Fluids Phan-Thien/Tanner

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

One can also show that all one dimensional time-dependent perturbations of a steady multifluid flow exist for all times, and stay bounded—as in the case of one fluid. Similar results can be obtained for axisymmetric Poiseuille flows of several fluids. A similar study is also made for plane Poiseuille or Couette flows of several fluids having a Phan-Thien-Tanner constitutive equation [50]. 

Figure 9.7. Steady shear and normal stress data at 170 °C for two polyethylenes (a) low-density Resin 10 and (b) high-density Resin 22. The lines are the fit to the Phan-Thien/Tanner model, discussed below. Reprinted with permission from Tsang and Dealy, /. Non-Newtonian Fluid Mech., 9, 203 (1981). Copyright Elsevier.

Table 9.4. Components of the Phan-Thien/Tanner Fluid, Equation 9.23, for two-dimensional flows...

Figure 9.13. Comparison of the Phan-Thien/Tanner model to the extensional data of Meissner. Reprinted with permission from Phan-Thien and Tanner, J. Non-Newtonian Fluid Mech.,...

Figure 10.1. Computed extrudate swell of a Phan-Thien/Tanner fluid with = 0 for various values of e. e = 0 corresponds to a Maxwell fluid. Reprinted by permission of Oxford University Press from Tanner, Engineering Rheology, 2nd ed., Oxford, New York, 2000, p. 429.

Here we have removed the overbars from the averaged quantities and made use of the fact that trr = ree. (This equality does not hold for hoUow-fiber spinning.) The steady-state equations for each mode of a Phan-Thien/Tanner fluid (Table 9.4) are as follows ... 

Figure 10.3. Draw ratio at z = 5dc as a function of dimensionless force for Maxwell and Phan-Thien/Tanner fluids. Reprinted with permission from Keunings et al., Ind. Eng. Chem. Fundam., 22, 347 (1983). Copyright American Chemical Society.

Figure 11.3. Critical draw ratio as a function of De = Xvo/L for a single-mode Phan-Thien/Tanner fluid with various values of e and Reprinted from Chang and Denn, Proc. 8th International Congress on Rheology, Naples, Italy, Vol. 3,1980, p. 9.

Figure 11.6. Amplitude ratios for PET spinniag, Dr = 100 response of the takeup area to various input disturbances, (a) Newtonian flnid (b) two-mode Phan-Thien/Tanner fluid. (Note the different vertical scales on the two figures.) Reprinted from Devereux, Computer Simulation of the Melt Fiber Spinning Process, Ph.D. dissertation, U. Cahfomia, Berkeley, 1994.

Phan-Thien, N. and Tanner, R.T., 1977. A new constitutive equation derived from network theory, Non-Newtonian Fluid Mech. 2, 353-365. 

Phan-Thien N, Atkinson JD, Tanner RI (1978) J Non-Newt Fluid Mech 3 309... 

Y. Fan, R. I. Tanner and N. Phan-Thien, Fully developed viscous and viscoelastic flows incurved pipes, J. Fluid Mech. 440, 327-57 (2001) S. C. R. Dennis andN. Riley, On the fully developed flow in a curved pipe at large Dean number, Proc. R. Soc. London Ser. A. 434, 473-8 (1991) W. M. Collins and S. C. R. Dennis, The steady motion of a viscous fluid in a curved tube, Q. J. Mech. Appl. Math. 28, 133-56 (1975). 

The term rheology dates back to 1929 (Tanner and Walters 1998) and is used to describe the mechanical response of materials. Polymeric materials generally show a more complex response than classical Newtonian fluids or linear viscoelastic bodies. Nevertheless, the kinematics and the conservation laws are the same for all bodies. The presentation here is condensed one may consult other books for amplification (Bird et al. 1987a Huilgol and Phan-Thien 1997 Tanner 2000). We begin with kinematics. 

Tanner et al. (2010a) have extended the above results to concentrated regimes by using the Roscoe procedure (Roscoe 1952, also see Phan-Thien and Pham 2000). In concentrated suspensions, some of the fluid is trapped between particles, and hence Roscoe (1952) suggested that the increment of small amount of volume fraction d(p results in an effective increase of concentration of d(p/ — (p/(p ), which is called the crowding function, where (j is the maximum volume fraction. We use N ) as an example to describe the procedure. From Eq. 5.61, one has... 

Evans DJ, Holian BL (1985) The Nose-Hoover thermostat. J Chem Phys 83 4069-4074 Fan XJ (2006) Numerical study on some rheological problems of fiber suspensions. Ph.D Thesis, Sydney University. See also, Fan XJ Phan-Thien N, Tanner RI (2008) Numerical study on some rheological problems of fiber suspensions, VDM Verlag, Saarbriicken, Germany Fan XJ, Phan-Thien N, Zheng R (1998) A direct simulation of fiber suspensions. J Non-Newtonian Fluid Mech 74 113-136... 

Phan-Thien N, Fan XJ, Tanner RI, Zheng R (2002) Folgar-Tucker constant for a fiber suspension in a Newtonian fluid. J Non-Newtonian Fluid Mech 103 251-260 Phan-Thien N, Fan XJ, Zheng R (2000) A numerical simulation of suspension flow using a constitutive model based on anisotropic interparticle interactions. Rheol Acta 39 122-130 Phan-Thien N, Graham AL (1991) A new constitutive model for fiber suspensions flow past a sphere. Rheol Acta 30 44-57... 

N. Phan-Thien, X. Fan, R. I. Tanner and R. Zheng, Folgar-Tucker constant for a fibre suspension in a Newtonian fluid. Journal of Non-Newtonian Fluid Mechanics, 103, 251-260 (2002). 

Phan-Thien, N. Sugeng, F. Tanner, R.I. J. Non-Newtonian Fluid Mech. 1984, 24, 301-325. 

Xue S C, Phan-Thien N and Tanner RI (1995), Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method . Journal of Non-Newtonian Fluids Mechanics, 59,191-213. 

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