Flow of a Non-Newtonian Fluid

Example 5.4 Flow of a Non-Newtonian Fluid. Write a general MATLAB function for solution of a boundary value problem by the shooting method using the Newton s technique. Apply this function to find the velocity profile of a non-Newtonian fluid that is flowing through a circular tube as shown in Fig. E5.4a. Also calculate the volumetric flow rate of the fluid. The viscosity of this fluid can be described by the Carreau model [5]  

The momentum balance for this flow, assuming the tube is very long so that end effect is negligible, results in 

Therefore, Eq. (1) is a second-order ordinary differential equation, which should be solved with the following boundary conditions  

Method of Solution First we define the following two variables  

In order to obtain the canonical form of Eq. (2), we apply the following transformation  

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For laminar flow of a non-Newtonian fluid, the wall shear stress can be expressed in terms of K and ri as... 

The modelling of the flow of a non-Newtonian fluid through a packed bed follows a similar, though more complex, procedure to that adopted earlier in this chapter for the flow of a Newtonian fluid. It first involves a consideration of the flow through a cylindrical tube and then adapting this to the flow in the complex geometry existing in a packed bed. The procedure is described in detail elsewhere(24,25). 

E. Broyer, C. Gutfinger, and Z. Tadmor, Evaluating Flows of a Non-Newtonian Fluid by the Method of Equivalent Newtonian Viscosity, AIChE J., 21, 198-200 (1975). 

An analogy between the drag characteristics for pipe flow of a non-Newtonian fluid and the power characteristic suggests the function /(Re, m) to be... 

D.O. Olagunju, Asymptotic analysis of the finite cone-and-plate flow of a non-Newtonian fluid, J. Non-Newtonian Fluid Mech., 50 (1993) 289-303. 

Solve the flow of a non-Newtonian fluid in a pipe, following the example, for pressure drops of 10, 10, 10, and 10 Pa. The parameters are tjq = 0.492 Pa s, A = 0.1 and n = 0.8. Plot the shear rate as a function of radial position. Calculate the average velocity. Plot the shear stress as a function of radial position. How do these curves change as the pressure drop is increased ... 

Polymer viscosity is strongly shear dependent. If we use the bulk viscosity measured at different shear rates to describe the flow behavior in porous media, our first task is to calculate the shear rate which is equivalent to that in the bulk viscometer. To do that, we start with the capillary flow of a non-Newtonian fluid. 

Let us consider a steady-state axisymmetric flow of a non-Newtonian fluid in a straight horizontal circular tube of radius a. The coordinate Z is measured along the tube axis and is directed downstream. We restrict our consideration to the hydrodynamically stabilized flow far from the input cross-section, where the streamlines are parallel to the tube axis. In this case, the pressure increment decreases with increasing Z, and the pressure gradient is negative and constant,... 

Now let us consider a steady-state hydrodynamically stabilized flow of a non-Newtonian fluid through a plane channel of width 2h. Let us introduce Cartesian coordinates X, with X-axis directed downstream along the lower wall and with coordinate measured inward the channel along the normal to this wall (0 < < 2h). Since the problem is symmetric about the midline = h, it suffices to consider the lower half of the region, 0 < < /i. 

Convective mass and heat transfer to a plate in a longitudinal flow of a non-Newtonian fluid was considered in [443]. By solving the corresponding problem in the diffusion boundary layer approximation (at high Peclet numbers), we arrive at the following expression for the dimensionless diffusion flux ... 

Melton and Malone (44, 45) developed an expression for turbulent flow of a non-Newtonian fluid through tubulars that was modeled after the Bowen relationship. Reidenbach et al. (11) modified the relationship to account for the changing density of foam as it travels through the tubulars. Equation 28 was developed for non-Newtonian fluids however parameters were developed to account for Newtonian fluid flow behavior. Table V shows the parameters for water. 

To illustrate the application of the generalized Newtonian fluid relation, we consider the steady fully developed flow of a non-Newtonian fluid in a circular pipe (Fig. 9.1.2). From an elemental force balance on a cylindrical fluid element of radius r and length Ax, we have on equating the pressure force to the shear force... 

The basic theory of flow of a non-Newtonian fluid in a compressional rheometer is due to Scott [S5], whose analysis is in terms of a power law fluid (Fig. 16). One presumes the flow is basically laminar shearing, with the fluid being driven radially outward from the approaching disks. By continuity at any radius r... 

The velocity profile can be calculated for laminar flow of a non-Newtonian fluid to show that the velocity profile for a Newtonian fluid given in Eq. (2.9-9) can differ greatly from that of a non-Newtonian fluid. For pseudoplastic fluids (n < 1), a relatively flat velocity profile is obtained compared to the parabolic profile for a Newtonian fluid. For n = 0, rodlike flow is obtained. For dilitant fluids (n > 1), a much sharper profile is obtained and for n = oo, the velocity is a linear function of the radius. 

The only difference for the flow in porous media between a Newtonian fluid and a non-Newtonian fluid is that the effective viscosity is not the same. Hence, it becomes necessary that a suitable effective viscosity be defined for the flow of a non-Newtonian fluid [52]. Constitutive equations for the gas-liquid and liquid-solid drag forces are given by Iliuta and Larachi [45,... 

If it is assumed that the influence of pressure and temperature on the viscosity of the polymer solution is independent of shear rate, then this experimental technique results in values of the viscosity of the polymer solution as a function of pressure, temperature, and shear rate. Finally, the consistency of this overall data analysis technique can be checked. The computer simulation presented in the previous section was used to predict the flow of a non-Newtonian fluid which obeys the truncated power-law model. The parameters in this model were determined by coupling the high and low shear rate measurements of viscosity. Once the viscosity of the polymer solution has been determined as a... 

Fenner, R. T., Extruder Screw Design, Ilifle Books, London, 1970. MacSporran, W. C., Comments on generalized Couette flow of a non-Newtonian fluid in annuli, Ind. Eng. Chem. Fundam. 27 98-99 (1982). 

McEachern, D. W., Axial laminar flow of a non-Newtonian fluid in an annulus, AIChEJ. 72(2) 328-332 (1966). 

Wagner, M. G. and J. C. Slattery, Slow flow of a non-Newtonian fluid past a droplet, AIChE J. 77 1198 (1971). 

Kamisli, E, 2006. Laminar flow of a non-Newtonian fluid in channels with wall suction or injection. Int. 

Sharma, H.G., Singh, K.R., 1987. Heat transfer in the laminar flow of a non-Newtonian fluid in a porous annulus by the method of quasi-linearization. Int. J. Heat Mass Transf. 30,1227-1231. 

Taklifi, A., Aliabadi, A., 2012. Analytical solution of unsteady MHD periodic flow of a non-Newtonian fluid through a porous channel. J. Porous Media 15, 1051-1059. 

Schechter R S (1961) On the steady flow of a non-Newtonian fluid in cyUnder ducts , AIChE Journal, 1,445-448. 

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