Coriolis and Centrifugal Forces

In most textbooks the apparent forces, like the Coriolis and the centrifugal forces, are derived with the help of the framework of classical mechanics of a point particle. To examine the elementary mathematical operations involved in Newtonian mechanics, for example, we describe the motion of a material particle by the Newton s second law of motion. The Newtonian frame of reference adopted is henceforth named O. The moving relative reference frame is designated 6. The basic task is thus to transform the Newton s second law of motion as formulated in an inertial frame of reference into a relative rotating frame of reference. 

Let the translative velocity (i.e., the relative movement without any rotating component) of the reference frame 6 relative to O be designated vq, and the angular velocity of O relative to O denoted by as sketched in Fig. 7.15. In the inertial frame in which are fixed orthogonal unit vectors, we define the vector of 

The vector r, shown in Fig. 7.15, can be determined in both coordinate systems 

A change of r with respect to time seen from O then occurs as a result of both change of the components r, fy and in the moving relative frame and change of the unit vectors and e  

The time derivatives of the unit vectors referred above are defined by (similar to a planar movement of a solid body)  

It turns out that two fictitious forces occur in the momentum equation when written in cylindrical coordinates. The term pmeVrlr is an effective force in the 0-direction when there is flow in both the r- and 0-directions. The term pvg jr gives the effective force in the r-direction resulting from fluid motion in the 0-direction. These terms do not represent the familiar Coriolis and centrifugal forces due to the earth s rotation. Instead, they arise automatically on transformation of the momentum equations from Cartesian to cylindrical coordinates and are thus not added on physical grounds (kinematics). Nevertheless, the pv /r term is sometimes referred to as a Coriolis force and the pv gjr term is often called a centrifugal force. It is thus important to distinguish between the different types of fictitious forces. 

In most textbooks the apparent forces, like the Coriolis and the centrifugal forces, are derived with the help of the framework of classical mechanics of a point particle. 

The virtual Coriolis and centrifugal forces due to the earth s rotation are normally added to ge and gr, as shown in (7.115). 

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When the isobars are curved, an additional force, a centrifugal force outward from the center of curvature, enters into the balance of forces. In the case of curvature around low pressure, a balance of forces occurs when the pressure gradient force equals the sum of the coriolis and centrifugal forces (Fig. 17-12) and the wind continues parallel to the isobars. In the... 

Gradient wind Wind resulting from a balance between the horizontal components of the pressure gradient, Coriolis, and centrifugal forces. 

In the friction layer where the isobars are curved, the effect of frictional drag is added to the forces discussed under gradient wind. The balance of the pressure gradient force, the coriolis deviating force, the centrifugal force, and the frictional drag in the vicinity of the curved isobars results in wind flow around low pressure and high pressure in the Northern Hemisphere, as shown in Fig. 17-16. 

The first pseudo force, Fi, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By definition, this force is perpendicular to the plane where vectors Vi and o are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second fictitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force... 

In this balance p is the static pressure, xtj is the stress tensor, and pgt is the gravitational body force. Ft is an external body forces component it can include forces from interaction between phases, centrifugal forces, Coriolis forces, and... 

Two terms in Eqs. (17) and (18) are worthy of special note. In Eq. (17) the term pvj/r is the centrifugal force. That is, it is the effective force in the r direction arising from fluid motion in the 0 direction. Similarly, in Eq. (18) pvrvg/r is the Coriolis force, or effective force in the 0 direction due to motion in both the r and 0 directions. Both of these forces arise naturally in the transformation of coordinates from the Cartesian frame to the cylindrical polar frame. They are properly part of the acceleration vector and do not need to be added on physical grounds. 

Quadratic and cubic potential constants have been obtained from IR frequencies, isotopic shifts, inertial defects, Coriolis constants and centrifugal distortion, assuming the geometry from microwave data. The quadratic force field is characterized by four symmetry force constants F which are related to the inner force constants by the following equations... 

The Coriolis force acts on a moving object on a rotating body such as the Earth and centrifuge bowl. It was first analyzed by a French engineer and mathematician, Gaspard de Coriolis (1835) [1]. 

There have been a large number of determinations of molecular force constants, mean amplitudes of vibration, bond asymmetry parameters, Coriolis coupling constants (and inertia defects) and centrifugal distortion constants [146,152,259,271,304,581,840,1221,1222,1278,1312, 1416,1448,1449,1549,1550,1575-1578,1587,1618,1671,1682,1806,1807,1858,1931,1961,1984,2021,2045, 2108,2109-2111,2167a], as well as a determination of the atomic potential energy distribution... 

For mathematical convenience, boundary conditions and initial conditions must be prescribed. For the simple marine propeller problem, a Lagrangian viewpoint was adopted. The frame of reference was attached to the propeller so that the propeller was fixed but the vessel was rotating. The boundary condition was then a zero velocity on the impeller, while the vessel wall rotated at -Qimpdier- The free surface was considered to be fiat, therefore the normal velocity was zero and a shear-free condition was assumed. It should be noted that in the Lagrangian viewpoint, the frame of reference is in rotation. The fluid is therefore subjected to a constant acceleration and the momentum conservation equation [Eq. (6)] must be modified to account for centrifugal forces and Coriolis forces.An advantage is, however, that the flow can be solved numerically at steady state provided the flow is fully periodic, which limits the computational efforts significantly. 

A comparison of equations (7.103) and (7.104) shows that the Newton s second law of motion in the inertial frame O is identical in form to that in O except that the latter formulation contains several additional fictitious body forces. The term —2mil x v is the Coriolis force, and —mQ x (f2 x r) designates the centrifugal force. No name is in general use for the term — x r. The acceleration —ao compensates for the translational acceleration of the frame. 

The centrifugal microfluidic platform uses inertial and capillary forces on a rotating microstructured substrate for liquid actuation. Relevant inertial (pseudo-) forces include the centrifugal force, Euler force and Coriolis force. The substrate is often disk-shaped. Liquid flow is possible in two dimensions but with the limitation that active liquid transport is always directed radially outwards. Active components can be limited to one rotational axis. 

Fig. 20 Centrifugal micromixer, (a) Micromixer device and its components, (b) Simulation of the mixing process shows the effect of the Coriolis pseudoforce (Fc) in folding the interface of the two liquids. Simultaneously, the centrifugal force (F ) drives the liquid toward the outlet (Reproduced from [186] with permission. Copyright Wiley-VCH)...

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