Flow of Incompressible Fluid

The data on pressure drop in irregular channels are presented by Shah and London (1978) and White (1994). Analytical solutions for the drag in micro-channels with a wide variety of shapes of the duct cross-section were obtained by Ma and Peterson (1997). Numerical values of the Poiseuille number for irregular microchannels are tabulated by Sharp et al. (2001). It is possible to formulate the general features of Poiseuille flow as follows  

Equation (3.4) reflects the dependence of the friction factor on the Reynolds number, whereas Eq. (3.5) shows conformity between actual and calculated shapes of a micro-channel. Condition (3.5) is the most general since it testifies to an identical form of the dependencies of the experimental and theoretical friction factor on the 

Basically, there may be three reasons for the inconsistency between the theoretical and experimental friction factors (1) discrepancy between the actual conditions of a given experiment and the assumptions used in deriving the theoretical value, (2) error in measurements, and (3) effects due to decreasing the characteristic scale of the problem, which leads to changing correlation between the mass and surface forces (Ho and Tai 1998). 

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In Chap. 3 the problems of single-phase flow are considered. Detailed data on flows of incompressible fluid and gas in smooth and rough micro-channels are presented. The chapter focuses on the transition from laminar to turbulent flow, and the thermal effects that cause oscillatory regimes. 

This chapter has the following structure in Sect. 3.2 the common characteristics of experiments are discussed. Conditions that are needed for proper comparison of experimental and theoretical results are formulated in Sect. 3.3. In Sect. 3.4 the data of flow of incompressible fluids in smooth and rough micro-channels are discussed. Section 3.5 deals with gas flows. The data on transition from laminar to turbulent flow are presented in Sect. 3.6. Effect of measurement accuracy is estimated in Sect. 3.7. A discussion on the flow in capillary tubes is given in Sect. 3.8. 

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

In capillary flow with a distinct meniscus separating the regions of pure liquid and pure vapor flows, it is possible to neglect the change in densities of the phases and assume po and pu are constant. For flow of incompressible fluid (p = const., p = 0, duildx = 0) the substitution of (11.8) in Eqs. (11.1-11.3) leads, in a linear approximation, to the following system of equations... 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

Relatively straightforward solutions can be obtained for the viscous flow of incompressible fluids in ducts of constant cross-section flowing at relatively low velocities. 

L = linear dimension of bed of powder parallel to direction of flow of air (commonly known as height of powder bed) r = viscosity of air at its temperature at time of determination q = rate of flow of incompressible fluid through bed of powder... 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

Kaganov, S. A., On steady-state laminar flow of incompressible fluid in a plane channel and in a circular cylindrical tube with regard to heat friction and dependence of viscosity on temperature, J. Appl. Mech. Techn. Phys., No. 3, 1962. 

Nikitin, N. V., Spectral finite-element method for the analysis of turbulent flows of incompressible fluid through tubes and channels, Comput. Math, and Math. Phys., Vol. 34, No. 6, pp. 785-798, 1994. 

Also, the usefulness of the corrected Bernoulli equation in solving problems of flow of incompressible fluids is enhanced if provision is made in the equation for the work done on the fluid by a pump. 

Equation (4.32) is a final working equation for problems on the flow of incompressible fluids. 

FLOW OF INCOMPRESSIBLE FLUIDS IN CONDUITS AND THIN LAYERS 85... 

Form-friction losses in the Bernoulli equation. Form-friction losses are incorporated in the term of Eq. (4.32). They are combined with the skin-friction losses of the straight pipe to give the total friction loss. Consider, for example, the flow of incompressible fluid through the two enlarged headers, the connecting... 

Flow of incompressible fluid through typical assembly. 

PoiseuilWs Law Poiseuille flow is the steady flow of incompressible fluid parallel to the axis of a circular pipe or capillary. Poiseuille s law is an expression for the flow rate of a liquid in such tubes. It forms the basis for the measurement of viscosities by capillary viscometry. 

The topic of flow through an effectively infinite system of particles belongs to the more general domain of flow through porous media (C12, C13, PIO, R5, S3, S4). In the absence of physicochemical interaction between the particles and fluid, the slow quasi-static Newtonian flow of incompressible fluids through such media is governed by Darcy s law. This linear phenomenological law has the form... 

For the flow of incompressible fluids, the shear stress at the wall, could be represented in terms of the friction factor / by... 

Constantinescu VN (1995) Steady parallel flow of incompressible fluids. In Ling EF (ed) Laminar viscous flow. Springer, New YotIc, p 123... 

An alternative molecular weight-sensitive detector is the on-line viscometer. All current instrument designs depend upon the relationship between pressure drop across a capillary through which the polymer sample solution must flow and the viscosity of the solution. This relationship is based upon Poiseuille s law for laminar flow of incompressible fluids through capillaries ... 

The Lattice Boltzmann Method (LBM), including the method Cellular Automaton (AC), present a powerful alternative to standard apvproaches known like "of up toward down" and "of down toward up". The first approximation study a continuous description of macroscopic phenomenon given for a partial differential equation (an example of this, is the Navier-Stokes equation used for flow of incompressible fluids) some numerical techniques like finite difference and the finite element, they are used for the transformation of continuous description to discreet it permits solve numerically equations in the compniter. 

Fort the solution of complex problems of dynamics of fluids, exist traditionally two kinds of points of view the first is macroscopic, which is considered continuous, with an ap>proach of differential equations in p>artial derivatives, for example of Navier-Stokes equations used for flow of incompressible fluids and numerical techniques for its solution. The second pwint of view is microscopic it has its basis in kinetics theory of gases and statistical mechanics. 

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