Velocity field, nonisothermal

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

Although there is no immediately useful information that we can glean from (2-56), we shall see that it provides a constraint on allowable constitutive relationships for T and q. In this sense, it plays a similar role to Newton s second law for angular momentum, which led to the constraint (2 41) that T be symmetric in the absence of body couples. In solving fluid mechanics problems, assuming that the fluid is isothermal, we will use the equation of continuity, (2-5) or (2-20), and the Cauchy equation of motion, (2-32), to determine the velocity field, but the angular momentum principle and the second law of thermodynamics will appear only indirectly as constraints on allowable constitutive forms for T. Similarly, for nonisothermal conditions, we will use (2-5) or (2-20), (2-32), and either (2-51) or... 

Fig. 10.69 Locations of the thermocouple and pressure transducers used by Ishikawa et al. (113). Numerical velocity, pressure, and temperature field at cross-sections A-A, B-B, and C-C. [Reprinted by permission from T. Ishikawa, S. I. Kihara, and K. Funatsu, 3-D Numerical Simulations of Nonisothermal Flow in Co-Rotating Twin Screw Extruders, Polym. Eng. Set, 40, 357 (2000).]...

Fig. 14.8 Simulation results for velocity, temperature, axial normal stress, and crystallinity fields for low-speed spinning of nylon-6.6. [Reprinted with permission from Joo et al., Two-dimensional Numerical Analysis of Nonisothermal Melt Spinning with and without Phase Transition, J. Non-Newt. Fluid Mech., 102, 37-70 (2002).]...

The morphology and properties of TLCPs can be significantly influenced within the spinline. In this processing regime, a nonisothermal uniaxial elongational flow field predominates. The morphology developed is maintained in the final extrudate due to solidification prior to take-up. The stretching flow results from differences between the take-up velocity and the... 

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