Basic Equations of Fluid Flow

Conservation Laws. The fundamental conservation laws of physics can be used to obtain the basic equations of fluid motion, the equations of continuity (mass conservation), of flow (momentum conservation), of... 

Techniques for measuring rheological properties of polymeric materials have been well described previously by others (e.g., Whorlow, 1980 Macosko, 1994). The text by Van Wazer et al. (1963) is still a valuable reference that explains in detail many facets of earlier attempts to measure rheological properties of polymeric materials as well as basic equations of viscometric flows. The unique nature of fluid foods prompted this author to review both the rheological properties of fluid foods and their measurement about 30 years ago (Rao, 1977a, 1977b). Subsequent efforts on rheology of foods include those of Rao (1992, 2005) and Steffe (1996). 

At the pore scale, the basic equations for fluid flow are the fundamental equations of motion. (And, once velocity is obtained it can be used in the convection terms of the energy- and mass-conservation equations.) Two significant hurdles exist when working with these equations. The first is the complexity of the boundary conditions. The second is the very small length scales over which numerical solutions can be obtained. 

This is the basic equation of fluid statics, also called the barometric equation. It is correct only if there are no shear stresses on the vertical faces of the cube in Fig. 2.1. If there are such shear stresses, then they may have a component in the vertical direction, which must be added to the sum of forces in Eq. 2.1. For simple newtonian fluids, shear stresses in the vertical direction can exist only if the fluid has a different vertical velocity on one side of the cube from that on the other side (see Eq. 1.5). Thus this equation is correct if the fluid is not moving at all, which is the case in fluid statics, or if it is moving but only in the X and y directions, or if it has a uniform velocity in the z direction. In this chapter, we apply it only when a fluid has no motion relative to its container or to some set of fixed coordinates. In later chapters, we apply it to flows in which there is no motion in the z direction or a motion with a uniform z component. 

The basic equations of fluid dynamics include the conservation of mass, momentum, and energy. The conservation of momentum is expressed by the Navier-Stokes equations, which determine the velocity field in a flow. 

Wilhelm Nusselt (1882-1957), a German engineer, published in 1915 his pioneering work "The basic law of heat transfer (Nusselt, 1915), in which he derived the dimensionless numbers directly from the differential equations of fluid flow and heat transfer. In this paper he states (see Topic 3.2.2) ... 

In this appendix, we will derive the essential equations for swirling flow from the basic equations of fluid mechanics the Navier-Stokes equations. The Navier-Stokes equations are derived in most textbooks on fluid mechanics, for instance Bird et al. (2002). 

Pressure drop on the tube-side of a shell and tube exchanger is made up of the friction loss in the tubes and losses due to sudden contractions and expansions and flow reversals experienced by the tube-side fluid. The friction loss may be estimated by the methods outlined in Section 3.4.3 from which the basic equation for isothermal flow is given by equation 3.18 which can be written as ... 

The theoretical and numerical basis of computational flow modeling (CFM) is described in detail in Part II. The three major tasks involved in CFD, namely, mathematical modeling of fluid flows, numerical solution of model equations and computer implementation of numerical techniques are discussed. The discussion on mathematical modeling of fluid flows has been divided into four chapters (2 to 5). Basic governing equations (of mass, momentum and energy), ways of analysis and possible simplifications of these equations are discussed in Chapter 2. Formulation of different boundary conditions (inlet, outlet, walls, periodic/cyclic and so on) is also discussed. Most of the discussion is restricted to the modeling of Newtonian fluids (fluids exhibiting the linear dependence between strain rate and stress). In most cases, industrial... 

Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restrictedTcase is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. 

In order to explain the basic properties of compressible flow, we will look at a two-dimensional, steady, boundary layer flow of a pure fluid. In a change from the previous discussion, in addition to the density, the viscosity and thermal conductivity will also be locally variable. The continuity equation is... 

Because of the fact that the capsules in HCP have no wheels and are strongly affected by the buoyancy and hydrodynamic lift, the fluid mechanics of HCP is far more complicated than that of PCP. However, extensive research has been conducted in HCP since 1960 by more than 30 researchers in about 10 nations to understand the basic nature of HCP flow and to derive the necessary equations for the design of HCP. In what follows, only the most pertinent theories and equations needed for the design of commercial HCPs are discussed. Readers interested in knowing more about the theory of HCP should read the vast body of papers and reports published they total well over 100. 

To use Eq. (8.23), the integral must be evaluated, which requires information on the path followed by the fluid in the machine from suction to discharge. The procedure is the same whether the compressor is a reciprocating unit, a rotary positive-displacement unit, or a centrifugal unit, provided only that the flow is frictionless and that in a reciprocating machine the equation is applied over an integral number of cycles, so there is neither accumulation nor depletion of fluid in the cylinders otherwise the basic assumption of steady flow, which underlies Eq. (4.32), would not hold. 

The basic equations for mass transfer were derived in Chapter 3. The overall rate of change in local concentration depends on both the rate of diffusion and the rate of fluid flow (see Table 3.2) ... 

If all the fluid flow is in the horizontal direction, then the basic equation of... 

In the region ou tside the boundary layer, where the fluid may be assumed to have no viscosity, the mathematical solution takes on the form known as potential flow. This flow is analogous to the flow of heat in a temperature field or to the flow of charge in an electrostatic field. The basic equations of heat conduction (Fourier s law) are... 

Conduction. Heat flow by conduction is a result of transfer of kinetic and/or internal energy between molecules in a fluid or solid. The basic equation of conductive heat transfer is Fourier s law... 

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