Start-up flow

For the start up flow case, we obtain the following set of curves of the dimensionless velocity profiles evolution shown in Figure 1, where the increase in the wall velocity with time can be clearly observed. The three dimensional plot for the velocity distribution is given in Figure 2 for the periodic case, and we can observe the quasi-steady-state (periodic state) establishment, and the time variation of the dimensionless slip velocity. 

Solve for the start-up flow of the power-law fluid and problem described in Problem 10.2. How long does it take to reach steady state ... 

D. Start-Up Flow in a Circular Tube - Solution by Separation of Variables... 

D. START-UP FLOW IN A CIRCULAR TUBE - SOLUTION BY SEPARATION OF VARIABLES... 

In this section, we consider one example of this type of problem, namely, the start-up from rest of the flow in a circular tube when a nonzero pressure gradient is suddenly imposed at some instant (which we shall denote as i = 0) and then held at the same constant value thereafter (t > 0). In Section G, we consider the start-up of simple shear flow between two infinite plane boundaries. Other relatively simple examples of start-up flows are left to the reader to solve (for example, see Problem 19 at the end of this chapter). 

The evolution of u with respect to 1 is plotted in Fig. 3-12. As in previous examples of start-up flows, we see that the momentum from the lower wall propagates across the gap by means of diffusion in a dimensionless time interval 1 = 0(1), or, in dimensional terms,... 

Now the problem defined by (3-258) and (3-259) is just the transient start-up flow due to a suddenly imposed constant pressure gradient that was already solved in Section C of this chapter. The final steady-state solution is the steady Poiseuille flow profile ... 

The problem of start-up flow for a circular cylinder has received a great deal of attention over the years because of its role in understanding the inception and development of boundary-layer separation. An insightful paper with a comprehensive reference list of both analytical and numerical studies is S. I. Cowley, Computer extension and analytic continuation of Blasius expansion for impulsive flow past a circular cylinder, J. Fluid Mech. 135, 389-405 (1983). 

The deformation of dispersed drops in immiscible polymer blends with the viscosity ratio X = 0.005-13 during extensional flow was studied by Delaby et al. [1994, 1995]. In the latter paper, the time-dependent drop deformation during a start-up flow at constant deformation rate was derived. The model is restricted to small drop deformations. 

Nagai R. 1999. Modeling Slot Coating Start-Up Flow. [LES]... 

This chapter is devoted to the molecular rheology of transient networks made up of associating polymers in which the network junctions break and recombine. After an introduction to theoretical description of the model networks, the linear response of the network to oscillatory deformations is studied in detail. The analysis is then developed to the nonlinear regime. Stationary nonhnear viscosity, and first and second normal stresses, are calculated and compared with the experiments. The criterion for thickening and thinning of the flows is presented in terms of the molecular parameters. Transient flows such as nonhnear relaxation, start-up flow, etc., are studied within the same theoretical framework. Macroscopic properties such as strain hardening and stress overshoot are related to the tension-elongation curve of the constituent network polymers. 

In the experiments, two main types of time-dependent flows have been studied start-up flows and stress relaxation. In the start-up flow experiments, shear flows with constant shear rates and elongational flows with constant elongational rates are started in the system in equilibrium under no external force, and the time-dependent stress build-up in the system is measured. In the stress relaxation experiments, constant deformations are applied to or removed from the system, and the time-dependent relaxation of the stress is measured. In this section, we study these two types within the framework of transient network theory. 

Chapter 9 presents the transient network theory of associating polymer solutions, which is the other one of the two major theories treated in this book. It studies the dynamic and rheological flow properties of structured solutions from a molecular point of view. Thus, linear complex modulus, nonlinear stationary viscosity, start-up flows, and stress relaxation in reversible polymer networks are studied in detail. 

In a third test, a step fimction shear rate is suddenly applied att = 0 (start-up flow). In this case, the shear stress a is measrmed as a fiinction of time and the shear stress growth coefficient (stressing viscosity) +( ) can be calculated ... 

In case of start-up flow from the rest state, a numerical solution of ODE set (11.4) and (11.5a)-(11.5c) should be obtained using the following initial conditions ... 

The ultimate goal of simulations was to show a possibUity to describe experimental data along with determining constitutive parameters. These parameters established for steady shearing are then used for calculating the evolution of director, shear stress, and first normal stress difference during relaxation and start-up flow. 

To resolve the problem of choosing the initial conditions for director in start-up flow, we preliminarily fitted the experimental data for stresses in steady shearing with the following adjustment of parameters to also describe the stress relaxation. In this case, parameters of the evolution equation for director, along with its orientation in steady shearing, were also established. Calculating then the orientation of director during stress relaxation, we found its final orientation at the rest state, which was taken as initial director value in the start-up flow. 

Figures 11.11 and 11.12 describe the evolution of normalized shear stress o (y,t)/a and first normal stress difference Nj (y, t)/Ni with strain yt for PSHQ9 in start-up shear flow. The flow temperature was 130 °C and shear rate was y = 1 s . The experimental data are shown by dots and simulated curves by dashed line. The simulated overshoots for shear and normal stresses are the same as those for experimental data, but the overshoots in calculated curves occur at a relatively low yt and are very narrow. One can attribute such large values of o (Y,t)/a ratios, characteristic for liquid crystalline polymers, to the existence of a lot of polydomains in the nematic state when the start-up flow initiated. Recall that the theory used for simulation utilizes monodomain approach, whereas PSHQ9 exhibits polydomains in nematic state in start-up flow. Thus, deviation of simulated results from experimental data seems reasonable.

Entezam M, Khonakdar HA, Yousefi AA, Jafari SH, Wagenknecht U, Heinrich G. Dynamic and transient shear start-up flow experiments for analyzing nanoclay localization in PP/PET blends correlation with microstructure. Macromol Mater Eng 2012 298 113-26. 

A sample that is initially rectangular will become longer, and its thickness (in the direction) will decrease, while its width (X2 direction) remains constant. To generate this deformation, normal stresses must be applied in both the Xj and X2 directions. This is very difficult deformation to generate. There are two measurable quantities, the net tensile stresses, (Til < 33 22 - < 33> for start-up flow these are ... 

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