UFDC Home  Search all Groups  World Studies  Federal Depository Libraries of Florida & the Caribbean   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
kAwAT0 13 4
,'O 7' / , NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1342 SPIRAL MOTIONS OF VISCOUS FLUIDS* By Georg Hamel INTRODUCTION The equations for the plane volume are, after elimination of stream function by which the v = x 3y motion of viscous fluids of constant the pressure and introduction of the velocity components v = y 6x are expressed, reduced to the one equation t + 6 a y 3 o A Ax at ox 6y ay 6x therein a cific mass indicates the ratio between viscosity coefficient and spe p, and A signifies the Laplace operator. This equation is satisfied by all potential motions L0 = 0 however, this fact is of little significance since viscous fluids adhere to solid walls and, from wellknown considerations of function theory, there cannot exist a potential motion which would do so. Otherwise, properly speaking, only Poiseuille's laminar motion is known as exact solution of equation (I) and that solution does not even show the sig nificance of the quadratic terms because they identically disappear there. "Spiralf6rmige Bewegungen zaher Flissigkeiten." Jahresber. d. deutschen Math. Ver. 25, 1917, pp. 3460. NACA TM 1342 Under these circumstances it seems perhaps useful to know a few more exact solutions of equation (I) for which the quadratic terms do not disappear; such solutions will be indicated below according to two methods. In both cases, one deals with motions in spiralshaped streamlines (which are observed frequently). Third, we shall, in addition, investigate the neighborhood solu tions to pure radial flow. FIRST PART We raise the question: Are there solutions of equation (I) which are not potential motions for which, however, the stream paths are the same as for a potential motion whereas the velocity distribution is to be different? We shall be able to indicate such solutions, in fact all of them: the streamlines are logarithmic spirals (including concentric circles and pure radial flow); for the velocity distribution, one arrives at an ordinary differential equation which for pure radial flow leads to elliptic functions. In the discussion, the influence of the quadratic terms becomes manifest in a considerable difference between inflow and outflow (see paragraphs 7, 8, and 9). We require, therefore, solutions s of equation (I) for which = f(,= ) and AQp = 0, but not A4 = 0. The latter condition excludes f' '( ( ) = 0 We limit ourselves to steady motions = 0. at 1. The calculation becomes clearer if first the auxiliary problem has been solved: NACA TM 1342 Transformation of equation (I) into isometric coordinates, that is, such curvilinear coordinates c,X that P + ix:' = w(x + iy) = w(z) Let us thus assume * = (Cp, x.) &x = 6x 6y If one denotes aviation 2 )+ 2 692 6/2 by A'*, there results first, with the abbre dw2 dz AWi = a' With the double integral extended over an arbitrary region, one has 6 A o A a A J x 6y yx fj (A') 5 JJ [_ oc 57/ .)dx dy = i dA dl drP dX a(Q A') ox Since, however, xp 6x ox oy Sa ax oy ox dw =2 dzi 8x ate a' ) a~  6Xdx dy a5CPJx y NACA TM 1342 there follows A Y = Q2 by bx 3P 6ai ca' O'ai' 6> 6c +A(4 n n (6i 1nT 8 In Qa 64 6x P 2 dw SIn = 2 R n 2R d In dw = 2R dz2 2P p dz dw dz (dw\2 \dzJ and 6 In q = 2 1R In dw 6x 6x dz S2 3 J In dw = 2J d In dW = 2J 8P dz dw dz are valid. If one puts the analytic function of 2 dw 2 d a + bi (dw\2 dz one obtains 6 A 6t 6x 6y 6 A 3x 6y ax  dw dz (C 6 6: 6x 6cp)JI + A'(a 6 A 6~ 6x 5y However, d2 dz (dw2 dz  + oX NACA TM 1342 Finally, there results A At = Q A'(Q A'') = Q2' A'* + Q A'Q A'' + 2 Q A = Q2A A'* A+  + 2 a p a 'b)j LA I ( 6x5\ ^ + A'* A ix dx/ In Q = In dw 2 dz is a harmonic function, thus A' In Q = O hence, AL'Q = ( In Q 2 (+ In = 2 + b2 Q \ )p + x j Thus one obtains as the result of the conversion isometric coordinates C,X for steady motion Am dx 0'p ax  A'' 6* +A't a +b b  ax YP ax W/] of equation (I) to = A' A'' + A'+(a2 t b2) + 2( A, a ( fYP 3 A' b) X Y, (II) therein, a + bi is the analytic function d2 dw 2 dz (d/)2 \dz/ (w = v2 + iX, z = x + ly) and A' denotes the operator 2 a3f 62 6C'2 6 NACA TM 1342 2. We return to the question on page 2: must be a mere function of 9> 4 = f(cP) If derivatives, with respect to q, are denoted by primes, equation (II) becomes f"f'b = a f + f''(a2 + b2) + 2f'''a (III) f may depend only on cP but must not depend on X. This is certainly possible if a and b do not depend on X, thus, since a + bi is an analytic function of CP + Xi, do not depend on 9 either, if a + bi is, therefore d2w a + bi = 2 dz C dz that is, constant. We shall see later (paragraph 3) that this is the only possibility. From a and b being constant, there follows w = 2 in (z o) + wo a + bi thus, after introduction of the polar coordinates z z0 = redi S= 2 2 (a In r + b)) + CP a + b NACA TM 1342 Thus, the streamlines q = const are identical with the logarithmic spirals a In r + bO = const a = 0 signifies pure radial flow, b = 0 flow in concentric circles. The velocity distribution, however, is given by equation (III): the radial component is 3l 2b 1 r 6 r 63 a2 + b2 r the circular component c P 2a f, r Tr 3r a2 + b2 r a tb r consequently 2 f' I the magnitude of the a2 + b r fore, f' must disappear on solid walls. velocity. There Without restriction of the generality, one may presuppose left hand spirals so that r increases with 6, thus a and b have dif ferent signs; since, furthermore, (4 + ix) is an analytical function just as T + iK., and equation (III) is actually invariant with respect to a simultaneous signI change of iP,a,b, one may presuppose a 0 b 0 Therefore, positive velocity components 3 and  r 6be r f' >0 signify for outflow, in contrast for f' <0 inflow. iTranslator's note: The original says "time change," obviously a misprint. NACA TM 1342 Since 9' may be replaced by cT, one may in addition impose a con dition on the constants a and b. 3. We now want to conduct the proof that on the basis of our require ments a and b must be constant, that therefore the flows in loga rithmic spirals are the only ones the flow patterns of which correspond to a potential motion without themselves being a potential motion. If a and b were not constant, the analytical function a + bi would produce a conformal transformation of the mP + iXplane; by virtue of equation (III) which with the abbreviations A f f I B f' 20 fIV f , (f' = 0 is excluded) may also be written a2 + b2 2A(C)a 2B(q)b + C(P) = 0 (III') the circles (equation (III')) would correspond to the straight lines ( = const in this transformation. These circles would therefore have to form an isometric curve family. However, if the family of curves g(a,b,T) = 0 (III') is to be an isometric one so that AP = 0, the function the equation g must satisfy ,gg.,2 23g (ga + ,b) pc a2 + 2) = 0 (IV) a2 2 oa db /2 and this equation must either be identically satisfied, or be a conse quence of equation (III'). NACA TM 1342 One has ga = 2(a A), gb = 2(b B), Ag = 4, ga = 2A', gqb = 2B' gq = 2A'a 2B'b + C', g = 2A''a 2B''b + C" ga2 gb2 = 4(a A)2 + 4(b B)2 = 4(A2 + B2 C) Thus, equation (IV) is quadratic in a and b; an easy calculation shows the result that the quadratic terms are automatically eliminated. There fore, the coefficients of the two terms must be zero whence follow three conditions 0 A'' C' 2AA' 2BB' _B'' C' 2AA' 2BB' A' 2C 9 B' 2 2 C A B2 C A B C'' C' 2AA' 2BB' C' C A2 B2 Hence, there follows further that A', B' furthermore that B' C A2 B2 must be constant. The final condition yields C = alB + 0 , C', must be proportional and A = y 1B 8 or with l 2o 71  7 2o fIV = af'f" + Bf'' f"' = yf'f'' + 8f" and NACA TM 1342 which, integrated, yields f"l =1 af'2 + Of' + e 2 and f"' = I yf.2 + 6f' + 2 Comparison of the two values for f''' results in 1 af'2 + Of' + f"f = 2 yf' + b which must be son requires identical with the preceding value of f'. 7 =0 a=O The compari e = r7I thus C =A2 = 2 constant and f" = 5f' + T. The second condition B B' const A2 + B2 C B2 however would yield S= const and this together with f' = 5f' + f'2 would result in the contradiction f' = const therewith, the proof has been produced. 4. We now turn to the determination of the velocity according to differential equation (III) which may be integrated once and assumes, after introduction of the quantity proportional to the velocity at unit distance u = f'(P) 0 = 62 NACA TM 1342 the form u' + 2au' + u a2 + b2) b u2 + C = 0 2a This equation is identical with a damped oscillation which takes place under the influence of the potential 3 + a2 + b2)u2 + Cu 60 2 We start with the limiting cases: 1. The streamlines are concentric circles: b = 0. Then u C + eam(A + BT) a2 and, because of S= 2 In r a u = const + r2(A + B1 In r) whereby, the velocity distribution 2 u ar is given. The exact solution of Conette's case is also contained therein: the three constants here are determined from the two limiting values of the velocity and from the fact that in case of a full turn around the circular annulus, the pressure must revert to its initial value. An easy calculation yields B1 = 0 and thus v = r2 r12) (More details on the determination of the pressure are seen in para graph 10.) NACA TM 1342 2. The flow is purely radial: a = 0. The differential equation reads u" = b2u b u2 + C = 0 2o and leads to elliptic functions u' = j u 3 + 3obu2 + const u + const = el u 2 u)e u) where the three e's are only subject to the one condition el + e2 t e3 = 3ab but otherwise are still at disposal. Since, according to the remark on page 8 one relation between a,b is still unused, it will be expedient to put b = 2 so that one obtains, according to page 7 Cj = 5 Then the conditional equation for the e reads el + e2 + e2 = 60 and one has U' = el u)(e2 u)(e3 u) NACA TM 1342 thus u = 20 +P 0. O0) g2,; where SOg2,g3 are the three integration constants. For the pressure (see paragraph 10) there results the equation m.p + 1 v2 = ff f 2 f 1 (1 f2 2ofT 6p\u 2 r ,2 r2 f' r2\2 its uniqueness is a priori ensured, thus does not determine here any of the constants. Discussion of the Padial Flow 5. The condition e + e + e = 60 (1) 1 2 j requires at least one e to have a negatively real part, for instance R(el) R(e2) ? R(e) then with the equality sign being same real part. Furthermore, since this (a) for three real e's m R(e3) 20 valid only when all three e's have the part is real, there must apply either < u 5 e < 2o e2 $ u 5 el 14 NACA TM 1342 (b) for one real e om u e e where, however, this e may be positive. Furthermore, two possible types of flow must be distinguished: 1. Either there are no solid walls, thus a source or sink in an unlimited fluid. Then u must be a periodic function of P, with a period which is an integral part of 2n. u = w is excluded, u = 0 need not occur. Therefore, this case can occur only for three real e's, and e2 5 u < el must be valid. 2. Or there are two solid walls, for instance for b = 0  = d1 (which may also be equal 2x); then at these walls u u = 0. and for must be (a) In case of three real e's there must be, additionally e2 < e >0 and either e2 u < 0 0 < u < e 1 (b) In the case of one real e, this e must be positive and 0 < u e One remembers, furthermore, that according to page 7, paragraph 2, u > O signifies outflow, u < 0 signifies inflow; so that one has inflow in the case of 2(a)(a), and outflow in the case of 2(a)(0), and 2(b) above. For the case 1, both cases may occur. NACA TM 1342 First Case: Free Flow 6. One must assume a = 0 for u = e2 and has therefore e ) e2)(u e3) Hence, there must with n bein an 2 with n being an int (e u)(u e2)(u By the known substitution u = e2 + (el e2) sin2 equation (2) becomes 2 e2 e3 d+ i + 2sin22 32 n 3a n If one now introduces the mean velocity2 u = (e 1+ e2) and the velocity fluctuation2 S= el e2 2At the distance  e3) V3a n 2 el e2 e2 e3  2 r = 1. NACA TM 1342 there becomes because of equation (1) e2 e = 6 +3u &>0 2 3 m 2 thus 60 + 3um 2 8 2 d2 1 t at 0 1 +X2sin2* 2 n From this, one may draw several interesting conclusions. One has 1+ X 2sin2 r I 0 dJ + \1 +X2sin2* \Il i2(1 + cos 2*) 2j ( J 10 4 d1 1 + X2cos2 + >2 4 0 d2 l+1 02 0 6a + 2u 2 5 \Fo NACA T 1342 thus 2f o It O0 1+ X2sin 2 where 0 < e 1 Thus the relation (2') between of x2 Urmb,n,a reads, due to the significance 2 1 n 6o 6a + 3u (1 ) m 2 1 n2 60 + 3um 1 25 = 60  with j being a proper fraction. Since, furthermore (2") 12 % ds > O \1 + X2sin2 I2 0 X d* I\f 1 X22 = sinh X 2 thus becomes arbitrarily large with increasing X, one has lim E = 0, thus, liam = 1 U=mO 1 1 2 2 NACA TM 1342 From equation (2'') there follows u > 2a ) which, with u = 1, gives as the minimum value 3 The mean inflow velocity is therefore considerably limited upward, the more so, the easier movable the fluid. However, this is the only restriction: If u are selected so that 2 60 + 3um > 60 " 4 there exists, certainly, a pertaining 8. For if I 8 increases from zero to the value 2 and the integer 6a + 3um, 1 12g 2 lies between zero and 60 + 3um (because for the second value 2 9 1 2 becomes infinite and, hence, n" = 1) so that certainly sometime 2 25 2 2 becomes equal to 6a + 3u 6a which is presupposed to be positive. 24 One sees, furthermore, that for a prescribed fluctuation 8 and for a prescribed fluctuation must increase to infinity with the mean velocity period number n the 5 the period number n um. Second Case: Outflow Between Solid Walls 7. The cases 2(a)(p), and 2(b) may be summarized thus du \/(e u)(u2 + 2u + u o~~? NACA TM 1342 19 e > 0 is the maximum velocity (at the distance r = 1); because of 2a = e e = 60 + e and p = e2e3 > 0, otherwise, however, arbitrary u2 + 2au + 0 may for prescribed e assume all values from u2 + 2ou to o, so that 1 = 2 du 0 j(e u)(u2 + 2au + ) appears not at all restricted downward, but upward restricted by e 1,max 21 _d v 0 V(e u)u(u + e + 60) since e dn 0 (e u)u one has 3ma = 21n/ 3' (4) l,max V2e(l e) + 120 where E signifies a positive proper fraction. For the outflow, the width of the wall opening appears therefore restricted, according to the preceding equation, by the maximum value e of the velocity. For small velocity and large viscosity, the maximum lies near n, otherwise, however, lower; with increasing e it drops below all limits. NACA TM 1342 If, therefore, an angle opening smaller than n is prescribed, it permits an outflow only up to a certain maximum value. If a greater outflow quantity is prescribed, the jet will, therefore, actually prob ably separate from the walls. Also, there is, of course, for any prescribed angle 01 a flow possible where partly inflow partly outflow occurs. Third Case: Inflow Between Solid Walls 8. There remains the case 2(a)(a) e u O0 2 all three roots e real, e 3, e2 negative, el positive. Here l iP e2 0 e2 c ^e0 du u e)(u e)(el u) du  2au ) where 2a =(e + e3) = 60 + e p = ee < 1 3 otherwise, however, arbitrary. Thus, the angle 01 may be made arbi trarily small for prescribed e2. On the other hand, however, it may also be made arbitrarily large: one takes, for prescribed e2, the  e2)(u2 NACA TM 1342 negative value e3 sufficiently close to e2, as far as this is not made impossible by el < 0. The sole relation between the e el + e2 + e3 = 6a however, results with el > 0 in e2 e > 60 If e2 ? 3a, e3 may actually be assumed arbitrarily close to e2. If the maximum inflow velocity is larger than 3a, any angle 61 is possible between the solid walls. If, however, e2 < 3c, say e3a, where E is a positive proper fraction, only e3 = e + (6 3E)o = e2 + el + 6(1 c)o is possible and 0 61 fe2 + (6 3e)O + e l (e u) attains its highest value for el = 0 \j r 1 ,max (u + 3Eo)u + (6  3e)O](u) \ 3E(2 + rr 1 ( + TI) I u e2 u NACA TM 1342 where I and 5 are positive proper fractions. Thus, the maximum of lI is larger than n. When the maximum inflow velocity is smaller than 3a, the angle openings of the solid walls also may attain any magnitude up to n. Flow in Spirals 9. Because of the damping 2au' (see paragraph 4, page 11), a periodic solution, aside from u = const, is not possible. A free motion in logarithmic spirals is always a potential motion. In contrast, there exist other flows on logarithmic spirals between solid walls. In order angle , one a manner that to have, for r = const, the variable q agree with the may furthermore prescribe for the constants a,b, in such one obtains 2b  2 b2 a + b thus b = 1 + \ a2 a must be a proper fraction, otherwise it remains arbitrary. Equation III, once integrated, (see page 10) then reads 2 u' + 2au' + 02u + C = 0 4a 2 2. where 0 = 2b = 2 T2 21 a < 4, but > a NACA TM 1342 The velocity at unit distance is 2 \fa2 + b2 2 u ia2 1+7 1 ia u is, therefore, the velocity at the distance r _2 _2 If one first omits the damping, one has exactly the same case one had before except that \6o instead of is in front of the square root (see page 12). The relation for the e remains the former one. Since P2 < 4, the angle opening is increased by this influence 91. The damping, however, takes effect in the same sense. Nevertheless, the main result remains correct. For outflow the admissible angle opening ia is restricted by the maximum flow velocity in such a manner that it tends toward zero when this velocity increases beyond all limits. If one puts u = vea the above differential equation becomes v" +(p2 a 2)v + ve + Cea =0 S' 4a NACA TI 1342 If cp = 0 is assumed to be the location of the maximum v0 for v, multiplication by 2v' and integration yields v2 + (32 a)(v2 v02) 2 v +  0 2 amP i am veadv + 2C eadv = 0 0 From the corresponding equation for u u'2 + 4a u u' du + u0 2 2 2) (u3 u03) + 2C(u uO) = 0 one can see that for equal u0 the ucurve will be the steeper, l therefore the smaller, the larger C. For value close to u0 this is immediately clear from the differential equation, for in case of u' = 0, u'' will be the smaller, the larger C, thus lu'l the larger; from the preceding equation one may see, however, that for larger C U u,2 + 4a u u"0 u u' du Ju has the higher value. Hence, follows directly for the (u' > 0) that always Iu'I has the higher value when value. For if u' would once reach for the initially (C smaller) the value of the steeper curve, ascending branch C is the larger flatter curve u u' du would have to u0 SuO have for the former the smaller, thus u' du the higher value u which for u' > 0 immediately leads to contradiction since, up to then, that is between u and uO, u' had been the smaller value. If, how ever, u' < 0, one has in case of a variation of the C by bC u u 4a 1 f u' du = 2 AC UO u u u NACA TM 1342 or, since A',2 = Alu'l(21u'l + Alu' I) lu'l (21u'l + Alu') + UO Alu'l dn+2 AC UO u u0 uJ If there were at one point A u'I = 0, then at this point the first term would, for fixed AC, decrease with decreasing u, that is, go over from positive to negative values; the second term also would decrease since the part of the integral I Au' du supervening with decreasing u would be negative. This is impossible, however, since the sum of both terms is supposed to be constant 2 AC. Therewith, it has been generally proved that the angle opening al decreases with increasing C (for fixed u0); since the maximum possible 9l is desired, the minimum admissible value for C may be assumed. This value is determined from v'2 > 0 PO aP> LA2 2V 2 2\ 2 vYO 2 am 2C e dv a 2v2 v2) v ye dv V V whence, one may see that the minimum admissible value of C is zero or negative3. The above inequality must be valid for all v's between v0 and zero and for the positive and negative m attained. One may write them C 2 2) m )e 2 a(2 + 9 O) 20 > Q3 a) v0+ v)eaO v02 +o v )e + V 6a o According to page 22, 02 a2 > 0. NACA TM 1342 where p0,cp2 are certain mean values. One must further note that, for v = VO, I01 and IT2 I must be zero whereas they have maximum values for v = 0. The severest restriction is due to the absolutely smaller value of the right side, thus the minimum admissible C is given by 2C=_(2 a2 voeaP O 12 a(P 2'+c0') 20C p a2)ve 60 e CPO'C 2' are positive and the maximum of c0 and 2 which occur for v = 0. Consequently, one has for the maximum possible l v'2 = (2 a2) v2 v2 eacPO' T eacdv + 2a v 3 e (P2 a2) 0v2 v) v 0 v)e a 'P0 o2 3 v3 aT 2 1 2 a cP 2af ( ' 10 2 3 3) 2 v 1 v02o 0 v)e z Oj Since CPQ' > cp and P2' > P2 v'2 >V v) v k a( + (v )e ] NACA TM 1342 and since cP2 < 91 v > ( v)v 2 a2)+ (v + V)ea whence follows 23t 91 < 2 2 a2+ v0(1 + e)eaB1 (Compare formula (4), page 19). Hence follows that with increasing vo, thus also with increasing uo, 1a must drop below all limits: a certain width of the spiral permits only a limited outflow velocity. SECOND PART 10. Since the only spiral motion, possible without walls, of the type used so far, lead to be a potential motion, exact steady and non steady twodimensional motions in free spirals will be investigated according to another method. In polar coordinates the differential equation (I) reads SA 15A+ .\ = a A A at r\ r 6(P 6r) where lA 1 82 A = r + 2 r or or r2 cP2 28 NACA TM 1342 Obviously this equation permits solutions which are linear in I? S= u + cpx in order to make the velocity which has the components v = r r and v =u Cp 6r  c 6r unique and thus enable a free motion, C must statement, the differential equation becomes SAu )t be constant. +X KAu = aA A r or with u= 1 = ru r 6r 6r Here it is necessary also in a motion about the singular pressure becomes evident. to investigate the pressure lest perhaps point r = 0 a multivaluedness of the Now one may write the equations of motion the pressure in the following form v 2 atA  = At d* + dx  ;by without elimination of xA L 6x or because of the invariance of the last term 3 aA q = A* d* + dr  r 2P By this "(a A 6 *^ Ardr rr NACA TM 1342 Hence, follows + 1 v = At r L a t = X Au ar 6 + r bp 2 6/ cP r\ ot/ (r By virtue of the differential equation for u the right side stant; thus it must be zero to make the pressure in case of a about r = 0 revert to its former value, so that one obtains 2 r Au r 2u Au = 0 6r a 6r 6t a which by introduction of r 6u = v assumes the form 6r  1 + 62u at 6r is con revolution a r _r ar 6r Steady Motions 11. The solution independent of t is 2+2 u = clr when '> 2, otherwise, when a + c2 In r + c3 2 3 a=  a u = c1(in r)2 + c2 In r + c 30 NACA IM 1342 If one disregards the trivial case of potential motion, a spiral motion the velocity of which disappears at infinity exists when A + 1 < 0 a that is, X < a, thus a sufficiently strong inflow takes place. The spirals then have the form ++2 1P u = Clr + C2 In r + C 2 3 If 2 + 2 < 0, they approach at infinity the logarithmic spirals; near the sink, in contrast, they converge considerably less pronouncedly toward the sink point, and the vortex velocity is considerably higher than in case of potential flow in logarithmic spirals. Unsteady Motions 12. If one uses the formulation v = entXn(r) one obtains from equation (V) X 1 ++ X ' X = 0 1n r oan n a X = thus, with the abbreviation X = 1 + 2 2o n +=r (J r NACA TM 1342 where the J are the Bessel functions r) = const rX + n + a 4 1(1 + x) n2 4 a ) 1 2'(l + X)(2 + x) If X does not happen to be an integer, rXJ and rXJ_ regarded as independent solutions. may be 13. Similarly to the case of the heat conduction equation there exist also of equation (V) integrals which show for r = 0 and t = 0 an indeterminate point. Since the differential equation (V) remains unchanged if v is multiplied by an arbitrary factor, r by a similar factor, t by its square, there must exist solutions of the form v = ract Ow = rat w(z) After substitution, one obtains for w the, differential equation w" + w1 + 1 w 2 + (VI) = 0 When does this equation permit a solution of the form w = epz A simple calculation yields P = 1 NACA !M 1342 and then either a =0 or a = 2x 3 = x 1 One thus has two simple integrals of the required type 1 r v = tXle and v = r2ktlke 1 r2 4a t for X = 0, both are transformed into the known integral of the heat conduction equation. Let us continue the discussion of the differential equation (VI). The singular point z = 0 is a determinate point. The determining equation reads p2 + p(a ) + a2 2xa 0 4 and has the roots P = p a = 1 2 2 2 NACA TM 1342 so that generally there exist developments of the form XaM w = z 2l + c z + c2 z aC w = z 2(1 thus v = r2kt 1 2 1 +c1, z c z2 + + c 'z + c2'Z2 + . 2 4 + c r + c r + 14wt 2 2 (4ht) and B+ 2 4 v = t 2 + c, r + C, r + 2 1 4at 2 (42t) with the power series continuously converging since singular point of the differential equation. z = 0 is the only If one assumes p = 0, that is, if one desires solution of equa tion (V) of the form ral I r2 an integration by definite integrals is possible. The differential equation (VI) reads after introduction of the roots p ,p 12 z2 d2 + z dl p + z + wpp = 0 dz2 dz\ 1l 2 12 dz (VI') 34 NACA TM 1342 The connection with Gauss' equation for the hypergeometric function can be easily recognized. If one makes the Euler transformation W = e3(1 ) y(s) ds with the integral extended over a suitable closed path, one finds for y a differential equation which may be satisfied for n = pl by y = +p2 by y=s and for n = p by y l+ by y = s Therefore w = e3fl  v = e3(l '' M1 ~( IP1 1+p Ss 2ds SP 1+p )s Ids z/ are integral of equation (VI'). The integrals are extended best over a path which leads from R(s) = +o around the points s = 0 and a = z back to R(s) = +0. Since fe3(z )Plsl+Pd and NACA TM 1342 is analytically regular in the neighborhood of z = 0, there is wl = C1 e31 2 = C2 e( P1 l+p  s)P s +p2ds S P2 sl+pds One can show that v=oo raC 1w 1 + C2 w 2 [ lklWV c tw are the general solutions of equation (V) and likewise are represented by definite integrals in closed form. I shall perhaps refer back to this and to the connection with the representation and the development in terms of Bessel functions elsewhere. THIRD PART Neighborhood Solutions to Radial Flow 14. We shall first look for steady neighborhood solutions to the radial flow (pages 12 and 13) by putting = = f(m) + p(I,r) where p is assumed to be a small quantity, the square of which is neglected. We then obtain for f the former equation f(CV) + 4f' + 2 f'f = 0 NACA TM 1342 with f' = u and integrating once '' + 4u + 1u2 + C = 0 o For p we obtain SAp + u Ap 2u' 6P at r Sr 4 kp r u" ap a A Ap r3 or a I8 r r r r r 6r 6r Since the differential equation in when r and p each are multiplied by exist solutions of the form the steady case remains unchanged an arbitrary factor, there must p = r'w(') One obtains for w the differential equation wIV + w' 2X2 4 + 4 + Zu u) + 2u'w' + a a a w( k43 + 2X2 +22 u + u" = 0 o a We are particularly interested in free flows and thus in tions in ''. periodic solu As concerns the uniqueness of the pressure (see paragraph 10, page 28), one obtains by a simple calculation, the condition 2 02 w[u ( 2) ]dCP = 0 where + 1 2 r2 acp2 (VII) NACA TM 1342 On the other hand, there follows from the above differential equation itself, by integration over the interval from 0 to 2x with assump tion of periodicity .r2x X2(A 2) w[u (A 2)o dc = 0 '0 so that in general the uniqueness of the pressure follows from the peri odicity except for the case when X = 2. For'free flow, u is an integral w part have the same period as part of 2n. itself is a periodic function of q'; the period of 2A. However, it is not necessary that w u; but this period must likewise be an integral Since u"' = C 4u 1 u2 a as well as u'2 = 2Cu 4u2 2 u3 + D = e u)u e2u e3) may be rationally expressed by u, it will be useful to introduce u instead of 'r as independent variable in equation (VII). Because of w' = dw u du w' = dw u'2 + d u du2 du wi" = d3w u'3 + 3 du3 IV = d4w u'4 + 6 d3w u'2u" + 3 du du3 d2w u'ui + dw u', du2 du d2w u,2 + 4 d2w u'u''' + dw uV S2 2 du du du 38 NACA TM 1342 and because of urV = 4 u)u 2 u,2 u'u' = (4 2 u'2 Sa \ a all coefficients of the new equation are integral and rational in u; indicating the degree, one writes them R6 d + R5 d3w+ Rdw +R dw + R2w = 0 (VII') du du du2 with R 6= 4 = 2 (e u)2(u e2)2(u e )2 9a From the form (VII) one can see that w possesses singularities only where they occur for u, thus certainly not in the real part of 9 (which is of interest); equation (VII') shows that, as a function of u, w becomes singular only at the branch points el,e2,e . Since R5 = 6u'2u is divisible by (el u)(u e2)(u e3), the points el,e2,e3 are determinate points, and since the degree of the coefficients decreases steadily by 1 with the order of the deriva tives, u = m also is a determinate point; the differential equa tion (VII') belongs to the Fuchs class. A wellknown calculation yields as the four roots of the determining equation for the points e the values p = 0 p =1 p =1 p 1 2 3 2 4 2 Although, therefore, two root differences here are integral, no loga rithmics appear in the developments: For from the form (VII) there fol lows that at the points P for which u becomes = e, where, therefore, u and u' are regular functions of P, w also must be such a func tion, whereas In(u e) does not possess this regular character. NACA TM 1342 Therefore, the solutions of equation (VII') have at every point e the form u = Pl(u e) + /u eP2(u e) other singularities do not exist in a finite domain. For u = = there results the determining equation (2p2 + 3) (2 + j + ) =0 which has the roots 1 =1 =_2 which are independent of X, and the roots } = i Wi 41 4 4 which are dependent on X. Continuation 15. Solutions with the real period 2n (this period must be present at least in case of free flow) will exist only for certain X. In analogy with Hermite's method for Laine's differential equation, one can proceed as follows: If wlw 2W3,W4 are a fundamental system of equation (VII), the w(cp + 2x) are expressed homogeneously linearly by the w 4 w (C + 2x) = Z ,w() (v = 1,2,3,4) u=1 40 NACA TM 1342 There certainly exist periodic functions of the second kind, that is, there exist solutions w for which w(cP + 2) = aw(c) This a is a root of the equation of the fourth degree  a a12 13 a22 a a23 a32 a33 a43 a24 a a34 a44 a If a periodic solution is to obtains for X the equation exist, a = 1 must be a root, and one D(1,X) = 0 The characteristic exponents which were calculated suggest the attempts of putting w = u + const, w = \u e and w= ea u)( e Elementary calculation yields the following particular solutions: 1. The trivial possibility w = u for k = 0 2. w = u for k = 2, that is p = ar2u I = f(cp) + a~r2f'(cp) all a21 a31 D(a; X) = = 0 NACA TM 1342 where a must be small and therefore with the same approximation = f(p + ar2) so that the streamlines are approximately the spirals Cp = 'T ar2 Remains the same elliptic function discussed before in the case of f' remains the same elliptic function discussed before in the case of radial flow. It is true that this flow since this is precisely the exceptional the condition for the uniqueness of the satisfied for w = u. now cannot exist as free flow, case X = 2 (see page 37); and pressure can certainly not be 3. w = u t 30, when X = 1 and C = 30 whence for el e2 < CJ3 no contradiction results. 4. w = u e2 for X = 1 and e2 = 0. This solution has a period twice that of u; likewise, w = el u for X = 1 and 5. w = e u)(u e3) or w and e2 = 0 or el = 0. This solution (u e2)(u e) when X = 1 too has a period twice that of u. The large X may be easily calculated approximately from equa tion (VII). For such large X there is in first approximation wIV + 2X2w,, + X4w = 0 that is, w = e1ki' (we restrict ourselves to the periodic solutions), so that is the period. The large setapart kvalues are therefore approximately integral. Finally, one case may be calculated quite elementarily: the case when u is constant, the basic flow therefore an all around uniformly distributed flow. e = 0. 42 NACA TM 1342 This case is also of significance for the more general one since 4 according to a wellknown theorem by Cauchy and Boltzmann the period of w in first approximation is obtained if the constant mean value is inserted for the periodic u, under the presupposition that the larger fluctuation el e2 be sufficiently small. For constant u there follows from equation (VII), page 36 wV + W" (22 4 + 2 u) + 4 + 3 + 4X2 + 2 2 u) = 0 thus with the formulation v = euip Sa a 4 2(2X2 2K + 4 + 2 u) + x4 4X3 + 4X2 + 2 22 K u =0 This equation has four roots P = X p2 = (A 2)2 2 u a so that all integral positive and negative K are possible (potential motions) as well as all K which are calculated from K = 2 + + 4  2 2 Fta) with integral w. For the case u = const, that is: 16. For the radial flow which is uniform all around, the unsteady neighborhood solutions also can be given. Boltzmann, Ges. Abh., Bd. 1, S. 43. NACA TM 1342 The differential equation now reads (see page 36) 8Ap u6p= + o Ap at r 6r One may integrate it either by means of the formulation Ap = e kt+ni w(r) (n integral) and thus arrives at the differential equation 2 d2w dr2 u 1 H G dw + d r dr which may be solved (compare page 32) by Bessel functions, or by means of the formulation Ap = enim r2) aot = eniQrmw(z) whereby one obtains for w(z) the differential equation z2'' + zw'(m + 1  For z = 0 this equation has tion has the real roots u + z + w 2 = 20 4 h a determinate point, the determining equa p mui +1 2 lu2 2 4a 2 4 2 aT n+ w = 0 r2 ci NACA 'M 1342 By introduction of the roots pl and p2 the differential equation assumes the form zw' + zw'l P+ z) + PlP2 =0 This is, however, exactly the differential equation (VI') of page 33 so that everything said about it there is also valid here. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACALangley 11653 1000 C6 r ff > L a G z i Ga . .2 ca Pk 2: z* 0 w* w  CU.ln BZ a .a d CD k C. S a bs c 4 o a; .a4lH A e ani ad adM 0 s .a G a o ..a 4. U 0 0G 'a 0 E t T(~ .~ U p 2~ C.,l .La 4i~ 0 o c5 5 a 0, 0 w 0. W. x zC ba m m v C 4 d c.cI Z00.n r '.4L. l cp CD 0 ed 5' ;. " Ga S GaT S" S < 3 " cu V 0. G S ldc 12 I. l.0 N ,r u, ! S 3 B > E r S06 0 2a f z z s S ;W. aE MMC I .S o I I 0 '.f0Gas'.! oj c a 0 b D OL L.'. a^ ^ ~ E c m 8 0 o g.M . Ma a~ Lm o o 0 G 2U G S'ap>Cs6'S~C1 .0 Ga J Ei ::, a no E L. U 0 w U a a 5 I Ga rA >n EfM o .0 a 0 C6 C 0 G = o G vi 8 Ga 0 Id 0d c a ra o ce G E L.a CL 0n3 0 0 .0 Io u Ua w a 2 cj 2 om ~* I S 0. 0  0 '1  s E I. 0 0 0 , 0 0 o <. S .. 3o 045 0 ,,.pl o s aU igg c.. m E i 0 u. 0 l :1L. o w 0 10 0m I N o ow 9 S I rt s< rt 3 E., 3 3 0 L I 1 P4< N S a.  5 0. S o r wS w 2 u < iSI^ U I z g V 5 Ss z a < w s t^ rl 0 r B 4CNOI wr n E f  . a I 04C I% *t CD Es ~ E~Bfl: o; 00; t t I It UNIVERSITY OF FLORIDA 3 1262 08105 814 0 