Reynolds number vanishingly small

For large Reynolds numbers fie — oo all terms with the factor 1 /fie and even more those with 1/fie2 will be vanishingly small. After neglecting these terms and returning to equations with dimensions, we obtain the boundary layer equations, which were first given in this form by L. Prandtl (1875-1953) [3.4] in 1904 ... 

Experimental studies have been made of the impaction of nearly monodi-sperse sulfuric add particles in the. size range 0.3 < dp <. 4 pm on a wire 77 m in diameter over a Reynolds number range 62 to 500 (Ranz and Wong, 1952 Wong and Johnstone, 1953). In Fig. 4.7, these data are compared with numerical computations based on inviscid flow theory for point particles (Brun el al., 1955). Agreement between measured and calculated values is fair. The experimental results fall somewhat below the calculated values for 0.8 < Stk < 3, At values of Stk above about 3, the data fall above theory. Dam were not taken at sufficiently small values of Stk to test the theoretical value (Stkiznt = 1/8) al which the efficiency is expected to vanish. The results of these measurements are not directly applicable to high-efficiency fibrous filters, which usually operate at much lower Reynolds numbers based on the fiber diameter. 

The reader may note that the preceding developments actually provide a basis for calculating corrections to the force on bodies, 3 D, because of a number of different kinds of weak departures from the creeping-flow limit for Newtonian fluids. One additional example is the effects of weak non-Newtonian contributions to the motion of a body. In this case, for vanishingly small Reynolds numbers, we can symbolically write the equation of motion in the form... 

Particles of finite size or with a density different from that of the surrounding fluid (e.g. liquid droplets or dust particles suspended in a fluid), due to their inertia and non-vanishing size, have an instantaneous velocity that is somewhat different from the local velocity of the fluid. Therefore such inertial effects can have a significant influence on the distribution of suspended particles. If the Reynolds number based on the size of the particle and its velocity relative to the fluid is small, the flow around the particle can be approximated... 

When the Reynolds number is not vanishingly small, Xl depends on the Reynolds number, too, and the dependence is easily determined using CFD. For example. Fig. 8.4 shows the results for a 3 1 contraction in a pipe. Note that Xl is approximately constant for Re < 1. The departure from a constant is due partially to extra viscous dissipation at higher Reynolds number, but mostly due to kinetic energy changes. 

If the motion is steady when viewed from an inertial reference frame, the term 5v/5i vanishes identically. Providing that the remaining nondimensional terms remain finite as i -> 0, these then reduce to Stokes equations at sufficiently small Reynolds numbers. In dimensional form, Stokes equations are... 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is... 

Even though the species velocity vectors v are nonzero for reactants and products that diffuse toward and away from the internal catalytic surface, it is customary to neglect convective mass transfer within the pores. In other words, the Reynolds number is vanishingly small and diffusion dominates convective transport. Under these conditions, the dimensionless mass transfer eqnation for component i reduces to... 

Most of the microfluidic devices consist of microchannel of different dimensions interconnected with each other. In the case of two straight channels of different dimensions connected to form one long channel, the translation invariance, that is, fully developed flow assumption, is broken. Hence, the expressions for the ideal Poiseuille flow no longer apply. However, it is expected that the ideal description is approximately correct if the Reynolds number (Re) of the flow is sufficiently small. This is because a very small value of Re corresponds to a vanishing small contribution from the nonlinear term (V - V)F in the Navier-Stokes equation (N-S equation), a term that is strictly zero in ideal Poiseuille flows due to translation invariance. 

It should be mentioned that this treatment for the terms that vanish under certain conditions is proper. It is similar to the treatment of neglecting the mass of the fluid when solving fluid dynamic problems for low-Reynolds number flows, where mass exists, but just too small to have any significant influence. 

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