Fluid flow incompressible flows

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. 

When the fluid is incompressible, flow rate changes at point 1 propagate instantaneously to any other point in the pipe. For compressible fluids such as gases, the simple expression of (6-33) may not be accurate. Note that use of a constant time delay implies constant flow rate. 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

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. 

Kheshgi, H. S. and Scriven, L. E., 1985. Variable penalty method for finite element analysis of incompressible flow. Int. J. Numer. Methods Fluids 5, 785-803. 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Example 1 Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity Vj is given and the pressures pi and measured. 

Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can he solved using the mechanical energy balance. In a well-made venturi, viscous losses are neghgihle, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

Example 4 Plnne Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to he found. This example is found in most fluid mechanics textbooks the solution presented here closely follows Denn. 

Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases. Liquid flows are normally considerea incompressible, except for certain calculations involved in hydraulie transient analysis (see following) where compressibility effects are important even for nearly incompressible hquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro Dynamics and Thermodynamics of Compre.ssible Fluid Flow, vol. 1 and 11, Ronald Press, New York [1953]) and Zucrow and Hofmann (G .s Dynamics, vol. 1 and 11, Wiley, New York [1976]). 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

This chapter is a brief diseussion of large deformation wave codes for multiple material problems and their applications. There are numerous other reviews that should be studied [7], [8]. There are reviews on transient dynamics codes for modeling gas flow over an airfoil, incompressible flow, electromagnetism, shock modeling in a single fluid, and other types of transient problems not addressed in this chapter. 

This example demonstrates the dimensioning of a duct with a frictional incompressible fluid flow. Now the Bernoulli equation can be written as... 

For an irrotational, incompressible, and frictionless fluid flow there exists a scalar velocity potential 4> such that the velocity vector V is... 

The steady, laminar, incompressible fluid flow in cyclone collectors is governed by the Navier-Stokes equations ... 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

D. Givoli, J. E. Flaherty, M. S. Shephard. Parallel adaptive finite element analysis of viscous flows based on a combined compressible-incompressible formulation. Int J Numer Meth Heat and Fluid Flow 7 880, 1997. 

Integrating Equation 4-116 assuming incompressible drilling fluid flow (p is constant) and after simple rearrangements yields the pressure loss across the bit Apj (psi) which is... 

The Navier-Stokes equations are the fundatnenta equations describing incompressible, fluid flow see Chapter 9. 

Flashing liquids, 134-146 Flow coefficients, Gv, for valves, 81 Friction loss, 68 Incompressible fluid, 71 Laminar flow, 77, 78, 86 Liquid lines, chart, 92 Long natural gas pipe lines, 120 Non-water liquids, 99 Pipe, 71... 

For an incompressible fluid flowing in a horizontal pipe of constant cross-section, in the absence of work being done by the fluid on the surroundings, the pressure change due to frictional effects is given by ... 

Obot NT (1988) Determination of incompressible flow friction in smooth circular and nondrcular passages. A generaUzed approach including validation of the century old hydraulic diameter concept. Trans ASME J Fluid Eng 110 431-440... 

In our analysis, we discuss experimental results of heat transfer obtained by previous investigators and related to incompressible fluid flow in micro-channels of different geometry. The basic characteristics of experimental conditions are given in Table 4.1. The studies considered herein were selected to reveal the physical basis of scale effect on convective heat transfer and are confined mainly to consideration of laminar flows that are important for comparison with conventional theory. 

If the pilot reactor is turbulent and closely approximates piston flow, the larger unit will as well. In isothermal piston flow, reactor performance is determined by the feed composition, feed temperature, and the mean residence time in the reactor. Even when piston flow is a poor approximation, these parameters are rarely, if ever, varied in the scaleup of a tubular reactor. The scaleup factor for throughput is S. To keep t constant, the inventory of mass in the system must also scale as S. When the fluid is incompressible, the volume scales with S. The general case allows the number of tubes, the tube radius, and the tube length to be changed upon scaleup ... 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

VAN Doormal, ). P., Raithby, G. D., Enhancement of the SIMPLE method for predicting incompressible fluid flows. Numerical Heat Transfer 7 (1984) 147-163. 

For a fan the changes in pressure and temperature are small enough that the incompressible fluid flow equations given for pumps may be used. In fact, for most gas systems, if the pressure drop is less than 40% of absolute upstream pressure the fluids can be treated as incompressible.30... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

Example 4-1 Manometer. The pressure difference between two points in a fluid (flowing or static) can be measured by using a manometer. The manometer contains an incompressible liquid (density pm) that is immiscible with the fluid flowing in the pipe (density pf). The legs of the manometer are connected to taps on the pipe where the pressure difference is desired (see Fig. 4-2). By applying Eq. (4-7) to any two points within either one of the fluids within the manometer, we see that... 

This is the microscopic (local) continuity equation and must be satisfied at all points within any flowing fluid. If the fluid is incompressible (i.e., constant p), Eq. (5-7) reduces to... 

Example 5-3 Diffuser. A diffuser is a section in a conduit over which the flow area increases gradually from upstream to downstream, as illustrated in Fig. 5-3. If the inlet and outlet areas (Ai and A2) are known, and the upstream pressure and velocity (Pi and V ) are given, we would like to find the downstream pressure and velocity (P2 and If ). If the fluid is incompressible, the continuity equation gives V2 ... 

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