MHD Flow of an Oldroyd B Fluid through Porous Medium in a Circular Channel under the Effect of Time Dependent Pressure Gradient

Transcription

1 American Jurnal f Fluid Dynamics 07, 7(): - DOI: 0.593/j.ajfd MHD Flw f an Oldryd B Fluid thrugh Prus Medium in a Circular Channel under the Effect f Time Dependent Pressure Gradient S. E. E. Hamza Physics Department, Faculty f Science, Benha University, Benha, Egypt Abstract In this paper the unsteady magnethydrdynamic (MHD) flw f Oldryd B fluid thrugh a tube filled with prus medium was investigated. A unifrm circular magnetic field was applied perpendicular t the flw directin. The fluid was driven by an axial time dependent pressure gradient which increased r decreased r pulsated with time. The gverning equatins were accurately slved taking in cnsideratin the mdified Darcy's law t cunt the resistance presented by the prus medium. The btained velcity prfiles had Bessel and mdified Bessel functins f zer rder representatins. The effects f the varius parameters n the flw characteristics such as nn Newtnian, magnetic and permeability parameters had been studied and discussed graphically. Figures were prvided t cmpare the velcity prfile f Newtnian fluid and Maxwell fluid with Oldryd fluid. It was fund that the velcity distributin increased with the increase f permeability parameter f the prus medium, but it decreased with the increased magnetic and frequency parameters. The flw was parablic fr small frequencies but it had bundary layer characters at large frequencies. Slutins f Maxwell and Newtnian fluids were appeared as limited cases fr the present analysis. Keywrds MHD flw, Oldryd B fluid, Unsteady channel flw, Pulsating flw, Prus medium. Intrductin MHD flw f nn Newtnian fluid thrugh pipes represents significant rle in different areas f science and technlgy such as petrleum industry, bimechanics, heat exchangers and s n. Due t the nnlinearity f the gverning equatins, exact slutins are difficult t be btained and are few in number under certain cnditins. As a cnsequence f different physical structures f nn Newtnian fluids, there is nt a single cnstitutive mdel which can predict all its salient features. Generally, there are three nn Newtnian fluid mdels, namely empirical mdels, differential mdels and integral mdels. But the mst famus amngst them are the last tw mdels. In this wrk, we will study the secnd mdel and cnsider its subclass knwn as Oldryd B fluid. The simplest subclass f differential type mdels is the Maxwell fluid. This mdel can describe the fluid in terms f its relaxatin parameter, while there are n any infrmatin abut its retardatin parameter. The Oldryd B mdel, n the ther hand, has a measurable retardatin time and can relate the viscelastic manner f dilute plymeric slutins under * Crrespnding authr: salah.hamza@fsc.bu.edu.eg (S. E. E. Hamza) Published nline at Cpyright 07 Scientific & Academic Publishing. All Rights Reserved general flw cnditins. Fetecau [] has prvided analytical slutins fr the flw f a nn Newtnian fluid in pipe like dmain. Yin and Zhu [] have studied the scillating flw f viscelastic fluid in a pipe with the fractinal Maxwell mdel. Khan et al. [3] have investigated the hydrmagnetic rtating flws f Oldryd B fluid in a prus medium. He btained exact slutins by using the Laplace transfrm technique and discussed the effects f relaxatin and retardatin parameters n fluid mtin. Khan and Zeeshan [4] studied the MHD flw f an Oldryd B fluid thrugh a prus space induced by tth pulses. It is well knwn that thery f time dependent pressure gradient flw thrugh channels may help in understanding many technlgical prblems such as pumping mechanisms used fr transprting petrl thrugh pipes. These mechanisms prduce, in general, pressure gradient flw which is nt cnstant but changes in sme way abut a nn zer mean. The unsteady flw f a fluid in cylindrical channels has applicatins in the chemical fd and petrleum industries. Darke [5] has studied unsteady viscus incmpressible flw thrugh circular and elliptic tubes under the influence f peridic pressure. Batchelr [6] studied the starting flw in a circular pipe which is initially at rest and set in mtin by pressure gradient suddenly impsed and maintained by external means. The laminar flw f an unsteady viscus liquid thrugh a circular

2 S. E. E. Hamza: MHD Flw f an Oldryd B Fluid thrugh Prus Medium in a Circular Channel under the Effect f Time Dependent Pressure Gradient cylinder under the influence f expnentially decreasing pressure gradient was studied by Ra and Murthy [7]. Ramkissn et al. [8] have studied unsteady viscelastic pipe flw thrugh circular pipe under the influence f peridic pressure gradient. A nn Newtnian fluid mdel that has attracted many researches is the Oldryd B fluid as this is fund t be a better mdel fr physilgical fluids. S, several researchers have studied Oldryd B fluid flws under different cnditins [9-3]. Pulsating flw is cmpsed f tw superimpsed cmpnents: steady cmpnent and scillatin cmpnent (time varying). Pulsating flw is appeared in natural systems e.g. respiratry system and circulatry system and in engineering systems e.g. pumps and pulse cmbustrs. Using a finite difference methd, Balmer and Firina [4] btained a numerical slutin fr a pwer law fluid by cnsidering the pressure pulse as a superpsitin f steady cmpnent and scillatin cmpnent. The cmputatinal study presented by Hung [5] is cnsidered ne f the excellent study in pulsatile nnlinear flw. Rahaman and Ramkissn [6] studied the unsteady axial viscelastic pipe flws under the influence f peridic pressure gradient. Bestman and Njku [7] cnstructed a slutin fr hydrdynamic channel flw f viscus fluid induced by tth pulses. Als, Ghsh and Debnath [8] cnsidered the hydrdynamic channel flw f a tw phasic fluid particle system induced by tth pulses and btained a slutin using the methd f Laplace transfrms. Datta and Dalal [9, 0] discussed the pulsatile flw and heat transfer f dusty fluid in a channel and in an annular pipe emplying the methd f perturbatin. The study f flw thrugh prus medium is f fundamental imprtance in gemechanics, bimechanics and industry e.g. flw f water thrugh rcks, regulatin f skin and filtratin f fluids. Prus medium is a material frmed f slid matrix with bres. This medium is characterized by its prsity which is a rati between the vid space and the ttal vlume f the medium and by its permeability which is a measure f the flw cnductivity in the prus medium. Earlier studies f flw in prus medium have taken in cnsideratin the Darcy law which relates between the flw velcity and the pressure gradient acrss the prus medium. In recent years, the flw f fluids thrugh prus medium has becme an imprtant tpic because it is used in suctin f crude il thrugh the pres f the rcks, here Darcy's law has a effective rle. Elshehawey et al. [] had studied the peristaltic transprt in an asymmetric channel thrugh a prus medium. Ghsh and Sana [] discussed the unsteady flw f Oldryd B fluid induced by rectified sine pulses. Jythi et al. [3] studied the pulsatile flw f Jeffrey fluid in a circular tube lined with a prus material frm inside. During recent years, the study f viscelastic fluids flw in the presence f magnetic field has applicatins in many areas including the medical therapy f bilgical fluids especially bld [4]. Hamza [5] studied Maxwell's fluid flw thrugh circular pipe under the influence f peridic pressure gradient and in the presence f axial magnetic field. Misra et al. [6] had investigated a mathematical mdeling f bld flw in prus vessel having duble stensis in the presence f external magnetic field. Eldabe et al. [7] studied the pulsatile flw f nn Newtnian fluid with heat and mass transfer thrugh a prus medium in a channel under the effect f unifrm magnetic field. The results f this paper may help in the field f medical research. The applicatin f magnetic field culd be useful in the treatment f sme healthy prblems such as cardivascular diseases and bld flw in crnary arteries when fatty plaques f chlesterl and bld clts are frmed in the lumen f the crnary artery. Als by using this apprpriate magnetic field, it is pssible t cntrl bld pressure e.g. pr circulatin. The pulsating flw passing thrugh arteries attracted the attentin f researchers fr a lng time due t its great imprtance in medical sciences. Under nrmal cnditins, bld flw in the human circulatry system depends upn the pumping actin f the heart and this prduces a pressure gradient acrss the arterial netwrk. Elshehawey et al. [8] studied MHD flw f bld under the effects f bdy acceleratin cnsidering the bld as an viscelastic fluid. Chaturani and Palanisamy [9] studied pulsatile flw f bld thrugh a rigid tube under the influence f bdy acceleratin as a Newtnian fluid. Elsud et al. [30] studied the interactin f peristaltic flw with pulsatile cuple stress fluid. The mathematical mdel that cnsidering a viscus incmpressible cuple stress fluid between infinite parallel walls n which a sinusidal travelling wave is impsed. Elshehawey et al. [3] investigated the pulsating flw f bld thrugh a prus medium under peridic bdy acceleratin, but the arterial MHD pulsating flw f bld under peridic bdy acceleratin has been studied by Das and Saha [3]. Assuming bld as a viscus fluid, the effect f unifrm transverse magnetic field n pulsating mtin f bld passing thrugh an axial symmetric tube was analysed by Sanyal and Biswas [33]. It is clear that, analysis f the flw prperties f nn Newtnian fluids are very imprtant in fluid dynamics field because f their technlgical applicatin. In additin, the effects f magnetic field n nn Newtnian fluids als have great imprtance in engineering applicatins. Mrever, MHD flws in prus medium have applied in a wide fields e.g. develpment f energy generatin systems, astrphysics and gephysical fluid dynamics. Therefre, the aim f this wrk is t btain analytic slutins fr MHD flw f an Oldryd B fluid passing thrugh circular channel filled with prus medium. The fluid is mved by time dependent pressure gradient that increases r decreases r pulsates with time. A circular unifrm magnetic field is applied perpendicular t the directin f the flw. Cnsidering Darcy's law fr Oldryd B fluid, the slutins f the gverning equatins have been btained in an exact manner. With the help f these slutins, the effect f the nn dimensinal parameters n the velcity field has been

3 American Jurnal f Fluid Dynamics 07, 7(): - 3 discussed fr varius particular values f the invlved parameters. The btained results shw that, the flw is appreciably influenced by the nn Newtnian parameters f the fluid, the magnetic parameter, the permeability parameter f the prus medium and the frequency parameter.. Basic Equatins In prus medium, the mmentum and the cntinuity equatins fr unsteady MHD flw f incmpressible fluid are: V ρ V V p τ J B, () t V 0. () where, V, p, τ, J, B and are density, velcity, pressure, extra stress tensr, current density, magnetic field and Darcy's resistance f the prus medium respectively. Ntice that, J B is the Lrentz frce due t magnetic field and is the measure f the flw resistance ffered by the slid matrix. This study is based n Oldryd B mdel f viscelastic fluids. The prperties f this fluids are specified by three cnstants: dynamic viscsity, relaxatin time parameter and retardatin time parameter respectively. The extra stress tensr τ fr Oldryd B fluid is satisfied by the fllwing cnstitutive equatin, Bird [34]: τ μd d, (3) τ where d is the rate f defrmatin tensr and d is the upper cnvected derivative f d : d d V d L d (L d) t the tensrs L and d are defined as: T, (4) L V, d (L L ), (5) where the symbl "T" ver any tensr dentes the matrix transpse. Therefre, Eq. (3) takes the frm: τ τ V τ L τ (L τ) t d μd V d Ld (Ld) t T T T. Nte that, the mdel (3) is reduced t the viscus (Newtnian) fluid when 0, t Maxwell fluid when 0 and t Oldryd B fluid when 0 (6). The Lrentz frce due t magnetic field is: J B σ B V, (7) where is the electrical cnductivity f the fluid and B is the strength f the applied magnetic field. The Darcy's resistance fr Oldryd B fluid satisfies the fllwing expressin [35]: μυ V, (8) t k t where and k are the prsity and permeability cnstants f the medium, respectively. Fr the prblem under cnsideratin we assume that: V 0, 0, W(r, t), B 0, B, 0 z 0, 0,,, (9a) τrr τrθ τrz τ τθr τθθ τθz. (9b) τzr τzθ τzz With the velcity in Eqs. (9), the cntinuity equatin is satisfied autmatically and Eqs. (7) and (8) take the frm: J B 0, 0, σb W, (0) μυ z W. () t k t Therefre, the mmentum equatin, Eq. (), in cmpnent frm is being: τ r rz τ r rz τ r rr τ r P σ B z rr 0, () W and the cnstitutive equatin, Eq. (6), gives: τ rr z W ρ, (3) t τ rr W W τrz μ, (4) t r r τ r θ τ r θ W τ θz 0 t r, (5) τ rz W W W τ rz τzz μ, (6) t r r rt τ τ θθ θz τθθ 0, (7) t τθz 0, (8) t τ zz τzz 0. (9) t Nte that, the treating f Eq. () gives τ rr f(t) / r while Eqs. (5), (7), (8) and (9) yields τ τ τ g(r) Exp( t / ) where f (t) is τ r θ θθ θz zz an arbitrary functin f time and g (r) is an arbitrary functin f r. The last cmpnents f the extra stress tensr,

4 4 S. E. E. Hamza: MHD Flw f an Oldryd B Fluid thrugh Prus Medium in a Circular Channel under the Effect f Time Dependent Pressure Gradient τ r θ, τ θθ, τ θz and τ zz decays expnentially with time. Therefre, the remaining field equatins can be expressed as: τ r rz τ r rz p σ B z W z W ρ, (3) t τ rz W W τ rz μ. (6) t r rt 3. Frmulatin f the Prblem In this wrk, we study the flw f an electrically cnducting Oldryd B fluid thrugh a circular cylinder f radius R filled with prus medium. We chse a cylindrical plar crdinate system (r,, z) in a way that z axis is parallel t the flw directin and there is n flw assumed in the azimuthal directin. This flw is axially symmetric and fully develped in a prus medium. A circular magnetic field B f strength B is applied n the azimuthal directin, Fig., and the mtin starts due t time dependent pressure gradient. In the analysis, we assumed that the induced magnetic field is negligible. Under the abve hyptheses, the unsteady flw f Oldryd B fluid can be described by Eqs. (3) and (6). The eliminatin f τ frm these tw equatins shws rz that, velcity field W (r, t) is gverned by the equatin: σb ρ W μ t t ρ W t ρ W W t r r r t z ρ (0) p, t z Using Eq. () fr Darcy's resistance expressin, we can replace the last tw terms in the LHS f Eq. (0) and rearranging the btained equatin we get: σb ρ W μ t t ρ μυ W t ρk W W t r r r W t ρ () p, t z Equatin () represents the gverning flw equatin with its bundary cnditins: W(0, t) W(R,t) 0, and 0. () r We rewrite the gverning equatins in dimensinless frm by defining the fllwing variables: r r R, μ ρr μt μl t, W ρr pr μ,. ρr W, (3) Using Eq. (3), the dimensinless gverning equatin and bundary cnditins are btained as fllws (fr simplicity we drp the dimensinless mark "*" fr simplicity): with W W W MW t t t r r r (4) W (t), K t t W(0, t) W(, t) 0, and 0, (5) r σ B R υr L p M,, (t). (6) μ K k p z The parameter M is knwn as the magnetic parameter and K is the permeability parameter. The values f M and K are index t the imprtance f magnetic frces and permeability f the prus medium respectively. When M 0, magnetic frces are absent. When M increases, the magnetic frce becmes increasingly imprtant. When the value K 0, the cylinder is blcked (slid); and when K (r / K 0 ) the cylinder becmes a hllw cylinder. Equatin (4) subject t the bundary cnditins, Eq. (5), is t be slved fr different types f pressure gradients, i.e. different frms f the functin (t). r W (r, t) B Prus medium Pressure gradient R z Figure. Schematic diagram f rhelgical flw thrugh a prus medium in a circular channel

5 American Jurnal f Fluid Dynamics 07, 7(): Slutin f the Prblem 4.. Pressure Gradient Increasing Expnentially with Time Let and assume that αt ( t) P e, (7) αt W(r, t) F(r)e, (8) where P and are knwn cnstants. Substituting Eqs. (7) and (8) int (4) and equating the harmnic terms, αt retaining cefficients f e, the crrespnding equatin becmes: F (α M)( α ) ( α F F P. (9) r ( α ) K ( α ) The bundary cnditins, Eq. (5), may be rephrased in terms f the new functins, with the result F() F (0) 0. (30) The slutin f Eq. (9) subject t the bundary cnditins (30) is: ( α)p F(r) ( α )β I 0 (βr), (3) I0 (β) where I 0 (x) is the mdified Bessel functin f zer rder, and β (α M)( α). (3) ( α ) K Therefre, the velcity field is given by: W(r, ( α)p I0 (βr) αt t) e. (33) ( α )β I0 (β) 4.. Pressure Gradient Decreasing Expnentially with Time The slutin f this case is btained by taking the functin (t) in the frm: with the assumptin α t (t) P e, (34) α t W(r, t) G(r)e. (35) The substitutin abut (t) and W (r, t) int (4) gives: G with (α M)( α ) ( α G G P, (36) r ( α ) K ( α ) G() G (0) 0. (37) The slutin f Eqs. (36) and (37) are given by: ) ) ( α)p G(r) ( α )β J 0 (β J 0 (β r) ), (38) where J 0 (x) is the Bessel functin f zer rder, and β (α M)( α). (39) ( α ) K Therefre, the velcity field is given by: ( α)p J 0 (β r) αt W(r, t) e. (40) ( α )β J 0 (β ) 4.3. Pulsating Pressure Gradient In this case the flw is generates impulsively due t csine pulses applied peridically n the psitive z directin. Equatin (4) must be slved subject t bundary cnditins (5) fr pulsating pressure gradient. S, let the pressure gradient r the functin (t) scillate with frequency ω and amplitude P, i.e. ( t) P csωt Re P e, (4) and assume the velcity functin has the frm: iωt iωt W(r, t) Re H(r)e, (4) where Re( ) represents the real part f the cmplex functin which fllws. The substitutin f Eqs. (4) and (4) int Eq. (4) gives: H (iω M)( iω ) H H P,(43) r ( iω) K ( iω) with its bundary cnditins: H() ( iω ) H (0) 0. (44) Equatin (43) can be easily recgnized as Bessel functin. Therefre, the slutin f Eq. (43) subject t the bundary cnditins (44) is given by: where ( iω) P H(r) ( iω )β β 3 I 0 (β 3r), (45) I0 (β 3) (iω M)( iω). (46) ( iω ) K 3 Hence, the velcity field is given by: ( iω )P I 0(β3r) iωt W (r, t) Re e. (47) ( iω)β3 I0(β3 ) 5. Results and Discussin In this wrk, analytic slutins fr the velcity field crrespnding t the unsteady MHD flw f Oldryd B fluids passing thrugh circular channel filled with prus medium have been established. It is wrth mentining that,

6 6 S. E. E. Hamza: MHD Flw f an Oldryd B Fluid thrugh Prus Medium in a Circular Channel under the Effect f Time Dependent Pressure Gradient if M 0 and / K 0 the slutins are reduced t thse crrespnding t Oldryd B fluid that flws in a hllw cylinder in the absence f magnetic field. The variatin f velcity field with r is calculated frm Eqs. (33), (40) and (47) fr different values f the included physical parameters. These variatins is shwn in Figs. t 0 at P and fr fixed,, M, K,, ω t. Initially, the effect f nn Newtnian parameters, and later n, the influence f magnetic and permeability parameters (M and K) n the velcity prfile are examined graphically. The effect f nn Newtnian parameters and is shwn in Figs. a, b, 5a, 5b and 8a fr fixed M, K, and ω t. We bserve that the velcity increases with increasing and decreases with increasing. In this sectin we display the effect f magnetic parameter n axial velcity prfiles. The influence f magnetic parameter M n velcity prfile is presented in Figs. 3a, 6a and 8b fr fixed,, K, and ω t. Frm these figures, it is nted that an increase in magnetic parameter M reduces the velcity prfile mntnically due t the effect f the magnetic frce against the directin f the flw. The influence f prus medium permeability K n the flw prfile is illustrated in Figs. 3b, 6b and 9a fr fixed values f,, M, and ω t. As expected frm the gverning equatins, the increase in the permeability f the prus medium K yields an effect ppsite t that f the magnetic parameter M. Figure. The increasing pressure gradient velcity field fr varius values f, and fr fixed M 0. 5, K 0, t 0. 3 and 3 in ] α, [ in Figure 3. The increasing pressure gradient velcity field fr varius values f M, K and fr fixed 3,, αt 0. 3, [ 0 M 0.5 in ] K in and

7 American Jurnal f Fluid Dynamics 07, 7(): - 7 Figure 4. The increasing pressure gradient velcity field fr varius values f and fr fixed 3,, 0. 5 M and K 0 Figure 5. The decreasing pressure gradient velcity field fr varius values f, and fr fixed M 0. 5, K 0, t. 4 and 3 in ] α, [ in Figure 6. The decreasing pressure gradient velcity field fr varius values f M, K and fr fixed 3,, αt. 4, [ 0 M 0.5 in ] K in and

8 8 S. E. E. Hamza: MHD Flw f an Oldryd B Fluid thrugh Prus Medium in a Circular Channel under the Effect f Time Dependent Pressure Gradient Figure 7. The decreasing pressure gradient velcity field fr varius values f and fr fixed 3,, 0. 5 M and K 0 Figure 8. Pulsating pressure gradient velcity field fr varius values f and 3. 3 in ], M and fr fixed 0. 5, 8 K, ωt π/ 4, [ M 0. 5 in Figure 9. Pulsating pressure gradient velcity field fr varius values f K, and K 8 in ] ω t and fr fixed 3. 3, 0. 5, 0. 5 M, [ ωt 3π / 4 in

9 American Jurnal f Fluid Dynamics 07, 7(): - 9 Figure 0. Three dimensin pulsating pressure gradient velcity field fr fixed values f 3. 3 and ω 6 in ], 0. 5, 0. 5 ω in M, K 8, [ 3 Figure. Maxwell fluid Pulsating pressure gradient velcity field fr fixed values f M 0. 5, K, t 0. 3, [ ] α in and α 0. 4 in In case f pulsating pressure gradient, the variatin f velcity field with r is calculated frm Eq. (47) fr different values f ω t. It is shwn in Fig. 8 fr fixed, K, ω t and in Fig. 9 fr fixed, and M. We bserve that, the velcity decreases with increasing ω t. The three dimensinal plt f velcity W (r, t) with r and ω t is shwn in Fig. 4b fr increasing pressure gradient, in Fig. 7b fr decreasing pressure gradient and in 0a and 0b fr pulsating pressure gradients. Finally, the present mdel gives a mst general frm f velcity expressin frm which the ther mdels can be btained by prper substitutins. It is f interest t nte that the presented slutin expressins, (33), (40) and (47), are mre general and the results f several prblems (e.g. upper cnvected Maxwell and Newtnian fluids) can be recvered as special cases, see Fig. fr upper cnvected Maxwell fluid; i.e Cncluding Remarks The main bjective f this wrk is t examine basic MHD flw f an Oldryd B fluid thrugh circular channel filled with prus medium in the presence f time dependent pressure gradients. We cnsider the unsteady flw f Oldryd B fluid in three cases: (i) when the pressure gradient change expnentially with time (increasing r decreasing) and (ii) when the pressure gradient is pulsating. The gverning nnlinear partial differential equatins are slved analytically. In the first prblem we lked separately at the pressure gradient rising expnentially with time and falling expnentially wit time, i.e. the pressure gradient is prprtinal t e αt. The effects f magnetic and permeability parameters (M and K) n the velcity field are studied. Therefre, we cnclude the fllwing remarks:

10 0 S. E. E. Hamza: MHD Flw f an Oldryd B Fluid thrugh Prus Medium in a Circular Channel under the Effect f Time Dependent Pressure Gradient The slutin f the gverning equatins and hence the behaviur f the flw depend t a large extent n (α M)( α) where β. The psitive sign ( α ) K fr increasing pressure gradient and the negative sign fr decreasing pressure gradient. The slutins depend n the parameter in the same frm as the slutin fr the upper cnvected Maxwell fluid. Hwever, in the present case the parameter depend n,, M, in additin t K. An increasing in the magnetic parameter M reduces the velcity prfile due t the effect f magnetic frce against the flw directin. An increasing in the permeability K yields an effect ppsite t that f the magnetic parameter M. In the absence f the magnetic and permeability parameters ( M 0 and / K 0 ), the btained slutins reduce t thse crrespnding t the flw f α( α) Oldryd B fluids with β. ( α ) Fr hllw cylinder ( / K 0 ) and if 0 then Lim β (α M)( α) which cincides with K Hamza [5] fr MHD Maxwell fluid. The results f Rahaman [6] fr upper cnvected Maxwell fluid in the absence f the applied magnetic field have been recvered by taking K, 0 and M 0 then Lim β α( α). K As a result f the present study and with the help f Eqs. (33), (40) and (47), we can intrduce the amplitude f the dimensinless velcity A, A, A fr increasing, decreasing and pulsating pressure gradients; respectively n the axis f the pipe ( r 0 ) as: ( α)p Ai ( α )β, (48) I0 (β ) ( α)p Ad ( α )β, (49) J 0 (β ) ( iω )P Ap Re, (50) ( iω )β I0 (β 3) 3 ACKNOWLEDGEMENTS I wuld like t thank Prf. Dr. A. Abu El Hassan Prfessr f theretical physics, Faculty f Science, Benha University, fr his helpful and valuable cmments. i d p REFERENCES [] C. Fetecau, Analytical slutins fr nn Newtnian fluid flws in pipe like dmains, Int. J. Nn Linear Mech., vl. 39(), pp. 5 3, March 004. [] Y. Yin, and K. Q. Zhu, Oscillating flw f a viscelastic fluid in a pipe with the fractinal Maxwell mdel, Appl. Math. Cmput. vl. 73(), pp. 3 4, Feb [3] I. Khan, K. Fakhar, and M. I. Anwar, Hydrmagnetic rtating flws f an Oldryd B fluid in a prus medium, Special Tpics and Review in Prus Media, vl. 3(), pp , 0. [4] M. Khan, and Zeeshan, MHD flw f an Oldryd B fluid thrugh a prus space induced by tth pulses, Chinese Physics Letters, vl. 8(8), 0: [5] D. G. Drake, Unsteady flw f classical viscus fluid thrugh a channel, Quart. J. Mech. Appl. Math., vl. 8, pp., 965. [6] G. K. Batchelr, An Intrductin t Fluid Mechanics, Cambridge University Press, 000. [7] P. Sambasiva Ra, and M. V. Ramana Murthy, The Mathematics Educatin, vl. XVIII, N., 984. [8] H. Ramkissn, C. V. Eswaran, and S. R. Majumdar, Unsteady flw f an elastic viscus fluid in tubes f unifrm crss sectin, Int. J. Nn Linear Mech., vl. 4(6), pp , 989. [9] T. Hayat, N. Ali, S. Asghar, and A. M. Siddiqui, Exact peristaltic flw in tubes with an endscpe, Appl. Math. and Cmput., vl. 8(), pp , Nv [0] M. Khan, T. Hayat, and S. Asghar, Exact slutin fr MHD flw f a generalized Oldryd B fluid with mdified Darcy's law, Int. J. Eng. Sci., vl. 44(5), pp , March 006. [] Bhagwan Singh, and N. K. Varshney, Effect f MHD visc elastic fluid (Oldryd) and prus medium thrugh a circular cylinder bunded by a permeable bed, Int. J. Math. Archive, vl. 3(8), pp. 9 97, August 0. [] S. Sreenadh, E. Sudhakara, and J. Prakash, MHD pulsatile flw f an Oldryd fluid in a channel f prus medium, Int. J. Adv. in Mech. and Autmbile Engg. (IJIME), vl. (), pp. 56 6, 04. [3] Santhsh Nallapu, and G. Radhakrishnamacharya, Jeffrey fluid flw thrugh prus medium in the presence f magnetic field in narrw tubes, Int. J. Eng. Math., vl. 04, pp. 8, June 04. [4] R. T. Balmer, and M. A. Firina, Unsteady flw f an inelastic pwer law fluid in a circular tube, J. Nn Newtnian Fluid Mech., vl. 7(), pp , 980. [5] T. K. Hung, Nnlinear characteristics f pulsatile flws, Prceedings f the 5th ICMMB, Blgnia, Italy, pp , 986. [6] K. D. Rahaman, and H. Ramkissn, Unsteady axial viscelastic pipe flws, J. Nn Newtnian Fluid Mech., vl. 57(), pp. 7 38, April 995.

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