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A FLUID MODEL FOR MULTIPLE-MIRROR AXIAL PLASMA FLOW PART I PRELIMINARIES by R. Bravenec Memrandum N. UCB/ERL M79/49 11 June 1979 ELECTRONICS RESEARCH LABORATORY Cllege f Engineering University f Califrnia, Berkeley 94720
A FLUID MODEL FOR MULTIPLE-MIRROR AXIAL PLASMA FLOW PART I PRELIMINARIES* R. Bravenec Department f Electrical Engineering and Cmputer Science Electrnics Research Labratry University f Califrnia Berkeley CA 9^720 ABSTRACT The fluid equatins are slved fr a straight magnetic field and a single mirrr under the assumptin f a steady-state, nedimensinal, isthermal flw. The latter assumptin is fund t be a pr ne during supersnic, shcked flw; therwise, it is accept able. Flw thrugh a magnetic mirrr is studied as a functin f the relative viscsity f the plasma and cmpared t fluid flw thrugh a V cnverging-diverging nzzle. Finally, applicatins t multiplemirrr flw are discussed. 'Spnsred by Department f Energy Grant N. DE-AS03-76F0003zf-PA# DE-ATOE-76ET53059. - 1
A FLUID MODEL FOR MULTIPLE-MIRROR AXIAL PLASMA FLOW PART I PRELIMINARIES I. INTRODUCTION This reprt lays the fundatin fr studying multiplemirrr axial plasma flw via a fluid mdel. The fluid equatins are slved fr a unifrm magnetic field and fr a cnverging-diverging field (a magnetic mirrr). The latter cnfiguratin is f interest because a multiple-mirrr device is a successin f single mirrrs. Our interest in a unifrm field is in studying the basic character istics f the equatins and their slutins. Frequently we will cmpare the plasma flw alng the mag netic field t actual fluid flw thrugh pipes f varying crss-sectin. The similarity is clear after making the fllwing assumptins: the plasma-vacuum interface is sharp and the plasma is f lw 3 > s that the magnetic field is cnsidered a knwn functin f z. Thrugh cnservatin f magnetic flux the plasma crss-sectin A is als knwn.1 Thus we may cmpare the plasma flw t that f a fluid flwing thrugh a pipe f crss-sectin A(z). Thrughut this reprt the flw is assumed steady, nedimensinal and isthermal. A steady-state results frm an arbitrary plasma surce at the system rigin and a sink at the exit. Radial mtin *Fr high 3, A(z) is nt s simply related t the vacuum magnetic field but must' be slved fr via pressure balance, as in Taylr and Wessn [1]. - 2 -
is neglected by making a "lng, thin apprximatin" with B large enugh t prevent any significant radial diffusin. Finally, isthermy riginates frm a large electrn thermal cnductivity which maintains Te(z)=T, a cnstant. This is cmmunicated t the ins via inelectrn cllisins n the time scale x. ^ x. *x» /m./m t.., where the le le i e ii superscript e dentes an ene/igy exchange time, and where t. and i e t.. are the in-electrn and in-in bcjouttojilyiq times, respectively. (See Trubnikv [2].) Thus, fr isthermal ins we require that x. «x, le s where xg is the time fr a fluid element t traverse the scale length f inhmgeneity f the fluid I. All ther electrn effects s are ignred in the analysis which fllws. With this assumptin the similarity between the plasma flw and that f a viscus fluid is even mre prnunced. In fact, the equatin f mtin used here differs frm the Navier-Stkes Equatin nly in the frm f the viscsity term. In a straight system they are identical. Sectin II" presents the fluid equatins and the derivatin f a secnd rder ODE fr the fluid velcity. This equatin is cast in dimensin less frm t select ut the imprtant parameters f the prblem and becmes the basis frmst f this reprt. Sectin III presents the analytical slutins f this equatin fr a unifrm magnet ic field. In Sec. IV an analytical slutin neglecting viscsity is btained fr a magnetic mirrr; hwever, this slutin des nt give transnic slutins (transitins frm subsnic t supersnic flw, r vice versa). The transnic case is discussed qualitatively, drawing heavily frm Shapir [3]. Finally, the mirrr-flw prblem, including - 3 -
viscsity, is slved numerically and the results cmpared t the inviscid case. Sectin V ummarizes the results and discusses the applica tins t multiple-mirrr flw. - h -
II. THE FLUID EQUATIONS The fllwing are taken directly frm Dawsn and Uman [k] ignring the electrstatic ptential and setting 3/3t = 0. ION EQUATION OF MOTION nm,v --&bti>** K'isf) +<-T«>i: <' > CONTINUITY EQUATION ^(nav) =0 (lb) PERPENDICULAR ION TEMPERATURE dti Ti-Tj 2vm /Hv/X2 T±f V a - +! L dz t.. \dz/ A dz + m i r7 [f<«>+t(t»e-t>] i ei "i 33f^[nA". dt{tl+2tj.,)] <lc> PARALLEL ION TEMPERATURE J dtl 2(Tl - Tjj ) 2v.m. dz x.. 3 11 /dv\ \dl) ' 2T» i dv d7 1 m r«+-l--s. r^(te-t') - 5 -
+ _la rna i(ti+2ti,] (ld) 3nA dz L i dz " J- J FLUX CONSERVATION B A = A^ -2- (ie) In the abve, n is the in density (assumed equal t the electrn density), A is the plasma crss-sectin, v is the z-cmpnent f the flw velcity (same fr ins and electrns), Tn ' and j» the electrn and in" temperatures (in energy units) i respectively paralare lei and perpendicular t the magnetic field, m. and m are the in and electrn masses, v. is the in bulk viscsity, x.. and x. are i 11 ei the in-in and electrn-in cllisin times, and k. is the in thermal i 2 cnductivity. The reader is referred t Dawsn and Uman fr a thrugh discussin f these equatins. Althugh we ultimately set T = Tn =T, Eqs. (1c) and (1d) are included t evaluate the last term f Eq. (1a). We cannt simply set T = Tn here since this term is fund t be f the same rder as the bulk viscsity term. Setting dt1 dz dtji' dz in Eqs. (1c) and (id), subtracting them frm each ther and then setting T - Tn - T thereafter (except in the difference T - Tn ), we btain 2 Thrughut this reprt all quantities subscripted with a zer are evalu ated at z = 0, i.e., A =A(0), B = B(0), etc. - 6 -
X1 I-' 1 T A d /V \ /0v T - Tn ~ x.. T ( J (2) 1 II 3 11 v dz \ /\ / Thus, the temperature difference is prprtinal t the small quantity x.., as is the in bulk viscsity v. = x.. T (3) 1 m. 11 v-" ( a f rder unity). Substituting Eqs. (2) and (3) int Eq. (1a), we have ydva _zl d(nt) } a _d_ dz m.n dz m.na dz \... 11 dz/, 3,... m.v dz dz \ A / 1 1 x ' 1 (T..TnA^)+lIlil " d l \ ( Next we define the fllwing dimensinless quantities "af- vi7" a5f <«sf- M5^= v (5) c s Here i^ is an apprpriate reference length in the prblem, i.e., the cell length f a multiple mirrr, and v = /T/m. is the sund speed s 1 r (als the in thermal speed). Substituting int Eq. (4) and using the 3/2 relatin x..=x/vs«t /n, where X is the in-in mean-free-path, we btain v±l=z±l*r +?lj>_±j_ I dv\.1^ _L J_ da d /v2\ tc, d? M2 n d5 c Mq na d? TdJ 3 *c Mq nv dc dc IT7 (6a) The cntinuity equatin becmes simply nav = 1 (6b) - 7 -
Using Eq. (6b) t eliminate n, Eq. (6a) becmes dv_j_j_ d(av) = ^J_ dc M2 av dc I M M_ c av d / dv\ a_ da d_ /v d? \adj 3d;d^ay (7) Equatin (7) is a secnd-rder ODE fr the nrmalized flw velcity v in terms f the knwn area functin a(?) and the tw parameters X / and M, hereafter referred t as the "flw equatin". Its c M slutin is cmpletely determined given the tw bundary cnditins v(0) and dv/d?(0) (r v(0) and v(l/& )), where L is the system c length). The first f each set is trivial since by definitin v(0) =1 (The bundary cnditin v(0) makes its appearance thrugh the para meter M.) Recall that the terms n the left-hand side represent the in inertia and scalar pressure, respectively. The term n the right represents the cmbined effects f pressure anistrpy and will here after be referred t as simply the "viscsity term." Fr v^dv/de; 2 2 ^d v/d? ^0(1), the relative rder f each term is easily seen t be inertia : pressure : viscsity = M ' : 1 : -t M & c Thus fr M,X /I «1 the pressure term is dminant. Hwever, fr c r ' v=1/m s that M=1, the rdering becmes 1 : 1 : (X / )(l/m ) s c that all terms may be cmparable. Fr future reference it is useful t rewrite the cnditin fr unifrm in temperature in terms f ur dimensinless variables. First it can be shwn that x = (l/m) U /% ) (z /x)x.. Since s seen A«1/n this can be written x = (l/m) U A ) U /l ) (x../n). Hence, S C O S C - 8 -
the cnditin fr isthermal ins becmes x A u O c M. i ^ /n\ Finally, we shuld nte that Eq. (4) differs frm that used by Miller [5], wh derives the frce equatin directly frm Braginskii [6]. The differences, which as f yet have nt been reslved, lie in the explicit frm f the viscsity term. Since this term is usually small, we will nt verly cncern urselves with this discrep ancy. J - 9 -
III. FLOW ALONG A UNIFORM MAGNETIC FIELD Setting a = 1 in Eq. (7) we have X 2 dv _]_ j_ dv av d v, * d? " M2 vd? I M.2 {3) M " cd? Ntice that the temperature anistrpy term has drpped ut, leaving us with the Navier-Stkes Equatin. Dividing by v and integrating nce, there results -' i(h - rr( -) M c where *;s < > Slving fr dv/d?, we btain. ir (, i (1.,\ +v. (10) dc A a I ' M \v /J This equatin can be integrated by first separating variables. Fr v«?*0 vdv M 2 /m ^ 1 X0 A.. 1 Cd?,,-x Mv - I M +TT -r~av,lv+-7- = r 2- (11) \ M j M X. a c / - 10 -
The results f the integratin depend n whether M <1 r M >1. Fr v'l K -'): «_ _L M (M +1) ' X 2a..,..2 the expressins are particularly simple3 a *c 1-M2 1/M' - v 1. \ 1/M - 1 '1 + u' - v> H In ' M < 1 (12a) C = a *cm2-1 f] + y» - v^ * 1 v- 1/M' jr M ln\ Vl-1/M2 2 O J, M > 1 (12b) where ^ 5 c am 2 1 - M (12c) There are tw distinct prfiles given by each f Eqs. (12), depending n the sign f V. Fr v^<0, Eq. (12a) gives the same prfile as (12b) except shifted in c. Chsing v' >0 in Eq. (12b) leads t an entirely unphysical prfile in which the velcity increases withut bund Thus, we are left with the tw cases M <1, v' > 0 and M >1, v1 <0 3 This inequality is impsed nly t btain the simple frms f Eqs. (12), and nly restricts ur placement f the rigin relative t the prfiles. - 11 -
As an example f the frmer, cnsider the chices M =0.5, v^o.01, a=1 fr which Eq. (12a) gives the prfiles shwn in Fig. 1(a) Here X /& is set t ne s that the units f c are mean-free-paths. c ^ r Superimpsed n the curve M = M v is that f n, btained frm the cn tinuity relatin nv = 1. The mst bvius feature f the curves is that M-*0 and n^00 at a pint c. t the left f the rigin, where mm c. = r In mm M (1-M2) y 1-M2 Althugh we have, impsed an upper limit n v" t btain Eqs. (12), slutins certainly exist fr v1 large. Fr example, shifting the curves f Fig. 1(a) t the right by an amunt less than \c,. rep resents such a slutin. The resulting flw is then characterized by an acceleratin and pressure drp within a few mean-free-paths int the system. Whatever the surce f the initial velcity gradient, the impr tant pint here is that bulk viscsity des nt lead t an increasingly accelerating flw, as is the case during fluid flw thrugh pipes with wall frictin [3]. Instead, it simply smths ut the initial gradients, causing the prfiles t asymptte t steady values. Nte that fr v1 =0 M arbitrary, Eq. (9) has the trivial slutin n = v=1, which may be easily seen frm Eq. (10). We have just shwn that the equatins have a preference fr such a unifrm slutin; thus we cnclude that the imp sitin f v* >0 des nt represent an easily realizable physical system. This cnclusin will be applied in the next sectin n flw thrugh a mag netic mi rrr. - 12 -
An example f the case M >1, v1 <0 is presented in Fig. 1(b) fr M =2.0, v»=-0.01, a=1. As required by basic fluid thery [3], the velcity decreases and the density increases during vis cus supersnic flw. These prfiles represent a shck frnt, resulting in a transitin t subsnic flw ( M = 0.5 ) ver a distance f abut fur mean-free-paths. Referring back t Eq. (12b), we see that in general the tw asympttes f v(c) are v - 1 and v. =1/M, such that max mm 2 the shck strength is v /v. =M. In a frame with X /% «1, Eq. max m in c (12b) predicts an almst discntinuus prfile. This is expected since the shck thickness depends n the magnitude f the viscsity term which in turn is prprtinal t X /I. Furthermre, n either side f the c shck dv/dc is very small. Thus, fr a fixed value f M, varying v1 within the range *, (M2-1)2 0 < v' < C 1 M (M + 1) has the effect f shifting the shck alng c (Letting \>' -+Q drives the shck tward +«.) This behavir will reappear in the fllwing sectin n mirrr flw. Befre cntinuing, we shuld realize that the previus results are nt t be interpreted t literally. First f all we have n justifi catin in accepting slutins which vary significantly ver a few meanfree-paths. This is a direct cnsequence f the limitatins f the fluid thery. Thus, we shuld be wary f the "bundary-layer effect" predicted by Eq. (12a) r the details f the shck structure f Eq. (12b). As t - 13 -
the latter, hwever, we are justified in accepting the values attained by the prfiles dwnstream f the shck. This is because the equatins cnserve such quantities as mass and mmentum flux [3]. That is, the shck strength predicted by Eq. (12b) is crrect under the assumptin f isthermy. Hwever, as mentined in the Intrductin, this assumptin is valid nly fr T (z)=t, a cnstant, and x. «x. Referring t e le s 3 Eq. (8) with I ^X, M^ 1 and n^ 1, the latter cnditin is badly vilated and isthermy is a pr assumptin. We will nt let this deter us, hwever, since ur ultimate intentin is nt t study shck frnts but subsnic, shck-free flw thrugn a multiple-mirrr, fr which is thermy is a fair assumptin. If desired we may imagine the temperature maintained unifrm acrss the shck by sme external means. 4? - 14 -
IV. FLOW THROUGH A MAGNETIC MIRROR Fr this sectin and the remainder f this reprt we use the area functin (C) = ^f+^fcs 2w5 (13) where Rs B /B., the mirrr rati. This functin is sketched fr nia j\ m i n R= 2 in Fig. 2. Substituting Eq. (13) int Eq. (7) results in an expres sin that can be slved easily nly in the limit X /l =0. In this c case, Eq. (7) becmes dv _ J_ d In av Slving fr dv/d?, dv 1 da d? 3dC(Mv)2-1 d4) This can be integrated t yield 2 v 2 - in av» 1 (15a) M Using Eq. (6b) we can write this in terms f n a 2'n2(l-^lnn) =1 (15b) - 15 -
Eqs. (15) are sketched in Fig. 3 fr R=2 and the tw chices (a) Mq =0.25 and (b) Mq =2.0. As expected the subsnic slutin is char acterized by a density minimum and a velcity maximum in the mirrr thrat, and the supersnic slutin by just the ppsite. Ntice further that fr a(c) peridic with perid unity the slutins are similarly peridic. This means that there is n verall density (pressure) drp acrss a successin f such mirrrs. We will return t this pint in the final sectin. Referring t Eq. (14), the slutin becmes singular at v=1/m (M=1), preventing a transnic slutin. We nw discuss qualitatively the family f slutins, bth subsnic and transnic, fr an inviscid fluid flwing thrugh a cnverg ing-diverging nzzle. We imagine being given the inlet pressure p and investigating the flw as a functin f the exit pressure p. Refer ences are repeatedly made t Fig. 4, taken frm Shapir, p. 140, which is fr adlabatic flw f a perfect gas. Initially, entirely subsnic flw results when p1 is reduced belw p, as in Fig. 3(a). This is shwn by case (a) f Fig. 4. Lwering p1 further, state (b) is even tually reached, where the velcity in the thrat has reached M=1 (snic flw), but the flw dwnstream is still subsnic. The flw is nw chked, meaning that n further reductin in p affects the flw upstream f the thrat r increases the fluid flux. Between cases (b) and (f) there exist n values f p. that result in smth shck-free flw. Case (f) crrespnds t a unique value f p1 fr which shck-free supersnic flw des exist in the diverging sectin. Cases (c) and (d) represent shck frnts invlving abrupt transitins frm supersnic t subsnic flw. As P1 is reduced frm that f case (b) the shck is seen' t prgressively - 16 -
mve dwn the nzzle. The flw appraches that f case (d), in which the shck ccurs at the exit. Except fr case (f), further reductin f the exit pressure results in blique shcks utside the nzzle, as represented by cases (e) and (g). These effects are tw-dimensinal and cannt be treated in ur simple thery. The flw characterized in Fig. 4 is that f an adiabatic, inviscid fluid. We wish nw t investigate the effects f isthermy and viscsity. Fr X /% «1 we expect Fig. 4 t be qualitatively duplicated since the viscsity term f Eq. (7) is then small. The slight amunt f viscsity simply gives structure t the shck frnts (Fig. 1(b)) s that n singularities ccur in the flw equatin. Fur thermre we expect that ur impsitin f cnstant temperature nly affects the magnitude f variatin f v and n, but nt the general behavir characterized in Fig. 4. The flw equatin (7) is slved numerically using an inte grating rutine devised by Gear and Hindmarsh [7]. The prgram requires a system f first-rder ODE's necessitating the definitin f a new va r iab 1e Eq. (7) then becmes w >L = Ida. [-LL*±/2\v^] 1 c 1 I"/,. t 1 \, 1 idal dc adcvl3aadc V+3a j VJ +a X~ iv LlV"FTVJ V mt I di"j (16b) - 17 -
In the fllwing, a is set t 0.96 [6]. The rutine further requires all bundary cnditins t be given at C= 0, namely v(0) and v'(0). (As mentined earlier, we have defined v(0) =1 s that v(0) appears thrugh the parameter M^.) Thus, given a value fr X /, the slu " c tin t the system f Eqs. (16) is cmpletely determined by the set M and v' h v' (0). T generate curves similar t thse f Fig. 4, X /% is c set t 0.01, s that the fluid is nearly inviscid. Frm the findings f the last sectin, v1 is set t zer since the plasma initially sees 1-7 a straight system da/d^(0)=0. Tw cases f entirely subsnic flw are presented in Fig. 5 fr M =0.25 and M =0.3. We immediately nte that the small amunt f viscsity shifts the velcity maximum and density minimum int the divergent sectin f the mirrr. As M is in creased t duplicate case (b) f Fig. 4, a value is reached fr which snic flw first ccurs at a pint slightly beynd the mirrr thrat. This situ atin is represented by the case labeled M =0.3175. As M is increased further, the flw becmes supersnic ver a shrt distance with a subse quent smth transitin back t subsnic flw. This behavir is inter preted as a weak shck develping in the divergent sectin f the mirrr. As Mq is increased still further, appraching snic flw in the thrat, the shck frnt grws in amplitude, steepens, and mves dwnstream. Curves fr which M = 1 in the thrat are presented in Fig. 5, crrespnding k The value f M in the mirrr thrat is simply estimated frm the curves. Hence, there is a small uncertainty invlved in the actual value. - 18 -
t M =0.3225. An interesting phenmenn ccurs if M is chsen s just slightly larger s that M is barely greater than unity in the thrat the slutin fr v diverges, indicating an unphysical slu tin. An example is the case labeled M =0.325. Thus, the numerical slutins preserve the phenmenn f chking n physical steady-state slutins fr M larger than a critical value. Furthermre, it appears that chking ccurs fr M=1 in the thrat even during isthermal vis cus flw. In rder t prceed further and generate curves analgus t cases (b) thrugh (f) f Fig. 4, we begin the integratin at C= 0.5 (in the mirrr thrat) where we may accurately impse the cnditin M=1. This leaves nly the quantity v'(.5) with which t determine the flw. As argued in the inviscid fluid case, the flw upstream f the thrat is unaffected by reducing the exit pressure. Thus, in the fllwing cases the prfiles upstream f the thrat are thse fr critical flw. It shuld be recalled that the relevant physical parameter that is being varied here is the exit pressure (density) via the bundary cnditin v'(.5). Thus, nly thse values f v'(.5) giving n(1.0)<n. (1.0) are cr 11 valid. This cnditin translates t v'(.5)>v'. (.5). It is fund en t that there exists a maximum value f v'(.5) abve which the slutin diverges. This valve, which we will dente by v' (.5), crrespnds t ' max ' v the instance f smth, shck-free supersnic flw, as in case (f) f Fig. 4. Furthermre vmax(-5) is fund t be nly slightly larger than v' :«.(*5), leaving us nly the restricted range v1. (.5)<v'(,5)<v' (.5) crit crit - - max This case is hereafter referred t as "critical flw". In the fllwing discussin its slutins n and v are subscripted with "crit". - 19 -
with which t wrk. In rder t generate a shck frnt and mve it dwn stream, v'(.5) must be chsen very near v1 (.5). T accmplish this, max r ' the prgram was mdified t search fr that value f v'(.5) which yields a specified value f n(1.0), the physical bundary cnditin. The results fr three chices f n(1.0) are presented in Fig. 5, labeled n(l.o) =0.85, 0.65, and 0.2. As evident frm the curves, the differ ences between their values f v1 (.5) are unnticable. The case n(1.0) =0.20 crrespnds t all but the "tail" f the shck having mved ut the system. Thus, we have seen that the bundary cnditin n dv/dc is nly imprtant in generating the flw prfiles dwnstream f the thrat in a chked system. Otherwise M has by far the dminant rle. We have als seen that the effect f v'(.5) is the same as in ur previus study f shcks in a straight field that is, it shifts the lcatin f the shck. Finally, we shuld mentin that isthermy is a fair assumptin nly fr the case labeled M =0.25. That this is s can be seen using 3 Eq. (8) with X /A =0.01, a / a (frm the curve fr M ), M*0.5 and n^o^ (wrst cases). Fr larger values f M, the scale lengths fr density and velcity variatins are t shrt fr in-electrn clli sins t maintain T. a cnstant. 1 T cmplete ur study f viscus flw thrugh a single mirrr, we nw cnsider the flw f a mre viscus plasma. T d this we increase vj the value f X / in Eqs. (7) and (16b). Hwever, we are limited in the fluid thery t X «Z, where as befre l is a typical scale s s length fr variatins in n, v r A. Analgus t Eq. (8) this cnditin can be written in terms f ur dimensin less variables as - 20 -
Fr M>j/nTTrnT, this cnditin is less stringent than that f Eq. (8). T begin, we define I in terms f A(z) fr which I /% «\. Fr s s c n=1, Eq. (17) then demands that X /l «. Chsing X /I =0.1 c c and vq =0, we present tw cases f purely subsnic flw in Fig. 6 as the curves labeled M =0.15 and M =0.25. We make the fllwing cm 3 parisns t Fig. 5: (1) the pints f velcity maxima and density minima are displaced farther dwnstream f the thrat. (2) Fr the same value f Mq (fr example, Mq =0.25 ) the pressure (density) drp acrss the system is greater. As Mq is increased, M=1 first ccurs at 5=0.675, pre sented by the curves labeled M =0.315. Increasing M further results in a regin f supersnic flw fllwed by a very slw transitin back t subsnic flw, such as the case labeled M =0.325. As M is in creased still further, the regin f supersnic flw increases until fr MQ =0.331 the flw dwnstream f the thrat is entirely supersnic. Thus, we see that shck-like behavir vanishes in such a viscus plasma. As Mq is increased further, the psitin f the velcity maximum and density minimum mves ut the exit. The curves fr M* 1 in the mirrr thrat & are given by thse labeled M =0.353. Unlike the case X /% =0.01, c n catastrphic behavir ccurs fr M>1 in the thrat. Referring back t Fig. 5, we ntice that fr M<1 in the thrat the velcity prfiles exhibit inflectin pints near the thrat. Hwever, fr M>1 jn tne Fr M=1 in the thrat it can be shwn using Eq. (16b) that this psitin is the inflectin pint. - 21 -
thrat, n such pint exists and the prfiles diverge. In the present case, hwever, the inflectin pint mves relatively slwly tward the exit as M is made greater than unity in the thrat. Tw examples are the curves labeled M =0.36 and M =0.4. In this sense the numerical slutins n lnger demnstrate an bvius chking effect. We instead resrt t intuitin and a cursry study f the flw equatin in asserting that chking still ccurs fr M=1 in the thrat, even with substan tial viscsity, and that the prfiles with M>1 here are unphysical. Since shck frnts d nt ccur there is n need t initialize the inte gratins in the thrat, where we specify M=1 and vary v'(.5). This is because we are able t generate all exist densities (pressures) dwn t that analgus t case (f) f Fig. 4 via the parameter M0 alne. Althugh the curves are much smther than thse f Fig. 5, isthermy is fund t be a pr assumptin fr all the cases shwn in Fig. 6. Even fr M =0.15 where %/% =0.6, M =0.3 and n-0.9, s c the inequality (8) is badly vilated. This is because X /I is a fac c tr f ten larger than that f Fig. 5, resulting in relatively few inelectrn cllisins. Finally we check if Eq. (17) is satisfied with &s/&c defined in terms f n and v, and taking int accunt the substantial decay f n alng? (which lengthens the mean-free-path). Because W& «s large where n is small, Eq. (17) is still fund t hld (althugh barely). - 22 -
V. CONCLUSIONS Cmparing Figs. 4 and 5, we see that the flw f a nearly inviscid plasma thrugh a magnetic mirrr is very similar t fluid flw thrugh a cnverging-diverging nzzle. Hwever, the flw f a mderately viscus plasma is quite different, in that shck frnts d nt develp. The fact that isthermy was a pr assumptin des nt hinder us in applying the results t subsnic multiple-mirrr flw, in which shcks are excluded and the flw is usually slwer. T see hw this is dne, cnsider a "half-system" with z = 0 crrespnding t the midpint f the multiple-mirrr and z = L an exit. In the limit X /% =0, Eqs. c ^ (15) give the slutin fr a(&) peridic With a(c) given by Eq. (13), Fig. 3(a) represents an example f the prfiles at any particular mirrr f the device. As mentined at that time, there is n change in the ver all levels f density and velcity frm mirrr t mirrr. Hwever, the inclusin f viscsity results in an verall decrease in density (pressure) and an acceleratin f the flw frm mirrr t mirrr, as evident frm the entirely subsnic prfiles f Figs. 5 and 6. This effect is mre dramatic with greater viscsity. Fr small pressure differences between the ends f ur half-system, the flw will be as described. As the pres sure difference is increased, snic flw and subsequent chking will first ccur in the exit mirrr. In physical systems in which the exit pres V? sure is lw, a chked flw is a reasnable assumptin, and M=1 in the exit mirrr thrat becmes a "natural" bundary cnditin. Since we are nt cncerned with the flw beynd the exit mirrr thrat, shck^ frnts d nt ccur in the multiple-mirrr. Furthermre, the flw is everywhere subsnic, adding t ur justificatin in assuming isthermy. These bservatins will be made mre quantitative in a frthcming reprt. - 23 -
REFERENCES [1] Taylr, J. B., Wessn, J. A., Nuclear Fusin 5. (1965) 159. [2] Trubnikv, B. A., "Particle Interactins in a Fully Inized Plasma", in Reviews f Plasma Physics, Vl. 1, Cnsultants Bureau, New Yrk, 1965. [3] Shapir, A. H., The Dynamics and Thermdynamics f Cm pressible Fluid Flw, Vl. 1, The Rnald Press, New Yrk, 1953. [4] Dawsn, J. M., Uman, M. F., Nuclear Fusin 5. (1965) 242. [5] Miller, G., Ls Alams Scientific Labratry Reprt LA- 7580-MS, Dec. 1978. [6] Braginskii, S. I., "Transprt Prcesses in a Plasma", in Reviews f Plasma Physics, Vl. 1, Cnsultants Bureau, New Yrk, 1965. [7] Hindmarsh, A. C, "Gear-Ordinary Differential Equatin System Slver", UCID-30001 Rev. 3, Lawrence Livermre Labr atry, Dec. 1974. - 24 -
- 3 i 2.5 0.5 M 2.0 M 0.4 0.3 0.2 V H 1.5 1.0 0.1 0.5 0 0 1 1 1 1 1 I k -2 1 0 2 3 4 5 6 min (a) 0.0 M (b) Fig. 1 Viscus flw alng a unifrm magnetic field ( c = z/x ) (a) Subsnic flw ( M =0.5 ) (b) Supersnic flw ( M =2.0 )
0.0 F'9' 2 The nrmalized plasma crss-sectin vs =z/a
0.0 Fig. 3 Inviscid flw thrugh a magnetic mirrr ( X /I =0 ) (a) Subsnic flw ( M =0.25 ) (b) Supersnic flw ( M =2.0 )
(cnstant) Lcus f states dwnstream f nrmal shck Distance alng nzzle VJ Fig. 4 Fluid flw thrugh a cnverging-diverging nzzle
M<1 M>1 2.0 Y M>1 M > M<1 0.0 Fig. 5 Viscus flw thrugh a magnetic mirrr ( X /I =0.01 ) c (a) nrmalized density vs =z/l (b) Mach number vs z, = z/l
M< M> 2.0-1.5 - y M>l M 0.5 - Y M<l 0.0 Fig. 6 Viscus flw thrugh a magnetic mirrr ( X_/A_ =0.1 ) (a) nrmalized density vs c=z/% c (b) Mach number vs = z/jl c