Hydrodynamic equations Reynolds

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

We developed the relationship between position-dependent viscosity q(z), n, and D(ti) via generalization of the hydrodynamic model (Reynolds equation) ... 

As a rule, the force of inertia is small in comparison with the viscous force, and therefore the hydrodynamic equations may be reduced to Stokes equations appropriate to hydrodynamics at small Reynolds numbers. 

Onishi, Y. Sone, Y. Kinetic theory of slightly strong evaporation and condensation - Hydrodynamic equation and slip boundary condition for finite Reynolds number -. J. Phys. Soc. Japan 47 (1979) 1676-1685. 

To calculate the particle velocities V (n = l,2,... N) we have to know the fluid velocity v(r) created by the external forces acting on the particles. In the usual condition of Brownian motion, the relevant hydrodynamic equation of motion is that of the low Reynolds number... 

One can expect the problems of this kind to be not simple technically, since they involve a free boundary whose shape has to be determined simultaneously with the computation of the flow field. We shall see, however, that the difficulty is not mere technical, but extends to the physics of the problem. Our aim is to understand this basic difficulty, and therefore we formulate the flow equations in the simplest possible way. First, we restrict to slowly moving fluids and neglect inertial effects. This is, in fact, not a serious restriction, since the Reynolds number relevant for the motion in the vicinity of the contact line should be based on the local film thickness, and goes down to zero as the contact line is approached. We shall also assume here that the fluid is incompressible. Thus, the hydrodynamic equations for the velocity field u x) are the Stokes and continuity equations ... 

The approach has proven to be advantageous in comparison with the conventional" method. Because the reduced equation is a special form of the Reynolds equation, a full numerical solution over the entire computation domain, including both the hydrodynamic and the contact areas, thus obtained through a unified algorithm for solving one equation system. In this way, both hydrod5mamic and con-... 

If a gas such as ammonia or CO2 (phase 1) is absorbing into a liquid solvent (phase 2), the resistance R2 is relatively important in controlling the rate of adsorption. This is also true of the desorption of a gas from solution into the gas phase. Usually R2 is of the order 10 or 10 sec. cm. h though the exact value is a function of the hydrodynamics of the system consequently various hydrodynamic conditions give a variety of equations relating R2 to the Reynolds number and other physical variables in the system. For the simplest system where the liquid is infinite in extent and completely stagnant, one can solve the diffusion equation... 

Since in hydrodynamic lubrication the friction force is completely determined by the viscous friction of the lubricant, the coefficient of friction can be calculated from hydrodynamics using the Navier-Stokes equations. This had already been done in 1886 when Reynolds published his classical theory of hydrodynamic lubrication [494], The friction force Fp between two parallel plates of area A separated by the distance d is given by ... 

Here x is a phenomenological parameter measuring the chirality and / is a size scale factor. Since here the Reynolds number is small ( 10 s), the Stokes equation can be used to get r = DS2. where D is the hydrodynamic drag coefficient and 2 is the rotational speed. The drag coefficient for a cylindrical object rotating about its axis with cross-sectional radius r and length L is D = 4ztT)r2L, where tj is the viscosity of the medium [19]. Therefore, D /3 and the rotational speed 2 of the rotor will scale as... 

See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

The MRS closures will attract most interest for use wherever MTE methods fail. For example, in flows with rotation the Coriolis terms enter the Rij equations, but drop out in the equation for Ru — q Therefore, an MRS method probably will be essential for including rotation effects, which are of considerable importance in many practical engineering and geophysical problems. Other effects that have not yet been adequately modeled and for which MRS methods may offer some hope include additive drag reduction, ultrahigh Reynolds numbers, separation, roughness, lateral and transverse curvature, and strong thermal processes that affect the hydrodynamic motions. 

If the characteristic linear dimension of the flow field is small enough, then the measured hydrodynamic data differ from those predicted by the Navier-Stokes equations [79]. With respect to the value in macrocharmels, in microchannels (around 50 microns of section) (i) the friction factor is about 20-30% lower, (ii) the critical Reynolds number below which the flow remains laminar is lower (e.g., the change to turbulent flow occurs at lower linear velocities) and (iii) the Nusselt number, for example, heat transfer characteristics, is quite different [80]. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value for the fully developed flow in the conventional channel. These effects have been investigated extensively in relation to the development of more efficient cooling devices for electronic applications, but have clear implications also for chemical applications. 

For low shear stresses in the dispersions, the characteristic velocity, of the relative particle motion is small enough for the Reynolds number, Re = pF L/ri, to be a small parameter, where L is a characteristic length scale. In this case, the inertia terms in Equations 5.247 and 5.249 can be neglected. Then, the system of equations becomes linear and the different types of hydrodynamic motion become additive e.g., the motion in the liquid flow can be presented as a superposition of elementary translation and rotational motions. 

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