Simplifying assumptions incompressibility

For liquids—and even for gases and vapors when the change in pressure is small—the fluid may be considered as incompressible for all practical purposes, and thus we may take wx = w2 = w as a constant. In turbulent flow the value of a is only a little more than unity, and, as a simplifying assumption, it will now be omitted. Then Eq. (10.5) becomes ... 

The simplifying assumptions for solving this flow problem are the same as those used in Example 2.5 for parallel plate flow, namely, we assume the flow to be an incompressible,... 

Example 7.3 Effect of Viscosity Ratio on Shear Strain in Parallel-Plate Geometry Consider a two-parallel plate flow in which a minor component of viscosity /t2is sandwiched between two layers of major component of viscosities /q and m (Fig. E7.3). We assume that the liquids are incompressible, Newtonian, and immiscible. The equation of motion for steady state, using the common simplifying assumption of negligible interfacial tension, indicates a constant shear stress throughout the system. Thus, we have... 

Solution This flow is z-axisymmetric. We, thus, select a cylindrical coordinate system, and make the following simplifying assumptions Newtonian and incompressible fluid with constant thermophysical properties no slip at the wall of the orifice die steady-state fully developed laminar flow adiabatic boundaries and negligible of heat conduction. 

From the standpoint of fluid mechanics the phenomena occurring in these devices can be classified under the usual headings of incompressible and compressible flow. In pumps and fans the density of the fluid does not change appreciably, and in discussing them, incompressible-flow theory is adequate. In blowers and compressors the density increase is too great to justify the simplifying assumption of constant density, and compressible-flow theory is required. 

The models also assume steady-state conditions and that gas flow behaves as a liquid (i.e. fhe gas or vapour is incompressible). These simplifying assumptions introduce uncertainty into the results of any modelling. However, as long as such factors are taken into accoxmt, simplified mathematical models are a useful aid to decision making and can act as a check on the results of more complex mathematical models. 

An alternative simplifying assumption is to treat the material as incompressible. This gives ... 

Using the same assumptions that were made in the vapor-layer model, the energy-conservation equation for the incompressible 2-D vapor phase can be simplified to a 1-D equation in boundary layer coordinates ... 

The only assumptions made in developing equation (2.18) are (1) that diffusion coefficient does not change with spatial coordinate and (2) incompressible flow. We will further simplify equation (2.18) in developing analytical solutions for mass transport problems. In some cases, all we need to do is orient the flow direction so that it corresponds with one of the coordinate axes. We would then have only one convection term. 

In view of the foregoing, a very close approximation for valve opening may be achieved by assuming an incompressible liquid in a rigid pipe. Such an assumption for instantaneous closure would indicate an infinite pressure rise, but no such result is possible for an instantaneous opening. Therefore in this simplified case we can employ the energy equation, but with a term added for the acceleration head. 

Starting with the open system balance equation, derive the steady-state mechanical energy balance equation (Equation 7.7-2) for an incompressible fluid and simplify the equation further to derive the Bernoulli equation. List all the assumptions made in the derivation of the latter equation. 

Throughout this text we will assume that polymer melts are incompressible liquids, by which we mean that the density never changes with position or time. This is clearly an approximation that must be relaxed in some applications - injection molding, for example, where the compressibility of the melt becomes important because of the extremely high pressures - but the incompressibility assumption will suffice for our purpose here. If the density never changes in time or space, rates of change with respect to these variables (i.e., derivatives) must be zero dp/dt = 0, dp/dx = 0, etc.), and the continuity equation simplifies to... 

A steady, time-independent solution is what we intend. For the plane Poiseuille flow, the most obvious kinematio simplification is to assume the flow is unidirectional (i.e. only the velocity component is non-zero). The second simplification is the assumption that the velocity component is independent from z (plane flow). Lastly, it can be assumed intuitively that is also independent from X, however, this property is also influenced by incompressibility. The stmcture of the flow is, therefore, simplified by using a solution in the form ... 

Experimental evidence has revealed a negligible change in volume occurring during the deformation. This behavior allows the modeling of mbber as an incompressible material. From one hand, this assumption simplifies the determination of equilibrium solutions, but, on the other hand, it makes the constitutive relation hard to implement in a numerical code. Therefore, both near-incompressible and incompressible materials will be considered in the following. 

Estimation of Injection Rates/Pressure Drop for Polymer Flooding. Injection rate is a critical variable in EOR processes. In this section, an approximate model is developed to predict the injection rate or the pressure drop during injection of polymer solution into a well. The well is assumed to be completed open hole or to have sufficient perforations to neglect the pressure drop across the perforations. The polymer solution flows radially away from the wellbore. Fluids are considered to be incompressible. Singlephase flow is used to simplify model development. These assumptions can be removed for specific applications. 

As stated earlier, our goal is to derive general equations that relate these quantities to rheological variables such as shear stress, shear rate, and normal stress differences. Based on the direction of the imposed velocity, the cylindrical symmetry of the flow geometry, and the assumptions that the fluid is incompressible and flow occurs under isothermal conditions, the equations of continuity (Appendix 8.A) yield v = Vg(r,z)eg. Neglecting inertia, the differential linear momentum conservation equations (Appendix 8.B) can be simplified to give... 

The complex flow was simplified by the assumption that the screw chaimel is fully filled with a steady isothermal flow of an incompressible fluid. The Reynolds number of the flow is very small. We ignored the mass force and inertia force, since they are not to be compared with the big viscous force. 

---