- Email: [email protected]
Possible mechanism of the roughness formation on a liquid layer caused by a high heat flux
Fusion Engineering and Design 19 (1992) 183-191 North-Holland
183
Possible mechanism of the roughness formation on a liquid layer caused by a high heat flux K. F u j i m u r a "~, M. Ogawa a a n d M. Seki b
Japan Atomic Energy Research bzstitute Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan b JAERI, Uchisaiwai-cho 2-2-2, Chiyoda-ku, Tokyo 100, Japan Submitted 10 April 1992, accepted 6 July 1992 Handling Editor: M. Ohta
Linear stability of a thin melt layer on metal walls after a plasma disruption was investigated in order to clarify the mechanism of the roughness formation which is observed in an experimental work by Ogawa et al. (1992). Stability characteristics in terms of a thermocapillary (Marangoni) instability are found to be consistent with the experimental results. A relationship between the critical Marangoni number and the crispation number parametrized by the Prandtl number and the Bond number was obtained to predict the occurrence of the roughness formation on the resolidified surface.
1. Introduction
In a thermonuclear fusion reactor such as the International Thermonuclear Experimental Reactor (ITER) [1] or the Fusion Experimental Reactor (FER) [2], plasma disruptions cause exposure of plasma-facing components (hereafter referred to as PFCs) to severe heat load, high-energy particles, and intense electromagnetic force. The heat load is evaluated to range from 10 to 20 M J / m 2 for the duration time of 0.1-3 ms in the ITER design [1]. Surface materials of PFCs melt, evaporate, sublimate, or fracture because of the heat load. The materials are thus eroded by plasma disruptions so that a life time prediction of PFCs is one of the key issues in the design of a thermonuclear fusion reactor. Some experimental studies [3-8] on thermal shock, thermal deformation, and erosion during plasma disruptions have been performed in order to understand thermal-structural characteristics and to predict the durability of PFCs. Madarame et al. [9] and Ogawa et al. [10] reported thermal-structural behaviors of metals subject to high heat flux. They found that the metal Correspondence to: Dr. Kaoru Fujimura, Department of High Temperature Engineering, Tokai Research Establishment, Japan Atomic Energy Research Institute, Tokai-mura, Nakagun, Ibaraki-ken 319-11, Japan.
after heating exhibits two features: a rough surface and a smooth surface. Ogawa et al. also reported that many visible cracks were observed on the rough resolidified surface after several irradiations, while little cracks were on the smooth resolidified surface. The formation of the roughness on the resolidified surface is not only physically an interesting phenomenon but also of importance from a point of view of the life time of PFCs associated with crack generations. We may list several possible mechanisms which cause the roughness formation on the resolidified surface: evaporation of selected elements involved in metal test pieces, the pressure of evaporated metal vapor, hydrodynamic instability, instability due to evaporation/condensation process, Rayleigh-Taylor instability, thermocapillary instability, etc. Fujimura [11] first emphasized that an instability driven by a temperature gradient of the surface tension, i.e., a thermocapillary (or Marangoni) instability, is one of the possible mechanisms to create the roughness on a melt layer. He carried out preliminary linear stability analyses of a melt layer on which an external body force acts in an arbitrary direction. From a hydrodynamic point of view, a similar problem, i.e., the linear stability of a liquid film down a heated inclined wall, was analyzed by Kelly, Davis and Goussis [12] in the long-wavelength limit. In these two cases, there are two driving mechanisms of instability for wave forma-
0 9 2 0 - 3 7 9 6 / 9 2 / $ 0 5 . 0 0 © 1992 - Elsevier Science Publishers B.V. All rights reserved
184
K. Fujimura et aL / Roughness formation on a liquid layer
tion: thermocapillary instability and hydrodynamic instability. The former is intrinsic for a liquid film on a heated or cooled horizontal wall while the latter is intrinsic for an isothermal liquid film down an inclined or vertical wall. Here we call wall surfaces perpendicular to the gravity as the horizontal and wall surfaces parallel to the gravity as the vertical. If both driving mechanisms are adequately taken into account in the analysis, the critical Marangoni number of the vertical case has to be zero [12], i.e., the hydrodynamic instability mechanism is dominant for the vertical film irrespective of the strength of the Marangoni effect. Madarame et al. [9] examined an effect of impurity contents in stainless steel on the roughness formation, and reported that small amounts of sulfur or oxygen have a strong effect on the formation. Madarame and Okamoto [13] carried out numerical analyses on the formation of the roughness on a vertical melt layer. Through numerical integration of model equations,
they concluded that an instability sets in if the Marangoni number M (a non-dimensional temperature gradient of the surface tension) is greater than some positive critical value, M~+ say. Although they tried to justify the positiveness of the Marangoni number by claiming that an increase of the amount of sulfur in stainless steel changes the sign of M from negative to positive, they unfortunately ignored the hydrodynamic instability mechanism in their model equations. Consider a horizontal melt layer. For M > M + = 80, a stationary convection with famous hexagonal pattern sets in, whose critical wavenumber is a c = 2. (In what follows, letters with an asterisk denote dimensional numbers while letters without asterisk denote non-dimensional ones.) The most unstable wavelength corresponding to this a e is much shorter than the typical wavelength of the experimentally obtained roughness on the resolidified surface of the melt layer [10]. We have to point out here that a horizontal melt layer is
(b)
(c)
!
I
10ram
(d)
(e)
Fig. 1. Topographic features of the rough surfaces with various heating durations. Peak heat flux: 80 MW/m 2, test piece: type-316. (a) 60 ms, (b) 70 ms, (c) 75 ms, (d) 120 ms, (e) 150 ms.
K. Fujimura et al. / Roughness forrnation on a liquid layer
known to be unstable to a traveling wave or a standing wave even for M < M~- < 0 [14] where M~- denotes the negative critical Marangoni number. The critical wavenumber for M < 0, as we will see later, ranges from 0.1 to 1 for stainless steel which can be comparable with the experimental one. The objective of the present paper is to demonstrate that the thermocapillary (Marangoni) instability is one of the main mechanisms to create the roughness on the resolidified surface and to obtain a criterion under which the resolidified surface becomes rough after a plasma disruption. We investigate the linear stability of a horizontal melt layer by solving the linear eigenvalue problem. The theoretical results are compared with the experimental ones obtained by Ogawa et al. [10] in the horizontal configuration. The present paper will provide a guiding principle for restraining the crack generation in metallic materials.
2. Experimental results In this section, we outline the experimental results of [10] on roughness formations on the resolidified surface after heating. For further details of the experiment, see [10]. The results will be compared with the linear stability characteristics in later sections. In the experiment, a metal test piece was heated by a hydrogen ion beam in a vacuum chamber. The mate-
185
rials used were stainless steel (type-316, type-304, type316F, and the Primary Candidate Alloy in Japan Atomic Energy Research Institute, PCA) and pure metals (aluminum, copper, molybdenum, and tungsten). Type-316F is a reference material of type-316 for research and developmentof the fusion reactor in JAERI. It involves less amounts of manganese, phosphorus, silicon, and carbon than the type-316. PCA is a modified type-316 which has a high void swelling resistance. Test pieces were originally fiat plates with diameters of 80-120 mm. The peak heat flux was ranging from 68 to 261 M W / m z and the heating duration was 40-250 ms. The heat flux had temporal as well as spatial variations. We summarize here the obtained results on the roughness formation on the resolidified surface. (i) Under common heating conditions, the resolidifled surfaces of stainless steel of type-316, PCA, and type-304 became rough, while the surface of stainless steel of type-316F remained smooth. (ii) For the stainless steels exhibiting a rough resolidifled surface (type-316, PCA, and type-304), the conditions of roughness formation did not depend on any heating conditions. For the aluminum test pieces, on the other hand, the resolidified surface remained smooth under relatively mild heating conditions, while it became rough under relatively intense conditions. The shape of the resolidified surface of aluminum thus depends on the heating conditions. The resolidified surfaces of copper,
ig. 2. Rough and smooth surfaces of aluminum test pieces. (a) 79 MW/m 2 and 60 ms, (b) 80 MW/m z and 100 ms.
186
K. Fujin~ura et aL / Roughness formation on a liquid layer
molybdenum, and tungsten test pieces were smooth under the experimental conditions performed. (iii) The roughness was not formed during the resolidifying process but during the initial stage of the melting process. In order to facilitate our understanding, we refer to fig. 1, which has already been presented in ref. [10]. Figures l a - e show topographic features of the rough surface for the type-316 stainless steel exposed to a common peak heat flux under different heating durations. If the heating duration is 60 ms, the surface has not melted yet. If the duration is 70 ms, many small mountains are observed on the surface. These small mountains form ridges with increase in heating duration (lc). If we further increase the heating duration, these ridges form several mountain chains (ld), and eventually become the outer rim (le). Figures 2a and b show topographic features of the aluminum test pieces exposed to the same peak heat flux. The heating dura-
(a) 22ms
(e) 34 ms
(~) 67ms
tions of figs. 2a and b are 60 ms and 100 ms, respectively. The surface of the test piece in fig. 2a melts but remains smooth, while the surface in fig. 2b becomes rough just like fig. le for the stainless steel. Figure 3 shows the surface of the melt layer observed with a high-speed video camera. These pictures were taken at a speed of 2000 frames per second. The test piece is type-304 stainless steel and the heating duration is 67 ms. The surface begins to melt at t* = 22 ms (fig 3a). The high-temperature parts in the melting region brighten so that they appear as white spots in the photo. The roughness is already observed at this stage. Evaporation is clearly observed for t * > 34 ms (fig. 3e). Note that the white foggy parts above the test piece show the vapor. The heating is stopped at t* = 67 ms (fig 3i). Judging from these photos, the initial roughness formation is considered to be due to some instability mechanism of the melt layer. In the following sections, we analyze the linear
(b) 24 ms
(c) 27ms
(.f) 38rns
(g) 40 ms
(h) 50 ms
(k) 68,ms
(t) 70.5ms
2~Omrn (j) 67,5rns
(d) 29.5ms
Fig. 3. Observation for surface of type-304 stainless steel test piece by high-speed video camera. Peak heat flux: 210 MW/m 2, heating duration: 67 ms.
K. Fujimura et al. / Roughness formation on a liquid layer
"4
.~_
g.
N~ ff
XXXXXX
X
X
X
X
X
X
I l i l i i
I
e..
.o
l
l
l
x~
,~, 0 e"0
8
XXXXXX
~~.~o
G o~
~d~l~
l!: ._
~
.
187
K. Fujimura et aL / Roughnessfomlation on a liquid layer
188
stability of the melt layer by restricting ourselves to the thermocapillary instability and compare the theoretical results with the experimental ones outlined above.
subject to the following boundary conditions:
(4)
w-to-c/=0, Cr[Pr-tto - (D z - 3aZ)]Dw + (Bo + az)a2"q = O,
3. Linear stability analysis
(D 2 + o~2)w -}- MazO - MotZr/= 0,
If the basic state, i.e., unperturbed melting and resolidifying process, changes very rapidly, the standard technique in linear stability analysis, i.e., the normal mode analysis, for an infinitesimal disturbance is invalid. Unfortunately, stability theory for such rapidly varying basic state has not been established yet except for rigorously periodic situations. Non-separability of the disturbance equations lets us treat partial differential equations other than ordinary differential equations. In order to avoid this difficulty; we assume here a slowly varying situation such that the time scale on which the basic state changes is completely separable from the one on which the disturbance linearly evolves. This situation enables us to make use of the method of multiple scales. At t h e lowest-order approximation, we then obtain linearized disturbance equations which are identical with the disturbance equations for the steady basic state problem. The unsteadiness is thus assumed to give a minor correction on the stability characteristics at higherorder approximations. The assumption will be justified in a later section by comparing the results of the analysis with the experimental results of Section 2. Because the disturbance equations for the steady basic state are well known, we only list the final form of the equations and explain how to solve them. Consider a liquid layer bounded by a horizontal rigid wall at z* = 0 and by a free surface centered around z* = h * . The gravity acts in the negative z*direction. Also assume that the layer extends to x* + oo and y* --* + oo Temperature of the rigid wall and the averaged one at the free surface are respectively maintained at To*- AT* and TO*, where we assume that AT* > 0. Appropriate non-dimensionalization, a shift of the z-coordinate, and the normal mode analysis
= /0(.0z)
exp[i(OtxX+Otey)+tot ] ,
(1)
yield the linearized disturbance equations of the form [ P r - lto - ( 0 2 - a2)] ( 0 2 - a2)w = 0,
(2)
[to - ( o
(3)
-
o = ,,.
(D + B )O - B~7 = O,
(5) (6) (7)
at z = 1 / 2 while
w=Dw=O=O
at z - - - 1 / 2 ,
(8)
where ~,, 7~, and ~ are respectively the disturbance vertical velocity, disturbance temperature, and location of the perturbed free surface, D = d / d z , and a z - a ~ + a~,. The above equations involve the following nondimensional parameters: the Prandtl number P r - u*/K*, the crispation number Cr=p*u*K*/(s*h*), the Bond number Bo = p * g * h * 2 / s *, the Marangoni number M = ( - d s * / d T * ) A T * h * / ( p * u * K * ) , and the Biot number B =cr*T*3h*/)t *, where p* is the density of the liquid, v* is the kinematic viscosity, ~.* is the thermal conductivity, K* is the thermal diffusivity, o-* is the Stefan-Boltzmann constant, g* is the acceleration due to gravity, and s* is the surface tension. Expansion of w and 0 in Chebyshev polynomials as
0
,,=00,,(z+l/2)
T,,(2z),
(9)
reduces the system of ordinary differential equations (2)-(8) to a system of algebraic equations for w,, and 0,, from which the eigenvalue to is determined as a function of parameters: co = to(a, Pr, M, Bo, B, Cr). Here T,(2z) denotes the Chebyshev polynomial of the nth degree. The factors (z + 1/2) z and (z + 1/2) are involved in the expansion functions so as to satisfy the homogeneous boundary conditions at the rigid wall, eq. (8). We applied a standard subroutine package based on the QR algorithm to solve the algebraic eigenvalue problem. We checked the accuracy of the numerical integration based on the expansion (9) by changing the value N which is the maximum degree of the Chebyshev polynomial involved in the expansion. The check clarified that N = 30 gives satisfactorily accurate stability characteristics for the present purpose.
4. Stability characteristics and comparison with the experimental results We tabulated some physical parameters involved in (2)-(8) in table 1 which are relevant to the typical
K. Fujimura et al. / Roughness formation on a liquid layer 0,9
........
I
........
0.1
a~r
0
=~-=-"~........ ~i....... ,,~
-0.I
",
I
........
SS(2)
" "" "
"
I ""
...
)
./
/
."
-0.4
,
0.001
, .......
, llltHI
"
,,: ......
0.01
I ,
0.1
,
--] stable
='"
, , .... ,
"
• --
"' '. . . .
1
]
,,.j
..Mo "
.
'q]-..
SS(l),
.
t ..~ u n s t a b l e
AJ([).
• ,"-,,
Y SS(2)
"
';?'-, "
,
-0.3
q
........ AI 2 ( )
. "~,::::~.Cu .
_._Mo........:_.--(~.
-02
.......
" ~. 1
,,,,,
10
Fig. 4. Linear growth rate a/r for stainless steel (SS(1) and SS(2)), aluminum (AI(1) and AI(2)), copper (Cu), and molybdenum (Mo) based on the non-dimensional parameters in table 1.
experimental data described in Section 2. Here, we define the energy density Q* by the peak heat flux integrated over the heating duration. Although we need to have the surface temperature and the thickness of the melt layer in order to evaluate the non-dimensional parameters in the table, it is not easy to obtain the data experimentally. Instead, we evaluated these data by solving a one-dimensional heat conduction equation taking account of an evaporation effect. The experimental data and the computational ones agree well with each other in the initial stage of melt-
189
ing. The agreement becomes worse after the initial stage because of a convection effect in the melt layer and an effect of vapor shielding. (See [10] for further details.) The parameters for stainless steel, SS(1), corresponds to fig. lb and the ones for aluminum, AI(1) and Al(2), correspond to figs. 2a and b, respectively. Utilizing the data of table 1, we evaluated the linear growth rate to, as a function of wavenumber t~ for stainless steel, copper, aluminum, and molybdenum as depicted in fig. 4. For each parameter, a pair of stable stationary modes existing in the low-wavenumber region merges into a pair of traveling waves in the higher-wavenumber region. The linear growth rate corresponding to the real part of the first eigenvalue asymptotically tends to zero from below in the longwavelength limit. It is therefore clear that the growth rate for a pair of traveling waves gives the information whether or not the liquid layer is unstable. The pair of traveling waves is composed of two traveling waves propagating in opposite directions with the same phase speed and having the same linear growth rate. Physically, therefore, a traveling wave propagating in one direction or a standing wave has to be attained. We cannot say anything about which wave is observable in real physical systems without analyzing the weakly nonlinear stage, which is out of scope of the present paper. As far as the data of table 1 are concerned, melt layers of stainless steel, SS(1), aluminum, AI(1), copper, and molybdenum are predicted to be stable while those of stainless steel, SS(2), and aluminum, Al(2), are
Fig. 5. Cross, sectional view of the resolidified layer. Peak heat flux: 80 MW/m 2, heating duration: 70 ms, test piece: type-316 stainless steel. See also fig. lb.
190
K. Fujinzura et al. / Roughness formation on a liquid layer
unstable. These are consistent with the experimental results outlined in Section 2 except for SS(1). SS(1) is predicted to be stable although a rough surface was obtained experimentally as is shown in fig. lb. This point will be discussed in the next paragraph. The peak of the linear growth rate predicts a physically realizable wavenumber, amax" According to the peak location for stainless steel, otto,x = 0.3. The wavelength of the roughness on the melt layer is thus predicted to be 2'n'/OtmaxXh*= 2 mm. Figure 5 shows a cross-sectional view of the resolidified surface for stainless steel. The photo shows that the typical wavelength is 1.6 ram, which is quite comparable with the theoretical prediction.This photo needs some discussion. According to the photo, the resolidified part seems to be localized, horizontally. This localization is considered to be due to the Marangoni effect. The melt layer is formed almost uniformly at the onset of melting. A thermocapillary wave is then created on the surface of the melt layer because of the instability of the layer. The melt layer with a uniform depth is thus replaced with a layer with sinusoidally changing depth. The strong surface tension tears the melt layer rapidly after the growth of the average depth of the melt layer. The melt layer is therefore isolated just prior to the resolidifying stage. The critical Marangoni number M~" and the corresponding wavenumber are depicted in fig. 6 as a function of the crispation number Cr parametrized by (Pr, Bo). It is known that the stability characteristics are not sensitive to the Blot number. We find that the critical wavenumber changes substantially as a function of both the crispation number and the Prandtl number, but is insensitive against the Bond number Bo. Utilizing this figure, one may predict the linear stability characteristics for melt layer dynamics. Beyond the curves for M~-, i.e. M < M~- < 0, the layer is unstable to a traveling wave or standing wave disturbance. The experimental parameters of table 1 are plotted in the figure. The experimental parameter for stainless steel SS(1) is just below the critical curve while the one for SS(2) is far beyond the curve. The parameter of AI(1) is far below the critical curve while the one of AI(2) is again far beyond the curve. Moreover, the parameters for copper and molybdenum are far below the critical curve. These data are completely consistent with the experimental results except for SS(1). Let us now discuss this disagreement. In order to obtain the parameters, we evaluated the surface temperature and the thickness of the melt layer from the numerical integration of the one-dimensional heat conduction equation as described above. These data are not accurate after the initial stage. If we increase the value of the dura-
103
(er,8o)=(o.t,io"2) ~____......~/ •
•
unstable ~ , q --ds/dt
_Mc0o_2 to..2)
-
s
lO2 1
I
i
i
i
,
,
,,I
,
t
i
i
i
i
.
i
t
ii
0.8 0.6
%
Ih----~,01
0.4 0.2 0
t
10-5
i
i
i
i
i l t l
10"4
i
.
.
.
.
10-3
Cr Fig. 6. Linear critical Marangoni number and wavenumber as functions of the crispation number parametrized by the Prandtl number and the Bond number. The critical conditions are almost independent of the Biot number. Beyond the curves of the critical Marangoni number, a melt layer is unstable, o: SS(1), e: SS(2), rn: AI(1), I1: AI(2), A: Cu, , : Mo.
tion time by 20%, then the parameter (M, Cr) for stainless steel moves to the vicinity of the position of SS(2) which is obviously in an unstable region. Judging from the figure, a correction less than 5% is enough to obtain the unstable situation. Of course, any experimental work as well as the assumption of a slowly varying basic state causes some amount of error. These errors may be much greater than 5%. We conclude that the disagreement for the SS(1) is mainly due to the error in the evaluation of the parameters. We further indicated in fig. 6 by three arrows the directions of the change in parameter space (M, Cr) resulting from an increase of the heating duration (At*), the heat flux (q*), the surface tension s*, or the temperature gradient of the surface tension - d s * / d T * . The increase of At*, q*, or - d s * / d T * exerts a destabilizing effect while the increase of s* exerts a stabilizing effect. It is well known that a surface tension s* in molten iron increases if the amount of impurity from group VI B is reduced. Thus the reduction of the impurity exerts a stabilizing effect. It is also clear that any melt layer of metallic material can in principle be unstable depending on the value of the parameters At* or q*.
K. Fujimura et aL / Roughness formation on a liquid layer
We now conclude that the linear stability analysis based on the assumption of slow variation is valid for the present problem. Moreover, the roughness formation can be explained by the thermocapillary instability mechanism. Because of the linearity assumption, the present analysis can only predict the primary roughness formation under weak heating conditions. It is invalid for nonlinear evolution involving a doubling or merging of several wavy structures. On the linear basis, however, all the terms are properly involved in the disturbance equations which is consistent with the lowestorder approximation of the multiple scaling expansions. Madarame and Okamoto's numerical analysis is complementary to our analysis for the horizontal configuration because their analysis can treat unsteady as well as nonlinear effects.
5. Conclusions
We analyzed the thermocapillary instability of a horizontal melt layer with negative Marangoni number. In the analysis, we assumed the slowly varying situation in order to adopt the multiple scaling approach to the unsteady melting and resolidifying problem. As far as the objective of the present paper is concerned, there is no need to proceed to any higher-order approximation although the higher-order approximation for the present analysis is straightforward. Despite the substantial assumptions on the basic state and experimental conditions computed from the one-dimensional heat conduction equation, the present theoretical analysis successfully explains the experimental results of [10] not only qualitatively but also quantitatively. The experimental findings about stainless steel and aluminum, (i) and (ii) in Section 2, are well reproduced by the analysis. Moreover, the predicted wavelength is comparable with the experimentally obtained wavelength for the roughness on the resolidified surface. Any metallic melt layer can be unstable without changing the sign of the Marangoni number. We provide fig. 6 for the critical Marangoni number as a function of Cr parametrized by Pr and Bo. Once the parameters (M, Cr, Pr, Bo) are prescribed, then one may roughly judge from the figure whether or not the roughness is formed on the resolidified surface.
191
References
[1] IAEA, ITER Conceptual Design Report, IAEA/ITER/ DS/No. 18 (1991). [2] S. Matsuda et al., Conceptual design study of the Fusion Experimental Reactor (FER), Proc. 13th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Washington DC, IAEA-CN-53/G-2-2 (1990). [3] J.R. Easoz et al., Thermomechanical testing of first wall test pieces in ESURF, Nucl. Technol. Fusion 4 (1983) 780. [4] C.D. Croessmann, G.L. Kulcinski and J.B. Whitley, Correlation of experimental and theoretical results for vaporization of simulated disruption, J. Nucl. Mater., 128/129 (1984) 816. [5] M. Seki et al., A simulated plasma disruption experiment using an electron beam as a heat source, J. Fusion Energy, 5 (1986) 181. [6] G. Rigon, P. Moretto and F. Brossa, Experimental simulation of plasma disruption with an electron beam, Fusion Engrg. Des. 5 (1987) 299. [7] M. Ogawa et al., Experimental observations of metal surfaces exposed to intense hydrogen beam; simulation of plasma disruption, J. Nucl. Sci. Technol. 26 (1989) 721. [8] M. Seki et al., Thermal shock tests on various materials of plasma facing components for FER/ITER, Fusion Engrg. Des. 15 (1991) 59. [9] H. Madarame et al., Effect of impurity content in stainless steel on resolidified surface condition after disruption load, Fusion Engrg. Des. (1989) 213. [10] M. Ogawa et al., Experimental study on melting and evaporation of metal exposed to intense hydrogen ion beam, Fusion Engrg. Des. 19 (1992) 193-202, next article in this issue. [11] K. Fujimura, in: Japanese Contributions to IAEA INTOR Workshop, Phase 2a, Part 2, eds. M. Seki et al., Research Report of Japan Atomic Energy Research Institute JAERI-M85-075, (1985) 93-107. [12] R.E. Kelly, S.H. Davis and D.A. Goussis, On the instability of heated film flow with variable surface tension, Proc. 8th Heat Transfer Conf., eds. C.L. Tien, V.P. Carey and J.K. Ferrell (Springer, 1986) p. 1937. [13] H. Madarame and K. Okamoto, Numerical simulation of surface tension induced melt layer movement after disruption load, Fusion Engrg. Des., in print. [14] M. Takashima, Surface tension driven instability in a horizontal liquid layer with a deformable free surface, I. Stationary convection; II. Overstability, J. Phys. Soc. Jpn. 50 (1981) 2745, 2751.
183
Possible mechanism of the roughness formation on a liquid layer caused by a high heat flux K. F u j i m u r a "~, M. Ogawa a a n d M. Seki b
Japan Atomic Energy Research bzstitute Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan b JAERI, Uchisaiwai-cho 2-2-2, Chiyoda-ku, Tokyo 100, Japan Submitted 10 April 1992, accepted 6 July 1992 Handling Editor: M. Ohta
Linear stability of a thin melt layer on metal walls after a plasma disruption was investigated in order to clarify the mechanism of the roughness formation which is observed in an experimental work by Ogawa et al. (1992). Stability characteristics in terms of a thermocapillary (Marangoni) instability are found to be consistent with the experimental results. A relationship between the critical Marangoni number and the crispation number parametrized by the Prandtl number and the Bond number was obtained to predict the occurrence of the roughness formation on the resolidified surface.
1. Introduction
In a thermonuclear fusion reactor such as the International Thermonuclear Experimental Reactor (ITER) [1] or the Fusion Experimental Reactor (FER) [2], plasma disruptions cause exposure of plasma-facing components (hereafter referred to as PFCs) to severe heat load, high-energy particles, and intense electromagnetic force. The heat load is evaluated to range from 10 to 20 M J / m 2 for the duration time of 0.1-3 ms in the ITER design [1]. Surface materials of PFCs melt, evaporate, sublimate, or fracture because of the heat load. The materials are thus eroded by plasma disruptions so that a life time prediction of PFCs is one of the key issues in the design of a thermonuclear fusion reactor. Some experimental studies [3-8] on thermal shock, thermal deformation, and erosion during plasma disruptions have been performed in order to understand thermal-structural characteristics and to predict the durability of PFCs. Madarame et al. [9] and Ogawa et al. [10] reported thermal-structural behaviors of metals subject to high heat flux. They found that the metal Correspondence to: Dr. Kaoru Fujimura, Department of High Temperature Engineering, Tokai Research Establishment, Japan Atomic Energy Research Institute, Tokai-mura, Nakagun, Ibaraki-ken 319-11, Japan.
after heating exhibits two features: a rough surface and a smooth surface. Ogawa et al. also reported that many visible cracks were observed on the rough resolidified surface after several irradiations, while little cracks were on the smooth resolidified surface. The formation of the roughness on the resolidified surface is not only physically an interesting phenomenon but also of importance from a point of view of the life time of PFCs associated with crack generations. We may list several possible mechanisms which cause the roughness formation on the resolidified surface: evaporation of selected elements involved in metal test pieces, the pressure of evaporated metal vapor, hydrodynamic instability, instability due to evaporation/condensation process, Rayleigh-Taylor instability, thermocapillary instability, etc. Fujimura [11] first emphasized that an instability driven by a temperature gradient of the surface tension, i.e., a thermocapillary (or Marangoni) instability, is one of the possible mechanisms to create the roughness on a melt layer. He carried out preliminary linear stability analyses of a melt layer on which an external body force acts in an arbitrary direction. From a hydrodynamic point of view, a similar problem, i.e., the linear stability of a liquid film down a heated inclined wall, was analyzed by Kelly, Davis and Goussis [12] in the long-wavelength limit. In these two cases, there are two driving mechanisms of instability for wave forma-
0 9 2 0 - 3 7 9 6 / 9 2 / $ 0 5 . 0 0 © 1992 - Elsevier Science Publishers B.V. All rights reserved
184
K. Fujimura et aL / Roughness formation on a liquid layer
tion: thermocapillary instability and hydrodynamic instability. The former is intrinsic for a liquid film on a heated or cooled horizontal wall while the latter is intrinsic for an isothermal liquid film down an inclined or vertical wall. Here we call wall surfaces perpendicular to the gravity as the horizontal and wall surfaces parallel to the gravity as the vertical. If both driving mechanisms are adequately taken into account in the analysis, the critical Marangoni number of the vertical case has to be zero [12], i.e., the hydrodynamic instability mechanism is dominant for the vertical film irrespective of the strength of the Marangoni effect. Madarame et al. [9] examined an effect of impurity contents in stainless steel on the roughness formation, and reported that small amounts of sulfur or oxygen have a strong effect on the formation. Madarame and Okamoto [13] carried out numerical analyses on the formation of the roughness on a vertical melt layer. Through numerical integration of model equations,
they concluded that an instability sets in if the Marangoni number M (a non-dimensional temperature gradient of the surface tension) is greater than some positive critical value, M~+ say. Although they tried to justify the positiveness of the Marangoni number by claiming that an increase of the amount of sulfur in stainless steel changes the sign of M from negative to positive, they unfortunately ignored the hydrodynamic instability mechanism in their model equations. Consider a horizontal melt layer. For M > M + = 80, a stationary convection with famous hexagonal pattern sets in, whose critical wavenumber is a c = 2. (In what follows, letters with an asterisk denote dimensional numbers while letters without asterisk denote non-dimensional ones.) The most unstable wavelength corresponding to this a e is much shorter than the typical wavelength of the experimentally obtained roughness on the resolidified surface of the melt layer [10]. We have to point out here that a horizontal melt layer is
(b)
(c)
!
I
10ram
(d)
(e)
Fig. 1. Topographic features of the rough surfaces with various heating durations. Peak heat flux: 80 MW/m 2, test piece: type-316. (a) 60 ms, (b) 70 ms, (c) 75 ms, (d) 120 ms, (e) 150 ms.
K. Fujimura et al. / Roughness forrnation on a liquid layer
known to be unstable to a traveling wave or a standing wave even for M < M~- < 0 [14] where M~- denotes the negative critical Marangoni number. The critical wavenumber for M < 0, as we will see later, ranges from 0.1 to 1 for stainless steel which can be comparable with the experimental one. The objective of the present paper is to demonstrate that the thermocapillary (Marangoni) instability is one of the main mechanisms to create the roughness on the resolidified surface and to obtain a criterion under which the resolidified surface becomes rough after a plasma disruption. We investigate the linear stability of a horizontal melt layer by solving the linear eigenvalue problem. The theoretical results are compared with the experimental ones obtained by Ogawa et al. [10] in the horizontal configuration. The present paper will provide a guiding principle for restraining the crack generation in metallic materials.
2. Experimental results In this section, we outline the experimental results of [10] on roughness formations on the resolidified surface after heating. For further details of the experiment, see [10]. The results will be compared with the linear stability characteristics in later sections. In the experiment, a metal test piece was heated by a hydrogen ion beam in a vacuum chamber. The mate-
185
rials used were stainless steel (type-316, type-304, type316F, and the Primary Candidate Alloy in Japan Atomic Energy Research Institute, PCA) and pure metals (aluminum, copper, molybdenum, and tungsten). Type-316F is a reference material of type-316 for research and developmentof the fusion reactor in JAERI. It involves less amounts of manganese, phosphorus, silicon, and carbon than the type-316. PCA is a modified type-316 which has a high void swelling resistance. Test pieces were originally fiat plates with diameters of 80-120 mm. The peak heat flux was ranging from 68 to 261 M W / m z and the heating duration was 40-250 ms. The heat flux had temporal as well as spatial variations. We summarize here the obtained results on the roughness formation on the resolidified surface. (i) Under common heating conditions, the resolidifled surfaces of stainless steel of type-316, PCA, and type-304 became rough, while the surface of stainless steel of type-316F remained smooth. (ii) For the stainless steels exhibiting a rough resolidifled surface (type-316, PCA, and type-304), the conditions of roughness formation did not depend on any heating conditions. For the aluminum test pieces, on the other hand, the resolidified surface remained smooth under relatively mild heating conditions, while it became rough under relatively intense conditions. The shape of the resolidified surface of aluminum thus depends on the heating conditions. The resolidified surfaces of copper,
ig. 2. Rough and smooth surfaces of aluminum test pieces. (a) 79 MW/m 2 and 60 ms, (b) 80 MW/m z and 100 ms.
186
K. Fujin~ura et aL / Roughness formation on a liquid layer
molybdenum, and tungsten test pieces were smooth under the experimental conditions performed. (iii) The roughness was not formed during the resolidifying process but during the initial stage of the melting process. In order to facilitate our understanding, we refer to fig. 1, which has already been presented in ref. [10]. Figures l a - e show topographic features of the rough surface for the type-316 stainless steel exposed to a common peak heat flux under different heating durations. If the heating duration is 60 ms, the surface has not melted yet. If the duration is 70 ms, many small mountains are observed on the surface. These small mountains form ridges with increase in heating duration (lc). If we further increase the heating duration, these ridges form several mountain chains (ld), and eventually become the outer rim (le). Figures 2a and b show topographic features of the aluminum test pieces exposed to the same peak heat flux. The heating dura-
(a) 22ms
(e) 34 ms
(~) 67ms
tions of figs. 2a and b are 60 ms and 100 ms, respectively. The surface of the test piece in fig. 2a melts but remains smooth, while the surface in fig. 2b becomes rough just like fig. le for the stainless steel. Figure 3 shows the surface of the melt layer observed with a high-speed video camera. These pictures were taken at a speed of 2000 frames per second. The test piece is type-304 stainless steel and the heating duration is 67 ms. The surface begins to melt at t* = 22 ms (fig 3a). The high-temperature parts in the melting region brighten so that they appear as white spots in the photo. The roughness is already observed at this stage. Evaporation is clearly observed for t * > 34 ms (fig. 3e). Note that the white foggy parts above the test piece show the vapor. The heating is stopped at t* = 67 ms (fig 3i). Judging from these photos, the initial roughness formation is considered to be due to some instability mechanism of the melt layer. In the following sections, we analyze the linear
(b) 24 ms
(c) 27ms
(.f) 38rns
(g) 40 ms
(h) 50 ms
(k) 68,ms
(t) 70.5ms
2~Omrn (j) 67,5rns
(d) 29.5ms
Fig. 3. Observation for surface of type-304 stainless steel test piece by high-speed video camera. Peak heat flux: 210 MW/m 2, heating duration: 67 ms.
K. Fujimura et al. / Roughness formation on a liquid layer
"4
.~_
g.
N~ ff
XXXXXX
X
X
X
X
X
X
I l i l i i
I
e..
.o
l
l
l
x~
,~, 0 e"0
8
XXXXXX
~~.~o
G o~
~d~l~
l!: ._
~
.
187
K. Fujimura et aL / Roughnessfomlation on a liquid layer
188
stability of the melt layer by restricting ourselves to the thermocapillary instability and compare the theoretical results with the experimental ones outlined above.
subject to the following boundary conditions:
(4)
w-to-c/=0, Cr[Pr-tto - (D z - 3aZ)]Dw + (Bo + az)a2"q = O,
3. Linear stability analysis
(D 2 + o~2)w -}- MazO - MotZr/= 0,
If the basic state, i.e., unperturbed melting and resolidifying process, changes very rapidly, the standard technique in linear stability analysis, i.e., the normal mode analysis, for an infinitesimal disturbance is invalid. Unfortunately, stability theory for such rapidly varying basic state has not been established yet except for rigorously periodic situations. Non-separability of the disturbance equations lets us treat partial differential equations other than ordinary differential equations. In order to avoid this difficulty; we assume here a slowly varying situation such that the time scale on which the basic state changes is completely separable from the one on which the disturbance linearly evolves. This situation enables us to make use of the method of multiple scales. At t h e lowest-order approximation, we then obtain linearized disturbance equations which are identical with the disturbance equations for the steady basic state problem. The unsteadiness is thus assumed to give a minor correction on the stability characteristics at higherorder approximations. The assumption will be justified in a later section by comparing the results of the analysis with the experimental results of Section 2. Because the disturbance equations for the steady basic state are well known, we only list the final form of the equations and explain how to solve them. Consider a liquid layer bounded by a horizontal rigid wall at z* = 0 and by a free surface centered around z* = h * . The gravity acts in the negative z*direction. Also assume that the layer extends to x* + oo and y* --* + oo Temperature of the rigid wall and the averaged one at the free surface are respectively maintained at To*- AT* and TO*, where we assume that AT* > 0. Appropriate non-dimensionalization, a shift of the z-coordinate, and the normal mode analysis
= /0(.0z)
exp[i(OtxX+Otey)+tot ] ,
(1)
yield the linearized disturbance equations of the form [ P r - lto - ( 0 2 - a2)] ( 0 2 - a2)w = 0,
(2)
[to - ( o
(3)
-
o = ,,.
(D + B )O - B~7 = O,
(5) (6) (7)
at z = 1 / 2 while
w=Dw=O=O
at z - - - 1 / 2 ,
(8)
where ~,, 7~, and ~ are respectively the disturbance vertical velocity, disturbance temperature, and location of the perturbed free surface, D = d / d z , and a z - a ~ + a~,. The above equations involve the following nondimensional parameters: the Prandtl number P r - u*/K*, the crispation number Cr=p*u*K*/(s*h*), the Bond number Bo = p * g * h * 2 / s *, the Marangoni number M = ( - d s * / d T * ) A T * h * / ( p * u * K * ) , and the Biot number B =cr*T*3h*/)t *, where p* is the density of the liquid, v* is the kinematic viscosity, ~.* is the thermal conductivity, K* is the thermal diffusivity, o-* is the Stefan-Boltzmann constant, g* is the acceleration due to gravity, and s* is the surface tension. Expansion of w and 0 in Chebyshev polynomials as
0
,,=00,,(z+l/2)
T,,(2z),
(9)
reduces the system of ordinary differential equations (2)-(8) to a system of algebraic equations for w,, and 0,, from which the eigenvalue to is determined as a function of parameters: co = to(a, Pr, M, Bo, B, Cr). Here T,(2z) denotes the Chebyshev polynomial of the nth degree. The factors (z + 1/2) z and (z + 1/2) are involved in the expansion functions so as to satisfy the homogeneous boundary conditions at the rigid wall, eq. (8). We applied a standard subroutine package based on the QR algorithm to solve the algebraic eigenvalue problem. We checked the accuracy of the numerical integration based on the expansion (9) by changing the value N which is the maximum degree of the Chebyshev polynomial involved in the expansion. The check clarified that N = 30 gives satisfactorily accurate stability characteristics for the present purpose.
4. Stability characteristics and comparison with the experimental results We tabulated some physical parameters involved in (2)-(8) in table 1 which are relevant to the typical
K. Fujimura et al. / Roughness formation on a liquid layer 0,9
........
I
........
0.1
a~r
0
=~-=-"~........ ~i....... ,,~
-0.I
",
I
........
SS(2)
" "" "
"
I ""
...
)
./
/
."
-0.4
,
0.001
, .......
, llltHI
"
,,: ......
0.01
I ,
0.1
,
--] stable
='"
, , .... ,
"
• --
"' '. . . .
1
]
,,.j
..Mo "
.
'q]-..
SS(l),
.
t ..~ u n s t a b l e
AJ([).
• ,"-,,
Y SS(2)
"
';?'-, "
,
-0.3
q
........ AI 2 ( )
. "~,::::~.Cu .
_._Mo........:_.--(~.
-02
.......
" ~. 1
,,,,,
10
Fig. 4. Linear growth rate a/r for stainless steel (SS(1) and SS(2)), aluminum (AI(1) and AI(2)), copper (Cu), and molybdenum (Mo) based on the non-dimensional parameters in table 1.
experimental data described in Section 2. Here, we define the energy density Q* by the peak heat flux integrated over the heating duration. Although we need to have the surface temperature and the thickness of the melt layer in order to evaluate the non-dimensional parameters in the table, it is not easy to obtain the data experimentally. Instead, we evaluated these data by solving a one-dimensional heat conduction equation taking account of an evaporation effect. The experimental data and the computational ones agree well with each other in the initial stage of melt-
189
ing. The agreement becomes worse after the initial stage because of a convection effect in the melt layer and an effect of vapor shielding. (See [10] for further details.) The parameters for stainless steel, SS(1), corresponds to fig. lb and the ones for aluminum, AI(1) and Al(2), correspond to figs. 2a and b, respectively. Utilizing the data of table 1, we evaluated the linear growth rate to, as a function of wavenumber t~ for stainless steel, copper, aluminum, and molybdenum as depicted in fig. 4. For each parameter, a pair of stable stationary modes existing in the low-wavenumber region merges into a pair of traveling waves in the higher-wavenumber region. The linear growth rate corresponding to the real part of the first eigenvalue asymptotically tends to zero from below in the longwavelength limit. It is therefore clear that the growth rate for a pair of traveling waves gives the information whether or not the liquid layer is unstable. The pair of traveling waves is composed of two traveling waves propagating in opposite directions with the same phase speed and having the same linear growth rate. Physically, therefore, a traveling wave propagating in one direction or a standing wave has to be attained. We cannot say anything about which wave is observable in real physical systems without analyzing the weakly nonlinear stage, which is out of scope of the present paper. As far as the data of table 1 are concerned, melt layers of stainless steel, SS(1), aluminum, AI(1), copper, and molybdenum are predicted to be stable while those of stainless steel, SS(2), and aluminum, Al(2), are
Fig. 5. Cross, sectional view of the resolidified layer. Peak heat flux: 80 MW/m 2, heating duration: 70 ms, test piece: type-316 stainless steel. See also fig. lb.
190
K. Fujinzura et al. / Roughness formation on a liquid layer
unstable. These are consistent with the experimental results outlined in Section 2 except for SS(1). SS(1) is predicted to be stable although a rough surface was obtained experimentally as is shown in fig. lb. This point will be discussed in the next paragraph. The peak of the linear growth rate predicts a physically realizable wavenumber, amax" According to the peak location for stainless steel, otto,x = 0.3. The wavelength of the roughness on the melt layer is thus predicted to be 2'n'/OtmaxXh*= 2 mm. Figure 5 shows a cross-sectional view of the resolidified surface for stainless steel. The photo shows that the typical wavelength is 1.6 ram, which is quite comparable with the theoretical prediction.This photo needs some discussion. According to the photo, the resolidified part seems to be localized, horizontally. This localization is considered to be due to the Marangoni effect. The melt layer is formed almost uniformly at the onset of melting. A thermocapillary wave is then created on the surface of the melt layer because of the instability of the layer. The melt layer with a uniform depth is thus replaced with a layer with sinusoidally changing depth. The strong surface tension tears the melt layer rapidly after the growth of the average depth of the melt layer. The melt layer is therefore isolated just prior to the resolidifying stage. The critical Marangoni number M~" and the corresponding wavenumber are depicted in fig. 6 as a function of the crispation number Cr parametrized by (Pr, Bo). It is known that the stability characteristics are not sensitive to the Blot number. We find that the critical wavenumber changes substantially as a function of both the crispation number and the Prandtl number, but is insensitive against the Bond number Bo. Utilizing this figure, one may predict the linear stability characteristics for melt layer dynamics. Beyond the curves for M~-, i.e. M < M~- < 0, the layer is unstable to a traveling wave or standing wave disturbance. The experimental parameters of table 1 are plotted in the figure. The experimental parameter for stainless steel SS(1) is just below the critical curve while the one for SS(2) is far beyond the curve. The parameter of AI(1) is far below the critical curve while the one of AI(2) is again far beyond the curve. Moreover, the parameters for copper and molybdenum are far below the critical curve. These data are completely consistent with the experimental results except for SS(1). Let us now discuss this disagreement. In order to obtain the parameters, we evaluated the surface temperature and the thickness of the melt layer from the numerical integration of the one-dimensional heat conduction equation as described above. These data are not accurate after the initial stage. If we increase the value of the dura-
103
(er,8o)=(o.t,io"2) ~____......~/ •
•
unstable ~ , q --ds/dt
_Mc0o_2 to..2)
-
s
lO2 1
I
i
i
i
,
,
,,I
,
t
i
i
i
i
.
i
t
ii
0.8 0.6
%
Ih----~,01
0.4 0.2 0
t
10-5
i
i
i
i
i l t l
10"4
i
.
.
.
.
10-3
Cr Fig. 6. Linear critical Marangoni number and wavenumber as functions of the crispation number parametrized by the Prandtl number and the Bond number. The critical conditions are almost independent of the Biot number. Beyond the curves of the critical Marangoni number, a melt layer is unstable, o: SS(1), e: SS(2), rn: AI(1), I1: AI(2), A: Cu, , : Mo.
tion time by 20%, then the parameter (M, Cr) for stainless steel moves to the vicinity of the position of SS(2) which is obviously in an unstable region. Judging from the figure, a correction less than 5% is enough to obtain the unstable situation. Of course, any experimental work as well as the assumption of a slowly varying basic state causes some amount of error. These errors may be much greater than 5%. We conclude that the disagreement for the SS(1) is mainly due to the error in the evaluation of the parameters. We further indicated in fig. 6 by three arrows the directions of the change in parameter space (M, Cr) resulting from an increase of the heating duration (At*), the heat flux (q*), the surface tension s*, or the temperature gradient of the surface tension - d s * / d T * . The increase of At*, q*, or - d s * / d T * exerts a destabilizing effect while the increase of s* exerts a stabilizing effect. It is well known that a surface tension s* in molten iron increases if the amount of impurity from group VI B is reduced. Thus the reduction of the impurity exerts a stabilizing effect. It is also clear that any melt layer of metallic material can in principle be unstable depending on the value of the parameters At* or q*.
K. Fujimura et aL / Roughness formation on a liquid layer
We now conclude that the linear stability analysis based on the assumption of slow variation is valid for the present problem. Moreover, the roughness formation can be explained by the thermocapillary instability mechanism. Because of the linearity assumption, the present analysis can only predict the primary roughness formation under weak heating conditions. It is invalid for nonlinear evolution involving a doubling or merging of several wavy structures. On the linear basis, however, all the terms are properly involved in the disturbance equations which is consistent with the lowestorder approximation of the multiple scaling expansions. Madarame and Okamoto's numerical analysis is complementary to our analysis for the horizontal configuration because their analysis can treat unsteady as well as nonlinear effects.
5. Conclusions
We analyzed the thermocapillary instability of a horizontal melt layer with negative Marangoni number. In the analysis, we assumed the slowly varying situation in order to adopt the multiple scaling approach to the unsteady melting and resolidifying problem. As far as the objective of the present paper is concerned, there is no need to proceed to any higher-order approximation although the higher-order approximation for the present analysis is straightforward. Despite the substantial assumptions on the basic state and experimental conditions computed from the one-dimensional heat conduction equation, the present theoretical analysis successfully explains the experimental results of [10] not only qualitatively but also quantitatively. The experimental findings about stainless steel and aluminum, (i) and (ii) in Section 2, are well reproduced by the analysis. Moreover, the predicted wavelength is comparable with the experimentally obtained wavelength for the roughness on the resolidified surface. Any metallic melt layer can be unstable without changing the sign of the Marangoni number. We provide fig. 6 for the critical Marangoni number as a function of Cr parametrized by Pr and Bo. Once the parameters (M, Cr, Pr, Bo) are prescribed, then one may roughly judge from the figure whether or not the roughness is formed on the resolidified surface.
191
References
[1] IAEA, ITER Conceptual Design Report, IAEA/ITER/ DS/No. 18 (1991). [2] S. Matsuda et al., Conceptual design study of the Fusion Experimental Reactor (FER), Proc. 13th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, Washington DC, IAEA-CN-53/G-2-2 (1990). [3] J.R. Easoz et al., Thermomechanical testing of first wall test pieces in ESURF, Nucl. Technol. Fusion 4 (1983) 780. [4] C.D. Croessmann, G.L. Kulcinski and J.B. Whitley, Correlation of experimental and theoretical results for vaporization of simulated disruption, J. Nucl. Mater., 128/129 (1984) 816. [5] M. Seki et al., A simulated plasma disruption experiment using an electron beam as a heat source, J. Fusion Energy, 5 (1986) 181. [6] G. Rigon, P. Moretto and F. Brossa, Experimental simulation of plasma disruption with an electron beam, Fusion Engrg. Des. 5 (1987) 299. [7] M. Ogawa et al., Experimental observations of metal surfaces exposed to intense hydrogen beam; simulation of plasma disruption, J. Nucl. Sci. Technol. 26 (1989) 721. [8] M. Seki et al., Thermal shock tests on various materials of plasma facing components for FER/ITER, Fusion Engrg. Des. 15 (1991) 59. [9] H. Madarame et al., Effect of impurity content in stainless steel on resolidified surface condition after disruption load, Fusion Engrg. Des. (1989) 213. [10] M. Ogawa et al., Experimental study on melting and evaporation of metal exposed to intense hydrogen ion beam, Fusion Engrg. Des. 19 (1992) 193-202, next article in this issue. [11] K. Fujimura, in: Japanese Contributions to IAEA INTOR Workshop, Phase 2a, Part 2, eds. M. Seki et al., Research Report of Japan Atomic Energy Research Institute JAERI-M85-075, (1985) 93-107. [12] R.E. Kelly, S.H. Davis and D.A. Goussis, On the instability of heated film flow with variable surface tension, Proc. 8th Heat Transfer Conf., eds. C.L. Tien, V.P. Carey and J.K. Ferrell (Springer, 1986) p. 1937. [13] H. Madarame and K. Okamoto, Numerical simulation of surface tension induced melt layer movement after disruption load, Fusion Engrg. Des., in print. [14] M. Takashima, Surface tension driven instability in a horizontal liquid layer with a deformable free surface, I. Stationary convection; II. Overstability, J. Phys. Soc. Jpn. 50 (1981) 2745, 2751.