Inertial forces, Reynolds number

In the case of flow in microscale pathways, some important parameters in macroscale fluid mechanics remain important, but the features of the flow are very different. The influences of viscosity and surface effects are much more significant compared with fluid flow in traditional pipelines or channels. That is, the ratio of the inertial force to the viscous force (Reynolds number) is in general smaller than unity, and the ratio of the inertial force to surface tension (Weber number) is also small, so that accurate microfluidic metering is somewhat more difficult than metering for macroscopic flows. For gas flows, the effect of the Knudsen number, defined as the ratio of the molecular mean free path to the channel size, is high and cannot be ignored for microscale channels, and hence the fluid metering discussed here will be for liquids only. The Reynolds number Re, the Weber number We, and the Knudsen number Kn are expressed as follows ... 

Reynolds dumber. One important fluid consideration in meter selection is whether the flow is laminar or turbulent in nature. This can be deterrnined by calculating the pipe Reynolds number, Ke, a dimensionless number which represents the ratio of inertial to viscous forces within the flow. Because... 

For Reynolds numbers > 1000, the flow is fully turbulent. Inertial forces prevail and becomes constant and equal to 0.44, the Newton region. The region in between Re = 0.2 and 1000 is known as the transition region andC is either described in a graph or by one or more empirical equations. 

Based on such analyses, the Reynolds and Weber numbers are considered the most important dimensionless groups describing the spray characteristics. The Reynolds number. Re, represents the ratio of inertial forces to viscous drag forces. 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

Darcy s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and conecting the velocity with porosity yields a Reynolds number in the form ... 

The Archimedes number may be considered as a ratio of thermal buoyancy force to inertial force, while the Reynolds number may be looked upon... 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

For an incompressible viscous fluid (such as the atmosphere) there are two types of flow behaviour 1) Laminar, in which the flow is uniform and regular, and 2) Turbulent, which is characterized by dynamic mixing with random subflows referred to as turbulent eddies. Which of these two flow types occurs depends on the ratio of the strengths of two types of forces governing the motion lossless inertial forces and dissipative viscous forces. The ratio is characterized by the dimensionless Reynolds number Re. 

Reynolds number pLu Re = P- inertial force viscous force 1 10 1 ... 

Another complication, caused by stirring the two phases at the same speed, occurs when the two solutions have different viscosities, which is common for immiscible liquids. The key fluid flow parameter is the Reynolds number. Re, which is the ratio of inertial to viscous forces in the solution, as indicated by... 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

The Reynolds number Re = vl/v, where v and l are the characteristic velocity and length for the problem, respectively, gauges the relative importance of inertial and viscous forces in the system. Insight into the nature of the Reynolds number for a spherical particle with radius l in a flow with velocity v may be obtained by expressing it in terms of the Stokes time, t5 = i/v, and the kinematic time, xv = l2/v. We have Re = xv/xs. The Stokes time measures the time it takes a particle to move a distance equal to its radius while the kinematic time measures the time it takes momentum to diffuse over... 

The boiling Reynolds number or bubble Reynolds number (Re,) is defined as the ratio of the bubble inertial force to the liquid viscous force, which indicates the intensity of liquid agitation induced by the bubble motion ... 

At high Reynolds numbers (high turbulence levels), the flow is dominated by inertial forces and wall roughness, as in pipe flow. The porous medium can be considered an extremely rough conduit, with s/d 1. Thus, the flow at a sufficiently high Reynolds number should be fully turbulent and the friction factor should be constant. This has been confirmed by observations, with the value of the constant equal to approximately 1.75 ... 

In this form, u2JgL, the Froude number, can be viewed as a ratio of inertial to gravity forces Ps jPf is a ratio of particle to fluid inertial forces A UodpjP is the Reynolds number or ratio of particle inertial to fluid viscous forces and Pf u0 L[ p, a Reynolds number based on the bed dimensions and fluid density, is a ratio of fluid inertial to viscous forces. 

The use of a Reynolds number based on relative velocity rather than superficial velocity in setting these limits was suggested by Horio (1990). In setting viscous or inertial limits, it is the interphase drag which is characterized as being dominated by viscous or inertial forces. The particle inertia is important even if the interphase drag is viscous dominated. This is because of the typically large solid-to-gas density ratio. 

For steady injection of a liquid through a single nozzle with circular orifice into a quiescent gas (air), the mechanisms of jet breakup are typically classified into four primary regimes (Fig. 3 2)[4°][41][22°][227] according to the relative importance of inertial, surface tension, viscous, and aerodynamic forces. The most commonly quoted criteria for the classification are perhaps those proposed by Ohnesorge)40] Each regime is characterized by the magnitudes of the Reynolds number ReL and a dimensionless number Z ... 

Reynolds Number Re = pL oA) / Hl Compare inertial force to viscous force Madejski [4011... 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

The Reynolds number is the ratio of inertial to viscous forces and depends on the fluid properties, bulk velocity, and boundary layer thickness. Turbulence characteristics vary with Reynolds number in boundary layers [40], Thus, variation in the contributing factors for the Reynolds number ultimately influences the turbulent mixing and plume structure. Further, the fluid environment, air or water, affects both the Reynolds number and the molecular diffusivity of the chemical compounds. 

At higher Reynolds numbers, the inertial forces become increasingly predominant and the drop will be distorted from the spherical shape. 

Flows can be classified into two major categories (a) incompressible and (b) compressible flow. Most hqnids fall into the incompressible-flow category, while most gases are compressible in nature. A perfect fluid can be defined as a flnid that is nonviscous and nonconducting. Fluid flow, compressible or incompressible, can be classified by the ratio of the inertial forces to the viscons forces. This ratio is represented by the Reynolds nnmber (Nji,). At a low Reynolds number, the flow is considered to be laminar, and at high Reynolds numbers, the... 

These groups have a definite, important, physical meaning. The Reynolds number is the ratio of inertial forces to viscous forces, the Sherwood number the ratio of mass transfer resistance in fluid film to mass transfer in bulk fluid, and Schmidt number the ratio of momentum diffusivity to mass diffusivity. 

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