Isothermal Newtonian fluids

According to the lubrication approximation, we can quite accurately assume that locally the flow takes place between two parallel plates at H x,z) apart in relative motion. The assumptions on which the theory of lubrication rests are as follows (a) the flow is laminar, (b) the flow is steady in time, (c) the flow is isothermal, (d) the fluid is incompressible, (e) the fluid is Newtonian, (f) there is no slip at the wall, (g) the inertial forces due to fluid acceleration are negligible compared to the viscous shear forces, and (h) any motion of fluid in a direction normal to the surfaces can be neglected in comparison with motion parallel to them. 

A full analytical solution of the cross channel flow vx(x,y) and vy x, y), for an incompressible, isothermal Newtonian fluid, was presented recently by Kaufman (18), in his study of Renyi entropies (Section 7.4) for characterizing advection and mixing in screw channels. The velocity profiles are expressed in terms of infinite series similar in form to Eq. 6.3-17 below. The resulting vector field for a channel with an aspect ratio of 5 is shown... 

Fig. 6.11 Vector field of the cross-channel flow of an incompressible isothermal Newtonian fluid in a channel with an aspect ratio of 5. [Reprinted by the permission from M. Kaufman, Advection and Mixing in Single Screw Extruder—An Analytic Model, The AIChE Annu. Tech. Conf. Meeting Proc., San Francisco (2003).]...

The screw extruder is equipped with a die, and the flow rate of the extruder as well as the pressure rise at a given screw speed are dependent on both, as shown in Fig. 6.16. The screw characteristic line at a given screw speed is a straight line (for isothermal Newtonian fluids). This line crosses the abscissa at open discharge (drag flow rate) value and the ordinate at closed discharge condition. The die characteristic is linearly proportional to the pressure drop across the die. The operating point, that is, the flow rate and pressure value at which the system will operate, is the cross-point between the two characteristic lines, when the pressure rise over the screw equals the pressure drop over the die. 

Parallel-Plate Flow of Newtonian Fluids A Newtonian polymeric melt with viscosity 0.21b(S/in2 and density 481b/ft3, is pumped in a parallel-plate pump at steady state and isothermal conditions. The plates are 2 in wide, 20 in long, and 0.2 in apart. It is required to maintain a flow rate of 50 lb/h. (a) Calculate the velocity of the moving plate for a total pressure rise of 100 psi. (b) Calculate the optimum gap size for the maximum pressure rise, (c) Evaluate the power input for the parts (a) and (b). (d) What can you say about the isothermal assumption ... 

We now derive the RTD function/(f) dt and the strain distribution functions / ) dy for isothermal Newtonian fluids in shallow screw channels. 

Next we derive a simple theoretical model to calculate the passage-distribution function (PDF) in a SSE11 (25), assuming isothermal Newtonian fluids. We examine a small axial section of length A/, as shown in Fig. 9.16. 

The first milestone in modeling the process is credited to Pearson and Petrie (42—44). who laid the mathematical foundation of the thin-film, steady-state, isothermal Newtonian analysis presented below. Petrie (45) simulated the process using either a Newtonian fluid model or an elastic solid model in the Newtonian case, he inserted the temperature profile obtained experimentally by Ast (46), who was the first to deal with nonisothermal effects and solve the energy equation to account for the temperature-dependent viscosity. Petrie (47) and Pearson (48) provide reviews of these early stages of mathematical foundation for the analysis of film blowing. 

This is the Navier-Stokes equation of motion for an incompressible, isothermal Newtonian fluid. Two comments are in order with regard to this equation. First, we shall always assume that p and // are known-presumably by independent means - and attempt to solve (2-89) and the continuity equation, (2 20) for u andp. Second, the ratio pjp, which is called the kinematic viscosity and denoted as v, plays a fundamental role in determining the fluid s motion. In particular, it can be seen from (2 89) that the contribution to acceleration of a fluid element (Du/Di) that is due to viscous stresses is determined by v rather than by ti. 

Physical properties of a fluid can be described within the context of transport analogies for all the transport processes. Numerical solutions to fluid dynamics problems require that the viscosity and the density p are known. Under isothermal conditions, if the fluid is Newtonian and incompressible, both of these physical properties are constants that depend only on the fluid, not the flow conditions. The viscosity p, is the molecnlar transport property that appears in the... 

The physical and mathematical description of the ribbon extrusion process was first given by Pearson [24], who simplified the conservation equations by using a onedimensional, isothermal, Newtonian fluid approach, and neglected the effects of polymer solidification. As in the case of blown film processes, several modifications and models have been proposed for the ribbon extrusion process (Table 24.2). 

The theoretical studies of draw resonance were initiated by Kase and Yoshimoto [102] and Pearson and Matovich [103]. These authors found that the critical drawdown ratio for isothermal Newtonian fluids is about 20. Fisher and Denn analyzed both isothermal and nonisothermal flows of spinning of viscoelastic fluids [104,105]. Figure 3.17 gives the results for the isothermal case, where... 

There are three non-dimensional parameters expected to affect spreading and flow characteristics during an isothermal, Newtonian filling the Reynolds, Bond, and Capillary numbers. A fourth non-dimensional value, the contact angle, is a characteristic of the experimental apparatus (i.e., the contact angle depends on the fluid injected and the mold surface.) Thus, a method needs to be developed to determine the experimental parameters i.e., the gap-width, L, the volume flow rate, Q, and the properties of the filling fluid, to achieve a prescribed set of non-dimensional parameters. To obtain this solution, the Reynolds, Bond, and Capillary numbers are defined as... 

We assume isothermal flow and a Newtonian fluid. (Isothermality is the more serious of the two assumptions. Even if the physical properties of the melt are independent of temperature, we must deal with the possibility of solidification of the melt near the cold mold faces during filhng.) For radial flow we have a single velocity component in cylindrical coordinates, Vr, and the continuity equation from Table 2.1 is... 

Squeeze flow between parallel plates was analyzed in Section 6.3 as an elementary model of compression molding. In that treatment we were able to obtain an analytical solution to the creeping flow equations for isothermal Newtonian fluids by making the kinematical assumption that the axial velocity is independent of radial position (or, equivalently, that material surfaces that are initially parallel to the plates remain parallel). In this section we show a finite element solution for non-isothermal squeeze flow of a Newtonian hquid. The geometry is shown schematically in Figure 8.16. We retain the inertial terms in the Navier-Stokes equations, thus including the velocity transient, and we solve the full transient equation for the temperature, including the viscous dissipation terms. The computational details. 

Yeow (1976) theoretically analyzed the instabilities due to axisymmetfic disturbances in an isothermal Newtonian fluid and presented neutral stability curves in the space Wf (= H(/Ro) and Br and for various values of the parameter X (X = Z/Ro, which is the dimensionless freeze line). Kanai and White (1984) experimentally studied the stability of nonisothermal film blowing of viscoelastic melts, such as LLDPE, LDPE, and HDPE, and their results are shown in Figures 9.27 and 9.28. LDPE is more stable than LLDPE and HDPE, which is in accord with LDPE s strain hardening... 

Figure 6 Predicted stability contours by the Zatloukal-Vlcek model for Newtonian fluid, isothermal conditions 6a) Effect of Newtonian viscosity 6b) Effect of melt strength 6c) Effect of die radius 6d) Effect of freeze line height 6e) Comparison between experimentally determined stability points and theoretical stability contours.

Bell, B.C. and Surana, K. S, 1994. p-version least squares finite element formulations for two-dimensional, incompressible, non-Newtonian isothermal and non-isothcmial fluid flow. hit. J. Numer. Methods Fluids 18, 127-162. 

Note that for a Newtonian fluid, n = 1, so for the isothermal case, equation (5.82) becomes. 

Consider isothermal laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

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