An Application of Bessel Functions: Study of Transient Flow in a Cylindrical Pipe

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1 Jouna of Mathematics and System Science 6 (16) 7-79 doi: / /16..4 D DAVID PUBLISHING An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe A. E. Gacía, Lu Maía Gacía Cu and Hécto Luna-Gacía Áea de Física Atómica Moecua Apicada, CBI, UAM-A, Avenida San Pabo 18, Coonia Reynosa, Acapotaco, México, D.F., México. Received: Octobe 14, 15 / Accepted: Novembe 1, 15 / Pubished: Febuay 5, 16. Abstact: The Navie-Stokes equations fo a fuid fow ae appied to a pipe. Unde conditions of symmety these ae educed to the we-known equation of eoth modified Besse, which is soved to find the veocity pofie and shape of the axia veocity. 1. Intoduction The fuids tanspot such as wate, oi, etc. is done though pipes, so it is impotant to study the fow chaacteistics unde diffeent conditions such as fow ate, pipe sie and then obtain mathematica expessions fo the veocity pofie, the axia veocity, fow ate, etc. Hee unde symmety conditions, the esuting diffeentia equations can be esoved accuatey in tems of Besse functions. This wok is divided in five pats: This intoduction, section iustates the method fo which we obtain the modified Besse equation fom the Navie Stokes equations; in section 3 we esove the Besse diffeentia equation; section 4 is an appendix with the most impotant chaacteistics about the modified Besse functions used in this atice. Finay, section 5 pesents concusions and futue pojects.. Navie-Stokes Equations Suppose a ong cicua cyindica pipe of adius R. We conside an incompessibe, isothema Newtonian fow (density ρ = constant, viscosity µ = constante ), u = u, u, u in tems of with a veocity fied ( φ ) Coesponding autho: Hécto Luna-Gacía, Áea de Física Atómica Moecua Apicada, CBI, UAM-A, Avenida San Pabo 18, Coonia Reynosa, Acapotaco, México, D.F., México. E-mai addess: ghm@coeo.ac.uam.mx. cyindica coodinates (, φ, ) equation fo incompessibe fow is [1, ] 1 u 1u u. The continuity (1) The -component of the Navie-Stokes equations is witten as u u u u u u u t 1 dp u 1u 1 u u d () In ou deveopment of fow in the pipe, consideing the veocity components with axia symmety, that is, they ony depend on the adius and time, we have,, u ut,, u ut, u u t (3) Without changes in, the continuity equation fo incompessibe fow, becomes ( u ) = (4) u = f φ,, t. Howeve, whee we must have, ( ) unde consideation of axia symmety u f ( t) f ( t) Moeove, u =. = = at = R, fo which

2 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe 73 f ( t ) must be eo. It foows that thee is no adia veocity, that is, u( t, ) =. By simia aguments to those above, we find that u = u t,. u φ =, and ( ) Theefoe, Navie-Stokes equation is witten as u 1 dp u 1u t d (5) dp If the pessue G( t) =, is peiodic with d fequency w, we can wite: ( ) G t = Re = i( wt) Ge (6) whee Re denotes the ea pat. If this gadient has an infinite peiod, we can take the Fouie tansfom. Simiay, fo the Fouie seies expansion of -component of the veocity, we have u ( t, ) = Re ue = i( wt) (7) Substituting equations 6 and 7 into equation 5 and matching hamonics, one has u 1 u iwu G (8) If it is assumed that, the fuid does not sip on the suface, u R and a eguaity condition on the cyinde axis, i.e. u max. The steady state component is simpy Poiseuie fow [1] u 1 u G and the equation 8 can be witten as d u 1 du G (9) (1) d d The soution of the equation 1 is u GR 4 R 1 whee the maximum speed is given fo u GR 4 (11) max (1) The veocity distibution is in the fom of a paaboa, with the fastest veocity in the vetex and fiction cause the veocity decease outwads. The figue 1 shows the veocity pofie in the steady state. The fow ate is cacuated as foows Q u da d R 4 GR GR 1 4 R 8 (13) whee the intega extends ove the entie coss sectiona aea A, of the pipe. Q The specific fow ate, q, is A q Q R GR 8 (14) In Dacy's equation, the fow ate has the fom dp Q ka (15) d whee the constant k is the hydauic conductivity, A epesents the coss-sectiona aea of the pipe, and is the axia diection. Compaing equation 13 to equation 15, one has that dp G (16) d and R k (17) 8 fo the hydauic conductivity, that is a popety of both the fuid and the pipe.

3 74 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe Fig. 1 Veocity pofie in a pipe o adius R fo the steady state. On the othe hand, the tems time independent u with obey the equation d du d d i w u G Reaanging tems, equation 18 is witten as d u 1 du w G i u d Then, changing obtain d (18) (19) x vaiabes, finay we R d u du wr R G x x ix u x dx dx () This equation is the desied diffeentia equation to descibe the tansient state of the fuid in the pipe, and in the next section, we wi esove it. 3. Modified Besse Diffeentia Equation The homogeneous equation fom equation is d u du wr x x ix u dx dx (1) which is equation of eoth modified Besse and its soution is witten as 1 1 u x AI i W x BK i W x () whee A, B ae abitay constants, and I y, K y ae the modified Besse functions of the fist and second kind of ode eo, defined fo the equations 47 and 5. In addition, the paamete W is the Womesey dimensioness numbe [4], and it is a measue of the ation of the tems time dependent of the momentum equation to the viscous pat, given fo

4 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe 75 w w W R R (3) The paametes, 4W ae the dynamic viscosity and the kinetic Reynods numbe [5] espectivey. On the othe hand, the paticua soution of the equation is G u x i (4) w The fu soution is witten as u x 1 1 G (5) AI i W x BK i Wx i w Since the veocity is finite at and K becomes infinite at this vaue, physicay acceptabe soution is I. Then, the soution is G (6) 1 u x AI i W x i w Appying the bounday conditions, u x1 in the equation 6, we obtain G i w A (7) I i W 1 Substituting equation 7 in equation 6 and eaanging tems, we obtain 1 G I i Wx u xi 1 w 1 (8) I i W Using the eation of Besse functions given by equation 43 in equation 8, one has 3 J i Wx G u xi 1 w 3 (9) J i W whee J y is the Besse function of the fist kind and ode eo. Then fo a fow of axia symmety of an isotopic, incompessibe and Newtonian fuid without extena foces, we have deived an anaytica soution fo amina fow puses in a pipe; this is often efeed to as Womesey fow. The axia veocity as a function of adia position and time t is given by GR u t, 1 4 R 3 J i W (3) G R iwt Re i 1 e 3 1 w J i W whee we used the equations 6, 7 and equation 9. Note that in equation 3, G is the time iwt independent pessue gadient, and Ge is the time dependent pessue gadient. The figue show the veocity pofies fo diffeent components. Using the equations 43 and 47, the equation 3 can be witten as be W ibei W GR G R R iwt u t, 1 Re i 1 e 4 R (31) 1 w be W ibei W whee be and bei, ae the ea and imaginay pats of Iy Jiy, and thei vaues ae given by the equations 48 and 49. u t, is witten as Finay, afte doing some ageba, the equation fo the veocity

5 76 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe 1 GR G u t wt 4 R w, 1 sin 1 G cos wt R R w be W bei W bei W be W be W bei W G sin wt R R bei W bei W be W be W be W bei W (3) 1 w In paticua, the axia veocity u, t, sin is given by bei Wcoswt be Wsin wt (33) 1 w 1 w be W bei W GR G G u t wt 4 Fig. Veocity pofies in a wate pipe of adius R fo each component.

6 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe 77 Fig. 3 The sinusoida axia veocity in a wate pipe of adius R fo each component. In equations 3 and 33 is omitted the subscipt eo in be and bei functions. In figue 3 it can be seen the axia veocity time dependent, i. e. the wavefom fo each component. 4. Appendix Besse functions ae one of the most impotant functions in physics and mathematics. The Besse s diffeentia equation [6] is d y dy x x x y (34) dx dx Thus, when ν is not an intege we may wite the soution of the equation 34 in the fom u x AJ x BJ x (35) whee A, B ae abitay constants and J x is known as the Besse function of the fist kind of ode ν, and is defined by the equation [7] x 1 J x F1 1; x 1 4 whee F, ; ; x 1 x x 4 1! 1 1 (36) is the hypegeometic function of two paametes and one paamete. Howeve, when the numbe is an intege n, the compete soution is whee Yn Jn y x CJ x DY x (37) n x is defined by the equation 36 and x is given by n 1! 1 x n n 1 n n 1 1 n 1 x Yn x og x Jnx n!!! (38)

7 78 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe The function Yn x so defined is known as Besse function of the second kind of ode n o Newmann function; the constant is known as Eue s constant, and [6] is given fo the equation 1 s (39) s1 Othe equation impotant is known as modified Besse diffeentia equation d y dy x x x y (4) dx dx This equation 4 can be tansfom into the equation 34, when epacing x by ix. Howeve, this eads to a compex soution of the equation 34. Simiay, to what happened with the soution of equation 34, thee ae two possibe genea soutions fo the equation 4, that depend if an intege is o not. When is not an intege the soution of this equation is whee the function that y x AI x BI x (41) x I is defined by the equation x 1 I x F1 1; x 1 4 (4) Compaing equation 4 with equation 36 we see I x i J ix (43) a esut which might have been conjued fom the diffeentia equation 4. If is an intege n, the genea soution of the equation 4 is whee the function n y x AI x BK x (44) n Kn x is defined by the equation 1 Kn x 1 og x In x n1 n 1 1 n1! x! n1 1 n 1 n 1 x 1! n! The functions In x, Kn n (45) x defined by the equations 4 and 45 espectivey ae known as modified Besse functions of the fist and second kind of ode n. A paticuay impotant case is when n. In this case the soution is whee y x AI ix BK ix 3 AJ i x BK ix I y and K (46) y ae the modified Besse functions of the fist and second kind of ode eo. It is common to intoduce two new functions bei x which ae [7, 8] espectivey ben x and n the ea and imaginay pats of In ix, i. e. I ix be x ibei x (47) In equation 47 is omitted the subscipt eo in be and bei functions. Fom definition given in the equation 44 and s s 1 1 bex x (48) s! s 4 s s1 1 1 x (49) s 4 bei x s 1! In simia way the functions ken x and x ae defined to be espectivey the ea and kein

8 An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe 79 imaginay pats of the compex function Kn ix, i. e. K ix ke x ikei x, (5) whee is omitted the subscipt eo in keix, and thei expessions ae given by and 1 ke x og x be x 1 x beix 4 1! ke x and 1 keix og x beix be x 1 1! x 1 (51) (5) Finay, it is notewothy that these fou functions ae vey usefu in appications to engineeing pobems. 5. Concusions Was successfuy appied to the soution of the modified Besse equation to find the veocity pofie and the axia veocity fo the tansient fow of a fuid in a pipe. Futhemoe, the Fouie tansfom is used in the anaysis of the above soution to appy the supeposition. Based on this wok we hope to attack othe fows whee the density is not constant, but whose expession aows us accoding to the symmety of the pobem, expessing the Navie-Stokes equations in a Besse equation, using the Fouie tansfom and obtaining infomation about the tansient fow. Refeences [1] Potte, Wigget, Ramadan, Mechanics of Fuids, 4th Ed., Cengage Leaning, 1, [] F.M. White, Viscous Fuid Fow, nd Ed., McGaw-Hi, Inc., 1991, [3] Steete V., Wyie E. B., Keith W. B., Mecánica de fuidos 9th ed., McGaw-Hi,. [4] R. Womesey, "Method fo the cacuation of veocity, ate of fow and viscous dag in ateies when the pessue gadient is known," J. Physio. 17(3), 1955, [5] T. S. Zhao and P. Cheng, Int. J. Heat and Fuid Fow Vo.17, 1996, [6] Dennis G. Zi, Waen S. Wight, Diffeentia Equations with Bounday-Vaue Pobems, 8th ed., Cengage Leaning, 13, [7] Ian N. Sneddon, Specia Functions of Mathematica Physics and Chemisty, Second Edition, Intescience, Inc., A Division of John Wiey & Sons, Inc.1961,19-41 (hypegeometic functions) (Besse functions). [8] Mathews J., Wake R., Mathematica Methods of Physics, th ed., Addison Wesey, 197,