Fluids power-law

In the previous section, it was discussed that polymer melts are pseudo-plastic fluids. The fact that the polymer melt viscosity reduces with shear rate is of great importance in the extrusion process. It is, therefore, important to know the extent of 

The viscosity at very low shear rates is essentially independent of shear rate. Thus, the fluid behaves as a Newtonian fluid at low shear rates. The low shear rate plateau value ri is often referred to as the low shear limiting Newtonian viscosity. The high shear rate plateau value ri , is often referred to as the high shear limiting Newtonian viscosity. This value is difficult to determine experimentally because the effects of pressure and temperature become very pronounced at these high shear rates (over 10- s ). 

Equation 6.23 can be used if the shear rate is positive throughout the flow channel being considered. If the shear rate changes sign at some point in the flow channel, a more general power law equation should be used  

Another form of the power law equation that is used quite often is Y = (p T P x 

Power law equations (Eqs. 6.23-6.24) can be used to describe simple viscometric flow, i. e., flow with velocity components in only one direction. For more complicated flow situations, a more general power law expression should be used. In order to do this, the rate of deformation tensor Ay has to be introduced. The components of A- in Cartesian coordinates are  

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After the substitution for Ai and A2 into Equation (5.74) the pressure potential equation corresponding to creeping flow of a power law fluid in a thin curved layer is derived as... 

Chaturani, P, and Narasimman, S., 1990. Flow of power-law fluids in cone-plate viscometer. Acta Mechanica 82, 197-211. 

For a power law fluid, Eq. (6-4), with constant properties and n, the flow rate is given by... 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

For non-Newtonian (power law) fluids in coiled tubes, Mashelkar and Devarajan Trans. Inst. Chem. Eng. (London), 54, 108-114 [1976]) propose the correlation... 

Laminar Flow A mathematically simple deviation from uniform flow across a cross section is that of power law fluids whose linear velocity in a tube depends on the radial position = r/R, according to the equation... 

Note that not all veloeity profiles in a tubular reaetor are parabolie. Power law fluids have the profile... 

It is worth noting that since a Newtonian Fluid is a special case of the Power Law fluid when n = 1, then the above equations all reduce to those derived in Section 5.3(a) if this substitution is made. 

In many practical situations involving the flow of polymer melts through dies and along channels, the cross-sections are tapered. In these circumstances, tensile stresses will be set up in the fluid and their effects superimposed on the effects due to shear stresses as analysed above. Cogswell has analysed this problem for the flow of a power law fluid along coni-cylindrical and wedge channels. The flow in these sections is influenced by three factors ... 

Example 5.13 Derive an expression for the flow length of a power law fluid when it is injected at constant pressure into a rectangular section channel assuming... 

Solution (a) As shown earlier the flow rate of a power law fluid in a rectangular section is given by... 

Example 5.14 A power law fluid with constants i]q= 1.2 x lO Ns/m and n = 0.35 is injected through a centre gate into a disc cavity which has a depth of 2 mm and a diameter of 200 mm. If the injection rate is constant at 6 X 10 m /s, estimate the time taken to fill the cavity and the minimum injecdon pressure necessary at the gate for (a) Isothermal and (b) Non-isothermal conditions. 

It is now necessary to derive an expression for the pressure loss in the cavity. Since the mould fills very quickly it may be assumed that effects due to freezing-off of the melt may be ignored. In Section 3.4(b) it was shown that for the flow of a power law fluid between parallel plates... 

Derive expressions for the velocity profile, shear stress, shear rate and volume flow rale during the isothermal flow of a power law fluid in a rectangular section slit of width W, depth H and length L. During tests on such a section the following data was obtained. 

Polyethylene is injected into a mould at a temperature of 170°C and a pressure of 100 MN/m. If the mould cavity has the form of a long channel with a rectangular cross-section 6 mm X 1 mm deep, estimate the length of the flow path after 1 second. The flow may be assumed to be isothermal and over the range of shear rates experienced (10 -10 s ) the material may be considered to be a power law fluid. 

A power law fluid with the constants rjo = 10 Ns/m and n = 0.3 is injected into a circular section channel of diameter 10 mm. Show how the injection rate and injection pressure vary with time if. 

Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

Chapter 4 describes in general terms the processing methods which can be used for plastics and wherever possible the quantitative aspects are stressed. In most cases a simple Newtonian model of each of the processes is developed so that the approach taken to the analysis of plastics processing is not concealed by mathematical complexity. Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

For a Power law fluid flow, the following formulas can be used Pipe flow... 

For annular flow of Bingham plastic and Power law fluids, respectively, PpLv T L... 

The critical velocities for the Bingham plastic and Power law fluids can be calculated as follows ... 

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