Mechanics fluid

Fluids are important to biological organisms because they fill interior spaces and completely surround every organism. Water and air are the two most important fluids for biological systems air because it supplies oxygen, acts as a sink for excess carbon dioxide, and can dry the organism and water because it acts as a solvent, attaches to dissolved molecules (see Section 3.2), and is relatively thick. 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

Pressure is the force per unit area exerted by or on a fluid. In a static fluid the pressure increases with depth, but according to Pascal s principle it is the same in all directions at any given depth. Pressure may be specified as either absolute, or gauge, the relationship between the two being  

The governing equation for the pressure within a fluid at any depth h is 

If Y can be considered to be constant, the fluid is said to be incompressible and F.quation 2-46 can be solved to yield 

If the gas behavior deviates markedly from ideal, the real gas law can be w ritten as Ps 

In this chapter, dimensional analysis is applied to several problems associated with the flow of fluids. In stud5dng this material, attention should be directed not to the fine details but to the underlying philosophy and the manner in which dimensional analysis can be used to knit together a vast number of problems which, at first, appear to be independent. What Galileo has to say concerning fluid statics is also considered. 

One of the most important analytic solutions in the study of bubbles, drops, and particles was derived independently by Hadamard (HI) and Rybczynski (R5). A fluid sphere is considered, with its interface assumed to be completely free from surface-active contaminants, so that the interfacial tension is constant. It is assumed that both Re and Rep are small so that Eq. (1-36) can be applied to both fluids, i.e., 

The boundary conditions require special attention. Taking a reference frame fixed to the particle with origin at its center, they are 

The internal motion given by Eq. (3-8) is that of Hill s spherical vortex (H6). Streamlines are plotted in Figs. 3.1 and 3.2 for k = 0 and k = 2, and show the fore-and-aft symmetry required by the creeping flow equation. It may also be noted in Fig. 3.2 that the streamlines are not closed for any value of k, the solution predicts that outer fluid is entrained along with the moving sphere. This entrainment, sometimes known as drift, is infinite in creeping flow. This problem is discussed further in Chapter 4. 

The solution given by Eqs. (3-7) and (3-8) is derived using only the first four boundary conditions (L3) i.e. without considering the normal stress condition, Eq. (3-6). The modified pressures can be obtained from Eq. (1-33) and are given by 

The pressure distribution given by Eq. (3-9) is an odd function of 0, so that the particle experiences a net pressure force or form drag. Integration of the pressure over the surface of the particle leads to a drag component given by 

This article is intended to provide a useful first understanding of flow phenomena and techniques and to provide an entry to more precise and detailed methods where these are required. Although the main concern is the proper design and operation of plant equipment, the importance of preservation of the environment is recognized. Thus data from the fields of meteorology and oceanography are occasionally needed by the technologist (see also Flowl asurel nt Fluidization). 

The most useful mathematical formulation of a fluid flow problem is as a boundary value problem. This consists of two main parts a set of differential equations to be satisfied within a region of interest and a set of boundary conditions to be satisfied on the surfaces of that region. Sometimes additional conditions are also of interest, eg, when one is investigating the stability of a flow. 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

General Equation of Motion. Neglecting relativistic effects, the rate of accumulation of mass within a Cartesian volume element dx-dy-dz must equal the sum of the rates of inflow minus outflow. This is expressed by the equation of continuity  

Kirk-Othmer Encyclopedia of Chemical Technology (4th Edition) 

Friction Loss. Flow, turbulent or streamline, may be represented by a single mathematical formula, except that a different friction factor is used for each region (streamline or turbulent) of flow. Fanning s equation (originally derived for turbulent flow) for flow in a circular conduit is 

Throughout years of industrial development this equation has been found to be valid for the flow of all kinds of fluids in circular conduits. For sections other than circular, the friction loss is usually computed by ui g the hydraulic radius m which is defined as the cross-sectional area divided by the wetted perimeter  

Tablb 13-1. Equivalent Lengths (Approximate) op Pipe Fittings and Restrictions fob Nominal Pipe Sizes In. (Largest Equivalent 

Equipment Diameters of straight pipe Equipment Dimneters of straight pipe 

The equivalent length L is equal to the total length of straight pipe plus the lengths of pipe that are equivalent to the restrictions caused by elbows, valves, bends, etc. Approximate equivalent lengths expressed as diameters of strmght pipe are shown, for turbulent flow, in Table 13-1. 

---

Theoretical models of the film viscosity lead to values about 10 times smaller than those often observed [113, 114]. It may be that the experimental phenomenology is not that supposed in derivations such as those of Eqs. rV-20 and IV-22. Alternatively, it may be that virtually all of the measured surface viscosity is developed in the substrate through its interactions with the film (note Fig. IV-3). Recent hydrodynamic calculations of shape transitions in lipid domains by Stone and McConnell indicate that the transition rate depends only on the subphase viscosity [115]. Brownian motion of lipid monolayer domains also follow a fluid mechanical model wherein the mobility is independent of film viscosity but depends on the viscosity of the subphase [116]. This contrasts with the supposition that there is little coupling between the monolayer and the subphase [117] complete explanation of the film viscosity remains unresolved. 

Landau L D and Lifshitz E M 1959 Fluid Mechanics (London Pergamon) ch 17... 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Landau L D and Lifshitz E M 1959 Fluid Mechanics (Reading, MA Addison-Wesley) eh 2, 7, 16, 17. (More reeent editions do not have ehapter 17.)... 

If a flow satisfies the condition of zero vorticity, that is, the velocity field v is such that V X V = 0, then there exists a function v such that v = Vv. In that case, one can describe the fluid mechanical system with the following Lagrangean density... 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

Aris, R., 1989. Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Publications, New York. 

Herschel, W.H. and Bulkley, R., 1927. See Rudraiah, N, and Kaloni, P.N. 1990. Flow of non-Newtonian fluids. In Encyclopaedia of Fluid Mechanics, Vol. 9, Chapter 1, Gulf Publishers, Houston. 

Pearson,. I.R.A., 1994. Report on University of Wales Institute of Non-Newtonian Fluid Mechanics Mini Symposium on Continuum and Microstructural Modelling in Computational Rheology. /. Non-Newtonian Fluid Mech. 55, 203 -205. 

Bird, R.B., Armstrong, R. C. and Hassager, O., 1977. Dynamics of Polymer Fluids, Vol 1 Fluid Mechanics, Wiley, New York. 

Fluid mechanics Fluid mixing Fluids, ferromagnetic Flumann... 

---