EFFECTS OF LASER RADIATION AND NANO POROUS LINING ON THE RAYLEIGH TAYLOR INSTABILITY IN AN ABLATIVELY LASER ACCELERATED PLASMA. N. Rudraiah Natinal Research Institute r Alied Mathematics (NRIAM), 49/G, 7th Crss, 7th Blck (west), Jayanagar, Bangalre 560 08. And UGC-CAS in Fluid Mechanics, Deartment Mathematics Bangalre University, Bangalre-560 00.
. INTRODUCTION Fr eicient extractin Inertial Fusin Energy (IFE) it is essential t reduce the grwth rate surace instabilities in laser accelerated ablative surace IFE target. The llwing three dierent tyes surace instabilities are bserved : Rayleigh Taylr Instability (RTI) Kelvin Helmhltz Instability (KHI) Richtmyer Meshkv Instability (RMI) At resent the llwing mechanisms are used t reduce the RTI grwth rate.
Gradual variatin density assuming lasma as incmressible hetergeneus luid withut surace tensin. Assuming lasma as cmressible luid withut surace tensin. IFE target shell with am layer. Numerus numerical and exerimental data r RTI grwth rate at the ablatin surace r cmressible luid its. n A l g ε l L β l v a (.) Rudraiah (00) derived an analytical exressin l n l β l v a B (.)
r a target lined with rus layer cmrising nantube, cnsidering viscus incmressible luid. Here Β δ h γ is the Bnd number, γ is the surace tensin, δ g ( ρ ρ ), ( ) α σ 4 α σ l ( ), v l ρ a 4 α σ α σ Β, and ρ are the β density rus lining and luid resectively. Density in the range 5< ρ <0 kg/cm, 5 k (. -.7) 0 in r am metal and k ε (0.06 0.07) 6 (. 0. 48) 0 in r alxite metal and
Chsing suitable values r the cnstants A, ε and β ne can it the available data. Authrs A ε β n m Takabe et al (985) 0.90 0.0.00 0.45 n bn Lindl et al (995).00.0.00 Betti et al (995) 0.98.0.70 Kilkenny et al (994) 0.90.0.00 Knauer et al (000) 0.90.0.0 Rudraiah (00).00.0 0.75.86 0.79 n bm (α 0., σ 4) 0.6 n bm (α 4, σ 0)
. MATHEMATICAL FORMULATION. Fig. Physical Cniguratin
The cnservatin mmentum: ( ) H J Kq q q q t q h r r r r r r r µ µ ρ (.) The cnservatin mass r cmressible Bussinesq luid 0 q r (.) ( ) { } [ ] T T T T T X α ρ ρ The cnservatin energy ( ) ( ) ( ) I n T X T q t T M X ˆ ) ( ) ( κ r (.)
where [ ] H q E J h h r r r r µ σ Current density, 0 0 E, t H E, H r r r r X ρ ρ ρ ρ Density, e X µ µ µ µ Viscsity, h h h X µ µ µ µ Magnetic ermeability h h h h X σ σ σ σ Electrical cnductivity
K µ ρ Cb X q k k κ κ e κ ( ) ( C ) ρ C m ρ * ( )* ρ C ( ) ( )( ) ε ρ C ε ρ C s X 0 r, shell r rus lining. Bundary Cnditins du u v 0 at y 0 and α σ u at y dy. DISPERSION RELATION, WITH LASER RADIATION In this sectin we derive the disersin relatin as well. as the temerature distributin incrrating the laser radiatin eect.
. Disersin relatin { Mch[ M ( y )] ασsh[ M ( y )] ασ [ Sh( My ) Sh( M )] Mch( M )} P [ Mch( M ) ασsh( M )] u where M H h h σ µ µ is the Hartman number; σ h M (.) is the k rus arameter; Csh (θ) and Sinh (θ) are dented by Ch (θ) and Sh (θ) resectively. v() ασ [ Ch( M )] ( ασ ) MSh( M ) M Ch( M ) [ ] M MCh( M ) ασsh( M ) x (.) n (.) n b β lv a where n is the grwth rate, l is the wave number,
M β 6α σ ( ch M ) ( M th M ) α σ ( M ) th M M chm (.4) α σ ( ch M )] [ M th M α σ th M ] M ch M a cnstant, v a is the velcity lw acrss the ablative rnt given by α σ ( ch M ) M th M α σ th M M ch M l v a l δ (.5) th M B M ( α σ ) M B δ h γ ( ) is the Bnd number, δ r θ θ where suix dentes the values θ at y, and l l n b B δ (.6)
when M 0, Eq. (.) reduces t Rudraiah (00) n n b β l v a (.7) where α σ 4 α σ β 4 α σ l (.8) v a l δ (.9) ( α σ ) B In the absence nanstructure rus lining, (k i.e., σ 0) the grwth rate (.) tends t where n M ( M th M ) ( M th M ) n b β l v a β (.) M v a M th M (.0) (.) In the case using Eq. (.9) which is ( n ) b ; we have
n ( ) n b β l v a β M ( M 6α σ ( chm ) thm ) α σ ( M ) thm α σ M M chm α σ ( chm )] [ M thm α σ thm ] M chm ( α σ thm ) va is the same as Eq. (.5). Temerature distributin Fr the shell ilm T T Ω y v a κ I Ω e (.) y y Fr the rus silicn layer T Ωy 0 κ ± I Ω e (.4) y
where κ and κ are the thermal diusivity r shell ilm and rus silicn lining. The selectin a articular sign in Eq. (.4) will deend n the hysical situatin. I we chse sitive sign, then θ will be negative imlying energy will be lst. The rblem cnsidered in this aer requires the additin energy t use BT. Fr this we have t chse negative sign in Eq. (.4) t ensure sitive θ. In this aer, we cnsider the llwing tw cases Case : The luid in the shell ilm and rus silicn layer is hmgeneus and incmressible with temerature, T, in the rus silicn is assumed t be cnstant which may be higher than luid temerature. Case : The luid in the shell ilm as well as in rus silicn is assumed t satisy Bussinesq arximatin with varying temerature T and T.
Case : Hmgeneus luid v a θ y R a θ y N e Ω y (.5) δ h where Ra is the Rayleigh number because δ κ µ has the dimensins ρ αt g T I Ω µ N and v δ T a is given with θ θ δ () that is δ is cnstant. Eqn. (.5) is slved using the llwing tw set bundary cnditins. Set : θ at y 0, and θ θ at y (.6) Set : θ at y 0, and ( θ ) dθ B i at y (.7) b dy
Where B i h c κ h is the Bit number, h c is the heat transer ceicient rm the rus silicn layer int shell-ilm, θ b is the temerature at y. θ ( by ) e ( b ) e Ω y by NRae a ( y) a a e a Ω ( Ω b) ( Ω y ) e (.8) a a ( Ω ) e θ, a Ω NR ( Ω b) a, b v a R a θ y y NR a ( Ω b) e Ω b a e b (.9) Similarly the slutin Eqn. (.5), satisying the bundary cnditins (.7), is
where θ B θ a y) a ( NR Ω a ( Ω y ) ( ) N R e by a e ( Ω ) e ( Ω b) NR a Ω e Ω b ( Ω b) Βi( Ω b) ( Ω b) ( b ) e (.0) a ( Ω b) NR ae Bi b ( θ B ) e b( Ω b ) b, a [ b ] Β ( e b i ) Case : Bussinesq luid Fr the shell ilm θ v a y R a θ y Ne Ω y (.)
Fr the rus silicn layer 0 θ y N e Ω y (.) where N I Ω κ T h The bundary cnditins n θ and θ are θ y θ y 0 as y, θ at y B i ( ) θ, B ( θ ) at y θ y i 0 (.) (.4) θ and θ are the values and θ at y θ P
The slutin Eq. (.) and Eq. (.) satisying the abve bundary cnditins are ( ) ( ) ( ) ( ) ( ) y b b y b b i y e e b a e e b B e a Ω Ω Ω θ θ and ( ) y i e e N e B N Ω Ω Ω Ω Ω θ Frm these we have () ( ) ( ) ( ) Ω Ω Ω Ω Ω b i i e b e a B e a a e B N a θ θ δ
4. CONCLUSIONS The linear RTI in an IFE target mdeled as a thin electrically cnducting luid ilm in the resence transverse magnetic ield lined with an incmressible electrically cnducting luid saturated nanstructured rus lining with unirm densities is investigated using nrmal mde analysis. The main bjective this study is t shw that the tw mechanisms, having a suitable strength magnetic iled and suitable rus material made u nanstructure lining, reduce the grwth rate ablative surace IFE target cnsiderably cmared t that in the absence these tw mechanisms. The disersin relatins given by Eqs. (.) t (.7) are analgus t the ne given by Takabe et al., (985) r cmressible nn-viscus nnelectrically cnducting luid
as shwn in Eqn. (.). The disersin relatin (.7) cincides with the ne given by Rudraiah (00) in the absence Magnetic ield. The disersin relatin given by Eqn. (.6) cincides with the ne given by Babchin et al., (98) in the absence magnetic ield (M 0) and the nanstructure rus lining (σ 0). Setting n0 in Eqn. (.), we btain the cut wave number, l ct, abve which MRTI mde is stabilized and which is und t be l ct δ B (4.) Fr hmgeneus luid δ and r Bussinesq luid σ θ θ. The maximum wave number, l m, btained rm n Eqn. (.) by setting 0, is l
l m B ( θ ) θ l ct (4.) The results given by Eqs. (4.) and (4.) are als true even r the cases in the absence bth nanstructure rus lining and magnetic ield. The crresnding maximum grwth rates, dented by suix m, rm Eqs. (.) t (.0) are B 4 B 4 n m r ( ) B 4 ασ 4( ασ ) θ θ (4.) B 4 n m r ( θ θ ) ασ 4 ( ασ ) (4.4) B 4 B 4 n m ( ) r θ θ (4.5)
B n bm r ( ) B θ θ (4.6) where M ( M thm ) M (4.7) M M α σ ( ch M M th M ch M M ( M α σ th M ) ) (4.8) Frm these, we get G G n (4.9) m 0m nbm m n n m b m ( 4 ασ ) ( ασ ) 4 (4.0)
G m n n m b m ( M tanh M ) M (4.) G m n n m m ( )( ασ ) 4 (4 ασ ) (4.) G 5m n ( n ) b m m G 4m n n ( α σ ) (4 α σ ) m m M ( ) ( ) ( α σ ) M ( ChM ) α σ thm M ChM thm α σ M (4.) (4.4) It will be interest t cmare these results with thse given in Eqn. (.) by Takabe et al., (985) r cmressible luid. In their case
0. 8g ( l ) ct Ta and ( m ) β v a The crresnding n m is ( l ) 0.8g ct Ta l Ta (4.5) 4 β v 4 ( n ) ( ) ( ) m 0.45 g 0. 45 n Ta m b Ta l (4.6) Ta ( n ) ( ) g b T a m Ta a l (4.7) and the quantities with suix T a crresnd t thse given by Takabe et al., (985). Using Eqn. (4.5) Takabe et.al. (985) have shwn that the maximum grwth rate was reduced t 45% their classical result given by Eqn. (4.6).
Frm Eqn. (4.8), Rudraiah, (00) has shwn that in the absence magnetic ield and in the resence rus lining, the reductin maximum grwth rate deends n the characteristics α and σ rus lining. Fr the tyes rus material, namely, ametal, α takes the value 0. and σ ranges rm 4 t 0 and r alxite materials α 4 and σ ranges rm 4 t 0 (see the exeriments Beavers and Jseh (967). Then r α 0. and σ 4 Rudraiah (00) has shwn that the maximum grwth rate given by Eqn. (4.8) has been reduced t 78.57% the classical value given by Eqn. (4.6). In the resence magnetic ield and absence rus lining it is clear that the maximum grwth rate given by Eqn. (4.9) deends n the Hartman number M. We nte that the rati deends urely n nanstructured rus lining when M0, the rati given by Eqn. (4.8) will deend nly n the values M in the absence rus lining where as ther ratis deend n M, α and σ.
Fig : The Grwth Rate n versus wave number r M and r dierent Bnd numbers B
The relatin (.) is ltted in Fig. which is r the grwth rate n versus the wave number l r M, α 0., and σ 4 r dierent values B. Frm this ig. we cnclude that the erturbatin the interace having a wave number smaller than l are amliied when ct δ > 0 ( i. e., ρ < ρ ) and the grwth rate decreases with a decrease in B imlying increase in surace tensin. That is, increase in surace tensin makes the interace mre stable even in the case electrically cnducting luid. Similar behavir is bserved r M> r ixed values α and σ and und that increase in σ is mre signiicant than an increase in M in reducing the grwth rate.
475 45 σ0 σ4 σ0 75 0 5 0 5 0 M Fig. Ablative surace Temerature r dierent θ b
Table 4: The values θ b M0 M M0 M0 0 8.64 40.98 496.79 498.86 4 40.504 4.8 496.756 498.865 8 46.56 4.44 496.77 498.868 44.549 47.08 496.786 498.87 6 40.5 44.47 496.799 498.874 0 44.99 444.45 496.8 498.876
500 M0 450 M M0 M0 400 50 0 5 0 5 0 σ Fig 4. Ablative surace Temerature θ b r dierent M.
Table 5: The values M 0 8. 40.5 44.99 40.9 4.4 444.5 5 486.9 487. 487.5 0 496.7 496.8 496.8 5 498.86 498.86 498.88 0 499.64 499.64 499.65 θ B The ablative temerature given by equatin (.0) is cmuted r dierent values M and σ. The results θ B vs M r dierent values σ are ltted in ig. and θ B vs σ r dierent values M are drawn in ig. 4. Frm these igures, we cnclude that r small values M and σ, θ B increases slwly and saturates r larger values M and σ θ b
Acknwledgement: This wrk is surted by IAEA, Vienna under IFE-CRP rject N. IND 54. Its inancial surt is grateully acknwledged. Reerences:. Babchin, A. J. Frenkel, A. L., Levich, B. G., Shivashinske, G.I., (98), Phys., Fluids, 6, 59.. Betti, R., Gncharv, V.N., McGry, R.L. and Verdn, C. P., (995), Phys., Plasmas, 844.. Kilkenny, J.D., Glendinning, S.G., Hann, S.W., Hamurel, B.A., Lindl, J.D., Munr, B.A., Remingtn, S.V., Weber, J.P., Kanauer, J.P. and Verdn, C.P., (994), Phys., Plasma, 79. 4. Knauer, J.P. et al (00), Phys., Fluids, 7(), 8. 5. Lindl, J.D. (995), Phys., Plasma, 9. 6. Rudraiah, N. (00), Fusin Sci. and Tech., 4, 07
I. KHI at the ablative surace lined with nan structured rus layer in a ully develed twhase cmsite layer using BJR cnditin. In the third year the rject, in cntributin 4, we have cnsidered KHI in a sarsely acked rus lining, where the Brinkman equatin is valid and the interace between the ilm and the rus lining is assumed t be a regular surace and using Residual shear cnditin. In these rblems the thickness the rus layer was absent. In many ractical alicatins including IFE, it is imrtant t ind the eect the thickness rus lining. This can be dne using Rudraiah (985) bundary cnditin. As the thickness becmes very large Rudraiah cnditin tends t Beavers-Jseh (BJ - 967) cnditin. Hence in the literature Rudraiah cnditin is dented by BJR cnditin. In the irst quarter the urth year the rject, we rse t investiage RTI in a inite thickness rus layer using BJR cnditin with the bjective redicting the eect the thickness rus lining n the reductin grwth rate KHI. This eect is imrtant in the design eective IFE target.
II. Kelvin-Helmhltz Instability at the ablative surace using external cnstraints magnetic ield and rus lining. In the remaining quarters the urth year the rject, we rse t investigate the eect magnetic ield n the reductin grwth rate using the llwing cases: Case : Eects densely acked rus lining in the resence a magnetic ield n KHI grwth rate using BJ cnditin. Case : Eects sarsely acked rus layer in the resence magnetic ield n the KHI grwth rate using residual shear cnditin. Case : Eects sarsely acked inite thickness rus lining in the resence a magnetic ield n the KHI grwth rate using BJR cnditin. This cnditin redicts the eect thickness rus lining.
Case 4: Eects magnetic ield and rughness the ablative surace n the reductin KHI grwth rate. The results btained in this case will be useul t take care the rughness the IFE design. The rblem sed abve invlves ur cases and each case take cnsiderable time because we have t slve lasma equatins in thin ilm and rus lining using nrmal mde analysis, Kinematics cnditin in the resence surace tensin and dynamic cnditin. In irst quarter urth year we rsed t investigate the rblem sed abve and in the remaining three quarters we rse t investigate cases and 4 sed in rblem abve. I time ermits, we initiate the third tye instability, namely Richtmyer and Meskv instability rsed in ur sectin n Objectives. This instability is als imrtant in the design IFE target. Here als, we rse t study the eects nan structure rus lining and the external magnetic ield using cases t 4 rsed in rblem abve.
Table (a):values G mi r dierent values M and σ 4, α 0. M G m G m G m G m G m4 G m5 0.5 0.796 0.7857 0.909 0.986 0.805 0.986.0 0.60 0.7857 0.75 0.765 0.8407 0.765.5 0.4657 0.7857 0.588 0.596 0.8806 0.596.0 0.546 0.7857 0.885 0.45 0.98 0.45.5 0.7 0.7857 0.906 0.46 0.960 0.46.0 0. 0.7857 0.8 0.700 0.954 0.700.5 0.687 0.7857 0.75 0.4 0.968 0.47 4.0 0.67 0.7857 0.407 0.740 0.979 0.740 4.5 0.7 0.7857 0.5 0.44 0.9777 0.44 5.0 0.094 0.7857 0.0960 0.0 0.980 0.98 5.5 0.0799 0.7857 0.08 0.07 0.985 0.07 6.0 0.0686 0.7857 0.0694 0.087 0.9876 0.087
Table (b): Values G mi r dierent values M and σ 0, α 0. M G m G m G m G m G m4 G m5 0.5 0.5896 0.650 0.909 0.944 0.6485 0.944.0 0.504 0.650 0.700 0.8000 0.7000 0.8068.5 0.4066 0.650 0.588 0.6506 0.7689 0.6506.0 0.0 0.650 0.900 0.500 0.800 0.55.5 0.50 0.650 0.906 0.40 0.867 0.40.0 0.00 0.650 0.00 0.00 0.8900 0.0.5 0.6 0.650 0.75 0.58 0.95 0.58 4.0 0.0 0.650 0.400 0.00 0.900 0. 4.5 0.095 0.650 0.5 0.75 0.950 0.75 5.0 0.090 0.650 0.0960 0.470 0.9600 0.474 5.5 0.0784 0.650 0.08 0.55 0.9664 0.55 6.0 0.0670 0.650 0.0690 0.070 0.970 0.080
Table (c): Values G mi r dierent values M and σ 0, α 0. M G m G m G m G m G m4 G m5 0.5 0.4775 0.5000 0.9090 0.9550 0.550 0.9549.0 0.40 0.5000 0.750 0.840 0.5880 0.84.5 0.50 0.5000 0.590 0.700 0.6640 0.705.0 0.860 0.5000 0.880 0.570 0.750 0.57.5 0.00 0.5000 0.90 0.460 0.790 0.4609.0 0.870 0.5000 0.0 0.70 0.880 0.7.5 0.50 0.5000 0.750 0.050 0.870 0.05 4.0 0.60 0.5000 0.40 0.50 0.8970 0.5 4.5 0.060 0.5000 0.50 0.0 0.960 0. 5.0 0.0890 0.5000 0.0960 0.790 0.900 0.787 5.5 0.0760 0.5000 0.080 0.50 0.940 0.58 6.0 0.0660 0.5000 0.0690 0.0 0.9500 0.0
Table (a): Values G mi r dierent values M and σ 4, α 4 M G m G m G m G m G m4 G m5 0.5 0.860 0.94 0.909 0.976 0.46 0.975.0 0.64 0.94 0.75 0.8986 0.695 0.8986.5 0.46 0.94 0.588 0.7978 0.448 0.7978.0 0.09 0.94 0.885 0.6898 0.5 0.6898.5 0.79 0.94 0.906 0.5878 0.5950 0.5878.0 0.466 0.94 0.8 0.498 0.6579 0.498.5 0.4 0.94 0.75 0.46 0.700 0.46 4.0 0.058 0.94 0.407 0.598 0.754 0.598 4.5 0.0907 0.94 0.5 0.08 0.7868 0.08 5.0 0.078 0.94 0.0960 0.659 0.846 0.659 5.5 0.0679 0.94 0.08 0.0 0.87 0.0 6.0 0.0594 0.94 0.0694 0.0 0.8560 0.0
Table (b): Values G mi r dierent values M and σ 0, α 4 M G m G m G m G m G m4 G m5 0.5 0.64 0.68 0.909 0.9744 0.875 0.9744.0 0.48 0.68 0.75 0.9050 0.95 0.9050.5 0.7 0.68 0.588 0.809 0.406 0.809.0 0.89 0.68 0.885 0.7050 0.4869 0.7050.5 0.64 0.68 0.906 0.605 0.5588 0.605.0 0.85 0.68 0.8 0.564 0.69 0.564.5 0.8 0.68 0.75 0.4404 0.6749 0.4404 4.0 0.0 0.68 0.407 0.768 0.787 0.767 4.5 0.0869 0.68 0.5 0.40 0.7444 0.40 5.0 0.075 0.68 0.0960 0.804 0.787 0.804 5.5 0.0656 0.68 0.08 0.44 0.8078 0.44 6.0 0.0575 0.68 0.0694 0.4 0.878 0.4
Table (c): Values G mi r dierent values M and σ 0, α 4 M G m G m G m G m G m4 G m5 0.5 0.58 0.59 0.909 0.9750 0.780 0.9750.0 0.5 0.59 0.75 0.907 0.88 0.907.5 0.08 0.59 0.588 0.80 0.986 0.80.0 0.84 0.59 0.885 0.70 0.479 0.70.5 0.584 0.59 0.906 0.6 0.545 0.6.0 0.55 0.59 0.8 0.56 0.608 0.5.5 0.58 0.59 0.75 0.4465 0.66 0.4465 4.0 0.099 0.59 0.407 0.86 0.705 0.86 4.5 0.0854 0.59 0.5 0.95 0.74 0.95 5.0 0.0740 0.59 0.0960 0.855 0.770 0.855 5.5 0.0646 0.59 0.08 0.490 0.7955 0.490 6.0 0.0567 0.59 0.0694 0.85 0.858 0.85