Content-Type: multipart/mixed; boundary="-------------9810301006300" This is a multi-part message in MIME format. ---------------9810301006300 Content-Type: text/plain; name="98-694.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-694.keywords" velocity dependence, aerodynamic drag, parachute ---------------9810301006300 Content-Type: application/postscript; name="v2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="v2.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58c Copyright 1986, 1994 Radical Eye Software %%Title: v2.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: /usr/local/lib/texmf/bin/wrapped/dvips v2.dvi -o %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.04.27:1517 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N 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y(c)m(o)m(e\016cient)p Fq(,)17 b Fm(C)282 2369 y Fj(D)318 2362 y Fq(,)f(whic)o(h)g(is)g(the)h (dimensionless)d(quan)o(tit)o(y)i(de\014ned)f(b)o(y)809 2494 y Fm(C)845 2501 y Fj(D)895 2494 y Fl(\021)1003 2461 y Fm(F)1035 2468 y Fj(D)p 953 2483 169 2 v 959 2511 a Fk(1)p 959 2520 20 2 v 959 2548 a(2)993 2531 y Fm(\032)8 b(A)g(v)1098 2517 y Fk(2)1127 2494 y Fm(:)745 b Fq(\(7\))0 2640 y(With)21 b(this)f(de\014nition)g(and)g(\(6\),)j(w)o(e)d(ha)o(v)o (e)g Fm(C)893 2647 y Fj(D)950 2640 y Fq(=)h(constan)o(t)o(,)h(i.e.,)f (the)g(drag)f(co)q(e\016cien)o(t)h(dep)q(ends)0 2700 y(only)15 b(on)f(the)h(shap)q(e)g(and)f(surface)g(c)o(haracteristics)g (of)h(the)g(b)q(o)q(dy)g(and)f(the)h(Reynolds)f(n)o(um)o(b)q(er.)19 b(Th)o(us)964 2760 y Fo(5)p eop %%Page: 6 6 6 5 bop 0 50 a Fq(in)17 b(\(3\))h(and)f(\(4\),)h(the)g(constan)o(t)f Fm(k)g Fq(=)725 30 y Fk(1)p 725 38 20 2 v 725 67 a(2)759 50 y Fm(C)795 57 y Fj(D)839 50 y Fm(\032)8 b(A)p Fq(.)25 b(F)l(urthermore,)15 b(the)j Fp(dynamic)i(pr)m(essur)m(e)f Fm(q)e Fl(\021)1834 30 y Fk(1)p 1834 38 V 1834 67 a(2)1868 50 y Fm(\032)8 b(v)1928 32 y Fk(2)0 110 y Fq(pla)o(ys)16 b(a)h(fundamen)o(tal)e(role)h(in)g(aero)q(dynamic)g(theory)i([7].)k(F)l (or)16 b(instance,)g(when)g(the)h(space)g(sh)o(uttle)0 169 y(Challenger)11 b(explo)q(ded,)i(it)g(w)o(as)e(op)q(erating)h(in)g (a)g(high)g(dynamic)f(pressure)g(regime.)19 b(V)l(ery)13 b(fast)g(aircraft)0 229 y(need)18 b(to)h(op)q(erate)g(at)g(high)e (altitude)i(\(where)f Fm(\032)h Fq(is)f(relativ)o(ely)g(small\))g(to)h (a)o(v)o(oid)e(excessiv)o(e)i(dynamic)0 289 y(pressure)c(and)g (catastrophic)h(damage)g(to)h(the)f(aircraft.)60 349 y(F)l(or)k(smo)q(oth)h(spheres)f(ha)o(ving)h(Reynolds)f(n)o(um)o(b)q (ers)f(in)i(the)h(range)e(10)1451 331 y Fk(3)1495 349 y Fq(to)i(3)14 b Fl(\002)g Fq(10)1703 331 y Fk(5)1726 349 y Fq(,)22 b(the)g(drag)0 408 y(co)q(e\016cien)o(t)13 b(is)f(appro)o(ximately)f(0)p Fm(:)p Fq(47,)i(while)f(for)h(Reynolds)e (n)o(um)o(b)q(ers)g(greater)h(than)g(3)t Fl(\002)t Fq(10)1722 390 y 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffff fffffffffffffffffffffffffffffffffffffffffffffffffdfdfdfdfdfdfdfdffffffff fffffffffffffffffffffffffdfdfdfdfdfdfdfdffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffefefdfc fcfdfefefffffffffffffffffffffffffffffffffdfdfdfdfdfdfdfdffffffffffffffff fffffefdfdfefffffffffffffffffffffdfdfdfdfdfdfdfdfefefdfcfcfdfefeffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff fffffffffffffffffffffffffffffffffffffffffffffffffffffefdfdfeffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffff showpage % stop using temporary dictionary end % restore original state origstate restore %%EndDocument @endspecial 198 1858 a Fr(Figure)h(1:)23 b(Dra)o(g)17 b(coefficient)g Fm(C)946 1865 y Fj(D)999 1858 y Fr(f)o(or)f(a)h(sphere) e(as)i(a)g(function)f(of)198 1917 y(Reynolds)h(number)g Fm(R)i Fr(\(fr)o(om)f(Figure)h(32)f(in)g Fq([11])p Fr(\))60 2093 y Fq(It)25 b(follo)o(ws)f(from)f(\(4\))j(and)e(\(7\))h(that)g(the) g(terminal)f(v)o(elo)q(cit)o(y)h(for)f(a)h(sphere)e(falling)h(in)h(air) f(is)0 2152 y(appro)o(ximately)781 2260 y Fm(v)805 2267 y Fk(T)850 2260 y Fq(=)903 2171 y Fh(s)p 953 2171 203 2 v 1014 2227 a Fq(2)p Fm(W)p 959 2249 191 2 v 959 2294 a(\032)8 b(C)1029 2301 y Fj(D)1073 2294 y Fm(\031)r(r)1126 2280 y Fk(2)1155 2260 y Fm(;)0 2391 y Fq(where)22 b Fm(W)29 b Fq(is)22 b(the)h(w)o(eigh)o(t)f(of)g(the)h(sphere,)f Fm(\032)h Fq(is)f(the)g(densit)o(y)g(of)h(air)e(at)i(sea)f(lev)o(el,)i (and)e Fm(r)i Fq(is)e(the)0 2451 y(radius)c(of)h(the)h(sphere.)29 b(The)20 b(densit)o(y)e Fm(\032)h Fq(is)g(a)h(complicated)e(function)h (of)h(temp)q(erature,)f(h)o(umidit)o(y)l(,)0 2510 y(and)d(pressure)e (\(whic)o(h)i(v)m(aries)g(with)h(altitude\))g(so)f(this)g(equation)g (is)h(only)f(an)g(appro)o(ximation.)60 2570 y(The)g(motion)g(of)g(a)h 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Fq(is)e(the)h(e\013ectiv)o(e)h(cross-)0 526 y(sectional)d(area)h (of)g(the)g(skydiv)o(er,)g(and)f(the)h(densit)o(y)f(of)h(air)f(is)h Fm(\032)e Fq(=)g(1)p Fm(:)p Fq(225)h(kg/m)1564 508 y Fk(3)1586 526 y Fq(.)26 b(Solving)17 b(for)h(the)0 586 y(drag)e(co)q(e\016cien)o(t)g Fm(C)382 593 y Fj(D)434 586 y Fq(w)o(e)h(obtain)756 668 y Fm(C)792 675 y Fj(D)841 668 y Fq(=)924 635 y(2)8 b Fm(W)p 900 657 136 2 v 900 702 a(\032)g(A)g(v)1005 685 y Fk(2)1003 717 y Fj(T)1055 668 y Fq(=)1117 635 y Fm(W)p 1113 657 62 2 v 1113 702 a(q)r(A)1181 668 y(;)691 b Fq(\(9\))0 785 y(where)16 b Fm(q)j Fq(is)d(the)h(dynamic)e(pressure)g(corresp)q(onding)f(to)j (terminal)e(v)o(elo)q(cit)o(y)l(.)60 845 y(If)g(a)f(72.6)g(kg)h(skydiv) o(er)f(carrying)f(a)h(19)h(kg)g(load)e(\(91.6)i(kg)g(=)f(867)g(N\))i (attains)e(a)g(terminal)g(v)o(elo)q(cit)o(y)0 905 y(of)22 b(49)h(m/s)e(\(in)h(the)h(b)q(elly-to-earth)e(p)q(osition\))h(and)f (has)h(a)g(cross-sectional)e(area)i(of)h(0.56)e(m)1857 887 y Fk(2)1879 905 y Fq(,)j(it)0 965 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF grestore %%EndDocument @endspecial 406 1724 a Fr(Figure)j(2.)34 b(F)o(or)o(ces)18 b(on)g(an)g(unpo)o(wered)f(air)o(craft)60 1878 y Fq(In)12 b(general,)g(ev)o(en)g(for)f(a)i(parac)o(h)o(ute,)e(the)h(equations)g (in)f(\(10\))i(do)f(not)g(ha)o(v)o(e)f(a)h(closed)g(form)f(solution.)0 1937 y(Ho)o(w)o(ev)o(er,)17 b(there)h(exists)g(a)g(closed)f(form)g (solution)g(in)g(one)h(remark)m(able)e(case)i(that)g(mo)q(dels)f(re-en) o(try)0 1997 y(of)g(the)g(space)f(sh)o(uttle)g(in)o(to)g(the)g(earth)h ('s)f(atmosphere.)k(W)l(e)d(discuss)e(this)h(example)g(in)g(Section)g (5.)0 2120 y Fs(5.)32 b(Re-en)n(try)22 b(of)e(the)i(Space)g(Sh)n (uttle.)30 b Fq(The)18 b(follo)o(wing)f(mo)q(del)h(pro)o(vides)f(a)i (reasonably)e(accu-)0 2179 y(rate)j(mo)q(del)f(for)h(a)f Fp(lifting)i(b)m(o)m(dy)p Fq(,)i(suc)o(h)18 b(as)i(the)g(space)f(sh)o (uttle)g(on)h(re-en)o(try)e(in)o(to)i(the)g(atmosphere,)0 2239 y(with)g(a)f(closed)h(form)f(solution.)30 b(This)19 b(remark)m(able)g(example)g(should)f(b)q(e)i(m)o(uc)o(h)e(b)q(etter)j (kno)o(wn)e(to)0 2299 y(mathematicians)c(and)h(can)g(easily)g(b)q(e)h (presen)o(ted)e(in)i(a)f(\014rst)g(course)g(on)g(di\013eren)o(tial)f (equations.)60 2359 y(During)c(m)o(uc)o(h)f(of)j(the)f(time)g(during)f (the)h(space)g(sh)o(uttle's)f(re-en)o(try)l(,)h(its)g(v)o(elo)q(cit)o (y)h(is)f(appro)o(ximately)0 2419 y(p)q(erp)q(endicular)h(to)j(a)f (line)g(connecting)g(the)g(sh)o(uttle)g(to)g(the)g(cen)o(ter)g(of)h (the)f(earth,)g(although)f(at)i(some)0 2479 y(instan)o(ts)e(the)h (angle)g(is)f(quite)i(large.)21 b(In)14 b(this)h(mo)q(del)f(w)o(e)h (assume)f(that)h(this)g(is)g(the)g(case)g(for)g(all)f(time.)0 2539 y(It)g(then)g(follo)o(ws)f(from)h(\(10\),)g(using)f Fm(\022)j Fq(=)d(0,)i(that)f(the)g(tangen)o(tial)g(v)o(elo)q(cit)o(y)g 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a(C)850 1684 y Fj(L)879 1646 y Fl(p)p 921 1646 82 2 v 31 x Fm(g)r(r)969 1684 y Fj(E)1008 1643 y Fm(t)f Fq(+)e(arctanh)1262 1573 y Fh(\022)1340 1610 y Fm(V)1369 1617 y Fk(0)p 1304 1632 124 2 v 1304 1646 a Fl(p)p 1346 1646 82 2 v 31 x Fm(g)r(r)1394 1684 y Fj(E)1434 1573 y Fh(\023)o(\023)1515 1643 y Fm(;)1861 1575 y Fq(\(12\))0 1798 y(where)k Fm(V)d Fq(\(0\))k(=)e Fm(V)341 1805 y Fk(0)364 1798 y Fq(.)21 b(Actual)15 b(space)f(sh)o(uttle)g(\015igh)o(t) g(test)h(data)g([5])f(sho)o(w)g(that)h(the)g(v)o(elo)q(cit)o(y)g (predicted)0 1858 y(b)o(y)j(this)f(simple)g(mo)q(del)g(is)g(reasonably) g(accurate)h(ev)o(en)f(though)h(it)g(is)f(based)g(on)h(man)o(y)f (simplifying)0 1918 y(assumptions.)60 1978 y(One)11 b(can)h(use)g (\(12\))h(to)f(estimate)g(the)g(maxim)o(um)e(acceleration)i(exp)q (erienced)g(b)o(y)g(the)g(space)g(sh)o(uttle)0 2038 y(up)q(on)k(re-en)o (try)l(.)0 2161 y Fs(6.)23 b(Pro)r(of)12 b(of)g(the)i(Pythagorean)f (Theorem)f(Using)h(Dimensional)h(Analysis.)30 b Fq(W)l(e)12 b(follo)o(w)f([1,)0 2221 y(p.)21 b(49])15 b(and)g(giv)o(e)g(an)g (insigh)o(tful)f(application)g(of)i(dimensional)d(analysis)h(to)i(pro)o (v)o(e)e(the)i(Pythagorean)0 2281 y(theorem.)60 2341 y(The)i(area)g Fm(A)g Fq(of)h(a)f(righ)o(t)f(triangle)h(is)g (determined)f(b)o(y)h(its)g(h)o(yp)q(oten)o(use)f Fm(c)h Fq(and,)g(for)g(de\014niteness,)0 2401 y(the)j(lesser)e(of)i(the)f (acute)h(angles)f Fm(\036)p Fq(:)29 b Fm(A)21 b Fq(=)f Fm(f)5 b Fq(\()p Fm(c;)j(\036)p Fq(\).)35 b(Since)19 b(the)i(units)f(of)g(area)g(are)h(the)f(square)g(of)0 2461 y(units)c(of)h(length,)g(dimensional)e(analysis)h(giv)o(es)g Fm(A)f Fq(=)f Fm(c)1061 2443 y Fk(2)1083 2461 y Fm(g)r Fq(\()p Fm(\036)p Fq(\).)24 b(The)16 b(altitude)h(p)q(erp)q(endicular)e (to)j(the)0 2521 y(h)o(yp)q(oten)o(use)12 b(\(see)i(Figure)e(3\))h (divides)g(the)g(basic)g(triangle)f(in)o(to)h(t)o(w)o(o)g(righ)o(t)f (triangles)g(that)i(are)f(similar)0 2580 y(to)20 b(it,)h(and)f(whose)f (h)o(yp)q(oten)o(uses)g(are)g(the)i(sides)e Fm(a)h Fq(and)g Fm(b)g Fq(of)g(the)g(original)f(triangle.)32 b(Their)19 b(areas)0 2640 y(are)d Fm(A)120 2647 y Fk(1)157 2640 y Fq(=)e Fm(a)236 2622 y Fk(2)259 2640 y Fm(g)r Fq(\()p Fm(\036)p Fq(\))j(and)f Fm(A)504 2647 y Fk(2)541 2640 y Fq(=)e Fm(b)615 2622 y Fk(2)637 2640 y Fm(g)r Fq(\()p Fm(\036)p Fq(\).)23 b(But)17 b Fm(A)e Fq(=)e Fm(A)1008 2647 y Fk(1)1042 2640 y Fq(+)e Fm(A)1129 2647 y Fk(2)1152 2640 y Fq(,)17 b(and)f(th)o(us)g Fm(c)1412 2622 y Fk(2)1434 2640 y Fm(g)r Fq(\()p Fm(\036)p Fq(\))e(=)g Fm(a)1621 2622 y Fk(2)1643 2640 y Fm(g)r Fq(\()p Fm(\036)p Fq(\))e(+)e Fm(b)1819 2622 y Fk(2)1842 2640 y Fm(g)r Fq(\()p Fm(\036)p Fq(\).)0 2700 y(Hence)17 b Fm(a)174 2682 y Fk(2)208 2700 y Fq(+)11 b Fm(b)279 2682 y Fk(2)315 2700 y Fq(=)j Fm(c)390 2682 y Fk(2)412 2700 y Fq(.)964 2760 y Fo(9)p eop %%Page: 10 10 10 9 bop 599 540 a @beginspecial 0 @llx 0 @lly 273 @urx 196 @ury 1504 @rwi @setspecial %%BeginDocument: triangle.Color-EPSF save 100 dict begin /bdf {bind def} bind def /xdf {exch def} bdf 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffff restore end restore showpage %%EndDocument @endspecial 658 660 a Fr(Figure)18 b(3:)25 b(right)18 b(triangle)1911 769 y Fd(\004)0 889 y Fs(7.)41 b(Conclusion.)29 b Fq(W)l(e)21 b(ha)o(v)o(e)g(discussed)e(mo)q(dels)h(of)h(motion)g(for) g(ob)s(jects)g(with)f Fp(smal)s(l)h Fq(Reynolds)0 948 y(n)o(um)o(b)q(ers)e(\()p Fm(R)k(<)f Fq(1\))g(and)f(for)h Fp(lar)m(ge)g Fq(Reynolds)e(n)o(um)o(b)q(ers)f(\()p Fm(R)k(>)p Fl(\031)f Fq(100\).)38 b(It)22 b(is)f(quite)h(di\016cult)f(to)0 1008 y(mo)q(del)16 b(the)g(motion)f(of)h(most)g(ob)s(jects)g(with)g (Reynolds)f(n)o(um)o(b)q(ers)e(in)j(the)g(in)o(termediate)f(range.)21 b(The)0 1068 y(mo)q(dels)c(w)o(e)h(ha)o(v)o(e)g(discussed)e(are)i (quite)g(p)q(opular)f(with)h(studen)o(ts)f(and)h(impress)e(up)q(on)h (them,)h(early)0 1128 y(in)f(a)h(di\013eren)o(tial)e(equations)h (course,)g(the)h(p)q(o)o(w)o(er)f(of)h(di\013eren)o(tial)e(equations)h (to)h(mo)q(del)g(non-trivial)0 1188 y(ph)o(ysical)f(phenomena.)24 b(W)l(e)18 b(applaud)e(the)i(trend)g(in)f(the)h(new)g(generation)f(of)h (calculus)f(and)g(di\013er-)0 1247 y(en)o(tial)f(equations)h(texts)g 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b(Douris,)e(George)h(Gallo)o(w)o(a)o(y)l(,)f(P)o (aul)h(Koning,)f(Mic)o(hael)0 1785 y(Mathews,)g(Colleen)g(McGrath,)g (Dan)g(P)o(o)o(yn)o(ter,)f(and)h(Jac)o(k)g(Rumple.)954 2760 y Fo(10)p eop %%Page: 11 11 11 10 bop 830 50 a Fr(References)0 130 y Fo([1])42 b(G.)15 b(Baren)o(blatt,)f Fc(Sc)n(aling,)i(Self-Similarity,)h(and)f(Interme)n (diate)f(Asymptotics)p Fo(,)h(Cam)o(bridge)c(Univ)o(ersit)o(y)i(Press,) 87 179 y(1996.)0 229 y([2])42 b(G.)14 b(Birkho\013,)f Fc(Hydr)n(o)n(dynamics)p Fo(,)g(Princeton)f(Univ)o(ersit)o(y)h(Press,)f (1960.)0 279 y([3])42 b(J.)16 b(Bo)o(yling,)h Fc(A)g(Short)h(Pr)n(o)n (of)f(of)g(the)h(Pi)f(The)n(or)n(em)f(of)i(Dimensional)g(A)o(nalysis)p Fo(,)f(Z.)g(Angew.)e(Math.)h(Ph)o(ys.)g Fb(30)87 329 y Fo(\(1979\),)g(531{533.)0 379 y([4])42 b(P)l(.)14 b(Brancazio,)f Fc(Sp)n(ort)i(Scienc)n(e)p Fo(,)e(Simon)f(and)h(Sh)o(uster,)f(1984.)0 428 y([5])42 b(H.)20 b(Compton,)h(et)g(al.,)h Fc(Shuttle)f(\(STS-1\))i (Entry)e(T)m(r)n(aje)n(ctory)f(R)n(e)n(c)n(onstruction)p Fo(,)i(AIAA)e(P)o(ap)q(er)f(No.)i(81-2459)87 478 y(\(1981\).)0 528 y([6])42 b(M.)17 b(Denn)o(y,)h Fc(A)o(ir)g(and)h(Water:)26 b(The)18 b(Biolo)n(gy)f(and)i(Physics)g(of)f(Life's)h(Me)n(dia)p Fo(,)f(Princeton)e(Univ)o(ersit)o(y)h(Press,)87 578 y(1993.)0 628 y([7])42 b(W.)14 b(L.)h(Hank)o(ey,)e Fc(R)n(e-Entry)j(A)n(er)n(o)n (dynamics)p Fo(,)c(IAA)i(Press,)e(1988.)0 677 y([8])42 b(S.)15 b(Ho)q(erner,)f Fc(Fluid-Dynamic)j(Dr)n(ag)p Fo(,)e(2nd)g(edition,)g(published)e(b)o(y)i(author,)g(1958,)i(a)o(v)n (ailable)d(b)o(y)i(calling)f(\(206\))87 727 y(576-3997.)0 777 y([9])42 b(D.)14 b(Hughes-Hallet,)e(A.)i(Gleason,)f(et)i(al.,)e Fc(Calculus)p Fo(,)g(Wiley,)h(1994.)0 827 y([10])21 b(T.)13 b(Knac)o(k)o(e,)h Fc(The)g(Par)n(achute)g(R)n(e)n(c)n(overy)g(Systems)i (Design)f(Manual)p Fo(,)f(P)o(ara)g(Publishin)o(g,)d(1992.)0 877 y([11])21 b(L.)14 b(Landau)g(and)f(E.)h(Lifshitz,)f Fc(Fluid)i(Me)n(chanics)p Fo(,)f(2nd)f(edition,)g(Butterw)o(orth)g(and) g(Heinemann,)e(1987.)0 927 y([12])21 b(B.)14 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Fo(\(1957\),)g(237{263.)0 1325 y([18])21 b(R.)14 b(S.)f(Shev)o(ell,)g Fc(F)m(undamentals)i(of)g(Flight)p Fo(,)e(Pren)o(tice-Hall,)f(1983.)0 1375 y([19])21 b(P)l(.)14 b(A.)g(Thompson,)d Fc(Compr)n(essible)k (Fluid)g(Dynamics)p Fo(,)f(McGra)o(w-Hill,)e(1972.)0 1425 y([20])21 b(S.)14 b(V)l(ogel,)g Fc(Life)h(in)h(Moving)f(Fluids:)20 b(The)15 b(Physic)n(al)g(Biolo)n(gy)f(of)i(Flow)p Fo(,)e(second)f (edition,)g(Princeton)f(Univ)o(ersit)o(y)87 1474 y(Press,)g(1994.)0 1524 y([21])21 b(R.)14 b(v)o(on)g(Mises)f(and)g(K.)h(O.)g(F)l(redric)o (hs,)d Fc(Fluid)k(Dynamics)p Fo(,)f(Springer-V)l(erlag,)d(1971.)0 1574 y([22])21 b(F.)14 b(M.)f(White,)h Fc(Visc)n(ous)i(Fluid)f(Flow)p Fo(,)f(2nd)f(edition,)g(McGra)o(w-Hill,)f(1991.)954 2760 y(11)p eop %%Page: 12 12 12 11 bop 546 50 a Fs(Biographic)19 b(sk)n(etc)n(h)i(of)d(Lyle)j(N.)e (Long)60 169 y Fq(Lyle)k(N.)g(Long)g(receiv)o(ed)f(his)g(D.Sc.)41 b(from)22 b(George)h(W)l(ashington)e(Univ)o(ersit)o(y)h(under)g(the)h (su-)0 229 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b(lnl@psu.psu)570 b(Email:)21 b(weiss@ma)m(th.psu.edu) 954 2760 y Fo(12)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9810301006300--