Oseen’s equations

A method for overcoming this paradox was proposed by Oseen [325] who suggested to approximate the inertia term by the expression (Uj V)V, since the difference between the flow velocity V and the incoming flow velocity Uj is small remote from the sphere. The system of Oseen s equations has the form... 

In the case of a spherical bubble in a translational flow at small Reynolds numbers, the solution of Oseen s equation (2.3.1) results is a two-term asymptotic expansion for the drag coefficient [476] ... 

Rather than obtaining accurate solutions to Oseen s approximate equation, Proudman and Pearson (P3) suggested a technique to obtain successive approxi-... 

Equation (2.20) also assumes laminar flow (Reynolds numbers less than about 0.1), i.e., low particle velocities, and a dilute suspension of particles that are large compared with the molecules of the fluid. For Reynolds numbers greater than about 0.1 but less than 1, Oseen s law is approximately ... 

Now we determine the scaling constant m by substituting (9-96) and (9-99) into the Navier-Stokes equation (9-75) and requiring that the limiting form of the resulting equation for Re -> 0 contain viscous terms and at least one inertia term, as suggested by Oseen s argument. On substitution of outer variables, (9-75) becomes... 

A similar calculation can in principle be carried out for a cylindrical collector. However, a difficulty arises in that there is no solution of the inertia free, Stokes equation for an infinite cylinder in an otherwise unbounded flow. This was already observed in Section 5.1. Nevertheless we can illustrate the low Reynolds number behavior by using the so-called Stokes-Oseen solution for uniform flow of velocity U past an infinite circular cylinder of radius a whose symmetry axis is perpendicular to the flow. Oseen s method accounts in an... 

By removing the inconsistency, Oseen thus placed Stokes result on a firmer mathematical footing. In effect, Oseen s solution provides a uniformly valid zero-order approximation to the solution of the Navier-Stokes equations throughout all space. Unfortunately, a stronger interpretation than this was ascribed to Oseen s solution. Prevailing opinion at the time, and indeed for many years after, held that Oseen s solution was, in fact, an asymptotically valid solution of the Navier-Stokes equations to 0(R). This led to the well-known Oseen correction of Stokes law, ... 

Proudman and Pearson obtained Oseen s result, Eq. (200). That they obtained the same result as did Oseen is, however, fortuitous. For Oseen s original velocity field does not agree to 0(R) with the correct asymptotic solution of the Navier-Stokes equations. Rather it is correct only to 0(R°). 

The radial component of the hydrodynamic force is denoted as Fsu The net hydrodynamic force is a sum of the external force acting on the particle from the liquid flowing around the obstacle, and the force of viscous resistance of the liquid film dividing surfaces of particle and cylinder. The external force can push the particle closer to or pull it away from the obstacle s surface. Note that the force of viscous resistance is negative. Next, denote as Fad the molecular force of the Van der Waals attraction. This force is directed along the perpendicular line from the particle to the symmetry axis of the cylinder. Since the Navier-Stokes equations in the Oseen s approximation are linear, the forces and velocity fields induced by them are additive. 

Extensive reviews of turbulent diffusion were provided by Levich andHinze. Tchen ° was the first investigator who modified the Basset-Boussinesq-Oseen (BBO) equation and applied it to study motions of small particles in a turbulent flow. Corrsin and Lumley pointed out some inconsistencies of Tchen s modifications. Csanady showed that the inertia effect on particle dispersion in the atmosphere is negligible, but the crossing trajectory effect is appreciable. Ahmadi and Ahmadi and Goldschmidt smdied the effect of the Basset term on the particle diffusivity. Maxey and Riley obtained a corrected version of the BBO equation, which includes the Faxen correction for unsteady spatially varying Stokes flows. 

The components E and were estimated as the solution of the stationary Navier-Stokes equations for liquid flow around a single fibre. The author then used what they claimed was Oseen s approximation to describe the velocity field components and Eg, as well as the hydrodynamic pressure P ... 

The viscosity of the medium is t, and 1 is the unit tensor. (The Oseen tensor is the Green s function for the Navier-Stokes equation under the conditions that the fluid is incompressible, convective effects can be neglected, and inertial effects coming from the time derivative can be neglected.)... 

Onsager and Fuoss viscosity equation, 125 order in liquids, 1 oriented molecules, 152, 155 0rsted s piezometer, 58 orthobaric density, 48, 327 Oseen correction for falling sphere equation, 87... 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

Hence, as Oseen noted, we cannot expect the Stokes solution to provide a uniformly valid first approximation to the solution of (9 75), but instead expect that it will break down for large values of r > 0(Re ). Thus Whitehead s attempt to evaluate the second term in the expansion (9 77) was unsuccessfiil for large r. Indeed, as noted earlier in conjunction with the thermal problem, it is not so much a surprise that we cannot obtain a solution for boundary condition (9 81) for r oo, in spite of the fact that the governing equation (9 80) is not a valid first approximation to the full Navier Stokes equation except for r < 0(1 /Re). 

The appearance of transcendental terms in R is in marked contrast to Goldstein s (GIO) formula for the force on a sphere to O(R ) (see footnote 23) when the Oseen, rather than Navier-Stokes, equations are taken as the basic equations of motion. Equation (212) represents an asymptotic expansion having the general form... 

The solution of the Rubinow-Keller problem had previously been attempted by Garstang (Gla) on the basis of the Oseen equations. His result for the lift force is larger than (216) by a factor of 4/3. But as Garstang himself pointed out, his result was not unequivocal. Rather, different results were obtained according as the integration of the momentum flux was carried out at the surface of the sphere or at infinity. Garstang s paradox is clearly due to the fact that the term U-Vv does not represent a uniformly valid approximation of the inertial term v Vv throughout all portions of the fluid, at least not to the first order in R. 

Stake s drag ignores inertial terms in the governing equations. Oseen [2] obtained the first inertial correction to the drag force in the form of... 

The hydrodynamic coupling tensor Xik given by the Oseen or the Navier-Stokes equations for Newton s law... 

Stake s law is applicable provided the Reynolds number. Re, is much less than unity. Oseen (1913) derived the following working equation for a sphere moving with a velocity V, which is applicable for a wide range of Re number... 

In the following we resume the notations used by Maxey and Riley (1983) and write the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations in a form that is quasi-identical to that of Maxey and Riley. This formulation is interesting insofar as it highlights the relative movement of the particle with respect to the fluid. The three components of the particle s relative velocity with respect to the fluid are obtained by solving the following differential equations ... 

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