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

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