Steady-flow Newtonian

In the fluid flow model, simulation is based on Darcy s law for the steady flow of Newtonian fluids through porous media. This law states that the average... 

In the steady flow of a Newtonian fluid through a long uniform circular tube, if ARe < 2000 the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material. 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

For almost steady flows one can expand yl1 or y about t t and obtain second-order fluid constitution equations in the co-deforming frame. When steady shear flows are considered, the CEF equation is obtained, which, in turn, reduces to the GNF equation for T i = 2 = 0 and to a Newtonian equation if, additionally, the viscosity is constant. 

Fig. E7.ll SDFs for fully developed Newtonian, isothermal, steady flows in parallel-plate (solid curves) and tubular (dashed curve) geometries. The dimensionless constant qp/qd denotes the pressure gradient. When qp/qd = —1/3, pressure increases in the direction of flow and shear rate is zero at the stationary plate qpjqd = 0 is drag flow when qp/qd = 1/3, pressure drops in the direction of flow and the shear rate is zero at the moving plate. The SDF for the latter case is identical to pressure flow between stationary plates. (Note that in this case the location of the moving plate at / — 1 is at the midplane of a pure pressure flow with a gap separation of H = 2H.)...

A. E. Green and R. S. Rivlin, Steady Flow of Non-Newtonian Fluids through Tubes, Quant. Appl. Math., 14, 299 (1956). 

Flow in a capillary can be maintained by a steady pressure difference Ap applied between inlet and outlet ends. We assume gravitational (and other external) forces to be negligible (true for a horizontal tube or for any tube with a large Ap). With the application of Ap, the fluid in the tube accelerates to a flowrate at which the viscous drag forces balance the applied pressure forces. For thin tubes the Newtonian acceleration forces are significant for only a brief moment before steady flow is achieved. 

Table 6.2 The momentum equations for a steady-flow, two-dimensional, incompressible, Newtonian fluid with constant properties in different coordinate systems. 

In order to explain dimensional analysis in chemical engineering, we present a typical problem of chemical engineering that requires an experimental approach. Consider the steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal and circular pipe which is heated from the outside. 

This chapter will be organized as follows. After a brief review of the classical differential models we will emphasize two important features of Maxwell type models (i.e., models without Newtonian viscosity), namely the instability to short waves and a transonic change of type in steady flows. 

Then we review the existence results of solutions known for steady flows and for unsteady flows. Afterwards we discuss the important topic of stability of flows. This issue is still wide open, in particular because of the presence of memory effects for viscoelastic fluids, contrary to the case of Newtonian fluids. 

The simplest case to consider is steady flow of a dilute suspension of Newtonian drops or bubbles in a Newtonian medium. If the capillary number y a / F is small, so that the drops or bubbles do not deform under flow, then at steady state the viscosity of the suspension is given by Taylor s (1932) extension of the Einstein formula for solid spheres ... 

To characterize Newtonian and non-Newtonian food properties, several approaches can be used, and the whole stress-strain curve can be obtained. One of the most important textural and rheological properties of foods is viscosity (or consistency). The evaluation of viscosity can be demonstrated by reference to the evaluation of creaminess, spreadability, and pourability characteristics. All of these depend largely on shear rate and are affected by viscosity and different flow conditions. If it is related to steady flow, then at any point the velocity of successive fluid particles is the same at successive periods of time for the whole food system. Thus, the velocity is constant with respect to time, but it may vary at different points with... 

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... 

Consider die simple case in which a low rate of mass transfer normal to die surface does not influence die velocity profile (vy 0), The velocity profile vj,y) in die film is determined by solving the z component of die eqnetion of motion for a Newtonian fluid in steady flow owing to iha ection of gravity. If the solid surface is inclined at an angle a with respect to die horizontal the momentum equation is... 

Consider a layer of a newtonian liquid flowing in steady flow at constant rate and thickness over a flat plate as shown in Fig. 5.18. The plate is inclined at an angle jS with the vertical. The breadth of the layer in the direction perpendicular to the plane of the figure is b, and the thickness of the layer in the direction perpendicular to the plate is 8. Isolate a control volume as shown in Fig. 5.18. The upper surface of the control volume is in contact with the atmosphere, the two ends are planes perpendicular to the plate at a distance L apart, and the lower surface is the plane parallel with the wall at a distance r from the upper surface of the layer. 

Conservation Equations. In the above section, the material functions of nonnewtonian fluids and their measurements were introduced. The material functions are defined under a simple shear flow or a simple shear-free flow condition. The measurements are also performed under or nearly under the same conditions. In most engineering practice the flow is far more complicated, but in general the measured material functions are assumed to hold. Moreover, the conservation principles still apply, that is, the conservation of mass, momentum, and energy principles are still valid. Assuming that the fluid is incompressible and that viscous heating is negligible, the basic conservation equations for newtonian and nonnewtonian fluids under steady flow conditions are given by... 

A somewhat harder problem is steady flow of an incompressible newto-nian fluid in sonJe duct or pipe which is of constant cross section but not circular, such asja rectangular duct or an open channel. The problem of laminar flow of a newtonian fluid can be solved analytically for several shapes. Generally the velocity depends on two dimensions. In several cases of interest, the problems can jbe solved by the same method we used to find Eq. 6.8, i.e., setting up a force balance around some properly chosen section of the flow, solving for the sh r stress, introducing the newtonian law of viscosity for the shear stress, and integrating to find the velocity distribution. From the velocity distribution the flow rate-pressure-drop relation is found. 

As an example of the use of the boundary-layer equations, we consider the simplest possible boundary-layer problem, the steady flow of a constant-density, newtonian fluid past a flat plate placed parallel to the flow at velocities... 

Equation 12.13 is known as the Blake-Kozeny equation or the Kozeny-Carman equations it describes the experimental data for steady flow of newtonian fluids through beds of uniform-size spheres satisfactorily for less than about 10. 

PoiseuiUe s Law The relationship between volumetric flow rate and pressure difference for steady flow of a Newtonian fluid in a long circular tube. 

Flow curves for L2, M2 and H2 are shown logarithmically mically in Fig. 13. The shear-rate or shear-stress region where the plots are represented by straight lines having a slope of 1 is the so called Newtonian region (Region II). The Regions of the steady flow behavior may be sensitive to the structural re-formation process after the cessation of the steady flow as mentioned above. 

As a first approximation of geometry, a Y bifurcation using straight rigid mbing may be used. Hie blood may be modeled as a Newtonian fluid without particles. As boundary conditions, the inlet flow may either be considered Poiseuille or uniform across the tube (if a sufficient entry length is included to produce a fully developed flow profile). Steady flow will be used for this model. 

Once the second model is functional, we will turn our attention to the material in the flow. A combination of Bingham and power-law fluid will be used to better model the non-Newtonian characteristics of whole blood. RBCs will not be added for this model because the fluid is now a reasonable approximation of large-vessel flow. We will still use steady flow in this model. 

The steady-flow viscosity of a 10 g.dL l solution of a, w -Mg dicarboxylato PBD in decahydronaphthalene (DHN) is clearly dependent on the Y shear rate when higher than 6 sec l (Figure 9). The observed deviation from the Newtonian behavior confers to the solution a rarely observed shear thickening (or dilatant) character. The substitution of DHN (e= 2.2) by chloroform (e= 4.4) is responsible for a large decrease in the steady-flow viscosity (ca. 350 times), which is now quite independent of the shear rate at least up to 150 sec l. 

Hagen-Poiseuille equation (Poiseuille equation) n. The equation of steady, laminar, Newtonian flow through circular tubes ... 

Another example is plane Couette flow with variable viscosity [13,14]. Consider the steady flow of an incompressible Newtonian fluid between two infinite parallel plates separated by a distance, a, as shown in Eigure 6.2. Each plate is maintained at a temperature Tq. The upper plate is allowed to move... 

---