Analytical solution for fully developed channel and pipe flow of Phan ...

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276 P. J. Oliveira and F. T. Pinho1.00.80.6u Nu0.40.20 2 4 6 8 10ε 1/2 DeFigure 1. Variation of the average velocity ratio ū N /ū with ɛ 1/2 De (solid line: plane flow; dashedline: axisymmetric flow; no symbols: linear PTT; symbols: exponential PTT).ɛ 1/2 De Channel Pipe ɛ 1/2 De Channel Pipe0.05 0.9746 0.9526 1.0 0.3134 0.24710.1 0.9132 0.8545 2.0 0.1858 0.14390.2 0.7695 0.6732 3.0 0.1342 0.10330.3 0.6515 0.5492 4.0 0.1059 0.081140.4 0.5628 0.4642 5.0 0.08778 0.067130.5 0.4953 0.4028 6.0 0.07519 0.057400.6 0.4427 0.3565 7.0 0.06589 0.050240.7 0.4007 0.3203 8.0 0.05873 0.044740.8 0.3663 0.2911 9.0 0.05303 0.040360.9 0.3376 0.2672 10.0 0.04839 0.03680Table 1. Numerical solution of equations (24) and (25). Values of ū N /ū.The normalized shear-rate and viscosity profiles are readily obtained from (20b) and(16):( ) () 2 ( ) 2 )˙γ(y)y(ūN y2κ ū/H = −ūN exp 8κ 2 ɛDe 2 (22)ū H ū Hand() 2 ( ) 2 )µ(˙γ)(ūN y= exp − 8κ 2 ɛDe 2 . (23)ηū HAgain, for the inverse problem of determining the pressure gradient for a given flux,the non-dimensional velocity profile is integrated across the channel or pipe sectionsto give the parameter ū N /ū. Here, also, different equations are obtained for the planar,1= 3 ()ū N exp (b(ū N /ū) 2 )1+ iπ1/2 exp (−b(ū N /ū) 2 ) erf (ib 1/2 ū N /ū)(24)2 ū b(ū N /ū) 2 2b 1/2 ū N /ū
1.0Fully developed flow of Phan-Thien–Tanner fluids 2770.8yH0.60.40.2UCM0.10.51.05.00 0.5 1.0 1.5u/uFigure 2. Velocity profile of the linear PTT fluid in channel flow as a function of the dimensionlessgroup ɛ 1/2 De (solid line: parabolic profile; dashed lines: ɛ 1/2 De =0.1, 0.5, 1.0 and 5.0).1.00.8yH0.60.4ChannelPipe0.20 0.5 1.0 1.5u/u2.0Figure 3. Comparison of the velocity profiles for the linear and exponential PTT model in channeland pipe flow (De = 2 and ɛ = 0.1) (solid lines: parabolic profile; dashed lines: linear andexponential PTT).and the axisymmetric cases1=2ūN ū(exp (b(ū N /ū) 2 )1 − 1 − exp (−b(ū )N/ū) 2 ). (25)b(ū N /ū) 2 b(ū N /ū) 2This time, these nonlinear equations are not amenable to an analytical solution andtherefore numerical methods are required. We have solved the above equations with
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