Chapter 4. Lubrication application: sphere approaching a wall. Some simple flow calculations. Pipe flow for a power-law fluid 2

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1 Chapte 4 Pipe flow fo a powe-law fluid length L me simple flow calculations Pipe flow fo a powe-law fluid Capillay heomety Bingham yield fluid in a Couette device Rod-climbing Unchanging flow field fo a second-ode fluid Conveging flow of igid-od suspension Spinning an Oldoyd-B fluid Axial momentum Powe-law fluid Flux Q Pessue dop p 0 = dp + 1 σ z adius a σ z = R σ wall with σ wall = pr 2L. σ z = κ γ n with γ = d Pipe flow fo a powe-law fluid 2 Lubication application: sphee appoaching a wall Integating w = σw 1 n R 1 n +1 1 n +1 κr 1 n + 1 n = 1 n<1 Gap Sphee adius a, minimum gap d h = d ad { Nea cente, low σ, high µ Nea wall, high σ low µ Hence volume flux Q = πr3 1 n pr n 2Lκ flattened pofile Al wie coating, film daining, dop speading & peistaltic pumping Mass flux Sphee appoaching at velocity W Powe-law flow dp d = 2πQ = π 2 W κ n [ 1 2 d ad n W n n ] 1+2n

2 Lubication application: sphee appoaching a wall 2 Capillay heomety Poblem: To find µ γ even though γ. Foce Mg = κ W d n ad a d n+1 2 π2 3n n 2n 0 Note integand like 3n at lage, need n > 1 3 fo lubication in gap to dominate. Student Execise Find velocity of a sphee falling in a tight tube filled with powe-law fluid. Hint: pπa 2 = ρ 4πa3 3 g 2+n n Hence Q = R = 0 R 0 = πr3 σ 3 w w 2π d γ π 2 d σw 0 γ wall = 1 σ 2 w = 1 πr 3 γσσ 2 dσ d dσ w as d = γ σ 3 w Q πr 3 3Q + σ w dq dσ w as σ Capillay heomety 2 Bingham yield fluid in a Couette device as σ w p γ wall = Q πr d ln Q, d ln P b Ω L Slope of plot ln Q vs ln p, = 1 if Newtonian, = 3 powe-law n = 1 3. Then the shea-ate dependent viscosity is found fom µ w = σ w γ w = pr 2L γ w Student Execise: Simila analysis fo a paallel plate heomete. θ-momentum Bingham fluid 0 = 1 d 2 2 σ θ d γ = 0 σ θ = σ Y + µ γ a σ θ = T 2πL 2 if σ < σ Y if σ > σ Y

3 Bingham yield fluid in a Couette device 2 Yields inside suface at 1. All yield Y > b 2. None yield Y < a = Y = 3. Patial yield a < Y < b In a < < Y yielding γ = d d u θ = σ Y µ uθ = σ Y µ T 2πLσ Y Y [ a ln ] a Bingham yield fluid in a Couette device 3 In Y < < b not yielding u θ Continuity of u θ at = Y gives = Ω Ω Y T Student execise Similaly in pipe flow Simila in squeeze film, although too difficult fo a few lectues. Rod-climbing fo a second-ode fluid Rod-climbing fo a second-ode fluid 2 Ω a z=h Second-ode fluid σ = pi + 2µE 2αE + βe E σ θ = µ γ Flow Newtonian u θ = Ωa2 γ = d d uθ = 2Ωa2 2 σ = p β γ2 σ θθ = p + 2α β γ 2 σ zz = p To find p and hence h

4 Rod-climbing fo a second-ode fluid 3 Radial momentum 0 = σ Vetical momentum Hence + σ σ θθ 2α γ2, last tem = σ = p β γ α γ2 = f z 0 = σ zz z ρg, with σ zz = 0 on z = h p = σ zz = ρg h z h = 1 ρg 2α + β Ω2 a 4 4. Could add suface tension and inetia = 8αΩ2 a 4 5 Unchanging flow field fo a second-ode fluid Second-ode fluid = Newtonian with small non-linea coection. Student execise Show 2E + 4E E = D Dt 2 u + u 2 u + E : E If ux,t and p 1 x, t satisfy Newtonian Stokes flow 0 = p 1 + µ 2 u and u = 0, then same ux,t with diffeent p 2 x, t satisfies Giesekus second-ode fluid equation with β = 4α and p 2 = p 1 α µ σ = 0 σ = p 2 + 2µE 2α E + βe E Dp 1 Dt + αe E Student Execise Unchanging flow field fo a second-ode fluid 2 Conveging flow of igid-od suspension Rheology: an anitopic viscosity in diection of ods/fibes p Simila esults with no estiction of α and β Plana flows Tanne & Pipkin unidiectional flows Langlois, Rivlin & Pipkin σ = pi + 2µ shea E + 2µ ext ppp E p In 2-D sink flow, adial flow u = f θ/ and ods align adially p = 1. with pessue gθ/ 2 the stess is σ = g 2 2µ s+µ e f 2, σ θ = µ s f 2, σ θθ = g 2 +2µ f s 2.

5 Conveging flow of igid-od suspension 2 Conveging flow of igid-od suspension 3 θ-momentum σ θ + 1 σ θθ θ g = µ s f + 2σ θ = 0 Newtonian flow has eciculation egion if angle > π Radial momentum σ + 1 σ θθ θ + σ σ θθ = 0 f µ e f = const µ s A compession in θ-diection of 1 + µ e /2µ s Newtonian Fluid Non Newtonian Fluid Rigid-od suspension, with the compession in θ-diection, has eciculation egion at angle = π Anitopy in heology leads to anitopy in flow Al 3D sink flow. Al flow ound a shap cone ods along steamlines. Spinning an Oldoyd-B fluid Spinning an Oldoyd-B fluid 2 z R w w da w da zz = A 1 τ A 1 = 2A zz 1 τ A zz 1 Volume flux Q = πr 2 w Tension, ignoing suface tension, gavity and inetia Fee suface σ = 0, p = µ + GA Oldoyd-B F = πr 2 σ zz σ = pi + 2µE + GA DA Dt = A u + ut A 1 τ A I Momentum equation σ zz = 3µ + GA zz A = F πr 2 = Fw Q This equation gives / which the can use in da.. / equations above.

6 Spinning an Oldoyd-B fluid 3 Spinning an Oldoyd-B fluid 4 Elastic limit µ/, GA GA zz Newtonian limit τ/ 1 A 1 τ, A zz 1 + 2τ σ zz 3µ + Gτ = Fw Q Fz wz w0 exp 3Qµ + Gτ substitute into fo with lution w da zz Fw Q = σ zz GA zz, o A zz F w GQ = 2A zz 1 τ A zz 1 small w = 2w 1 τ w w = w 0 + z, independent of F! τ Need stetch to avoid elaxation