Modified Theory of Laminar Flow Around Rigid and Liquid Cylinders

Transcription

1 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) Mdified The f Lamina Flw Aund Rigid and Liquid Clindes FRIEDRICH K. BENRA, and SIAVASH H. SOHRAB Depatment f Engineeing Sciences Univesit f Duisbug-Essen Lthast. GERMANY Rbet McCmick Schl f Engineeing and Applied Science Depatment f Mechanical Engineeing Nthwesten Univesit, Evanstn, Illinis 68 UNITED STATES OF AMERICA Abstact: - The mdified fms f the equatin f mtin and the Helmhltz vticit equatin ae slved f the classical pblem f Stkes flw acss a igid clinde. F such ceeping flws, the mdified equatin f mtin suggests the eistence f a clsed steamline aund the clinde within which the suface-geneated vticit will be cnfined. The slutin within such clsed egin is pesented that ma help in the eslutin f the well knw Stkes paad. The pblems f lamina flw acss and within a clindical liquid bd ae als eamined. F the fme, an appimate analtical slutin is pesented and qualitativel cmpaed with the numeical slutin f the Navie-Stkes equatin. F the latte, paallel t the classical Hill spheical vte, the slutin descibing tw clindical vte lines is pesented. Ke Wds: - The f lamina flw acss a clinde. Clindical vtices. Stkes paad. Intductin The univesalit f tubulent phenmena fm stchastic quantum fields t classical hddnamic fields esulted in ecent intductin f a scaleinvaiant mdel f statistical mechanics and its applicatin t the field f themdnamics [4] and invaiant fms f cnsevatin equatins [5, 6]. In the pesent stud, the mdified fm f the equatin is slved f the classical pblem f Stkes flw acss a igid clinde [7, 8] and the esults ae cmpaed with the numeical slutin f the Navie- Stkes equatin. Als, the pblems f flw acss and within a liquid clinde lcated in a unifm gaseus steam ae investigated. Invaiant Fms f the Cnsevatin Equatins in Chemicall Reactive Fields Fllwing the classical methds [-3], the invaiant definitins f the densit ρ, and the velcit f atm u, element v, and sstem w at the scale ae given as [4, 5] ρ = nm = m fdu, u = v () v =ρ m f d u u, w = v + () The invaiant definitins f the peculia and the diffusin velcities have been intduced as [4] V' = u v, V = v w = V' + (3) Fllwing the classical methds [-3], the scaleinvaiant fms f mass, themal eneg, linea and angula mmentum cnsevatin equatins at scale ae given as [5, 6]

2 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) ρ ε p π ( ρ v ) +. =Ω (4) ( v) +. ε = (5) ( pv) +. =. P (6) ( π v ) +. = invlving the vlumetic densit f themal eneg ε = ρ h, linea mmentum p = ρ v, and angula mmentum π chemical eactin ate, P = ρ h ω and is the patial stess tens []. Als, (7) Ω is the is the abslute enthalp, P = m ( u v )( u v )f du (8) In the deivatin f (6) we have used the definitin f the peculia velcit (3) alng with the identit VV = ( u v )( u v ) = uu v v j (9) i j i i j j i j i The summatin f (6) ve all the species esults in the classical fm f the equatin f mtin [, 3] ρw +.( ρww) =. P wee w = w = v + () is the mass-weighted aveage velcit and the ttal mitue stess tens is [, 3] P= P = m ( u v )( u v )f du () The lcal velcit v tems f the cnvective velcities [5] in (4)-(6) is epessed in w and the diffusive V v = w + V g, Vg = D ln( ρ) (a) v = w + V tg, Vtg = α ln( ε) (b) v = w + V hg, Vhg = ν ln( p) (c) v = w + V hg, Vhg = ν ln( π) (d) whee (V g, V tg, V hg) ae espectivel the diffusive, the them-diffusive, the linea hd-diffusive velcities. Substitutins fm (a)-(d) int (4)-(7), neglecting css-diffusin tems and assuming cnstant tanspt cefficients with Sc = P =, esult in [5, 6] ρ t T t ρ D ρ =Ω +w. (3) T α T = h Ω /(ρc p) +w. (4) v +w. v ν v = p/ρ v Ω/ρ (5) ω ω Ω +w. ω ν ω = ω. w ρ (6) An imptant featue f the mdified equatin f mtin (6) is that it is linea since it invlves a cnvective velcit w that is diffeent fm the lcal fluid velcit v. Als, the last tems f the equatin (5) and (6) epesents a suce (sink) f espectivel linea and angula mmentum that ae induced b ethemic (endthemic) chemical eactins. 3 Mdified Equatin f Mtin f Stead Flw Acss a Clinde F unifm flw past a clinde the nndimensinal stead fms f (3)-(6) in clindical cdinates and in the absence f eactins Ω = educe t v w v wv v w + = v v v v p + + w w w v + v + v = v v v v v p w w (7) (8) ωz ωz ωz ω z + = + (9) v v v + + = ()

3 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) whee (v, v, w ) = (v, v, w )/w, p = p /( ρw ), and w is the unifm steam velcit. Als, the new cdinates (,,) = (,, ) /( ν / w ) elate t the classical nes (,, ) = (,, )/R as (,,) = (,, )Re and the Renlds numbe is R = R /( ν / w ) = Re/ () whee R is the clinde adius. 4 Mdified The f Lamina Flw Inside and Acss a Liquid Clinde The imptant classical pblems f spheical and clinde flws [7, 8] using the invaiant fms f cnsevatin equatins (7)-() wee subjects f ealie investigatins [9-]. F the pblem f lamina flw inside f a liquid clinde in unifm etenal gaseus steam w = cs, w = sin () it was shwn that the eact slutin f the mdified Helmhltz vticit equatin (9) esults in the steam functin [] i = [( / R) ]sin (3) whee = i i / ν. The adial and angula velcit cmpnents within the clinde ae given b (3) as v = [(/R) ]cs (4) v = [3( / R) ]sin (5) On the suface f the liquid clinde the gas velcit will be assumed t be equal t that f the liquid and hence v(r) =, v(r) = sin (6) The velcities (), (4), and (5) ae net substituted int the adial (7) and the angula (8) mmentum cnsevatin equatins t aive at the pessue distibutin within the liquid clinde p p (cs sin R is = c ) p = p (cs sin ) (8) (7) whee p c is the pessue at the cente =. The pessue n the suface f liquid clinde becmes is c Als, the aial vticit assciated with the intenal flw field (4)-(5) is 8 sin ω z = (9) R whee ω =ω ( ν /w ). It is emphasized that the z z slutins (3) and (9) satisf the cmplete mdified fm f the Helmhltz vticit equatin (9) including the cnvective tems. Similal, the slutins (4), (5) and (7) satisf the cmplete mdified fms f the equatins f mtin (7) and (8). The pessue field (8) helps t cnfine the vticit (9) t the intei f the clinde. In tems f the Catesian cdinates = cs, = sin (3) the steam functin (3) will assume the fm = [( + )/ R ] (3) i Sme f the steamlines calculated fm (3) with R = using Mathematica [3] ae shwn in Fig Fig. Steamlines f flw within a liquid clinde in a unifm gaseus steam fm (3). The tw clindical line vtices shwn in Fig. ae analgus t the Hill spheical vtices [4] fmed within a liquid dplet in unifm flw []. It is inteesting t nte that the steamlines f the inne slutin (3) f > R ae qualitativel cnsistent with the epected steamlines f the ute slutin t be discussed in the fllwing. Because the mdified Helmhltz vticit equatin (9) is linea, it is pssible t empl the esult (3) t find the steamlines f flw within tw cncentic clindical liquid bdies that ae made f immiscible fluids. T shw this, the aial vticit is epessed in tems f the steam functin ω z = (3) with the Laplacian peat = + + (33)

4 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) such that the Helmhltz vticit equatin (9) assumes the fm ( ) w. = (34) In view f the lineait f (34), the pducts f the steam functins f flw within cncentic liquid clindes f adii R and R fm (3) = ( / R )sin (35a) = ( / R )sin (35b) esult in a new steam functin 3 = = ( /R )( /R )sin (36) that als satisfies (34) and the bunda cnditins 3(R ) =, 3(R ) = (37) Sme f the steamlines calculated fm (36) epessed in tems f Catesian cdinates + + = ( )( ) (38) 3 R R f cncentic liquid clindes with adii R =.6 and R = ae shwn in Fig Fig. Steamlines f flw within tw cncentic clindes made f immiscible liquids in a unifm gaseus steam fm (38). Net, the pblem f lamina ute flw f a viscus gaseus steam acss a liquid clinde in the Stkes limit Re is eamined. One ntes that the classical pblem f an ideal fluid flwing acss a igid clinde is equivalent t the pblem f a eal fluid flwing acss a liquid clinde because bth flw fields allw f a finite velcit (6) at the suface f the clinde. Hweve, in the fme pblem the finite gas suface velcit is due t the fact that bth gas and liquid mve tgethe with a finite tangential suface velcit while in the latte pblem ne allws f a finite gas slip velcit at the suface f the igid clinde. In a ecent stud [], it was shwn that with the bunda cnditins v = cs, v = sin (39a) = R v = v + sin = (39b) the mdified fm f the Helmhltz vticit equatin (9) leads t the slutin v [ (R/) ]cs = (4) v = [ + (R/) ]sin (4) cespnding t the steam functin f = [ (R / ) ]sin (4) It is nted that n the clinde suface while the adial velcit (4) vanishes, the angula velcit (4) assumes the finite value f v(r) = sin (43) that matches the esult (6) f flw inside the liquid clinde at = R as equied. The steam functin (4) epessed in Catesian cdinates = cs, = sin (44) assumes the fm R f = + (45) Sme f the steamlines f flw utside and inside f a liquid clinde calculated fm (45) and (3) ae shwn in Fig.3. The slutins (4)-(4) ae identical t the classical slutins f the ptential flw f an ideal fluid acss a igid clinde [7]. Theefe, these slutins wee cnsideed t nl satisf the Laplacian pat f the equatin f mtin, in the limit Re, i.e. f Stkes flw. The aial vticit f the etenal flw (4)-(4) vanishes identicall ω = (46) z Theefe, the ute flw (4)-(4) is vticit-fee and hence a ptential flw. It is then clea that the liquid-gas inteface with the velcit (39b) sepaates the vtical inne flw (4)-(5) fm the vticitfee ute flw (4)-(4). Thus, the suface f the liquid clinde, acting as an appaent velcit slip n a igid clinde, pvides the equied bunda cnditin f the eistence f the ute ptential flw.

5 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) Fig.3 Steamlines f liquid clinde in unifm gaseus flw calculated fm (3) and (45). Since the slutins (4)-(4) lead t the vanishing f the Laplacian tems, the equatins f mtin (7)-(8) educe t v w v w v p + = w v w v w v p w + + = (47) (48) B substituting f the velcities fm () and (4)- (4) int (47)-(48), ne aives at the pessue field utside f the liquid clinde R p = a + (cs sin ) (49) whee a is an abita cnstant. The value f a in (49) is detemined b the applicatin f the Benulli equatin between the clinde suface and the fa field bunda cnditin (39a) leading t p p = ( 4sin )/ (5) s in eact ageement with the classical esults [7]. The diffeence between the pessues (8) and (5) must be accunted f b the actin f sme tpe f suface tensin. It is imptant t emphasize that as ppsed t the classical the that assumed the Stkes limit, the esults (47)-(5) shw that (4)- (4) cespnd t the slutin f cmplete equatins f mtin (7)-(8). Althugh the esults shwn in Fig.3 ae in easnable qualitative ageement with the eact numeical calculatin f the Navie-Stkes equatin at etemel lw values f Re, the diffe cnsideabl at mdeate values f Re. The suce f this diffeence will nw be eamined. Fist ne ntes that with the nn-dimensinal cdinates intduced heein the Renlds numbe des nt ccu in the fmulatin f the pblem (7)-(). Similal, using the chaacteistic diffusin length =ν/w (5) H as the unit f length the stead dimensinless Navie-Stkes equatin assumes the fm. = p (5) v v v that is als fee fm Re as a paamete. Hweve, the Renlds numbe that is identified as the dimensinless clinde adius R nw ccus in the bunda cnditin (39b) R Re/ Rw / = = ν, v = v + sin = (53) Theefe, f the pblem f flw utside f a clinde f adius R ne must equie the cnditin Re/ (54) It is emphasized that since the diffusin length (5) invlves the tw mst imptant fluid and flw chaacteistic paametes ( ν,w ), it is cleal a me natual chice f the unit f length as cmpaed t the clinde adius R. Appling (5) t define = / H is equivalent t the stetching the classical cdinate = /R as = (Re/). Thus, the slutins f the Navie-Stkes equatin (5) will nl be valid f adial distances that eceed Re/ due t cnstaint (54) and bunda cnditin (53). Theefe, unde the mdified fmulatin the imptant paamete Re that plas a cental le in tansitin fm lamina t tubulent flw n lnge appeas in the equatin f mtin but assumes a less pminent psitin in the bunda cnditin. In paticula, in view f the absence f Re in (5), the classical methd f etaining nl the diffusin tem in the Navie-Stkes equatin in the limit Re t aive at the ptential flw slutins (4)-(4) will n lnge be applicable. With the mdified fms f the equatins f mtin n the the hand, f Stkes flw aund a clinde ne can take the limit f vanishing pessue gadient p and use the cnvective velcit () alng with the ptential flw slutin (4)-(4) t evaluate the w. v tems such that (7)-(8) esult in v v v v v + + R 3 (cs sin ) = (55)

6 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) v + v v + v + v 4R 3 cs sin = (56) The abve pcedue is nt applicable t the Navie- Stkes equatin f mtin (5) because this equatin nl invlves the lcal velcit v and nt a sepaate and distinguishable cnvective velcit w t allw f the appimate substitutin f the cnvective tems. It is nw pssible t detemine an appimate cectin t the classical ptential flw slutins (4)-(4) b slving (55)-(56) t btain the mdified velcit field R v = [ (R / ) ]cs (cs sin ) (57) v = [ + (R/) ]sin (58) that ma be cmpaed with (4)-(4). One ntes that the slutins (57)-(58) als satisf the cntinuit equatin () as equied. Hweve, the geneal slutins (57) (58) d nt satisf the bunda cnditin (39b) since the adial velcit des nt vanish at the clinde suface = R. Als, the flw field given b (57)-(58) is n lnge a ptential flw but has a finite aial vticit given b R z cs sin ω = (59) The vticit (59) culd be viewed as glball induced vticit since it iginates fm the glbal cnvective field w. v that epesents the last tems in (55)-(56). Althugh the ute flw field (57)-(58) is n lnge ptential, ne can use the elatin v = and (57) t aive at the mdified steam functin (6) R m = ( R / )sin (cssin ) (6) that in tems f the Catesian cdinates assumes the fm Re m Re = ( + ) 4 ( + ) (6) whee (,) = ( / R, / R ). Sme f the steamlines calculated fm (6) ae shwn in Fig.4a f R = that b () cespnds t the usual definitin f Re = based n the clinde diamete Fig. 4a Steamlines utside f liquid clinde in unifm gaseus flw at Re = fm (6). Als, calculatins epesenting the numeical slutin f the Navie-Stkes equatin ae shwn in Fig4b. Fig. 4b Steamlines f flw acss a igid clinde at Re = fm diect numeical slutin f the Navie-Stkes equatin. It is emphasized that the appimate slutins (57)- (58) ae nl valid f etemel lw velcit i.e. Stkes flw cnditin and d nt satisf the bunda cnditin (39b) n the clinde suface. Als, the esults in Fig4a cannt be diectl cmpaed with thse in Fig4b since the fme cespnds t flw aund a liquid clinde with suface velcit while the latte cespnds t flw aund a igid clinde with n-slip bunda cnditin. The absence f Re in (7)-(9) suggests that the slutins (4)-(5) and (4)-(4) shuld

7 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) emain valid f lamina flw at all scales such as lamina edd-dnamics LED, lamina clustednamics LCD, and s n [5, 6]. Fm epeimental bsevatins it is well knwn that as Re inceases multiple-scale phenmena aise leading t the bunda lae asmmet, and eventuall t bunda lae sepaatin and nset f secnda flws. Hweve, these me cmplicated aspects f the flw field at highe velcities will be addessed in the futue. Instead, attentin is net fcused n the pblem f Stkes flw ve a igid athe than a liquid clinde. 5 Mdified The f Flw Outside f a Rigid Clinde in Unifm Steam It is well knwn that the slutin f the pblem f viscus flw utside f a igid clinde in unifm steam encuntes difficulties leading t what is knwn as the Stkes paad [7]. The difficult is that thus fa a slutin that simultaneusl satisfies bth the fa field unifm velcit bunda cnditins as well as the n-slip bunda cnditins n the suface f the clinde has nt been pssible. The pblem was patiall eslved b the classical slutin f Oseen [5-6] that assumed a cnstant velcit cnvective tem. It is ve inteesting t nte that the vticit equatin cnsideed b Oseen [5] is indeed simila t the mdified fm f the Helmhltz vticit equatin (9). F a igid clinde the cnsevatin equatins (7)-() ae subject t the fa field cnvective velcit lae egin suunding the clinde and des nt leak int the ute vticit-fee dmain. Inteestingl, the numeical slutins f the Navie-Stkes equatin, a tpical eample is shwn in Fig.5, d nt eveal the pesence f such a clsed steamline and the assciated eciculatin zne in the paamete ange < Re <. A pssible easn f this behavi culd be that the Navie-Stkes equatin in the paamete ange < Re < nl eveals the flw field at a single scale within the hieach f statistical fields f diffeent scales [4, 5]. F eample, it is easnable t assciate the cnventinal fluid mechanics within the thin bunda lae with the scale f lamina cluste dnamics LCD with the velcities ( u, v, w ), the c c c c c length scales (l = λ = L = ) m, c,, e,, e e and the kinematic viscsit ν c = l c u c /3 = λ m v m /3 [4, 5]. Hweve, this wuld suggest that the flw field utside f this bunda lae shuld be assciated with the net lage scale f lamina edd-dnamics LED [5] with velcities (u e, v e, w e ), length scales 5 3 (l = λ = L = ) m, and the eddviscsit ν e = l e u e /3 = λ c v c /3 [5]. As a esult, if ne cnfines the equatin f mtin t a single scale, sa LCD, then the pesence f the bunda lae cannt be detected since the slutin is cnfined within this same bunda lae that ne wishes t eveal. w = cs, w = sin (63a) The Stkes paad is encunteed when ne wishes t simultaneusl satisf the fa field bunda cnditin (63a) and the bunda cnditins n the suface f the igid clinde f adius R = R v = v = (63b) In cmpaisn, the ptential flw slutins (33)-(35) that ae vticit fee, satisf the bunda cnditin (39b). The eaminatin f bth the steamlines as well as the is-vticit sufaces in the eal numeical calculatins f the Navie-Stkes equatin [7-9] suggest that as the steam velcit vanishes Re, the ze-level is-vticit suface assumes clindical gemet. It is theefe easnable t epect that at ve lw Renlds numbes, i.e. f Stkes flw, the vticit that is geneated n the clinde suface will be cnfined t a thin bunda Fig. 5 Calculated steamlines and velcit vects fm diect numeical slutin f the Navie- Stkes equatin. In view f the abve cnsideatins, ne cnsides that f a viscus fluid at ve lw velcities a bunda lae f thickness (R -R ) is fmed adjacent t the suface f a clinde f adius R. The thickness f this bunda lae is epected t be almst cnstant aund the clinde f the Stkes

8 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) flw Re being cnsideed heein. In such a mdel, the ute vticit-fee flw will be subject t the bunda cnditins identical t (39b) at the ute edge f this bunda lae = R such that slutins simila t (4)-(4) will emain valid. F flw within the bunda lae f thickness (R -R ), the cnvective tems f the mdified Helmhltz vticit equatin (9) ma be neglected such that it can be epessed in tems f the steam functin as ( ) = (64) F the finite-dmain (R -R ), we cnside the geneal slutin f (64) in the fm C = [Aln + B + + D]sin (65) that will be subject t the bunda cnditins n the igid clinde at = R and the ute edge f the bunda lae at = R = R =, / = (66a) = R =, / = sin (66b) Appling the bunda cnditins (66) t the geneal slutin (65) leads t a (a a ) B = a (a ) a (a ) A = a B(a ) a (67a) (67b) C= A+ B (67c) D = (A+ B) (67d) with the new paametes a, a, and a defined as a = R /R (68a) a = a lna+ a (68b) a = a + a (68c) It is emphasized hee that the size f the bunda lae and hence the adius is nt R knwn a-pii and must ultimatel be detemined as pat f the slutin. Hweve, f the pupse f illustatin nl we shw sme f the steamlines calculated fm (65)-(68) f the flw within the cncentic annula lae (R -R ) n a igid clinde in Fig Fig. 6 Steamlines f Stkes flw utside f a igid clinde in unifm steam fm (65). The calculated flw field shwn in Fig.6 is epected t nl be a valid mdel f bunda lae flw in the Stkes limit Re, since nl in such a limit the bunda lae thickness is epected t be cnstant and smmetic. Of cuse, the thickness f the bunda lae in Fig.6 is eaggeated f the pupse f illustatin. The gemet f the bunda lae will becme elngated and assumes inceasingl nn-cicula shape as Re is inceased simila t the gemet f is-vticit sufaces in the eact numeical slutins f the Navie-Stkes equatin shwn in [7-9]. The bseved absence f an enclsed egin f bunda lae in the slutin f the Navie-Stkes equatin discussed abve culd be futhe eamined in cnnectin t Fig.6. One ntes that, as the adius f the ute edge f the bunda lae in Fig.6 R is steadil inceased, the bunda lae flw becmes the ute flw aund a igid clinde subject t cmpatible fa-field bunda cnditin, v = v + sin = (69) Theefe, a seach within this flw field f the pesence f clindicall clsed steamline will nt be pssible at a single scale f sa LCD ( = c) itself. Hweve, if ne allws the pesence f a lwe scale f lamina-mlecula-dnamics LMD, then ne culd identif the lcal velcities ( vc, v c) as the cnvective velcities ( wm, w m) f the lwe scale ( = m, and b pcedues paallel t thse descibed abve eveal anthe inne bunda lae simila t Fig.6 at this lwe scale f LMD. Pssible eistence and natue

9 Pceedings f the 4th WSEAS Intenatinal Cnfeence n Fluid Mechanics and Aednamics, Elunda, Geece, August -3, 6 (pp95-3) f such enclsed flw eciculatin egins equie futhe futue eaminatin. 6 Cncluding Remaks The mdified fm f the Helmhltz vticit equatin was slved f the classical pblem f Stkes flw ve a igid clinde. It was suggested that a thin bunda lae is fmed aund the clinde that cnfines the suface geneated vticit and pevents its leakage int the ute ptential flw. The pesence f this bunda lae ma help eslve the classical Stkes paad f flw ve a clinde. The slutins f the mdified fm f the Helmhltz vticit equatin wee als detemined f flw inside and utside f liquid clinde lcated in unifm gaseus steam. The esults ma help the undestanding f vte dnamics in tubulent fields and the undestanding f evapatin/cmbustin f clindical egins f liquid/slid fuels that ma be encunteed in tubulent spa cmbustin. The geneatin f cascades f embedded clindical vtices aund cpe pais f clindical vte lines (Figs. and ) ma be identified as ne pssible mechanism f tubulent dissipatin. Refeences: [] de Gt, R. S., and Mazu, P., Nnequilibium Themdnamics, Nth- Hlland, 96. [] Schlichting, H., Bunda-Lae The, McGaw Hill, New Yk, 968. [3] Williams, F. A., Cmbustin The, nd Ed., Addisn-Wesle, New Yk, 985. [4] Shab, S. H., Rev. Gén. Them. 38, (999). [5], Pceeding f the 3st ASME Natinal Heat Tansfe Cnfeence,HTD- Vl. 38, 37-6 (996). [6] Shab, S.H., WSEAS Tansactins n Mathemathics, Issue 4, Vl.3, 755 (4). [7] Pantn, R. L., Incmpessible Flw, Wile, New Yk, 996. [8] Zdavkvich, M. M., Flw Aund Cicula Clindes, Ofd Univesit Pess, New Yk, 997. [9] Shab, S. H., Reactive Hddnamics in Rtating Spheical and Clindical Flws Futh Intenatinal Micgavit Cmbustin Wkshp, NASA, Ma 9-, 997, Cleveland, Ohi. [], Hddnamics f spheical flws and gemet f pemied flames nea the stagnatin-pint f aismmetic viscus cunteflws. Fifth Intenatinal Micgavit Cmbustin Wkshp, NASA, Ma 8-, 999, Cleveland, Ohi. [], WSEAS 7 th Intenatinal Cnfeence n Applied Mathematics, Ma -4, Cancun, Meic (5). [], IASME Tansactins 4, Vl., 634 (4). [3] Wlfam, S., and Beck, G., Mathematica, The Student Bk. Addisn Wesle, New Yk, 994. [4] Hill, M. J. M., Phil. Tans. R. Sc. A 85, 3 (894). [5] Oseen, C. W., Ak. Mat. Ast. Fs. 6, N. 9, (9). [6] Kaplun, S., J. Math. Mech.6, 595 (957). [7] Kawaguti, M. and Jain, P., J. Phs. Sc. Japan, 6 (966). [8] Kawaguti, M., J. Phs. Sc. Japan 8, 747 (953). [9] Takami, H., and Kelle, H. B., Phs. Fluid. Suppl II, 5 (969).