Two Dimensional Inertial Flow of a Viscous Fluid in a Corner

Transcription

1 Applied Mathematical Sciences, Vol., 207, no. 9, HIKARI Ltd, Two Dimensional Inetial Flow of a Viscous Fluid in a Cone A. Mahmood and A.M. Siddiqui Depatment of Mathematics Pennsylvania State Univesity, Yok Campus 03 Edgecomb Avenue, Yok, PA 7403, USA Copyight c 207 A. Mahmood and A.M. Siddiqui. This aticle is distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Abstact This pape investigates the inetial flow of a viscous fluid between two intesecting planes. The nonlinea equations descibing the flow ae fomulated, and analytical solutions fo the fist and second-ode inetial effects ae obtained by using the ecusive appoach due to Langlois. A detailed analysis eflecting the effects of the vaiation of the angle of the scape on the flow is pesented. In addition, pessue, and the tangential and nomal stesses ae also computed. Finally, the diffeence between inetial and non-inetial flow behavio is shown by sketching the gaphs of velocities, stesses and steamlines. Mathematics Subject Classification: 76D05, 76D07 Keywods: Ceeping flow, Cone, Viscous fluid, Scape, Inetial flow Intoduction Scaping of fluids fom a solid suface such as cleaning of pipes that conduit fluids used in the pocessing of vaious foods is a common situation in eveyday life and in industial pocesses. Taylo scape, the mathematical model used to epesent the dynamics of the scaping of a viscous fluid was oiginally poposed by Taylo [0]. This model studies the flow of a viscous fluid in a cone made up of two intesecting planes whee one moves steadily ove the othe without taking into account the inetia foce.

2 408 A. Mahmood and A. M. Siddiqui Related flow poblems have been investigated by seveal authos. Dean & Montagnon [4] studied the flow induced in a cone with fixed sides by a geneal motion at a lage distance fom the cone. Late Moffatt [9] consideed the simila type of poblem and showed that thee exists an infinite sequence of eddies of deceasing size and stength, povided that the cone angle is less than some citical value. The non-inetial flow of a shea thinning fluid between intesecting planes was examined by Mansutti and Rajagopal [8]. It was shown that the shap and ponounced bounday layes develop adjacent to the solid boundaies, even at zeo Reynolds numbe. In addition, Andeson and Davis [] studied the two-dimensional viscous flow of two fluids in a wedge made up by one igid and one stess fee bounday. It is woth noting that in almost all these investigations of the cone flows no attention is paid to the flow of viscous fluids with inetia. Though the cone flow of viscous fluids in the pesence of inetia foce is one of the inteesting poblems in fluid mechanics, but thee ae only a few papes which deal with the flow of viscous fluids with inetia. The effects of inetia in foced cone flows wee investigated by Hancock et al. [5]. They obtained the fist-ode inetial effect analytically in the fom of a egula petubation seies fo the steam function, and pesented steamline plots indicating the fist influence at distances fom the cone whee inetia foces become significant. Hills & Moffatt [6] consideed the effects of inetia fo a thee-dimensional flow in a cone induced by the otation in its plane of one of the boundaies. A local similaity solution valid in a neighbohood of the cente of otation was obtained, and it was shown that the inetial effects wee significant in a small neighbohood of the plane of symmety of the flow. In this pape we extend the wok of Hancock et al. [5] and obtain the secondode inetial effect analytically. In addition, we pesent the expessions fo pessue, and nomal and tangential stesses fo both inetial and non-inetial flows which wee absent in thei wok. We also study the effects of vaying the angle of the scape on the flow. The pesence of inetia foces give ise to cetain non-linea patial diffeential equations whose solution is obtained by employing the ecusive appoach intoduced by Langlois [7]. This appoach was basically poposed fo slow and steady viscoelastic flows fo which fluid inetia was neglected thoughout. Hee we tied to use this appoach fo the inetial flow and have shown that this appoach can be applied to flows including fluid inetia. Ou esults subsume Taylo s and Hancock et al. esults as special cases when the zeoth and fist ode solutions ae consideed espectively. Finally, the diffeence between inetial and non-inetial flow behavio is shown by sketching the gaphs of velocities, stesses and steamlines.

3 Two dimensional inetial flow Basic Flow Equations The basic equations govening the motion of an incompessible steady viscous fluid, neglecting themal effects and body foces ae as follows: divv = 0. (2.) ρ (V. ) V = divt, (2.2) Hee V is the velocity vecto, ρ is the constant density of the fluid, and T is the Cauchy stess tenso given by T = pi + µa (2.3) whee µ is the viscosity, p is the pessue, I is the unit tenso, and A = V + ( V) T (2.4) the fist Rivlin Eicksen tenso. Following the Langlois Recusive Appoach [7], we expand the velocity V, and pessue p in the following fom V = ɛv () + ɛ 2 V (2) + ɛ 3 V (3) (2.5) p = constant + ɛp () + ɛ 2 p (2) + ɛ 3 p (3) (2.6) whee ɛ is a small dimensionless constant, and neglect all tems of the fouth o highe ode in ɛ. Fom Eqs. (2.3) and (2.4), ignoing the constant pessue the stess tenso then takes the fom whee T = ɛ[ p () + µa () ] + ɛ 2 [ p (2) + µa (2) ] + ɛ 3 [ p (3) + µa (3) ] (2.7) A () = V () + ( V () ) T A (2) = V (2) + ( V (2) ) T A (3) = V (3) + ( V (3) ) T Using Eqs. (2.5), (2.6), and (2.7), the equation of continuity and momentum then becomes ɛ.v () + ɛ 2.V (2) + ɛ 3.V (3) = 0, (2.8) ɛ 2 ρ ( V (). V ()) + ɛ 3 ρ ( V (). V (2) + V (2). V ()) ) = ɛ (µ.a () p () +ɛ 2 ( p (2) + µ.a (2) ) + ɛ 3 ( p (3) + µ.a (3) ) (2.9)

4 40 A. Mahmood and A. M. Siddiqui 3 Fomulation of the poblem Conside the flow of a viscous fluid nea the cone made by two igid boundaies intesecting at a constant angle θ 0. Assume that one of the boundaies is sliding steadily with velocity ɛu at the othe. Both adial and tangential velocities ae futhe assumed to be zeo at the stationay bounday, we call the hoizontal moving bounday a plate and the fixed bounday at θ = θ 0, the scape. We choose the plane pola coodinates (, θ) to study the flow behavio of the fluid between the boundaies. The velocity field fo the poblem unde consideation then has the fom and the bounday conditions fo the poblem ae and V = u(, θ)e + v(, θ)e θ (3.) u = ɛu, v = 0 at θ = 0, (3.2) u = 0, v = 0 at θ = θ 0, (3.3) Again following the ecusive appoach, we assume the velocity components u and v as of the fom u = ɛu () + ɛ 2 u (2) + ɛ 3 u (3) (3.4) v = ɛv () + ɛ 2 v (2) + ɛ 3 v (3) (3.5) Eqs. (3.) (3.3) then takes the fom V = ɛ [ ] u () (, θ)e + v () (, θ)e [ ] θ + ɛ 2 u (2) (, θ)e + v (2) (, θ)e θ +ɛ [ ] 3 u (3) (, θ)e + v (3) (, θ)e θ (3.6) ɛu () + ɛ 2 u (2) + ɛ 3 u (3) = ɛu, ɛv () + ɛ 2 v (2) + ɛ 3 v (3) = 0 at θ = 0 (3.7) ɛu () + ɛ 2 u (2) + ɛ 3 u (3) = 0, ɛv () + ɛ 2 v (2) + ɛ 3 v (3) = 0 at θ = θ 0 (3.8) Equating coefficients of ɛ, ɛ 2, ɛ 3 fom Eqs. (2.8) (2.9), and (3.7) (3.8), we obtain the following fist, second, and thid ode bounday value poblems O(ɛ):.V () = 0. (3.9) µ.a () p () = 0 (3.0) u () = U, v () = 0 at θ = 0, (3.) u () = 0, v () = 0 at θ = θ 0, (3.2)

5 Two dimensional inetial flow 4 O(ɛ 2 ): and O(ɛ 3 ):.V (2) = 0. (3.3) ρ ( V (). V ()) = p (2) + µ.a (2) (3.4) u (2) = 0, v (2) = 0 at θ = 0, (3.5) u (2) = 0, v (2) = 0 at θ = θ 0, (3.6).V (3) = 0. (3.7) ρ ( V (). V (2) + V (2). V ()) = p (3) + µ.a (3) (3.8) u (3) = 0, v (3) = 0 at θ = 0, (3.9) u (3) = 0, v (3) = 0 at θ = θ 0, (3.20) 3. Fist Ode System (Non-inetial flow) Recalling that the fist ode velocity and pessue fields fo the poblem unde consideation have the fom V () = u () (, θ)e + v () (, θ)e θ, p () = p () (, θ) (3.2) Then equations (3.9) and (3.0) in tems of pola coodinates, assumes the fom whee u () + u() + v () p () p () = 0, (3.22) = µ Ω (), (3.23) = µ Ω(), (3.24) u () Ω () = v() + v() is the non-zeo component of the voticity tenso. Intoducing the fist ode steam function ψ () (, θ) such that u () = ψ () (3.25), v() = ψ(), (3.26) and then eliminating the pessue fom Eqs. (3.23) and (3.24), we end up with the following fist ode system fo ceeping viscous flow 4 ψ () = 0, (3.27)

6 42 A. Mahmood and A. M. Siddiqui with bounday conditions ψ () ψ () = U, = 0, ψ () ψ () = 0, at θ = 0 (3.28) = 0, at θ = θ 0 (3.29) 3.2 Solution of the fist ode system (Non-inetial flow) As discussed in [0], the solution of the system (3.27) (3.29) is given by whee ψ () = Uf (θ) (3.30) f (θ) = θ0 2 sin 2 {θ0 2 sin θ θ cos θ sin 2 θ 0 (θ 0 sin θ 0 cos θ 0 )θ sin θ} θ 0 (3.3) By using Eqs. (3.25), (3.26), and (3.3), the velocity components u (), v () and voticity Ω () in tems of f (θ), espectively, can be obtained in following foms: and whee u () = (Uf ) = Uf, v () = (Uf ) Ω () = U f + f = U g = Uf (3.32) (3.33) g(θ) = f + f. (3.34) The pessue field obtained fom Eqs. (3.23) and (3.24), has the fom p () (, θ) = p 0 µ Ug (3.35) whee p 0 is a positive constant detemined by conditions fa fom the cone. Hee we note that pessue becomes singula as 0. The nomal and tangential stesses T n () and T () t to the scape at distance fom the point of contact ae then obtained as T () n = µ Ug (3.36) and T () t = µ Ug (3.37) We now tun to examine the fist ode inetial effects on the flow nea the cone.

7 Two dimensional inetial flow Second Ode System (The fist inetial coection) We now conside the equations of motion (3.3) and (3.6), which include the inetia foce and study its effects on the flow unde consideation. Hee we assume that the second ode velocity and pessue fields ae of the fom V (2) = u (2) (, θ)e + v (2) (, θ)e θ, p (2) = p (2) (, θ) (3.38) Equations of motion (3.3) and (3.6) fo the second ode system then takes the fom whee p (2) p (2) 0 = u(2) + u(2) = ρv (2) Ω (2) µ = µ Ω(2) p (2) = p (2) + ρ 2 + Ω (2) v (2), (3.39), (3.40) + ρu (2) Ω (2), (3.4) ( (u (2) ) 2 + (v (2) ) 2) (3.42) is the modified pessue. Now, we eliminate the pessue fom Eqs. (3.40) and (3.4) by coss diffeentiation, and intoduce the second ode steam function ψ (2) (, θ) such that u (2) = ψ (2), v(2) = ψ(2), (3.43) by vitue of which we get the following second ode system of equations ν 4 ψ (2) = (ψ (), 2 ψ () ), (3.44) (, θ) ψ (2) ψ (2) = 0, = 0, ψ (2) ψ (2) = 0, at θ = 0 (3.45) = 0, at θ = θ 0 (3.46) 3.4 Solution of the Second ode system In ode to solve the second ode system (3.44) (3.46), we assume the solution of the fom ψ (2) = 2 U 2 ν f 2(θ) (3.47) Substituting values of ψ and ψ 2 fom Eqs. (3.30) and (3.47) into Eqs. (3.44) (3.46), we get the fouth ode odinay diffeential equation of the fom

8 44 A. Mahmood and A. M. Siddiqui f iv 2 + 4f 2 = β sin 2θ + β 2 cos 2θ + β 3 θ sin 2θ + β 4 θ cos 2θ, (3.48) whee β s ae the known constants and thei expessions will be given late. The coesponding bounday conditions on f 2 ae f 2 = 0, f 2 = 0 at θ = 0 (3.49) f 2 = 0, f 2 = 0 at θ = θ 0. (3.50) The solution of the Eq. (3.48) is then found by elementay techniques in the fom f 2 (θ) = R + R 2 θ + R 3 cos 2θ + R 4 sin 2θ + β 5 θ cos 2θ +β 6 θ sin 2θ + β 7 θ 2 cos 2θ + β 8 θ 2 sin 2θ, (3.5) whee R s and β s, depending upon angle θ 0, ae constants that ae given by, R = 2X(θ 0) 2X (θ 0 )θ 0 (β 5 X (θ 0 )) sin 2θ 0 2(X(θ 0 ) β 5 θ 0 ) cos 2θ 0, 4(cos 2θ 0 + θ 0 sin 2θ 0 ) R 2 = X (θ 0 ) + β 5 (X (θ 0 ) + β 5 ) cos 2θ 0 2X(θ 0 )) sin 2θ 0 ), 2(cos 2θ 0 + θ 0 sin 2θ 0 ) R 3 = 2X(θ 0) 2X (θ 0 )θ 0 (β 5 X (θ 0 )) sin 2θ 0 2(X(θ 0 ) β 5 θ 0 ) cos 2θ 0, 4(cos 2θ 0 + θ 0 sin 2θ 0 ) R 4 = β 5 X (θ 0 ) + (X (θ 0 ) β 5 ) cos 2θ 0 + 2(X(θ 0 ) β 5 θ 0 ) sin 2θ 0, 4(cos 2θ 0 + θ 0 sin 2θ 0 ) k = θ 0 sin θ 0 cos θ 0, β = sin4 θ 0 k 2 2θ 2 0 sin 2 θ 0 (θ 2 0 sin 2 θ 0 ) 2, β 2 = 2k θ 2 0 sin 2 θ 0, β 3 = 4k sin2 θ 0 β 5 = β β 4, β 6 = (θ0 2 sin 2 θ 0 ), β 2 4 = 2 sin4 θ 0 k 2 (θ0 2 sin 2 θ 0 ) ( 2 β ) 64 β 3, β 7 = 32 β 3, β 8 = 32 β 4, X(θ 0 ) = β 5 θ 0 cos 2θ 0 + β 6 θ 0 sin 2θ 0 + β 7 θ 2 0 cos 2θ 0 + β 8 θ 2 0 sin 2θ 0. X (θ 0 ) = β 5 cos 2θ 0 2β 5 θ 0 sin 2θ 0 + β 6 sin 2θ 0 + 2β 6 θ 0 cos 2θ 0 + 2β 7 θ 0 cos 2θ 0 2β 7 θ0 2 sin 2θ 0 + 2β 8 θ 0 sin 2θ 0 + 2β 8 θ0 2 cos 2θ 0. (3.52) Hee, we notice that though we have used a diffeent appoach to find the solution fo fist inetial coection, but ou esults match with those obtained by Hancock et al. [5]. The velocity components u (2), v (2) and voticity Ω (2) in tems of f 2 (θ), espectively, can be obtained by using Eqs. (3.43) and (3.47) in the following

9 Two dimensional inetial flow 45 fom: and u (2) = ( 2 U 2 ν f 2 ) = U 2 ν f 2, v (2) = ( 2U 2 f ν 2 ) = 2U 2 ν f 2 (3.53) Ω (2) = U 2 ν [4f 2 + f 2 ] (3.54) The second ode pessue field obtained fom Eqs.(3.40) (3.4), togethe with Eq. (3.42), then has the fom p (2) (, θ) = ρu [ 4 2 2f 2 ν f 2 f 2 f ] µu 2 2 ν ln [4f 2 + f 2 ] (3.55) whee f 2 is given by (3.5). Hee we notice that the second ode pessue field has a logaithmic singulaity as 0. The nomal and tangential stesses T n (2) and T (2) t ae calculated as T (2) n = p (2) 2 µu 2 ν f 2, (3.56) T (2) t = µu 2 ν f 2. (3.57) 3.5 Thid Ode System (The second inetial coection) In ode to obtain the second-ode inetial effect analytically, ecalling that the thid ode velocity and pessue fields fo the poblem unde consideation have the fom V (3) = u (3) (, θ)e + v (3) (, θ)e θ, p (3) = p (3) (, θ) (3.58) we wite the equations of motion (3.7) and (3.20) in the plane pola coodinates fom as whee p (3) p (3) is the modified pessue. 0 = u(3) + u(3) + v (3) = ρ [ v () Ω (2) + v (2) Ω ()] µ = µ Ω(3), (3.59) Ω (3), (3.60) ρ [ u () Ω (2) + u (2) Ω ()], (3.6) p (3) = p (3) + ρ ( u () u (2) + v () v (2)) (3.62)

10 46 A. Mahmood and A. M. Siddiqui Eliminating the pessue fom Eqs. (3.60) and (3.6) by coss diffeentiation, and then intoducing the thid ode steam function ψ (3) (, θ) such that u (3) = ψ (3), v(3) = ψ(3), (3.63) we get the following thid ode system of equations ν 4 ψ (3) = [ ] (ψ (), 2 ψ (2) ) + (ψ(2), 2 ψ () ), (3.64) (, θ) (, θ) with bounday conditions ψ (3) = 0, ψ (3) = 0, ψ (3) ψ (3) = 0, at θ = 0 (3.65) = 0, at θ = θ 0 (3.66) whee ψ and ψ 2 ae aleady obtained, and thei expessions ae given in Eqs. (3.30) and (3.47). 3.6 Solution of the thid ode system Fo the thid ode system (3.64), we assume the following fom of the steam function ψ (3) = 3 U 3 ν 2 f 3(θ) (3.67) Using (3.67) togethe with expessions fo ψ and ψ 2 fom Eqs. (3.30) and (3.47), the linea patial diffeential equation given by Eq. (3.64), afte a fai amount of calculations, is conveted to the fouth ode odinay diffeential equation of the fom f iv 3 + 0f 3 + 9f 3 = M cos 3θ + M 2 sin 3θ + M 3 cos θ + M 4 sin θ + M 5 θ sin 3θ +M 6 θ cos 3θ + M 7 θ cos θ + M 8 θ sin θ + M 9 θ 2 sin θ +M 0 θ 2 cos θ + M θ 2 sin 3θ + M 2 θ 2 cos 3θ, (3.68) whee M s ae known constants and thei expessions ae given in Eq. (3.72). Also, the bounday conditions (3.65) (3.66) ae educed to f 3 = 0, f 3 = 0 at θ = 0 (3.69) f 3 = 0, f 3 = 0 at θ = θ 0. (3.70) Afte a consideable amount of wok, solution of the Eq. (3.68) using bounday conditions (3.69) (3.70) is then obtained as f 3 (θ) = C cos θ + C 2 sin θ + C 3 cos 3θ + C 4 sin 3θ + M 3 θ cos 3θ + M 4 θ sin 3θ + M 5 θ cos θ + M 6 θ sin θ + M 7 θ 2 sin θ + M 8 θ 2 cos θ + M 9 θ 2 sin 3θ + M 20 θ 2 cos 3θ, (3.7)

11 Two dimensional inetial flow 47 whee constants C s and M s ae given by C = Y (θ 0 ) sin θ 0 3Y (θ 0 ) cos θ 0 + (M 3 + M 5 ) sin 2θ 0 4 sin 2 θ 0, C 2 = 3Y (θ 0 ) cos θ 0 sin 2 θ 0 + 3Y (θ 0 ) sin θ 0 ( 3 cos 2 θ 0 ) + (M 3 + M 5 )(cos 2 θ 0 2 cos 4 θ 0 + ) 4 sin 4 θ 0, C 3 = Y (θ 0 ) sin θ 0 3Y (θ 0 ) cos θ 0 + (M 3 + M 5 ) sin 2θ 0 4 sin 2 θ 0, C 4 = Y (θ 0 ) cos θ 0 sin 2 θ 0 + Y (θ 0 ) sin θ 0 ( 3 cos 2 θ 0 ) (M 3 + M 5 )(2 cos 4 θ 0 3 cos 2 θ 0 + ) 4 sin 4 θ 0 θ0 2 B = θ0 2, C = sin2 θ 0 k sin2 θ 0 θ0 2, D = sin2 θ 0 θ0 2 sin2 θ 0 M = 4CR 3 6Bβ 7 Cβ 6 Dβ 5 4Bβ 6 4DR 4, M 2 = 4Bβ 5 + Cβ 5 6Bβ 8 Dβ 6 + 4CR 4 + 4DR 3, M 3 = 4Bβ 6 + Cβ 6 Dβ 5 + 6Bβ 7 + 4CR 2DR 2 M 4 = 6Bβ 8 4Bβ 5 Cβ 5 Dβ 6 4BR 2 + 2CR 2 + 4DR, M 5 = 8Dβ 5 + 8Bβ 7 + 8Cβ 6 + 8Cβ 7 8Dβ 8, M 6 = 8Cβ 5 8Bβ 8 8Dβ 6 8Cβ 8 8Dβ 7, M 7 = 4Cβ 5 + 8Bβ 8 + 4Dβ 6 4Cβ 8 + 4Dβ 7, M 8 = 4Cβ 6 4Dβ 5 8Bβ 7 + 4Cβ 7 + 4Dβ 8, M 9 = 8Cβ 8 8Dβ 7, M 0 = 8Cβ 7 + 8Dβ 8, M = 2Cβ 8 + 2Dβ 7, M 2 = 2Cβ 7 2Dβ 8, M 3 = ( M 2 ) 48 2 M 6, M 4 = ( M + ) 48 2 M 5, M 5 = ( ) 6 4 M 7 M 4, M 6 = ( M 3 + ) 6 4 M 8, M 7 = 32 M 7, M 8 = 32 M 8, M 9 = 96 M 6, M 20 = 96 M 5, Y (θ 0 ) = M 3 θ 0 cos 3θ 0 + M 4 θ 0 sin 3θ 0 + M 5 θ 0 cos θ 0 + M 6 θ 0 sin θ 0 + M 7 θ 2 0 sin θ 0 + M 8 θ 2 0 cos θ 0 + M 9 θ 2 0 sin 3θ 0 + M 20 θ 2 0 cos 3θ 0, Y (θ 0 ) = M 3 cos 3θ 0 3M 3 θ 0 sin 3θ 0 + M 4 sin 3θ 0 + 3M 4 θ 0 cos 3θ 0 + M 5 cos θ 0 M 5 θ 0 sin θ 0 + M 6 sin θ 0 + M 6 θ 0 cos θ 0 + 2M 7 θ 0 sin θ 0 + +M 7 θ 2 0 cos θ M 8 θ 0 cos θ 0 M 8 θ 2 0 sin θ M 9 θ 0 sin 3θ 0 + 3M 9 θ 2 0 cos 3θ 0 + 2M 20 θ 0 cos 3θ 0 3M 20 θ 2 0 sin 3θ 0. (3.72) The velocity components u (3), v (3) and voticity Ω (3) in tems of f 3 (θ), espectively, can be obtained in following foms: and u (3) = ( ) 3 U 3 ν f 2 3 = 2 U 3 ν f 3, v (3) = ( ) 3 U 3 2 ν f 2 3 = 32 U 3 f ν 2 3, (3.73) Ω (3) = U 3 ν 2 [9f 3 + f 3 ], (3.74) whee f 3 is given by Eq. (3.7). The pessue field obtained fom Eq.(3.62) togethe with Eqs.(3.60) (3.6),

12 48 A. Mahmood and A. M. Siddiqui has the fom p (3) (, θ) = µu 3 [9f ν f 3 ] + ρu 3 ν [4f f 2 + f f 2 f f 2 + 2f f 2 ], (3.75) whee f, f 2, and f 3 ae given by Eqs. (3.3), (3.5), and (3.7) espectively. Hee we notice that unlike the fist and second-ode pessue fields, which have singulaities at = 0, the pessue field found hee is defined fo all. Finally, the nomal and tangential stesses, T n (3) and T (3) t to the scape ae obtained as and T (3) n = p (3) 4 µu 3 ν 2 f 3, (3.76) T (3) t = µu 3 ν 2 (f 3 3f 3 ). (3.77) 4 Combined solution fo the inetial flow In this section, we combine the expessions of steam functions, velocity components, pessue fields and, nomal and shea stesses fo all the odes found in pevious sections, to give the combined solution fo the inetial flow. Combining the expessions fo ψ (i) s, u (i) s, v (i) s, p (i) s, T (i) t s, and T n (i) s, ( i 3), found in pevious sections we get ψ = ɛψ () + ɛ 2 ψ (2) + ɛ 3 ψ (3), (4.) u = ɛu () + ɛ 2 u (2) + ɛ 3 u (3), (4.2) v = ɛv () + ɛ 2 v (2) + ɛ 3 v (3), (4.3) p = constant + ɛp () + ɛ 2 p (2) + ɛ 3 p (3), (4.4) T t = ɛt () t + ɛ 2 T (2) t + ɛ 3 T (3) t, (4.5) T n = ɛt n () + ɛ 2 T n (2) + ɛ 3 T n (3). (4.6) Taking ɛ, and making use of Eqs. (3.30), (3.32), (3.35), (3.36), (3.37), (3.47), (3.53), (3.55), (3.56), (3.57), (3.67), (3.73), (3.75), (3.76), and (3.77) we get ψ = Uf + U 2 2 u = Uf + U 2 v = (Uf + 2 U 2 ν f 2 + U 3 3 ν 2 f 3, (4.7) ν f U 3 ν 2 f 3, (4.8) ν f U 3 ν 2 f 3), (4.9)

13 Two dimensional inetial flow 49 p = p 0 µ Ug + ρu 4 2 ν 2 [ 2f2 2 + f 2 f 2 f ] + µu 2 ν ln [4f 2 + f 2 ] + µu 3 [9f ν f 3 ] + ρu 3 [4f f 2 + f f 2 f ν f 2 + 2f f 2 ], (4.0) ( U 2 T t = p 2µ T n = µ ( U g + U 2 ν f ν f U 3 ν 2 f U 3 ν ), (4.) ). (4.2) 2 [f 3 3f 3 ] whee f, g, f 2, and f 3 ae espectively given by Eqs. (3.3), (3.34), (3.5) and (3.7). Hee we notice that by taking ν = ρ µ non-inetial flow obtained by Taylo. = 0, we ecove the esults fo 5 Discussion on Nomal and Shea stesses Let T and T denote espectively, the components of the total stess paallel and pependicula to the plate. Then T and T ae given by T = T n cos θ 0 + T t sin θ 0, (5.) T = T n sin θ 0 T t cos θ 0. (5.2) whee tangential and nomal stesses, T t and T n espectively, ae given by Eqs. (4.) and (4.2). We now constuct tables of values to see the effects of T n, T t, T and T fo a set of values of angle θ 0 fo both non-inetial and inetial flows.

14 420 A. Mahmood and A. M. Siddiqui Table Table 2 θ o 0 T n 2µU T t 2µU T 2µU T 2µU θ o 0 T n 2µU T t 2µU T 2µU T 2µU Table and 2 espectively give the values of T n, T t, T, and T, divided by 2µU/ fo a set of diffeent values of θ 0 fo non-inetial and inetial flows. We can see that fo non-inetial flow, T deceases with an incease in θ 0, and attains its least value when θ 0 = π, but the least value of T fo inetial flow occus at θ 0 = 5π/8. It is inteesting to obseve that fo non-inetial flow, T does not change sign in the ange 0 < θ 0 < π. But fo the inetial flow, we can see that T changes sign in the ange 7π/2 < θ 0 < π/2. We futhe see that fo non-inetial flow, the only singulaity in the stess filed occus at θ 0 = 0, wheeas we have two such points fo inetial flow, i.e at θ 0 = 0, and at θ = π. This means that we cannot scap the fluid fo inetial flow if the scape is held at these two positions, as an infinite foce will be equied thee. A human intuitively holds the scape in this position to scape the fluid easily. But this type of analysis is necessay fo the engineeing point of view, because fo doing the wok on bulk basis, we need machines like obots, bulldozes etc. fo the scaping puposes. 6 Gaphs and Discussion In this section some gaphs ae displayed fo the behavio of velocity, stesses, and steamlines fo both inetial and non-inetial flows. Figs.(6.) and (6.2) pesents the behavio of velocity components u and v against the adial distance fo the inetial flow. It is obseved that the effect of the inetial foces nea the cone is small and the ise in the velocity components is less thee. While as we ecede the cone, the contibutions of the inetia foces comes into

15 Two dimensional inetial flow 42 play and consequently, and a much ise is seen in the velocity gaphs. Physically it is expected because the boundaies ae close to each othe nea the cone, and hence viscous foces dominate thee. Theefoe, fo low Reynolds numbe, the flow stats to ceep and so velocity deceases. In mathematical analysis, we noted that the velocity components fo the non-inetial flow become independent of the adial distance, so no change is noticed in the velocity against the adial distance fo the non-inetial flow. Howeve, the velocity components u and v ae dawn against the velocity of the moving bounday in Figs. (6.3) and (6.4). And it is obseved that, as expected, both the components incease in the negative diection with the bounday velocity U. ν = 0.2, θ 0 = π/3, U = 5 ν = 0.2, θ 0 = π/3, U = = 0 0 = = = u 0 v = = θ Figue 6.: vaiation of velocity component u with adial distance fo inetial flow θ Figue 6.2: vaiation of velocity component v vs adial distance fo inetial flow. θ 0 = π/3 θ 0 = π/ U = U = 5 u () 2 v () - 0 U = U = 5 U = U = θ Figue 6.3: vaiation of velocity component u () with U fo non-inetial flow θ Figue 6.4: vaiation of velocity component v () with U fo non- inetial flow.

16 t 422 A. Mahmood and A. M. Siddiqui 0 ν = 0.2, µ = 0.0, U = 0 ν = 0.2, µ = 0.0, U = T n 5 T () n θ 0 = π/6, π/4, π/3 4 3 θ 0 = π/6, π/4, π/ Figue 6.5: nomal stess vs angle θ 0 fo inetial flow at fixed bounday. Figue 6.6: nomal stess vs angle θ 0 fo non-inetial flow at fixed bounday. 0.5 ν = 0.2, µ = 0.0, U = 0.5 ν = 0.2, µ = 0.0, U = T t 0.25 θ 0 = π/6, π/4, π/3 T () 0.25 θ 0 = π/6, π/4, π/ Figue 6.7: tangential stess vs angle θ 0 fo inetial flow at fixed bounday. Figue 6.8: tangential stess vs angle θ 0 fo non-inetial flow at fixed bounday. In Figs. (6.5) and (6.7) nomal and tangential stesses at the fixed bounday fo the inetial flow ae sketched against angle θ 0. Hee we see that stesses decease as we incease θ 0. It is futhe noticed that vey close to the cone i.e fo 0, stesses shoots up vey apidly. It is a good ageement with mathematical analysis, because thee exist a singulaity at this point in the stess field. In Figs. (6.6) and (6.8) simila behavio has been seen fo tangential and nomal stesses fo non-inetial flow, but the diffeence is that, fo non-inetial flow, values of tangential stesses ae highe wheeas the values of nomal stesses ae smalle than those fo non-inetial flow, fo the simila kind of data.

17 Two dimensional inetial flow 423 Figue 6.9: steamlines with θ 0 = π/2; solid cuves ae steamlines of non-inetial flow ψ = constant; dashed lines ae the steamlines (including the fist inetial coection) ψ + ψ 2 =constant. Figue 6.0: steamlines with θ 0 = π/2; solid cuves ae steamlines ψ =constant of non-inetial flow; dashed lines ae the steamlines ψ + ψ 2 + ψ 3 = constant, including the fist and second inetial coections. Finally, Figs. (6.9) and (6.0) espectively compae the steamlines ψ = constant (solid cuves) fo non-inetial flow, and steamlines ψ +ψ 2 =constant (dashed lines) including the fist inetial coection, and ψ +ψ 2 +ψ 3 = constant (dashed lines) including the fist and second inetial coections. We obseve that the steamlines tend to get compessed towads the fixed bounday θ = θ 0, unde the influence of inetia. 7 Conclusion In the pesent wok, cone flow of a viscous fluid in the pesence of inetia has been analyzed, and the esulting nonlinea patial diffeential equations ae solved to obtain the expessions fo the velocity field, pessue field and stess field. By the compaison of inetial and non-inetial flows, we came to the conclusion that close to the cone the effects of inetia can be neglected. But as we move away, inetia foces dominate and cannot be ignoed, and this analysis is an attempt to thow some light on the impotance of inetia foces fo such flows. Refeences [] D.M. Andeson and S.H. Davis, Two-fluid viscous flow in a cone, J. Fluid Mech., 257 (993),

18 424 A. Mahmood and A. M. Siddiqui [2] G.K. Batchelo, An Intoduction to Fluid Dynamics, Cambidge Univesity Pess, 967. [3] R.B. Bid, R.C. Amstong and O. Hassage, Dynamics of Polymeic Liquids, Vol., John Wiley & Sons, 987. [4] W.R. Dean and P.E. Montagnon, On the steady motion of viscous liquid in a cone, Mathem. Poc. Camb. Phil. Soc., 45 (949), [5] C. Hancock, E. Lewis and H.K. Moffat, Effects of inetia in foced cone flows, J. Fluid Mech., 2 (98), [6] C.P. Hills and H.K. Moffatt, Rotay honing: a vaiant of the Taylo paintscape poblem, J. Fluid Mech., 48 (2000), [7] W.E. Langlois, A ecusive appoach to the theoy of slow, steady-state viscoelastic flow, Tansactions of the Society of Rheology, 7 (963), [8] D. Mansutti and K.R. Ramgopal, Flow of a shea thinning fluid between intesecting planes, Int. Non-Linea Mech., 26 (99), [9] H.K. Moffat, Viscous and esistive eddies nea a shap cone, J. Fluid Mech., 8 (964), [0] G.I. Taylo, On scaping viscous fluid fom a plane suface, in Miszellangen de Angewandten Mechanik, (ed. M. Schafe), Akademie-Velag, Belin, 962, Received: Januay 2, 207; Published: Febuay 4, 207