GENERALIZED SECOND GRADE FLUID PERFORMING SINUSOIDAL MOTION IN AN INFINITE CYLINDER

Transcription

1 Inernaional Journal of Mahemaics and Saisics Sudies Vol.5, No.4, pp.1-5, Augus 217 Published by European Cenre for esearch Training and Developmen UK ( GENEALIZED SECOND GADE FLUID PEFOMING SINUSOIDAL MOTION IN AN INFINITE CYLINDE Nazia Afzal Assisan Professor, Deparmen of Mahemaics, Governmen Faima Jinnah College for Women, Chuna Mandi, Lahore. ABSTACT: This paper shows he calculaion of velociy field and shear sress corresponding o Generalized second grade fluid performing sinusoidal moion. Shear sress is found by using D β Ga,b,c(.,.) = Ga,b+β,c(.,.) [4]. Velociy field obained by applying Laplace and Hankel ransforms. The soluion have been wrien in series form by using generalised funcion G.,.,.(.,) and Bessel funcions. KEYWODS: Velociy Field, Generalized Second Grade Fluid, Shear Sress INTODUCTION The exac soluion corresponding o he flow of fracional second grade fluid in circular cylinder were found and wrien under inegral and series form by using G.,.,.(., ) funcion and found Newonian and ordinary second grade fluid performing he same moion [1]. The aim of his paper is o calculae shear sress corresponding o non-newonian fluid by applying fracional derivaive and he resul menioned in absrac of his paper. The non-newonian fluids wih fracional derivaives have encounered a lo of success in describing complex fluid dynamics. The governing equaions corresponding o he moion of fluid are obained from hose of ordinary fluids by replacing inner ime derivaive by he so called iemann Liouville operaor D β defined by D β f() = 1 d f(τ) ɼ 1 β d ( τ) β dτ β < 1 = d d f() β = 1 An excellen discussion of fracional differenial equaion and a good hisory of fracional calculus is given by K. S. Miller [9] carl. F. Lorenzo [3] presened a very useful paper on fracional derivaive which are much flexible in describing visco elasic behaviour of fluids [6] M. Kamran [7] and A. Mahmood [8] presened exac soluion for unseady roaional flow of he generalized second grade fluid. M. Ahar [2] solved Taylor Couee flow of he generalized second grade fluid. Governing Equaions In his paper we consider he velociy V and he exra sress S of he form V = V(r, ) = ω(r, )e θ S = S(r, ) ISSN (Prin), ISSN (Online) 1

2 Inernaional Journal of Mahemaics and Saisics Sudies Vol.5, No.4, pp.1-5, Augus 217 Published by European Cenre for esearch Training and Developmen UK ( Where e θ is he uni vecor in he θ direcion of he cylindrical coordinae sysem. A = we have ω(r, ) = The governing equaions corresponding o such moion of ordinary second grade fluid are τ(r, ) = (μ + α 1 )( 1 ) ω(r, ) (1) r r ω(r,) = (θ + α 2 )( r 2 r r r 2) ω(r, ) (2) Where µ is dynamic viscosiy of he fluid and α = α 1 ρ is maerial consan; θ = μ ρ Kinemaic viscosiy of he fluid. Where ρ being is consan densiy and τ(r, ) = S rθ (r, ) is he shear sress. Governing equaions corresponding o fracional second grade fluids are obained by replacing inner ime derivaive w.r. by fracional derivaive D β, β >. ω(r,) τ(r, ) = (μ + α 1 D β )( 1 ) ω(r, ) (3) r r = (θ + α 1 D β )( r 2 r r r Flow hrough a circular cylinder wih a shear on boundary 2) ω(r, ) (4) Consider incompressible generalized second grade fluid a res, in an infiniely long cylinder of radius >. A ime = fluid is a res and a ime = + cylinder begins o roae and boundary of cylinder applies a sinusoidal shear sress on fluid. The fluid is gradually moved. The governing equaions as given by (3) and (4). Boundary condiions and iniial condiions are ω(r, ) = Where rε(, ] Calculaions of velociy field τ(, ) = (μ + α 1 D β )( r 1 r ) ω(r, ) (=r) = Ωsin(ω, ) wih >o Ω is consan Applying Laplace ransform o (3) and (4), and hen applying Hankel ransform and breaking ω H (r n, q) ino wo pars [1]. ω 1H (r n, q) = 1 μr2 ( J 1 (r n )Ωω n (q 2 + ω 2 ) ) (5) and ω 2H (r n, q) = J 1 (r n )Ωωq(1+qβ 1 αr n 2 ) μr n 2 (q 2 + ω 2 )(q+υr n 2 +αq β r n 2 ) (6) ISSN (Prin), ISSN (Online) 2

3 Inernaional Journal of Mahemaics and Saisics Sudies Vol.5, No.4, pp.1-5, Augus 217 Published by European Cenre for esearch Training and Developmen UK ( Applying inverse Hankel ransform o (5) and (6) and hen inverse Laplace ransform we ge he velociy field; ω(r, ) = Ωr3 2µ sin(ω) 2 J 1(rr n )Ω µr 2 x n J 1 (r n ) [ ( θr n 2 ) k sin(ωs)g 1 β,1 β βk,k +1 ( αr n 2, s)ds + αr n 2 ( θr n 2 ) k sin(ωs)g 1 β, βk,k +1 ( αr n 2, s)ds] Where generalized funcion Ga,b,c (d,) is defined by [3] equaions (97) and (11) (7) q b G a,b,c (d, ) = L 1 [ (q a d) c] = dk ɼ c+k ɼ c ɼ k+1. (c+k)a b 1 ɼ(c + k)a b e(ac b) >, d q a < 1 Calculaion of shear sress τ(r, ) = (μ + α 1 D β )( r 1 ) ω(r, ) r τ(r, ) = (μ + α 1 D β ) Ωr2 µ sin(ω) + 2 J 2(rr n )Ω µr n J 1 (r n ) [ ( θr n 2 ) k sin(ωs)g 1 β,1 β βk,k +1 ( αr n 2, s)ds + αr 2 n ( θr 2 n ) k sin(ωs)g 1 β, βk,k+1 ( αr 2 n, s)ds] = Ωr2 sin(ω) + 2 J 2(rr n )Ω (r n )J 1 (r n ) [ ( θr n 2 ) k sin(ωs)g 1 β,1 β βk,k +1 ( αr n 2, s)ds + ISSN (Prin), ISSN (Online) 3

4 Inernaional Journal of Mahemaics and Saisics Sudies Vol.5, No.4, pp.1-5, Augus 217 Published by European Cenre for esearch Training and Developmen UK ( αµr n 2 ( θr n 2 ) k sin(ωs)g 1 β, βk,k +1 ( αr n 2, s)ds] + α 1 ωr 2 µ [D β (sin(ω)] + 2 α 1 J 2(rr n )Ω μr n J 1 (r n ) ( ( θr n 2 ) k sin(ωs)g 1 β,1 βk,k +1 ( αr n 2, s)ds + α 1 αr 2 n ( θr 2 n ) k sin(ωs)g 1 β,β βk,k+1 ( αr 2 n, s)ds]) (8) Applying α and α 1 in (7 and 8) we ge, 1. Newonian ω(r, ) = Ωr3 2µ sin(ω) 2 J 1(rr n )Ω µr 2 x n J 1 (r n ) [ ( θr 2 n ) k sin(ωs)g 1 β,1 β βk,k+1 (, s) ds (9) τ(r, ) = Ωr2 sin(ω) + 2 J 2(rr n )Ω r n J 1 (r n ) [ ( θr n 2 ) k sin(ωs)g 1 β,1 β βk,k +1 (, s)ds] β 1 for ordinary second grade fluid (1) ω(r, ) = Ωr3 2µ sin(ω) 2 J 1(rr n )Ω x µ(r n2 )J 1 (r n ) [ ( θr n 2 ) k sin(ωs)g, k,k +1 (, s)ds] τ(r, ) = Ωr2 sin(ω) x 2 J 2(rr n )Ω r n J 1 (r n ) (11) ISSN (Prin), ISSN (Online) 4

5 Inernaional Journal of Mahemaics and Saisics Sudies Vol.5, No.4, pp.1-5, Augus 217 Published by European Cenre for esearch Training and Developmen UK ( [ ( θr 2 n ) k sin(ωs)g, k,k +1 (, s)ds] (12) CONCLUSION The velociy field and shear sress corresponding o generalized second grade fluid were calculaed and wrien in he series form wih he help of generalized funcion G.,.,.(.,). The velociy field was calculaed by applying Laplace ransform and Hankel ransform. Shear sress was calculaed by using D β Ga, b, c (., ) = G a, b+β, c (., ) Newonian and ordinary second grade fluid were found as a limiing case. EFEENCES [1] Afzal, N. and M. Ahar. Fracional second grade fluid performing sinusoidal moion in a circular cylinder. Inl. J. Scie. Engg. es. 6(6): [2] M. Ahar, M. Kamran, C. Feecau. Taylor-couee flow of generalized second grade fluid due o consan couple. Nonlinear analysis modeling and conrol. vol. 15, no. 1, 3-13, 21. [3] Carl F. Lorenzo, Tom T. Harly. Generalized funcion for he fracional calculus. NASA/TP /EVI. Ocober [4] L. Debnah, D. Bhaa. Inegral ransform and heir applicaion (Second ediion). Chapman and Hall/CC, 27. [5] M. Ahar, M. Kamran, and M. Imran. On he unseady roaional flow of frac ional second grade fluid hrough a circuler cylinder. Meccanica, vol. 81, no. 11, , 211. [6] M. Kamran, M. Imran, M. Ahar. Exac soluions for he unseady roaional flow of a generalized second grade fluid hrough circular cylinder Nonlinear anal ysis modeling and conrol. vol. 15 no.4, , 21. [7] A. Mahmood, N. A. Khan, I. Siddique, S. Nazir. A noe on Sinusoidal moion of a viscoelasic non-newonian fluid. Archives of applied mechanics. vol. 82 issue 5, , 212. [8] A. Mahmood, Saifullah, Q. ubab. Exac soluion for roaional flow of gener alized socond grade fluid hrough a circuler cylinder, Bulenenimul academiei de sine a republicii moldova, mahemaica, no. 3 (58), 9-7, ISSN , 28. [9] K. S. Millar, B. ose. An inroducion o he fracional Calculus and fracional differenial equaion, John Wiley and Sons [1] K.. ajagopal, and P. N. Kaloni. Coninuum Mechanics and is applicaions, Hemisphere Press, Washingon DC,USA, ISSN (Prin), ISSN (Online) 5