 1, Germany

Akademie Verlag ZAMM . Z. angew. Math. Mech. 73 (1993) 1, 58-61

SINGH, K. D.

Three-Dimensional Viscous Flow and Heat Transfer along a Porous Plate

MSC (1980): 76D10

1 . I n t r o d u c t i o n

The problem of laminar flow control is gaining considerable importance in recent years particularly in the fields of Aeronautical Engineering in view of its applications to reduce drag and hence to enhance the vehicle power requirement by a substantial amount. Several methods have been developed for the purpose of artificially controlling the boundary layer and the developments on this subject since World War 11, they have been reported by LACHMANN [I]. Theoretical and experimental studies have shown that the transition from laminar to the turbulent flow which causes the drag coefficient to increase may be prevented by the suction of the fluid from the boundary layer to the wall. The effects of different arrangements and configurations of the suction holes and slits have been studied extensively by various scholars. Most of the investigators have, however, confined themselves to the two-dimensional flows only.

GERSTEN and GROSS [2] have investigated the effect of transverse sinusoidal suction velocity which leads to a three dimensional flow over the flat surface. SINGH et al. [3] further extended this idea by applying the transverse sinusoidal suction velocity distribution fluctuating with time at the plate. The heat transfer aspect has also been studied in both of these investigations ignoring the heat due to viscous dissipation. However, there are a number of physical situations where the viscous dissipation heat is present even in subsonic flows of an incompressible viscous fluid. Also, in the case of fluids with high Prandtl number the heat due to viscous dissipation is always present even in slow motions.

Thus, the aim of this paper is to study the flow of a viscous incompressible fluid along an infinite porous plate with transverse sinusoidal suction in the presence of viscous dissipative heat.

2. Mathemat ica l ana lys i s

We consider a co-ordinate system with plate lying horizontally on x*-z*-plane. The x*-axis is taken along the plate, being the direction of the flow, and the y*-axis is taken normal to the plate directed into the fluid flowing laminarly with free stream velocity U. Since the plate is considered infinite in x*-direction, so all physical quantities will be independent of x*, however, the flow remains three-dimensional because of the variation of suction velocity distribution at the plate which is of the form:

v*(z*) = - V(1 + & cos xz* /L) ,

where V > 0 is the mean suction velocity, E( < 1) is a small quantity and L is the wave length of the periodic suction velocity. The negative sign indicates that the suction is towards the plate.

Denoting velocity components u*, v*, w* in the directions x*, y*, z* respectively and the temperature by T*, the flow is governed by the following equations:

an* aw* a p az*

au* au* a2u* aZu* u * - + w * - = v - + - , aY* az* (ay*. az*2) av* av* 1 ap* aZv* aZv*

v * - + f w * - - = - - - + v -+- aY * az* ay* (ay*2 a z * 2 ) l

8W* aw* 1 a p * azw* a z W *

aY* az* e az* ( a p az*2)’

- + - = 0 ,

v * - + w * - = - - - + v -+-

Short Communications

where

In these equations Y is the kinematic viscosity; Q is the density; p* is the pressure; C, is the specific heat at constant pressure; K is the thermal conductivity, and p is the viscosity.

u* = 0 ;

U* = u* 1'* - - -v, w* = o , T* = T*, as )'* + z .

The boundary conditions of the problem are

c* = - V(1 + 8 cos nz*/L), 1 (6)

1 ( 7 )

M.* = 0 , T* = T: at y* = 0 ,

P* = P*,,

The subscripts u' and z c denote physical variables at the plate and in the free stream respectively.

Introducing the following non-dimensional quantities in equa- tions (1) to (5)

y = y*iL , z = z*/L , u = u*/Lr , u = v*jU , M' = w*/U, x = V j U , p = p*/eU2, 0 = (T* - T*,)/(T:. - T Z ) , R = U L / v , P = p C p / K ,

where SI is the suction parameter; R Reynolds number; P Prandtl number, and E is the Eckert number, we get

E = UZ/C,(T$ - z),

(9)

(1 1)

(12) 20 ao I a20 E c - + \s- = - ?y 2z R P ( G + $ ) ' E "

where

The corresponding boundary conditions are reduced to

u = 0 , L = - r ( l + ccosxz), w = 0 , )

t 0 = 1 at ) ' = o , u = l , r = - z 1 w = o , p = pa ,

0 = 0 as y + x . 1 We now assume the main flow velocity u in the neighbourhood

of the plate of the following form:

u(y, 2 ) = uo(y) + EU1(4., z) + 0(E2). (14)

The similar expressions hold for other variables u, w, p and 0. When E = 0, the problem reduces to the two-dimensional flow with constant suction at the plate. In this case equations (8) to (12) are reduced to

.b = 0 , (15)

US + %Rub = 0 ,

06 + %RPflo = - E P u f ,

where the primes denote differentiation w.r.t. y, with corresponding boundary conditions

u , = O , B o = 1 at y = O , u o = l , B o = O as y + m .

The solutions of equations (15) to (17) under the boundary conditions (18) are

(19)

(20)

EP N=-, 2(2 - P)

When E 4 0, substituting (14) in equations (8) to (12) and comparing the coefficients of like powers of E, neglecting those of e2, we get the following equations as the coefficients of E with the help of equation (21):

aul awl ay aZ - + - = o ,

(25)

2E 8uo au (26)

The corresponding boundary conditions are

u1 = 0 , u, = -xcosnz , wi = 0 ,

u1 = o , u , = o , w1 = o , p , = o , O 1 = O as y + c o .

These are the linear partial differential equations which describe the three-dimensional flow.

In order to solve these equations we shall first consider the equations (22), (24) and (25), being independent of the main flow component u1 and the temperature field 0, . Wc assume c , , w , and p , of the form:

u1(y3z) = u11(Y)cosnz, (28)

w,(y,z) = - -u;,(y)sinnz, (29)

Pl(Y,4 = PIIQcosnz, (30)

(27) 0, = O at y = O ,

1

R

where prime denotes differentiation with respect to y. Equations (28) and (29) have been chosen so that the continuity equation (22) is satisfied. Substituting these equations into the equations (24) and (25) and applying the corresponding transformed boundary condi- tions, we get the solutions of I ) , , w1 and p, as:

60 ZAMM . Z. angew. Math. Mech. 73 (1993) 1

a21

n - 1 P l C v , Z ) = - e - n y cos R;? , (33)

(34)

(35) and substituting in equations (23) and (26), we obtain the following equations:

u; , + u R u ; ~ - R’u,~ = R u , , u ~ ,

Wil + aRPB;, - n2011 = RPol1Wo - ~ E P u ~ u ; , ,

(36)

(37) with the corresponding boundary conditions

where the primes denote differentiation w.r.t. y. Solving equations (36) and (37) under the boundary conditions (38) and using equations (34) and (35), we get

x cos 712, (39)

O~(Y,Z) = ~ aRP [” - { N + E(n + aR)} {e-iy - e-(n+2aR)y} 2-71 A 71

where

1 = 4 [aRP + ( ( E R P ) ~ + 4?r2}1/2], A = 4(71 + U R ) - P ( x + ~ G I R ) ,

B = 5A + 4aR - P ( l + 2aR), C = 31 + aR - P(1 + aR).

3. Resul ts a n d discussion

We now discuss the important flow characteristics of the problem. Knowing the velocity field, we can obtain the expressions for the shear stress components in the x* and z*-directions in the non- dimensional form as:

T x = - - @;v - @ ) y = o = 1 + E(l - F,(a, R)} cos 712,

where

(42)

(43)

F,(a, R) = 1 = 4 [aR + (a2R2 + 4 ~ ~ ) ’ / ~ ] . (44)

For G( = 1, I becomes f [R + ( R Z + 4 ~ ~ ) ” ~ ] and

(45)

reduces to the case of Ref. [2]. It is found that the expression for F, (1, R) in equation (45) differs

from that obtained by GERSTEN and GROSS [2] by a multiple of n due to some calculation mistake in their paper. The function F1 (a, R ) gives the correction to the quasi-two-dimensional value. The F2(a, R) in equation (44) being similar to the one obtained by GERSTEN and GROSS has not been discussed any more.

The heat flux at the plate in terms of Nusselt number Nu is given by

+ ~ { l - F,(u, R , P, E)} cos zz , (46) where

Pz(1 + N ) E”(1 + P )

x ( A - i, - 2aR) -

x (X - I - aRP) 2(l + N ) - 2E1 +- (1 - 71 - aRP) - -

R c

For E = 0 and a = 1,

+---- 1 + P R p P R 71 .I9

where

1 = [R + (R2 + 4 ~ ~ ) ~ / ~ ] .

It is noted that F3(l, R, P, 0) in equation (48) also differs from that obtained in Ref. [2] due to some calculation error there. The expression F,(a, R, P, E ) gives the correction to the quasi-two- dimensional value of Nusselt number.

It is easy to show that for E = 0 and P = 1 the Reynolds analogy holds:

Nu = 7 , .

The skin friction factor, F,(a, R) has been shown in Fig. 1. It is observed that F, tends to zero in the limiting case when R + co and the quasi-two-dimensional value is obtained exactly. When R + 0, F,(a, R) approaches the limiting value 1 but not to 0.318 as pointed out in Ref. [2]. In this case, the maximum correction to the quasi-two-dimensional value is 100% as against 32% given in Ref. [2]. It is also noted that, as the value of the suction parameter is increased, the skin friction factor F , decreases significantly.

In Fig. 2 the numerical values of F,(a, R, P , E ) have been plotted for different values of I, R, P and E. In order to be realistic, the values of Prandtl number are chosen as 0.71 and 7 approximately, which correspond to air and water respectively at 20 “C. Since the values of the Eckert number E are very small for an incompressible fluid, thus, to be more appropriate from the practical point of view, the values of E have been chosen as 0.01 and 0.05. Fig. 2 shows

Short Communications 61

that F , takes constant values for limiting values of R. When R + 0, F , + 1 which is obvious because there is no oscillatory flow. However, when R + x , F , + 0 which means that heat transfer approaches to quasi-steady value. It is interesting to note that F , decreases considerably with the increase of Prandtl number P or suction parameter x but increases with the increase of Eckert number E.

I. The function F , ( x , R) defined by eqn. (43)

0 5 0.71 005 0 5 7.0 0.01

io2 18' 1 R 10 d 2. The function F 3 ( g , R. P. €1 defined by eqn. (47)

References

1 LACHMAXS. G. V.: Boundary layer and flow control. Its principles and applications.

2 GIRSTFII. K.; GROSS.. J . F,.: Flow and heat transfer along a plane wall with periodic

3 SIXGH. P.: Srmw,& V. P.: MISKA, U. N.: Three dimensional fluctuating flow and heat

Vols. I. I I . Perganion Press, Oxford 1961.

suction, TAMP 25 (1974). 399 -408.

transferalon_eaplate~ithsuction,Int. J.HeatMassTransfer21 (1978),1117-1123.

Received June I , 1990

Address: Dr. K. D. SINGH, Department of Mathematics (D.C.C.); Himachal Pradesh University, Shimla-171 005, India

Akademie Verlag ZAMM . Z. angew. Math. Mech. 73 11993) 1, 61 - 6 3

GWEN, U.

On Elastic-Plastic Stresses of a Sphere with Linear Hardening in a Discontinuous Temperature Field

MSC (1980): 73E99, 731105

1. Introduct ion Discontinuous temperature fields are frequently encountered in many industrial applications. The simplest example is the case in which a part of the body is heated to a constant temperature T'", the remaining part having a temperature 7'"'. This case occurs when we are dealing with a thermal inclusion. Stress analysis in cylindrical and spherical coordinates under discontinuous temperature field has received considerable attention [l].

The behavior of elastic-plastic spheres with internal pressure and thermal loading has been extensively studied [2 - 51. The aim of this work is to develop analytical solutions for the elastic-plastic sphere with discontinuous temperature field. The sphere matcrial is assum- ed to be a linear work-hardening material that obeys Tresca yicld condition. A similar problem for the disk has investigated by this author [6].

2. Statement of the problem Let the temperature field inside a solid sphere with point symmetry (0 < r < a) be constant equal to To, the temperature outside being zero. This temperature field can be expressed by means of the Heaviside function as follows:

T = T,ij(a - I ) . (2.1) For sufficiently small values of the temperature, the sphere remains fully elastic and in this case the stresses can be obtained as follows:

(2.2) 1 %TOE

6 0 = - 3 ( 1 - v ) - I>ira - r ) + a3

(2.3)

where M is the coefficient of thermal expansion, b is the radius of the solid sphere. For the present problem the Tresca yield condition may be formally expressed as

uo - br = 6 0 , (2.4) where uo is the initial tensile yield stress. Since the Tresca yicld condition governs the behavior of the material of the sphere, the range of validity of the fully elastic solutions (2.2) and (2.3) is determined by that value of the temperature with which equation (2.4) is satisfied. Yielding begins at r = u, where the difference

by - b, = - is maximum. Therefore, equations (2.2) and (2.3)

remain valid for temperature below

%TOE

(1 - v)

(2.5)

3. Stresses and displacements in the elastic and plastic regions

For values of the temperature larger than that given by equation (2.5) a plastic region forms in the sphere, such that for a 5 r 5 e the sphere material is plastic, while for e 5 r 5 b it is still in an elastic state. The elastic-plastic interface radius e is a function of the temperature.

Firstly, consider the spherical plastic region a 5 r 5 Q. In this region, the equilibrium equation, the Tresca yield condition and the plastic flow rule apply. We assume that the sphere is made of a linear work-hardening material which obeys Tresca yield condition