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( x , t ) = [ ξ ( x , t ) , t ]

Physically this says that the value of the property at position x at time t is the value appropriate to the particle that is at x at time t . Conversely, the material description can be derived from the spatial one.

( ξ , t ) = [ ξ ( x , t ) , t ]

meaning that the value as seen by the particle at time t is the value at the position it occupies at that time.

t t x = derivative with respect to time keeping x constant

and

D D t t ξ = derivative with respect to time keeping ξ constant

Thus / t is the rate of change of as observed at a fixed point x , whereas D / D t is the rate of changed as observed when moving with the particle, i.e., for a fixed value of ξ . The latter we call the material derivative . It is also called the convected, convective, or substantial derivative and often denoted by D / D t . In particular the material derivative of the position of a particle is it velocity. Thus putting = x i , we have

v i = D x i D t = t x i ( ξ 1 , ξ 2 , ξ 3 , t )

or

v = D x D t

This allows us to establish a connection between the two derivatives, for

D D t = t ( ξ , t ) = t [ x ( ξ , t ) , t ] = t x + t i x i t ξ = t + v i x i = t + ( v )

Streamlines

We now have a formal definition of the velocity field as a material derivative of the position of a particle.

v ( x , t ) = D x ( ξ , t ) D t = x ( ξ , t ) t

The field line lines of the velocity field are called streamlines ; they are the solutions of the three simultaneous equations

d x d s = v ( x , t )

where s is a parameter along the streamline. This parameter s is not to be confused with the time, for in the above equation t is held fixed while the equations are integrated, and the resulting curves are the streamlines at the instant t . 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

D x D t = v ( x , t )

subject to x = ξ at t = 0 . 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 t 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 C is a closed curve in the region of flow, the streamlines through every point of C generate a surface known as a stream tube . Let S be a surface with C as the bounding curve, then

s v n d S

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

a = D v D t = v t + ( v ) v a i = v i t + v j v i x j

Notice that in steady flow this does not vanish but reduces to

a = ( v ) v for steady flow.

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.

Streaklines

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 t the streakline through a fixed-point y is a curve going from y to x ( y , t ) , the position reached by the particle which was at y at time t = 0 . A particle is on the streakline if it passed the fixed-point y at some time between 0 and t . If this time was t ' , then the material coordinates of the particle would be given by ξ = ξ ( y , t ' ) . However, at time t this particle is at x = x ( ξ , t ) so that the equation of the streakline at time t is given by

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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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