Next: Acceleration: Eulerian description Up: Kinematics of Fluids Previous: Methods of representation of

# Acceleration: Lagrangian description

Before proceeding with a discussion of acceleration it may be well to restate the definition of velocity as given essentially by Eq. (30) in a modified form.

Consider again a particular particle of fluid A whose trajectory is known. The position of the particle A at any instant is . In effect, the position at time t is a vector (from the origin to the point in question), which will be referred to as the radius vector of the particle and will be denoted by . Thus, the function describes the trajectory. This is illustrated for the two-dimensional case in Figure 1.04-1.

In this figure the radius vectors for the instnats and are indicated. The vector difference

also indicated in the figure, represents the net incremental displacement of the particle A in the elapsed time

It will be noted that (73a) can be rearranged so as to state that is the increment vector diplacement that must be added to to get the radius vector on the trajectory after an elapsed time .

If the time interval is reduced, the vector is accordingly reduced, and its orientation becomes more and more tangent to the trajectory curve. In addition, the magnitude ratio approaches a limit. This limit is the magnitude of the velocity and its direction is tangent to the trajectory. Thus, the velocity at the instant is

which, when applied to the radius vector at any instant t, an be written as

This relation is actually nothing more than a concise statement of Eqs. (30), representing in effect three scalar equations. The definition (75) holds equally well for three dimensions, but the geometrical interpretation is slightly more complex in respect to the concept of tangency to a skew curve in space.

We may now proeceed with the definition and interpretation of acceleration

in an analogous manner. The trajectory of a particular particle A is again depicted in two dimensions in Figure 1.04-2a. The reference axes are considered to be fixed in space. The velocity vectors at different points along the trajectory are indicated by the solid arrows. If the vectors are all referred to a common origin (as indicated in the adjoining Figure 1.04-2b) then the end points describe a curve which will be referred to as the hodograph of the velocity vectors. This curve is accurately defined by all the vectors applicable to the motion of particle A. The coordinates of the hodograph are nominally labeled x and y but should more accurately be labeled and , i.e. the components of the velocity in the x and y directions, respectively.

The vector difference

represents the net change in velocity for the elapsed time defined by Eq. (73b). As can be seen by comparison of Figures 1.04-2a and b, the direction of is not necessarily oriented along the trajectory. This would be true only if the trajectory is a straight line.

The limiting direction of , as is reduced to zero, is along the tangent to the hodograph curve at the instant . The limit of the ratio , if it exists, is the acceleration of the particle A at the instant . Thus, if we let represent the acceleration of the particle at any instant then

The acceleration can also be expressed directly in terms of the radius vector function which describes the trajectory of A, since it follows from Eq. (75) that

These definitions again hold equally well for three-dimensional vectors.

There are, categorically speaking, essentially two modes of acceleration which a particle can possess. These are, specifically: (1) acceleration along the trajectory, and (2) acceleration normal to the trajectory. The first merely represents the rate of change of the magnitude of the velocity. The second is associated with the rate of change of the direction of the velocity vector.

Let s be the path distance along the trajectory of particle A and ds a differential displacement between points (1) and (2) on the path. The velocities of particle A at points (1) and (2) are and (see Fig. 1.04-3). The differential change of the magnitude of , denoted by , is indicated in the inset of Fig. 1.04-3. It actually represents the projection of the differential vector change along the path. Thus, the acceleration along the path can be expressed by the derivative . It is positive if the magnitude of is increasing as the particle proceeds along its path and negative if the particle is slowing down. The component acceleration along the path can be expressed alternately by the forms

However, ds/dt is merely the magnitude of , thus

We can also refer to this acceleration as the tangential acceleration since it has a direction tangential to the path of the particle. It should be remarked that is not the same as . The latter quantity is the magnitude of the total acceleration while the tangential acceleration is merely a component of the acceleration representing the rate of change of the magnitude of the velocity.

An expression for the second mode of acceleration can be derived by again referring to Fig. 1.04-3. The second component of the differential vector change is and is directed normal to the velocity vector and hence normal to the path of the particle at the point in question. The differential angle represents the change in direction of the velocity in the distance ds along the trajectory. The magnitude of the normal component of acceleration of particle A is therefore

This can be expressed in the form

where is the curvature of the path of the particle. In Figure 1.04-3 the path is shown in two dimensions only. This may be regarded as the projection of the three-dimensional trajectory in the x-y plane. The projections in the x-z and y-z planes are, in general, curved paths as well. We may regard the trajectory as merely the common intersection of the three cylindrical surfaces formed by the extensions of the projected paths on the three reference planes (see Figure 1.04-4).

In general therefore the curvature K is to be interpreted as the rate of change of with s along the three-dimensional trajectory. It is possible to interpret the curvature as a vector normal to the path. This is discussed by Holmboe et al. (1945) (Arts. 7.04-7.09).

In terms of the radius of curvature of the path, the normal acceleration of the particle becomes

where

For the special case where the path of A is a plane circle then R is constant. If the speed is also constant then the acceleration is everywhere directed towards the center of the circular path and is of constant magnitude . The acceleration associated with this special type of motion is commonly referred to as centripetal acceleration. The latter term might also be used to apply to the normal acceleration in the general case, where it is to be kept in mind that R is not necessarily constant for the particular path being considered.

The normal acceleration occuring at different points along an example trajectory is shown by the arrows in Figure 1.04-5. The total

acceleration at any point is given by the vector sum of and as illustrated in Figure 1.04-6.

By way of emphasis, it should again be remarked that it is the curvature of the trajectories which determines the normal acceleration component of a particular particle. This must be kept in mind in later developments where streamline patterns are frequently the object of discussion. It is a rather common mistake to use the curvature of streamlines in order to ascertain the normal acceleration. This procedure is correct only if the pattern of flow within the fluid is steady.

Next: Acceleration: Eulerian description Up: Kinematics of Fluids Previous: Methods of representation of

Steve Baum
Mon Dec 1 08:50:29 CST 1997