Macroscopic flow equations, laminar

The Mean Velocity of Laminar Pipe Flow Use the macroscopic mass-balance equation (Eq. 2.4.1) to calculate the mean velocity in laminar pipe flow of a Newtonian fluid. The velocity profile is the celebrated Poisseuille equation ... 

The "laminar" macroscopic flow equations contain phenomenological terms which represent averages over the macroscopic dynamics to include the effects of turbulence. Examples of these terms are eddy viscosity and diffusivity coefficients and average chemical heat release terms which appear as sources in the macroscopic flow equations. Besides providing these phenomenological terms, the turbulence model must use the information provided by the large scale flow dynamics self-consistently to determine the energy which drives the turbulence. The model must be able to follow reactive interfaces on the macroscopic scale. 

The stability of flow in open channels has been investigated theoretically from a more macroscopic or hydraulic point of view by several workers (Cl7, D9, DIO, Dll, 14, J4, K16, V2). Most of these stability criteria are expressed in the form of a numerical value for the critical Froude number. Unfortunately, most of these treatments refer to flow in channels of very small slope, and, under these circumstances, surface instability usually commences in the turbulent regime. Hence, the results, which are based mainly on the Ch<5zy or Manning coefficient for turbulent flow, are not directly applicable in the case of thin film flow on steep surfaces, where the instability of laminar flow is usually in question. The values of the critical Froude numbers vary from 0.58 to 2.2, depending on the resistance coefficient used. Dressier and Pohle (Dll) have used a general resistance coefficient, and Benjamin (B5) showed that the results of such analyses are not basically incompatible with those of the more exact investigations based on the differential rather than the integral ( hydraulic ) equations of motion. The hydraulic treatment of the stability of laminar flow by Ishihara et al. (12) has been mentioned already. 

It turns out that Eq. (5-56) can also be applied to turbulent flow over a flat plate and in a modified way to turbulent flow in a tube. It does not apply to laminar tube flow. In general, a more rigorous treatment of the governing equations is necessary when embarking on new applications of the heat-trans-fer-fluid-friction analogy, and the results do not always take the simple form of Eq. (5-56). The interested reader may consult the references at the end of the chapter for more information on this important subject. At this point, the simple analogy developed above has served to amplify ouf understanding of the physical processes in convection and to reinforce the notion that heat-transfer and viscous-transport processes are related at both the microscopic and macroscopic levels. 

The discussion of modeling and simulation techniques for microreactors shows that the toolbox available at present is quite diverse and goes well beyond the standard capabilities of CFD methods available in commercial solvers. In micro-reactors, special methods needed for the modeling of noncontinuum physics play only a minor role and most of the effects are described by the standard continuum equations. However, even if the laminar nature of the flow somehow reduces the difficulty of simulation problems compared to macroscopic flows, there are a number of problems that are extremely difficult and require very fine computational grids. Among these problems is the numerical study of mixing in liquids that often suffers severely from discretization artefacts. 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pf/L/p, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro-/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term (pu Vu) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro-/ nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity... 

Equation 6 applies to the analysis of fluid behavior in tribology and microchannel flows this kind of flow is called Stokes flow. In this case, the velocities are free of macroscopic fluctuations at any point in the flow field and the flow is defined as laminar. 

Laminar flow sometimes known as streamline flow , this type of flow of a liquid or gas occurs when the fluid flows in parallel layers, with no macroscopic disruption between the layers (exchange at the molecular level via diffusion can still occur), in accord with the Poiseuille Equation for flow through a tube of radius r, (,g and length /t be (volume flow rate U = ir.rube/8-i1-/ be)- where y is the viscosity coefficient of the fluid laminar flow is to be contrasted with turbulent flow. 

As can be noted, the axial velocity does not depend on the radial coordinate, which is xmique for any macroscopic flow. This property simplifies a mathematical handling of the particle-transport equation, as will be discussed later. Another advantage of the rotating disk is that the flow remains steady and laminar for a Reynolds number Re = mR /v) as large as 10 [78]. In contrast, all of the remaining macroscopic flows discussed subsequently are approximate only, usually valid for a limited range of Reynolds numbers. 

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