Viscoelastic Catenary

Transcription

1 Viscoelasic Caenary Anshuman Roy 1 Inroducion This paper seeks o deermine he shape of a hin viscoelasic fluid filamen as i sags under is own weigh. The problem is an exension of he viscous caenary [1] and we refer o his problem as viscoelasic caenary. Viscoelasic filamens appear in applicaions such as fiber processing from mels and soluions, exensional rheomery ec. An undersanding of he dynamics of he viscoelasic caenary will herefore aid in beer design of such applicaions. Experimenal Observaions We invesigaed a Boger fluid composed of 0.05% w/w Polysyrene of molecular weigh x 10 6 dissolved in syrene oil. The relaxaion ime for his fluid is around 4 seconds and is zero-shear viscosiy, η 0, is 50 Pa.s. There is no shear hinning in he fluid over several decades of srain rae, and expecially in he regime of our experimens. We ook some fluid beween wo plaes and sreched ou in he horizonal direcion o shape i ino a hin filamen, h << L, where h is he hickness of he filamen and L is he lengh o which i is sreched. Figure 1) shows a snapsho of one such experimen wih h = 0.00 m, L = 0.05 m. Two problems emerged ou of his experimen ha need o be undersood. Firs is he problem of he viscoelasic caenary, wherein he fluid filamen sags under is own weigh and is shape evolves wih ime. Second, is wha we refer o as he chewing-gum problem. In his problem, fluid beween wo plaes is sreched ou ino a hin filamen and hen insananeously, he wo plaes are brough closer ogeher. This makes he filamen buckle in he direcion of graviy, hereby making a viscoelasic caenary o begin wih. Wha happens hen is, o our knowledge, a phenomenon unique o viscoelasic fluids only - he caenary sars moving upwards agains graviy, like a recoil. However, if he plaes are brough ogeher a a rae equivalen o he inverse of he relaxaion ime of he fluid, we do no see his recoil effec. We refer o his effec as he chewing-gum problem because we observed he effec for he firs ime in a chewing-gum. 166

2 x H Figure 1: Snapsho of a viscelasic caenary.in his case, he fluid used is a mixure of Polysyrene MW x 10 6 ) in syrene oil - a Boger fluid. The zero shear viscosiy is approximaely 50 Pa.s and here is no shear hinning for he shear raes under consideraion.squares in he background are 1 mm in dimension. 3 Governing Equaions All he dynamics of a viscoelasic filamen can be undersood by considering he simpler problem of a D shee. The analysis ha we will presen is applicable o many viscoelasic sysems ha do no shear-hin. However, for now consider a soluion of polymer molecules in a viscous solven. The governing equaions for he fluid are conservaion of mass, conservaion of momenum and he closure model o describe he polymer sress wihin he fluid:.u = 0 1) ρ u + u. u) = p + µ u +.τ ρg ) τ + u. τ u) T.τ τ. u) = 1 τ G) 3) λ where, he subscrips represen differeniaion wih respec o he subscriped variables. τ is he polymer sress ensor, G is he equilibrium polymer sress and λ is he relaxaion ime of he polymer molecules. Noe ha eqs. 1) and ) no closed wihou eq. 3) which describes he evoluion of polymer sress in he flow, referred o as he Oldroyd-B consiuive model. In order o make he above equaions dimensionless, we choose a velociy scale U and a lengh scale L. Then, he scaling for ime is L/U, where L is he lengh of he shee beween he clamps. We scale he pressure and polymer sress wih µ U L. We perform he 167

3 following expansion: u = u 0 + ɛ u + Oɛ 4 ) ɛv = v 0 + ɛ v + Oɛ 4 ) H = ɛh 0 + ɛ 3 H + Oɛ 4 ) h = ɛh 0 + ɛ 3 h + Oɛ 4 ) p = p 0 + ɛ p + Oɛ 4 ) τ = τ 0 + ɛ τ + Oɛ 4 ) 4) Then he governing equaions for a D shee become, ɛ u x + v y = 0 5) ɛ 4 Re u + ɛ uu x + vu y ) = ɛ p x + ɛ u xx + u yy + ɛ τ x 6) ɛ 4 Re v + ɛ uv x + vv y ) = ɛ p y + ɛ v xx + v yy ɛ 4 ϖ 7) ρg µu/l where, Re = ρul/µ is he Reynolds number and ϖ = is he dimensionless weigh variable. In he limi of Ca = µu/γ >> 1, he effecs due o surface ension can be ignored. So, we consider racion-free boundaries. A y = H ±h/, he kinemaic boundary condiion is:. v = H ± h ) + ɛ u H ± h ) x The sress boundary condiion resuls in he following wo equaions: ɛ p + u x + τ) H x ± h ) x + u y + v x ) = 0 9) ɛ u y + v x ) H x ± h ) x ɛ p + v y = 0 10) A leading order, O1), he incompressibiliy equaion reduces o: The x-momenum and y-momenum balances are respecively: 8) v 0y = 0 11) u 0yy = 0 1) v 0yy = 0 13) We assume ha he polymer sress ensor has only one non-zero componen, τ xx, where he superscrip refers o he componen of he sress ensor. Here onwards, we drop he superscrip and refer o τ xx as τ. The Oldroyd-B equaion for τ a his order is: τ 0 + v 0 τ 0y = 1 W i τ 0 G) 14) 168

4 We assume ha τ 0y = 0. Sreching he filamen embeds a sress wihin he fluid, τ 0 0, ) = τ 0 0). Boundary condiions a y = H ± h/ are: v 0 = H 0 ± h ) 0 15) Then we conclude ha h 0 = 0. Also u 0y + v = 0 16) v 0y = 0 17) v 0 = H 0 18) u 0 = H H y) + u 0 x) 19) where, u 0 x) is he velociy of he cenerline of he filamen, i.e. y = H 0. Inegraing Eq. 14) gives he equaion for he leading order polymer sress ha decays wih ime. τ 0 = G + τ 0 0) G) e /W i 0) A second order, Oɛ ), he incompressibiliy equaion and momenum balances yield, The Oldroyd-B equaion becomes v y = u 1) p u yy = u x + τ ) p 0y = v x u y 3) τ + v 0 τ y + τ W i = τ 0u u 0 τ 4) The kinemaic and sress boundary condiions a his order are: v = H ± h ) + u 0 H ± h ) u y + v x + p 0 H ± h ) = u + τ 0 ) H ± h ) 5) 6) p 0 = v y 7) Inegraing he y-momemum balance, Eq. 3) and applying he appropriae boundary condiion, Eq. 7), we can evaluae he leading order pressure. p 0 = u 8) To calculae v, we inegrae he second order incompressibiliy equaion, Eq. 1), v = H x y H) T y H) + ṽ 9) 169

5 where, ṽ = H + u 0 H and T = u + H H. Noe ha T is he viscous conribuion o he dimensionless ension in he viscoelasic shee. The second order horizonal velociy hen is u = H xx y H) 3 3 H H x + T x ) y H) τ y H) +kx)y H)+ũ x) 30) where kx) = H xx h 0 / + 3T + τ 0)H ṽ x and ũ x) is he consan of inegraion. A his order, he boundary condiions and he equaions impose he following solvabiliy condiions: h = ū 0 h 0 ) x 31) [4T + τ 0 ) h 0 ] x = 0 3) Eq. 3) is a saemen of ension balance. Ineria is oo small o appear a his order. So he caenary is in a quasi-saic balance. A he nex order, Oɛ 4 ) he incompressibiliy, momenum balances and he Oldroyd-B equaion are as follows. v 4y = u x 33) p x + u 4yy = Re u 0 + v 0 u 0y ) u xx τ x 34) The boundary condiions are, p y = Re v 0 ) + v xx + v 4yy ϖ 0 35) v 4 + v y H ± h ) = H 4 ± h ) 4 + u H ± h ) + u 0 H x ± h ) x 36) p 0 + u + τ 0 ) H x ± h ) x p + u x + τ ) H ± h ) + u 4y + v 4x = 0 37) u y + v x ) H ± h ) [p + p 0y H ± h )] + [v 4y + v yy H ± h )] = 0 38) Inegraing he y-momenum balance, we ge anoher solvabiliy condiion. For he sake of simpliciy, we assume ha h = 0 and ha τ = 0. Then, h 0 ReH 0 + h3 0 3 H xxx = 4T + τ 0 )h 0 H x ϖh 0 39) We can now rescale Eq. 39) o gain more insigh ino he problem. All lenghs are scaled wih L and ime wih 6µ/ρgh. The cenerline velociy a x = 0 and he ends of he caenary, x = ± 1/ is zero. So inegraing he firs solvabiliy condiion, we have ha ) 4T + τ 0 = 8 L/ H x ) L dx + τ 0 40) 0 170

6 which is a saemen of ension balance. The second solvabiliy condiion becomes, ) 1/ Re g H + ɛ3 3 H xxxx = H x ) dx + Λτ 0 ɛh xx 1 41) 0 ) where we have dropped he subscrip 0 from he equaion. Re g = ρgh L 6µ g is he appropriae Reynolds number, ofen referred o as he Galileo number in engineering circles. Λ = G ρgh, where G is he equilibrium polymer sress. Eq. 41), along wih he boundary condiions H±1/, ) = 0 and H x ±1/, ) = 0, describes he shape of he viscoelasic caenary as i sags under is own weigh. The second erm on he lef hand side, H xxxx is he conribuion from orque balance and is referred o as he beding erm. 4 Resuls and discussion The final equaion o be solved, Eq 41) is no, apparenly amenable o analyical soluions.however, some simplifcaions are in order. For he fluid filamens ha we consruced, Re g So we can enirely neglec he inerial erm. Also, a early imes, he sraigh filamen mus firs bend o begin he formaion of a caenary. Neglecing he non-linear viscous sreching erm, we have ɛ 3 3 H xxxx = Λτ 0 ) ɛh xx 1 4) where, he parameer Λ 5 and τ 0 can be evaluaed from Eq.0). As he caenary evolves, sreching will resul in ension due o viscous sresses and he non-linear sreching erm can no longer be ignored. A presen, we presen only hese hypoheses. We hope o examine hem in he process of solving Eq.41) numerically. We inend o aack chewing-gum problem using he framework ha we have developed for he viscoelasic caenary. I appears o be a special case of he caenary - one in which he iniial sae of he filamen is a caenary o begin wih. 5 Acknowledgemens I sincerely hank Jean-Luc Thiffeaul and L. Mahadevan for all heir help and guidance. This projec would have been impossible wihou hem. References [1] J. Teichman and L. Mahadevan. Viscous caenary. J. Fluid Mech., 478:71 80,