Magnus force

Magnus Force (Magnus Effect). A sideways thrust acting on a spinning projectile in flight because of the component of the air current acting perpendicular to the axis of the yawing projectile... 

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

This also describes a Magnus force-free flow, which is expressed by the following relationship ... 

Figure 2. Indicating the Magnus force and vortices surrounding a wing.

However, in a Beltrami field, the vorticity and velocity vectors are parallel or antiparallel, resulting in a zero Magnus force. The Beltrami condition (1) is therefore an equivalent way of characterizing a force-free flow situation, and vice versa. 

In 1954 Lust and Schluter [14] introduced force-free magnetic fields (FFMFs) into a theoretical model for stellar media in order to allow intense magnetic fields to coexist with large currents in stellar matter with vanishing Lorentz force. Notice should be taken that the Lorentz force is the electrodynamic analogue of the Magnus force alluded to above (see Fig. 6 and compare with Fig. 2). 

The first term on the right-hand side of Eq. (3.80) is the total drag force in the opposite direction of Up, including both Stokes drag and Oseen drag. The second term represents a lift force in the direction perpendicular to Up. Thus, the lift force or Magnus force for a spinning sphere in a uniform flow at low Reynolds numbers is obtained as... 

The ratio of the Magnus force to the Stokes drag can be expressed by... 

At high Reynolds numbers, the rotation of the sphere yields an asymmetric wake, as shown in Fig. 3.4. In this case, the theoretical analysis of the Magnus force and the drag force becomes rather complex because of the difficulties in obtaining the expressions for the pressure and velocity distributions around the surface of the sphere. Thus, the determination of the lift force as well as the drag force relies mainly on the empirical approach. 

For general particle motion formulation, additional forces such as the Saffman force, Magnus force, and electrostatic force should be included. Assuming that all forces applied on the moving particle are additive, the equation of motion of a particle in an arbitrary flow can be expressed by... 

It should be noted that the forces in Eq. (3.104) are not generally linearly additive. The drag force, Basset force, Saffman force, and Magnus force all depend on the same flow... 

Fe m F M Fs Lorentz force vector Magnus force vector Saffman force vector Pi Pi Stokes expansion Pressure of fluid Dipole moment vector... 

To simplify the following analysis, we assume that (1) the particles are spherical and of identical size (2) for the momentum interaction between the gas and solid phases, only the drag force in a locally uniform flow field is considered, i.e., all other forces such as Magnus force, Saffman force, Basset force, and electrostatic force are negligible and (3) the solids concentration is low so that particle-particle interactions are excluded. 

Fig. 5. Buoyant force acting on particles in a sheared flow (Magnus force). Left Particle with velocity relative to the surrounding fluid. Right Particle in a parabolic flow with no relative velocity.

A quahtative understanding of shear thickening is that hydro-dynamic forces (the Magnus force) drive particles close to physical contact such that hydrodynamic lubrication forces and frictional... 

If one finally considers that the Magnus force depends on the particles radius as the strong dependency of the onset of shear thickening on the particles effective radius is obvious without concern for which relation exactly holds. 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

Rigid spheres sometimes experience a lift force perpendicular to the direction of the flow or motion. For many years it was believed that only two mechanisms could cause such a lift. The first one described is the so-called Magnus force which is caused by forced rotation of a sphere in a uniform flow field. This force may also be caused by forced rotation of a sphere in a quiescent fluid. The second mechanism is the Saffman lift. This causes a particle in a shear flow to move across the flow field. This force is not caused by forced rotation of the particle, as particles that are not forced to rotate also experience this lift (i.e., these particles may also rotate, but then by an angular velocity induced by the flow field itself). 

Krahn [76] explained how the rotation of the sphere would cause the transition from laminar to turbulent boundary layers at different rotational velocities at the two sides of a sphere. The direction of the asymmetrical wake was explained based on the separation points for laminar and turbulent boundary layers. Krahn studied the flow around a cylinder. For a non-rotating cylinder the laminar boundary layer separates at 82° from the forward stagnation point, while the turbulent boundary layer separates at about 130°. Due to the rotation the laminar separation point will move further back, while the turbulent separation point will move forward. For some value of v qaa/v between 0 and 1 the laminar and turbulent separation points will be at equal distance from the stagnation point. The pressure on the turbulent side will be smaller than on the laminar side causing a negative Magnus force. 

Swanson [145] reviewed the investigations of the Magnus force, and presented experimental drag and lift coefficients for an infinite, rotating cylinder at different Reynolds numbers and velocity ratios. For velocity ratios less than 0.55, and Reynolds numbers between 12.8 x 10 and 50.1 x 10 the cylinder would experience negative lift. 

Saffmann [125] (p 394) compared the Magnus force developed by Rubinow Keller [123] with the shear force given above and showed that unless the rotational speed is much larger than the rate of shear k = dv /dy, and for a freely rotating particle 17 = the lift force due to shear is an order... 

The term Foo,er represents other forces, usually important for submicron particles and/or at specific conditions, for example, phoretic. Basset, Saffman, Magnus forces, etc. more details are given in [27] all these forces have been omitted in the present work for simplicity. 

The Lorentz force on a vortex at rest is Fl = Ps [i s ], where ps is the charge density of the condensate and Vg is the condensate velocity. This force gives the vortex a velocity normal to the current. In the rest frame of the vortex the superfluid now has an additional velocity -t y giving rise to an additional Lorentz force (the Magnus force) of Fi = -ps o[Vv x ]. The total force on the vortex is the sum of these Fl = p,s o[(t s - Vv) X ]. To complete the equation of motion for the vortex, a pinning force kr and damping rj are added, where r is the position of the vortex. At infrared frequencies the inertia of the vortex comes into play and a vortex mass can be included in the equation of motion ... 

The parameter a has been included to parametrize the strength of the Magnus force. 

It is noted that the buoyancy force is also included in Eq. (53) as one of the fluid-particle interaction forces. Due to the small particle size, the Saffman and Magnus forces are ignored in Eq. (53). 

Maxey and Riley pointed out that the Magnus force is of order and, for that reason, it is less important than the Saffman lift force. 

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