Zaleski 1 Joseph Zaleski Mah 451H Final Repor Conformal Mapping Mehods and ZST Hele Shaw Flow Inroducion The Hele Shaw problem has been sudied using linear sabiliy analysis and numerical mehods, bu a novel advanage of conformal mapping mehods is ha exac soluions are obainable. However much of he heory which deals wih conformal mapping is concerned wih he zero surface ension problem; as summarized in he end he non-zst problem is much more inracable. My main goal his semeser was o obain an undersanding of why he ZST soluions are no necessarily relevan, and surface ension plays a role in he problem. The following repor makes use of he process from Chaper in Dr. Cumming s hesis. Basic Equaions Using he process oulined in chaper one of Cummings (1996) we have he basic equaions for one phase and Hele-Shaw flow: p 0 in ( ) (1), where p is pressure, and is he varying fluid domain. Surface Tension BC: p Ton ( ) (), where T is surface ension and is curvaure Kinemaic BC: vn on ( ) (3) n However, under he assumpion ha curvaure is small (ha is, he fluid domain () is smooh and nowhere of order 1/T), and surface ension is negligible, we migh conjecure ha seing he righ side of eq. () equal o zero is a valid approximaion o he problem. Thus we have he zero surface ension boundary condiion: p 0 on ( ) (4) Polubarinova/Galin Mehod Nex, I ouline wha I will call he P-G approach. The crucial facor which allows his mehod o be used is ha equaion (1) says pressure p is harmonic on he fluid domain. Complex variables heory herefore implies ha p is he real or imaginary par of some complex funcion which is analyic. The key idea o his mehod is mapping a known region (in his repor, we use he uni disc as his region, bu for channel flows, he righ half plane may be more convenien) ino he evolving fluid domain via a ime dependan conformal map. We are a he libery o choose he region and mapping, bu for simpliciy, mapping he origin o he pressure source is convenien. If he fluid domain is simply conneced, he exisence of he map is known, by he Riemann mapping heorem (for his reason we also enforce ha he map be univalen, which prevens he border of he fluid domain from crossing iself). Furhermore, assuming ha he derivaive is real and posiive gives us a unique map. I define his map as w z(, ), where i are poins in he uni disc.
Zaleski 1 fig.1, Cummings (1996) Nex, on he fluid domain in he z=x+iy plane we will define he complex poenial of he flow o be: ( z, ) f ( z, ) i g( z, ), s.. analyic wihin Ω() (excep a sink/driving poins) and p { ( z, )} (5) Nex, we describe he asympoic behavior of he problem near singulariies in he fluid domain by making assumpions abou he behavior a he source or sink of he pressure field (or where fluid is injeced or sucked from, in he problem). Common assumpions for he sucion or blowing problem 1/ include: p ~ log ( x y ), as ( x, y) (0, 0), where 0 sands for source/sink srengh Clearly pressure blows up as we near origin, maching he physical naure of he problem. From his assumpion and using he definiion of he complex poenial, we can now obain he following relaion: { ( z)} p ~ log r. To saisfy his and have he correc singulariy, we can ake: ( z) ~ log z ( 5), since log z log r i. Wha we nex consider below, is complex poenial in he plane, and we define his funcion assuming ha he behavior near origin in he plane is idenical o ha in he z plane:
Zaleski 3 (, ) ( z(, )) log (6) Noe ha he ZST condiion says ha { ( z(, ))} 0 for z on, which corresponds o (, ) vanishing on he uni disc, 1. (hus he ZST condiion is saisfied in boh planes) Moving on, we wrie he KBC (3) as: KBC: n v, n where n / and vn / n n vn ( / ) ( / ) p p on ( ) (i) To reduce he above expression, on boundary =1, we have: '( ) px py '( z) w( ) w( ) 4 w( ) (ii) and p { ( ) } { } (iii) To calculae : w z w(, ) 0 w w (iv) w Now noe ha ( i) ( iii) ( ii) : 4 w( ) w( ) { } { } w and herefore ( iv) { } w w( ) Polubarinova-Galin eq: { w'( ) w ( )} on 1, ( 7) To summarize, we now have a relaion which mappings mus saisfy o be soluions o he ZST problem.
Zaleski 4 Examples The P-G equaion involves guessing he general form of a map, bu a variey of mappings exis. Ex 1: Take our iniial guess for he mapping o be he quadraic map, where a1, aare real valued funcions. z w a (, ) a1( ) ( ), i Subsiuing ino (7), and aking ino accoun ha he P-G eq. applies on 1, or in polar form e, we ge: i w'( ) a ( ) a ( ) a ( ) a ( ) e, 1 1 w ( ) a e a e w ( ) a e a e i i i i 1 1 P-G: { w'( ) w ( )}, i i i i i i i i { e ( a1( ) a( ) e )( a1 e ae )} { e ( a1a1 e a1a e aa1 aae )} i i { a1a1 a1a e aa1 e aa} { a1a1 aa a1a cos( ) aa1 cos i(...)} a1a1 aa ( a1a aa1 )cos And lasly we ge he following sysem of ODEs, by noing ha he erms mus vanish o mach he righ sid e: a1a1 aa a a a a 0 1 1 Using Malab and random iniial condiions, I solved his sysem (which can be manually inegraed):
Zaleski 5 1.5 uadraic map: soluions break down a a cusp 1 0.5 0-0.5-1 -1.5-1 -0.5 0 0.5 1 1.5 The previous plo shows behavior which makes he ZST problem no useful he soluions break down a he cusp poin (where he curvaure blows up, and he mapping is no longer univalen). I is also worh nohing ha various oher maps work, making an arbirary amoun of soluions possible, and also making he problem no really seem applicable o he real life case (see figure below, which displays a cusp problem on he lef from a polynomial map, and a logarihm map ha is a sli/finger model). The real life case also involves surface ension and unis, so of course i makes sense ha he ZST model may no accuraely predic resuls. The ZST problem is also ill posed, because as saed in one hesis I read, soluions wih close iniial condiions can eiher break down in finie or infinie ime. (Dallason, 013) Dallason, (013)
Zaleski 6 Conclusion To conclude, surface ension regularizes he Hele-Shaw problem, and makes i relevan using he mehod we oulined, i is possible o obain all sors of ineresing geomeric soluions, bu in realiy hese may no be useful. Surface ension also prevens cusps from forming in experimens. Oher complex variables mehods for analyzing he Hele Shaw problem (boh he ZST and NZST cases), such as he Schwarz funcion exis, bu I have no included hem, as hey are much more involved. However, hey confirm ha he NZST problem is generally much less easy o solve han he ZST problem. Furher research on he NZST problem is useful, because surface ension is wha keeps he problem from being undeermined. References Linda Cummings, Phd hesis, Oxford, 1996 Michael Dallason, Phd hesis, ueensland U. of Technology, 013 Code funcion Capsone1 clc clear all inegraionspan=[0,0.6] ICs=[1,0.1] [,P]=ode15s(@SysemofEqs,inegraionSpan,ICs); a1=p(:,1); a=p(:,); s=size(a) i=0:*pi/(s(1)-1):(*pi); q=size(i) circle=ranspose(i); x1=cos(circle); y1=sin(circle); for k=1:1:38 plo(a1(k).*x1+a(k).*(x1.^-y1.^),*a(k).*x1.*y1+a1(k).*y1,'r') hold on end ile 'uadraic map: soluions break down a a cusp' funcion aprime=sysemofeqs(,a) =5; a1=a(1);a=a(); aprime=[(-1*a1*/(*pi))/(a1^-4*(a^)); -*a*(-1*/(*pi))/(a1^-4*(a^))];
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