Laser induced convection instability

Volume 87A, number 3

PHYSICS LETJERS

28 December 1981

LASER INDUCED CONVECTION INSTABILITY K. ERNST and J.J. HOFFMAN1

Institute of Experimental Physics, Warsaw University, Warsaw, Poland Received 9 July 1981 Revised manuscript received 30 October 1981

We describe theeffect of convection instability induced in agas heated by laser light. Ordered spatial pattern in the form of rotating rings appears only above the threshold value oflaser pulse energy. The effect is visualized and monitored by means oflight scattering on laser snow particles.

Introduction. The problem of convection instability has been investigated and described by many authors (see e.g. review papers [1,2]). In a typical experimental situationa horizontal fluid layer is heated from below. When the temperature gradient reaches a critical value, a well ordered spatial pattern in the form of rolls or hexagones, depending on the nature of the boundaty conditions, emerges. This kind of instability is usually called Rayleigh—Bënard instability. In this caper we present our results on the convection instability observed in a quite different configuration. In our experiment a cylindrical cell containing CS2 vapour is illuminated by nitrogen laser beam directed horizontally along the axis of the cell. In such a systern, for CS2 vapour pressure higher than 1 Torr almost total absorbed energy is transferred into heat, The energy irradiated in the form of the fluorescence light coming from excited CS2 molecules is less than 1% of the total energy absorbed by the gas [3]. The heating of gas by light absorption in the volume illuminated by laser light and concentrated along the cell axis leads to temperature and density gradients pointing outward from that volume. Due to buoyancy force acting on the heated gas the convection pattern appears above the threshold value of the laser pulse energy. 1

From the Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland.

Apart from the geometrical configuration, heating the system by a nitrogen laser beam leads to two other essential modifications: (1) the gas in its convective motion penetrates the heated volume illuminated by laser light, (2) the heating of gas is not continuous but consists of series of periodic heat pulses with the period corresponding to the repetition rate of the laser. A method of visualization of the convective structure is strictly connected with the system used in our experiment. As it was described in ref. [4] a fotochemical reaction induced by nitrogen laser in CS 2 vapour leads to the formation of micron-size solid particles called also laser snow. The particles cause intense scattering of laser light and they can be easily observed by the naked eye. In this way any convective structure appearing in the cell can be observed and recorded very conveniently. Our experimental results have shown that the convection pattern in the form of rings is observed only above a threshold value of the laser pulse energy depending on the gas pressure. The phenomenon described below is an example of the formation of spatially ordered structures in the system driven away from equilibrium where the driving force is explicitly time dependent. This should be contrasted with Bénard convection where the driving force is static. We wanted to show also that the laser snow being itself an interesting object of studies can be also conveniently used for monitoring and visualizing various .phenomena occurring in gas systems,

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PHYSICS LETTERS

Experimental. The nitrogen laser beam had a pulse energy of about 1 mJ, a pulse width 6 ns and a maximum repetition rate 35 Hz. The intensity of the mcident light was changed by means of neutral glass filters of known transmission while the cross section of the nitrogen laser beam was determined by a circular aperture placed in front of the cell. In order to compensate the divergency of the laser beam and to increase its intensity the beam was slightly focused. The diameter of the beam passing through the cell was about 5mm. The cylindrical pyrex cell (length 10 cm and internal diameter 2 cm) was connected to a vacuum system and a container with liquid CS 2 allowing to fill up the cell to a given pressure. Typical pressure values in our experiment were of the order of 100 Torr. In order to visualize the convective pattern by means of light scattering on laser snow particles, a He—Ne laser was used as a monitoring beam. Its cross section was transformed by means of a set of lenses to a rectangular form with a tinckness of about 0.2 mm. The beam intersected the cell perpendicularly to its axis allowing us to visualize the convective pattern corresponding to the intersection plane. Such a twodimensional pattern was observed and recorded by means of a photo or movie camera looking along the cell axis through one of the windows and the filter absorbing laserany uvlight. In ordernitrogen to minimize kmd ofadditional heating and/or cooling the cell was placed inside a metal box provided with appropriate wmdows. Theoretical approach. In order to describe convection instability in a gas we used heat conduction equation (I). Navier—Stokes equation (2) and the continuity equation (3) in the Boussinesq approximation

28 December 1981

at constant volume, p0 the density and ci the volume expansion coefficient. The laser pulse energy adsorbed at time t = 0 is given by the last term of eq. (1) where A denotes the absorption, p the pressure, E the pulse energy, S0 the cross section of the laser beam and S(r) the beam intensity distribution. In order to analyseeqs. (1 )—(3) we first use dimensionless variables defined as follows: 2/i, ~~vv/R, r-9-rR, t—~tR T T = F~2I~R3. —

b

Then noting the cylindrical symmetry of the problem we consider the convective motion in the XY plane perpendicular to the cell axis only. The Yaxis is directed along the earth gravity g. Finally the current function H is introduced, through which the velocity field is expressed as follows: v

=

aH/ay,

v

=

—

Y

X

or 1

v

r~aH/ap,

r

v

—aH/ar.

(4)

Eqs. (1) and (2) can now be written in terms of H, the z-component of the vorticity vector (V X v)~ G and the dimensionless temperature F 2F— [FHI + ~Ra/~)S~r)~(t) (5) aF/at=Pr’V aG/at = \~2G [G,H] aF/ax (6) —

—

with G V2H

7

— —

—

[5], [61.

In the above [A,B] = (aA/ax)(8B/ay)— (aA/ay) X (aB/ax), Pr is the Prandtl number Pr = r/K and Ra the Rayleigh numberRa = (gaR3/~p)L~T with L1T

aT/at= KV2T—(v’ V)T+(ApE/pocvSo)S(r)~(t),

ApE/pocvS 0.

2u

au/at = vV V~u0.

—

(u V)u

—

p~Vp

+ a(T— T~)g,

(1) (2)

F(r,

~,

t)

=

F 0(r, t) + F1(r, t)sin ~

(3)

The boundary conditions on the walls of the cell are T(rR) Tb andu(rR)’~ 0, whereR is the radius of the cell. In the above K is the thermometric conductivity, i.’ the kinematic viscosity, cv the specific heat 134

Since F, G and H are functions of polar coordinates rand p we express them as follows:

+F2(r, t)cos2p+...,

(8)

-

H(r,p,t)H1(r,t)cosp+H2(r,t)sin2p+ G(r, p, t) = G1(r, t)cos p + G2(r, t)sin 2~+

...

,

(9) (10)

Volume 87A, number 3

PHYSICS LETTERS

Here

28 December 1981

Discussion. We analyse solutions of our equations

~ are Bessel functions of the firstkind and nth order. The boundary conditions F 0 and Ur = 0 for r = I are satisfied for aflk corresponding to kth zeros ofJ~. The condition = 0 leads to the following sequence of equations dJ (~kr) h,~1~(t) ~ = 0. (12) k r r=1 Substituting eqs. (8)—(1 1) into eqs. (5) and (6) and using the well known orthogonality conditions for tngonometric functions and Bessel functions we obtam the infmite set of equations for amplitudes fflk(t), gfl~(t)and hflk(t). In order to see how the convection pattern emerges it is sufficient to use modes with lowest possible values of n and with k = 1. Knowing, from eq. (7), that gfl~ = aflkhflk we have then

by means ofthe conventional phase diagram technique [5,7]. Fig. 1 shows such a diagram for our case. Arrows indicate the signs of derivatives df0 and df1. Points (f1 f0) where df0 = 0 are determined by the equation W0 = 0 (two dashed lines) and points for which df1 = 0 by the equation W1 = 0 (ellipse). Point A on the figure is a stable equilibrium point and the system returns to it when Q is smaller than a certain critical Q~value. For Q> Q~(e.g. point Q2) we have creation of a convection instability. In order to get physical solutions for Q> Q~higher modes have to be taken into consideration. In contradistinction to the Bdnard configuration the radial pressure gradient created in our case cannot be compensated by vertical gravity force. This means that a non-zero velocity field appears also when Q < Q~, but it does not lead to the formation of a spatially organized pattern. In our experiment we observed the convection pat-

d ~ ‘di’

~13)

atern certain in thethreshold form ofrotating value ofrings laserappearing pulse energy onlycorreabove

c3h21.111 ,(14)

sponding to our critical ~ No organized convective motion was observed for Q < ~ Photoregistrations of the observed images corresponding to three different Q values (pulse energies) are shown on fig. 2. Fig. 2a corresponds to Q much larger than the critical value Q~and the convection rings are very well pronounced. Fig. 2b corresponds to Q> Q~but close to it. Finally on fig. 2c Q is smaller than Qc and no convection rings are observed.

An

=

~I aflk(t)Jfl(aflkr)

(A~= F~,G~, H~)-

k

—

_~aoi~ r 2 ip~ ‘~‘0i+ ~ 1 h ii~ii + Q8(t), ~-

df11/dt = —(cr~1/Pr)f11 c2h11f01 —

—

(11)

2

dh11/dt = —a11h11 dh

/dt = 21

+ 2 h + 21 21

C4f01 C ~ 5’11

,

‘

(15) 1~ ‘

Here Q and C1 are positive coefficients given in terms of the Bessel functions integrals. Since the relaxation rates of h11 and f11 obey the inequalities

‘

~ ~ it is possible to eliminate adiabatically h11 [5]. We then obtain two nonlinear equations for amplitudes of the conduction mode f0 ~f01 and the convection modef1 ~f11 df0/dt = —(a~1/Pr)f0+ (C1C4/a~1)f0f1+ Q6(t)

df1/dt = —(a~1/Pr)f1+ (C2C4/ci~1)f~

—(C3C5/c41)f~=W1(f0,f1).

(18)

W0 and W1 are polynomials of the second order expressing the mode coupling (second order terms) and damping (linear terms), Q Ra is the heat induced by the laser pulse. ‘-‘

In conclusion we would like to point out three new

df1rO —

—

dt.,~O

°

~ (17)

=W0(f0,f1)+Q~(t),

,

,7/

/7~

~

-1 I

Fig. 1. Phase diagram (compare text for explanation).

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Volume 87A, number 3

PHYSICS LETTERS

_____

28 December 1981

aspects of our experiments: (1) the new configuration in which convection instability has been studied, (2) heating of gas by means of laser light absorption, (3) application of laser snow for visualizing and nionitoning the convection pattern. The authors wish to thankt.A. Turski for helpful discussions. References

,~

[1] P. Bergé, in: Fluctuations, instabilities and phase transitions, ed. T. Riste (Plenum, New York, 1975) p. 323. [2] C. Normand, Y. Pomeau and M. Velarde, Rev. Mod. -~s,w

I

__________________________________

Phys.49(1977)581. [3] L. Brus, Chem. Phys. Lett. 12(1971)116. [4] K. Ernst and J.J. Hoffman, Chem. Phys. Lett. 68 (1979)

[5] H. Haken, Synergetics (Springer, Berlin, 1977).

[6] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Oxford, 1961). [7] J. Maynard-Smith, Models in ecology (University Press, 1974).

~iII~

Fig. 2. Photoregistrations of convection patterns corresponding to different pulse energies. (a) Q a~Q~.(b) Q ~ ~ (c)

Q
136