Steady-flow calculation from creep

Figure 11.8 shows typical curves for Re/Rex as functions of t and M, calculated from Eqs. (11-31) to (11-36) for 7 = 2.65. Even for low Rex (curve 2), the velocity approaches the terminal value more rapidly than predicted by the creeping flow solution. At higher Rex the steady terminal velocity is approached more rapidly, but the value required to achieve a given fraction of Rcx increases with Rex- The trajectory is generally more sensitive to Rex than to 7 as shown by Fig. 11.9, where we have plotted the t and required to... 

Another consequence of the integral theorem (8-111) is that we can calculate inertial and non-Newtonian corrections to the force on a body directly from the creeping-flow solution. Let us begin by considering inertial corrections for a Newtonian fluid. In particular, let us recall that the creeping-flow equations are an approximation to the full Navier-Stokes equations we obtained by taking the limit Re -> 0. Thus, if we start with the ftdl equations of motion for a steady flow in the form... 

The steady-flow viscosity qo and the steady-state compliance can easily be determined from creep data in the region of linear viscoelastic behavior as shown-in Fig. 1-12, from equation 40 of Chapter 1, provided steady-state flow has been attained. However, it is easy to be misled into believing prematurely that the linear portion of the creep curve has been reached in general, it cannot be expected to become linear until the flow term t/vo is at least as large as the intercept / . It is always desirable to perform the recovery experiment shown in Fig. 1-12 to conflrm the calculation. 

FIG. 11-14. Creep measurements on a polystyrene with molecular weight 46,900, reduced from different temperatures as indicated to 100 C with shift factors calculated from steady flow viscosity. (After Plazek.25) Subscript p denotes multiplication by Tp/Topo. 

Equation 28 would imply that the creep compliance in excess of the steady-flow term increases without limit instead of approaching the customary limit J , and cannot be valid at extremely long times. However, in a range where the last two terms are of similar magnitude, it can provide a useful simplification in calculating 1)0 from creep measurements. It is not necessary to wait until J t) is a strictly linear function of time. A plot of J t) against is made, and Ja and are determined... 

A good diagnostic for creep and stress relaxation tests is to plot them on the same scales as a function of either compliance (J) or modulus (G), respectively. If the curves superimpose, then all the data collected is in the linear region. As the sample is overtaxed, the curves will no longer superimpose and some flow is said to have occurred. These data can still be useful as a part of equilibrium flow. The viscosity data from the steady-state part of the response are calculated and used to build the complete flow curve (see equilibrium flow test in unit hi.2). 

Any uncross-linked polymer system with a viscosity high enough to be studied by the methods described in this section will be very slow in coming to steady-state flow attempts to obtain jjo and 7° from linear plots such as Fig. 1-12 should be approached with caution and skepticism, and if possible tested by creep recovery experiments. From the methods described in Chapter 4, it is possible to calculate the retardation spectrum L from J t) even if one is not sure whether steady-state flow has been achieved. 

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