The Tangential Force Distribution on Inner Cylinder of Power Law Fluid Flowing in Eccentric Annuli with the Inner Cylinder Reciprocating Axially
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1 Open Journal of Flud Dynamcs, 2015, 5, Publshed Onlne June 2015 n ScRes. The Tangental Force Dstrbuton on Inner Cylnder of Power Law Flud Flowng n Eccentrc Annul wth the Inner Cylnder Recprocatng Axally Junhe Ma, Yu We, Janan Zhang, Hongjan Lu, Xanjun Meng Enhanced Ol and Gas Recovery Key Laboratory of Mnstry of Educaton, Northeast Petroleum Unversty, Daqng, Chna Emal: @qq.com Receved 12 May 2015; accepted 12 June 2015; publshed 15 June 2015 Copyrght 2015 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract Based on the governng equatons of the nner cynder of the unsteady flow of the power law flud n eccentrc annul wth the nner cylnder recprocatng axally n bpolar coordnate system, the calculaton formulae of tangental force were establshed, and the relevant numercal calculaton method was gven. Takng the aqueous soluton of partally hydrolyed polyacrylamdes (HPAM) for examples, the tangental forces were calculated by usng the formulae and numercal calculaton method mentoned above; the curves of the tangental force on the wall of the nner cylnder of HPAM aqueous soluton were plotted; and the effects on the tangental force of the flow behavor ndex of the power law flud, the stroke and the stroke frequency of the nner cylnder were analyed. Keywords Tangental Force, Power Law Flud, Eccentrc Annul, Recprocate Axally 1. Introducton At present n our country, there s few theoretcal research on the tangental force on the wall of the nner cylnder of the power law flud flowng n eccentrc annul wth the nner cylnder recprocatng axally. Ths force for sucker rod eccentrc wear research has an mportant nfluence. Therefore, research on the tangental force has a practcal sgnfcance. Based on the governng equatons of the nner cynder of the unsteady flow of the How to cte ths paper: Ma, J.Z., We, Y., Zhang, J.N., Lu, H.J. and Meng, X.J. (2015) The Tangental Force Dstrbuton on Inner Cylnder of Power Law Flud Flowng n Eccentrc Annul wth the Inner Cylnder Recprocatng Axally. Open Journal of Flud Dynamcs, 5,
2 power law flud n eccentrc annul wth the nner cylnder recprocatng axally n the bpolar coordnate system [1], the calculaton formulae of tangental force are establshed; the relevant numercal calculaton method s gven; and takng the aqueous soluton of partally hydrolyed HPAM for examples, the tangental forces are calculated and analyed n ths artcle. 2. Mathematcal Models 2.1. Assumed Condton Power law flud wth the nner cylnder recprocatng axally, wt ( ) s axal velocty, T s the moton cycle, R o s the radus of the nner cylnder of annul, R s the radus of the nner cylnder, e s the eccentrcty of the eccentrc annulus Governng Equatons The governng equatons of the unsteady flow of the power law flud n eccentrc annul wth the nner cylnder recprocatng axally n the bpolar coordnate system by Haqng Cu, Tao Gao [1]-[3]. 2 η w w C w ( I2) η ( I2) P ξ ξ ζ ζ c s ρ + = + t (1a) 2.3. Intal Condtons and Boundary Condtons By assumpton flow condtons can be obtaned the boundary condtons of dynamc flow equaton. where ( ) ( ξζ,,0) ( ξζ, ) w = w p (1b) (,, ) sn ( 2π ) w ξ ζ t = W ft (1c) ( t ) w ξ, ζ, = 0 (1d) o η I 2 s the vscosty functon of the power law flud; I 2 s the second nvarant of the component of the frst order Rvln-Ercksen tensor; ξ, ζ are the dpolar coordnates, and ξ < 0, ζ 0 ; ρ s the densty of the power law flud; t s the tme; p s the pressure gradent; wp ( ξζ, ) s the velocty dstrbuton of the power law flud n eccentrc annul wth the motonless nner cylnder; W s the ampltude of the velocty of the nner cylnder recprocatng axally; f s the stroke frequency of the nner cylnder Tangental Force on the Wall of the Inner Cylnder From the formulae of shear stress on the wall of the nner cylnder ch ξ cos ζ τ ( I2) w ξ = η C ξ τ ch ξ cos ζ η( I ) w = C ζ ζ 2 (2) (3) τ = τ + τ (4) ξ ξ the calculaton formulae of tangental force are gven as follows whch can be expressed as where 0 ( ( ) ) π F = 2 R τ ξ, ζ, t cos θ d θ (5) π F = 2R f ( ζ ) dζ (6) 0 184
3 f ( ζ ) shξ chξ cosζ w chξ cosζ w = η( I2) + η( I2). ( cosζ chξ ) C ξ C ζ (7) 3. Calculaton Method From the densty of the power law flud ρ, the flow behavor ndex of the power law flud n, the consstency coeffcent of the power law flud k, the radus of the nner cylnder of annul R o, the radus of the nner cylnder R, the eccentrcty of the eccentrc annulus e, the stroke S, the stroke frequency of the nner cylnder f and the pressure gradent P, the dstrbuton of the tangental force F can be calculated: frstly, through the equatons (1a), (1b), (1c) and (1d), the velocty dstrbuton of the power law flud at anytme by usng the fnte dfference method s calculated; then, through the formula (6), the tangental force by usng the numercal ntegraton method s calculated. 4. Calculaton Examples Take HPAM aqueous soluton as an example, the radus of the nner cylnder of annulus R o = m, the radus of the nner cylnder R = m, the pressure gradent of the aqueous soluton P = Pa/m and by usng the calculaton method mentoned above, the tangental forces on the wall of the nner cylnder of the HPAM aqueous soluton are calculated and the relevant dstrbuton curves are plotted as Fgure 1 to Fgure 4 (the mnus of the tangental force ndcates that the nner cylnder moves upwards). Under dfferent flow behavor ndex n, the curves of the tangental force to the cycle number N are shown as Fgure 1, where N= tt. It shows that, wth the eccentrcty, the stroke and the stroke frequency of the nner cylnder beng gven as constants, as the flow behavor ndex n ncreases, the peak values (or valley values) of the tangental force F change a lot. Under dfferent eccentrctes e, the curves of the tangental force to the cycle number are shown as Fgure 2. It shows that, wth the flow behavor ndex n, the stroke S and the stroke frequency of the nner cylnder f beng gven as constants, as the eccentrcty e ncreases, the peak values (or valley values) of the tangental force F change a lot. Under dfferent strokes S, the curves of the tangental force to the cycle number are shown as Fgure 3. It shows that, wth the flow behavor ndex n, the eccentrcty e and the stroke frequency of the nner cylnder f beng gven as constants, as the strokes S ncreases, the peak values (or valley values) of the tangental force F change a lot. Fgure 1. The curves of the dstrbuton of the tangental force of the HPAM aqueous soluton under dfferent flow behavor ndex (ρ = 998 Kg/m 3, k = Pa s n, e = m, S = 1.0 m, f = H). 185
4 Fgure 2. The dstrbuton of the tangental force of the HPAM aqueous soluton of qualty concentraton 0.1% under dfferent eccentrctes (ρ = 998 Kg/m 3, n = 0.556, k = Pa s n, S = 1.0 m, f = H). Fgure 3. The dstrbuton of the tangental force of the HPAM aqueous soluton of qualty concentraton 0.1% under dfferent strokes (ρ = 998 Kg/m 3, n = 0.556, k = Pa s n, e = m, f = H). Under dfferent frequences f, the curves of the tangental force dstrbuton to the cycle number are shown as Fgure 4. It shows that, wth the flow behavor ndex n, the eccentrcty e and the stroke S beng gven as constants, as the stroke frequency f ncreases, the peak values (or valley values) of the tangental force F change a lot. 5. Summary 1) Based on the governng equatons of the nner cynder of the unsteady flow of the power law flud n eccentrc annul wth the nner cylnder recprocatng axally n the bpolar coordnate system, the calculaton formulae of tangental force on the wall of the nner cylnder of the power law flud flowng n eccentrc annul wth the nner cylnder recprocatng axally n the bpolar coordnate system were establshed, and the relevant numercal calculaton method was gven. 186
5 Fgure 4. The dstrbuton curves of tangental force of the HPAM aqueous soluton of qualty concentraton 0.1% under dfferent frequency (ρ = 998 Kg/m 3, n = 0.556, k = Pa s n, e = m, S = 1.0 m). 2) Takng the HPAM aqueous soluton for examples, by usng the calculaton formulae and method mentoned above, the tangental forces of the HPAM aqueous soluton on the wall of the nner cylnder were calculated and the tangental force curves were plotted. 3) The effects on the tangental force of the flow behavor ndex of the power law flud, the eccentrcty of the eccentrc annulus, the stroke and the stroke frequency of the nner cylnder were analyed; the nfluence of the eccentrcty was more obvous. References [1] Cu, H.Q., Sun, Z. and Gao, T. (2003) Velocty Dstrbuton of Unsteady Flow of Non-Newtonan Flud n Eccentrc Annul wth the Inner Cylnder Recprocatng Axally. Journal of Hydrodynamcs, 18, [2] Yang, Y.J., Cu, H.Q. and Gao, T. (2004) Flow Rate Dstrbuton of the Unsteady Flow of Power Law Flud n Eccentrc Annul wth Inner Cylnder Recprocatng Axally. Journal of Daqng Petroleum Insttute, 28, [3] Meng, X.J., Lu, H.J., Zhang, H.Z., Gao, T. and Cu, H.Q. (2013) The Normal Force Dstrbuton on the Wall of the Inner Cylnder of the Power Law Flud Flowng n the Eccentrc Annul wth the Inner Cylnder Recprocatng Axally. Appled Mechancs and Materals, 487,
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