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FLUID MECHANICS - THEORY

   

Two basic and important field variables in the study of fluid mechanics are the velocity and acceleration of the fluid, and they are the focus of the discussion in this section. Both the velocity and acceleration equations are presented in Eulerian viewpoint.

     
    Velocity Field


Velocity Components

 


Velocity Field

 

Velocity is an important basic parameter governing a flow field. Other field variables such as the pressure and temperature are all influenced by the velocity of the fluid flow. In general, velocity is a function of both the location and time. The velocity vector can be expressed in Cartesian coordinates as

    V = V(x,y,z,t)
     = u(x,y,z,t) i + v(x,y,z,t)j + w (x,y,z,t) k
 

where the velocity components (u, v and w) are functions of both position and time. That is, u = u(x, y, z, t), v = v(x, y, z, t) and w = w(x, y, z, t). This velocity description is called the velocity field since it describes the velocity of all points, Eulerian viewpoint, in a given volume.

For a single particle, Lagrangian viewpoint, the velocity is derived from the changing position vector, or

     V = dr/dt = = dx/dt i + dy/dt j + dz/dt k

where r is the position vector (r = xi + yj + zk). Notice, the velocity is only a function of time since it is only tracking a single particle.

     
    Acceleration Field


Acceleration Field

 

 

Another important parameter in the study of fluid in motion is the acceleration. Acceleration is related to the velocity, and it can be determined once the velocity field is known. The acceleration is the change in velocity, δV, over the change in time, δt,

     a = [dV(t + δ) - dV(t)] / δt = δV/δt = dV/dt

But it is not just a simple derivative of just time since the velocity is a function time, AND space (x,y,z). The change in velocity must be track in both time and space. Using the chain rule of calculus, the change in velocity is,

     
or
     

This can simplified using u, v, and w, the velocity magnitudes in the three coordinate directions. In cartesian coordinates, the acceleration field is

 
 

This expression can be expanded and rearranged as

     

The acceleration equation can also be written in polar coordinates and are given in the Basic Equations appendix.

     
    Material Derivative

   

The time and space derivative used to determine the acceleration field from the velocity is so common in fluid mechanics, it has a special name. It is called the Material or Substantial Derivative and has a special symbol, D( )/Dt. For cartesian coordinates, it is

     

or in vector form,

 
 

where is the gradient operator. From the above equation, it can be seen that the material derivative consists of two terms. The first term  ∂t( )/∂t is referred to as the local rate of change, and it represents the effect of unsteadiness. For steady flow, the local derivative vanishes (i.e., ∂t( )/∂t = 0).

The second term, , is referred to as the convective rate of change, and it represents the variation due to the change in the position of the fluid particle, as it moves through a field with a gradient. If there is no gradient (no spatial change) then is zero so there is no convective change.

As an example, the acceleration field equation can be written as

     

     
   
 
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