Stokes flow with inertia

For low-viscosity liquids flowing in concentric cylinders, with the inner cylinder rotating at high velocities, we see the appearance inertia-driven secondary flow cells called Taylor vortices, see figure 8. The onset of these vorhces is controlled by the Taylor number which is given by 

CO is the rotational speed of the inner cylinder in rad/ s (= 2ti(I) where/is in revs, per second) and rj is the fluid viscosity. 

The onset of the vortices is governed by the critical Taylor number, Tc, which for small gaps 0.95) is about 1700, but for larger gaps is given by 

The cell pattern becomes wavy for a Taylor number 50% greater than the onset value, Tc, and above that value soon becomes turbulent. 

On the other hand, if in this flow geometry the outer cylinder is rotated and the inner is kept stationary, there is a sharp and catastrophic transition to turbulence (without any Taylor-Hke secondary flows) at a simple Re molds number. Re, of about 15,000, which is given by 

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Problem 8-5. The annular region between two concentric rigid spheres of radii a and A.a (with k > 1) is filled with Newtonian fluid of viscosity /i and density p. The outer sphere is held stationary whereas the inner sphere is made to rotate with angular velocity fl. Assume that inertia is negligible so that the fluid is in the Stokes flow regime. 

The spherical form of a drop or a bubble in Stokes flow follows from the fact that the flow is inertia-free. However, even for the case in which the inertia forces dominate viscous forces and the Reynolds number cannot be considered small, the drop remains undeformed if the inertia forces are small compared with the capillary forces. The ratio of inertial to capillary forces is measured by the Weber number We = p U a/a, where cr is the surface tension at the drop boundary. For small We, a deformable drop will conserve the spherical form. 

The flow is considered to be inertia free and to obey the Stokes equation, with the solid spherical particle taken to move within the cell with superficial velocity U. This is equivalent to using a coordinate system moving with velocity U. The appropriate boundary conditions are thus... 

Inertial Impaction and Bounce. Whenever fluid flow streamlines change direction it is possible that particles may be less able to alter direction because of their greater inertia. If a collection surface is nearby the particles may make contact with the target before becoming trapped in another flow streamline. The dimensionless term used to characterise the inertia of a system is called the Stokes number ... 

Because of the inertia of the fluid, the Stokes layer thickness, 5, is inversely related to the flow frequency, with S [Pg.120]

In contrast, at a = 13 (Figure 7.2b), which characterizes rest state flow in the aorta and trachea, the velocity profile of the pipe core is nearly uniform at all flow phases. Flow in the boundary layer is out of phase with that in the core, and flow reversals are possible in the Stokes layer. These changes in the velocity fields result from the inertia of the fluid, since as the flow frequency increases, less time is available in each flow cycle to accelerate the fluid... 

The behavior of continuum fluid flow is governed by the well-known Navier-Stokes equations. In the low-Reynolds-number regime, where the kinematic fluid viscosity V is large compared with the characteristic velocity of the flow u and inertia terms can safely be neglected, the Stokes equations can be deduced from the Navier-Stokes equations. The particle Reynolds number is defined for the particle radius a as ... 

The differ tial equations of Navier-Stokes are basic equations in fluid dynamics. By solving them together with the equation of continuity for each of the phases of a multi-phase flow stem with the corresponding boundary conditions, theoretically, it is possible to describe the hydrodynamic processes in all technical and natiue systems. The equations, written on the basis of a balance of die forces of viscosity, gravity and inertia, are as follows. 

The impaction of spherical particles depicted in Figure 9.1 depends on the fluid properties and the particle diameter and density as described by the Stokes number. Those particles with an aerodynamic diameter larger than a well-defined cutoff size will collide with an impaction plate due to their larger inertia. Smaller particles will follow the flow field and pass by the plate. The impaction of nanoflbers, however, depends on their length in addition to diameter and density. Cheng et al. [14] have outlined a theoretical approach to this and confirmed their theory via experimental studies. For nanofibers, the aerodynamic equivalent diameter is defined as... 

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