A.M.Mancho

Instituto de Matemáticas y Física Fundamental,

Consejo Superior de Investigaciones Científicas,

Madrid, Spain.

M. C. Navarro , H. Herrero

Departamento de Matemáticas,

Universidad de Castilla-La Mancha,

Ciudad Real, Spain.

S. Hoyas

Universidad Politécnica de Madrid,

Madrid, Spain.

INSTABILITIES IN A NON- HOMOGENEOUSLY HEATED FLUID

IN MARANGONI CONVECTION

PHYSICAL SETUP

dar /,0 ]1,0[z

Domain:

RELATED THEORETICAL WORKS

We have used numerical techniques developed and theoretically justified in these articles

Herrero, H; Mancho, A.M. On pressure boundary conditions for thermoconvective problems. Int. J. Numer. Meth. Fluids 39 (2002), 391-402.

H. Herrero, S. Hoyas, A. Donoso, A. M. Mancho, J.M. Chacón, R.F Portugues y B. Yeste. Chebyshev Collocation for a Convective Problem in Primitive Variables Formulation. J. of Scientific Computing 18 (3),315-318 (2003)

RELATED THEORETICAL WORKS

These numerical techniques have been applied to a similar problem with lateral constant temperature gradient.

Hoyas, S. ; Herrero, H.; Mancho, A. M. Bifurcation diversity in dynamic thermocapillary liquid layers. Phys. Rev. E 66 (2002), 057301-1-057301-4.

Hoyas, S. ; Herrero, H.; Mancho, A.M. Thermal convection in a cylindrical annulus heated laterally. J. Phys. A: Math. Gen. 35 (2002), 4067-4083.

Hoyas, S.; Mancho A.M.; Herrero, H. Thermocapillar and thermogravitatory waves in a convection problem. Theoretical and Computational Fluid dynamics 18 (2004), 2-4, 309-321.

Hoyas, S; Mancho, A.M.; Herrero, H.; Garnier, N.; Chiffaudel, A.; Benard-Marangoni convection in a differentially heated cylindrical cavity. Phys. of Fluids. 17, 054104-1,12 (2005).

Linear heating

RELATED EXPERIMENTAL WORKS

These experiments describe a similar problem with lateral constant temperature gradient.

R.J. Riley and G.P. Neitzel, Instability of thermocapillary-buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities, J. Fluid Mech. 359, 143 (1998).

N. Garnier, Ondes non lineaires a une et deux dimensions dans une mince couche de fluide, Ph.D. thesis, Université Paris 7, France, 2000.

J. Burguete, N. Mukolobwiez, F. Daviaud, N Garnier, A. Chiffaudel, Buoyant-thermocapillary instabilities in extended liquid layers subjected to a horizontal temperature, Phys. of Fluids 13 (10) 2773-2787 (2001).

-Pr

,

0,

2

2

uuuu

u

u

pt

t

BASIC EQUATIONS:

FORMULATION OF THE PROBLEM

No gravity effects

Pr => Prandtl number

Domain

BOUNDARY CONDITIONS:

for the velocity are,

M => Marangoni number

dar /,0 1,0z

1on ,0 M

,0 M

on 0,

0on ,0

1on ,0

zr

uu

ruuu

zuuu

zu

zrrz

zr

zr

z

BOUNDARY CONDITIONS:

for the temperature are,

B => Biot number

=> Gaussian width. Heating shape

S/Tu=> Quotient of lateral and vertical temperature differences

Regularity conditions at the origin

Multiparametric problem

, Pr, M, B, , S/Tu

Control Heat related parameters

parameter

on 0,

1on ,0

0on 1,1e

ee 2

2

22

1/

1/

)1(4/11

r

zB

zT

S

r

z

r

u

0on ,0

rX

1-s211-s1-s1

1211-s1

1-s1

1s2s1-s1-ss

121-s1s

1s

-Pr

)(

)(

-Pr

,

,

uuu

u

u

uuuuu

uu

u

ss

sss

s

ss

ssss

s

pc

b

a

cp

b

a

Stationary and axisymmetric,

Regularity conditions for the basic state at the origin are,

We solve the basic state with a Newton-Raphson iterative method.

The equations and boundary conditions are linearized at each step s, around solutions at step s-1

THE BASIC STATE

.0 ,0 XXt

,0 uupu rrzrr

111 , , ssssss pppuuu

0 ,0

0)(1//Pr - ,

0 0, ,

0)(1//Pr - 1,

0 0, 0, ,0 1,

)(1

0 ,0 0, ,0

2

2

r

p

rr

wur

ur

pr

rr

wz

pz

z

v

rM

z

uB

zwz

z

w

r

ru

rz

r

z

uu

u

uu

u

+

+

THE COLLOCATION METHOD

At each step unknowns are expanded in Chebyshev polynomials:

Basic equations are evaluated at collocation points

Boundary Conditions are evaluated at:

1

0

1

0 )()(),(

L-

l

M-

nnlnl

s zTrTxzrX

0p

With those rules we obtain 4xLxM equations and unknowns.

Some results are:

S/Tu = 0.001

S/Tu = 0.5

THE LINEAR STABILITY ANALYSIS

We perturb the basic state :

Regularity conditions at r=0 are:

The perturbation fields are expanded in Chebyshev polynomials,

A trick for m=1,

tmb zrxzrXzrX ie ),(),() ,,(

0;1 0;

1 0; i

0 0;

mpuuu

mpuuu

mr

p

rr

uuu

zr

zr

zr

1

0

1

0 )()(),(

L-

l

M-

nnlnl zTrTxzrx

2

0

1

0 )()(),(

L-

l

M-

nnlnl zTrTxzru

CONVERGENCE RESULTS

7 9 11 13

33 68.90035 68.64861 68.68878 68.71441

35 68.88943 68.64569 68.68735 68.71319

37 68.68648 68.64278 68.68559 68.71157

r-coordinate

z-coordinate

B=0.05, M=92*Tu, =10, S=1ºC, =0.8

The number of unknowns and equation are:

for m=1, 4xLxM+(L-1)xM

otherwise 5xLxM

=10, Pr=0.4

THE INFLUENCE OF THE HEAT PARAMETERS

The shape of the heating on the range {0.8-10}

THE STABILITY RESULTS

Biot number fixed to B=0.05,

=0.8

The influence of S/Tu

S/Tu ~ {0.001-1}

Thresholds

S/Tu

M

Patterns at critical thresholds

S/Tu=0

S/Tu=0.01

Patterns at critical thresholds

S/Tu=0.05 S/Tu=0.5

=10, B=0.05

THE INFLUENCE OF THE PRANDTL NUMBER

For Pr=0.1 thresholds diminish

S/Tu

M

Pr=0.4, B=0.05,

THE INFLUENCE OF ASPECT RATIO

at =2 thresholds are M~13000 for S/Tu =0.02

Patterns have wavenumber m=1

Pr=0.01, B=0.05 and S/Tu =-1

these waves are possible

COMPARISONS WITH EXPERIMENTS

N. Garnier y A. Chiffaudel, Eur. Phys. J. (2001)

=11.76, S/Tu~0.05, B=0.2, Pr= , M= 542

Hoyas, Mancho, Herrero, Garnier and Chiffaudel, Physics of Fluids, (2005)

= 33.4, S/Tu~1, B=0.2, Pr= , M= 642

COMPARISONS WITH EXPERIMENTS

= 52.9, S/Tu~ 0.6, B=0.2, Pr= , M= 508

Hoyas, Mancho, Herrero, Garnier and Chiffaudel, Physics of Fluids, (2005)

Non-homogeneous heating develop new instabilities on Marangoni convection. Some of them are also present in purely buoyant convection (see MC Navarro poster)

The shape of the heating has been shown to be less influencial than the ratio S/Tu

S/Tu

* may increase considerably instability thresholds

* cause spiral waves and other oscillatory instabilities.

* is on the origin of localized patterns mainly for large values. Once S/Tu is large enough, localized patterns are then due to combined effects of other parameters as Pr, B and

CONCLUSIONS