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Physically this says that the value of the property at position at time is the value appropriate to the particle that is at at time . Conversely, the material description can be derived from the spatial one.
meaning that the value as seen by the particle at time t is the value at the position it occupies at that time.
and
Thus is the rate of change of as observed at a fixed point , whereas is the rate of changed as observed when moving with the particle, i.e., for a fixed value of . The latter we call the material derivative . It is also called the convected, convective, or substantial derivative and often denoted by . In particular the material derivative of the position of a particle is it velocity. Thus putting , we have
or
This allows us to establish a connection between the two derivatives, for
We now have a formal definition of the velocity field as a material derivative of the position of a particle.
The field line lines of the velocity field are called streamlines ; they are the solutions of the three simultaneous equations
where is a parameter along the streamline. This parameter is not to be confused with the time, for in the above equation is held fixed while the equations are integrated, and the resulting curves are the streamlines at the instant . These may vary from instant to instant and in general will not coincide with the particle paths.
To obtain the particle paths from the velocity field we have to follow the motion of each particle. This means that we have to solve the differential equations
subject to at . Time is the parameter along the particle path. Thus the particle path is the trajectory taken by a particle.
The flow is called steady if the velocity components are independent of time. For steady flows, the parameter s along the streamlines may be taken to be and the streamlines and particle paths will coincide. The converse does not follow as there are unsteady flows for which the streamlines and particle paths coincide.
If is a closed curve in the region of flow, the streamlines through every point of generate a surface known as a stream tube . Let be a surface with as the bounding curve, then
is known as the strength of the stream tube at its cross-section S.
The acceleration or the rate of change of velocity is defined as
Notice that in steady flow this does not vanish but reduces to
Even in steady flow other than a constant translation, a fluid particle will accelerate if it changes direction to go around an obstacle or if it increases its speed to pass through a constriction.
The name streakline is applied to the curve traced out by a plume of smoke or dye, which is continuously injected at a fixed point but does not diffuse. Thus at time the streakline through a fixed-point is a curve going from to , the position reached by the particle which was at at time . A particle is on the streakline if it passed the fixed-point at some time between 0 and . If this time was , then the material coordinates of the particle would be given by . However, at time this particle is at so that the equation of the streakline at time is given by
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