Velocity Field

In a deformable system there are an infinite number of particles. In order to define the velocity of a particle we must define the position of the particles, spatial coordinates, at the specific time. Using this, the velocity of all particles can be as v(x, y, z, t) which is consisted of three components of velocity vx, vy, vz in x, y and z direction of coordinates, respectively. v is called field velocity vector. For steady flow, the value of the velocity at a position remains invariant with time, v = v(x, y, z) .


In investigation of the fluid motion, two procedures may be used to study fluid particles. We can stipulate fix coordinates xi, yi and ziin the velocity field functions and letting time pass. we can express velocity of function at any time passing this position as v(x1,y1,zi,i). Using this approach we study the motion of a continuous string of particles which pass the fixed point. This viewpoint is called the Eulerian viewpoint.

On the other hand, we can study the motion of a single particle in the flow by following the particle in its path. This approach means continuous variation of the x, y and z to locate the particle. This approach is called the Lagrangian viewpoint where v = v(x, y, z). The x, y and z can be expressed as a function of time x = x(t), y = y(t) and z = z(t) (figure 2.2).

The Lagrangian viewpoint is used mainly in particle mechanics. In continuum mechanics this method requires the description of motion of an infinite number of particles and thus becomes extremely cumbersome. The Eulerian viewpoint is easier to use in continuum mechanics because it is concerned with the description of motion at a fixed position.

In the partial time derivative, from the Eulerian point of view, suppose we are standing on a bridge and note how the concentration of fish (c) just below us changes with time. In this case, the position is fixed in space, therefore bydc we mean partial of c with respect to t, holding x, y and z constant. We can also assume ourselves on a boat and moving with around the river. The change of fish concentration with respect to time reflects the motion of the boat as well as time variation. This is called total time variation and can be a

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