Mechanics of Fluids by Oteh Uche

Author:Oteh, Uche [Oteh, Uche]
Language: eng
Format: epub
Publisher: Zencon Languages
Published: 2011-03-15T16:00:00+00:00

From the foregoing, it is obvious that dynamic similarity requires the existence of kinematic similarity, which on its own requires the existence of geometric similarity. One important feature common to all kinds of physical similarity is that for the two systems considered, certain ratios of like magnitude are fixed. Geometric similarity requires a fixed ratio of lengths, kinematic similarity, a fixed ratio of velocities; dynamic similarity, a fixed ratio of forces and so on. Whatever the quantities involved, the ratio of their magnitudes is dimensionless. A dimensionless parameter is any quantity, physical constant or group of quantities, formed in such a way that all units identically cancel. Consider a situation where another force is applied in a direction opposite F above, and which brings the fluid particles to rest (zero acceleration). This other force is termed the inertia force, which constitutes an additional force on the list of forces that influence the fluid flow. If the magnitudes and directions of the component forces acting on a fluid element are known, then the resultant force may be determined from a vector diagram. Since the resultant force is the inertia force, we may write

or

Since we are always concerned with dynamic similarity, let us turn our attention to some of the force ratios relevant to the study of dimensional analysis in fluid mechanics. The forces that influence the behavior of fluids arise in a number of ways. Depending on the particular situation, some of the forces disclosed earlier on come into play while others do not. Any of these forces, while acting on a particle of fluid produce a resultant force F, which causes an acceleration of the particle in the direction of the resultant force. This result in the flow pattern experienced generally by the particles of the fluid.

Let us express these forces in terms of the parameters arising from physical or dimensional arguments. For instance the inertia force exerted on a particle of the fluid may be expressed as , and since the velocity has dimensions , then . In the same manner other forces that may apply to a fluid can be expressed as

Viscous force

Pressure force

Elastic force

Surface tension force

Gravity force

To recapitulate, the forces that act on the fluid are those due to viscosity, pressure differences, compressibility or elasticity, surface tension, gravity and inertia forces. We have already indicated that for dynamic similarity the forces acting on any particle in one system bear the same ratio of magnitude to each other as the forces on the corresponding particle in the other system. Therefore we may proceed to inspect the several ratios of pairs of forces that influence the behavior of the fluid. Since the inertia forces directly influence the acceleration and hence the flow pattern of the fluid particles in every flow situation, let us set it apart and obtain the ratio of each of the other possible forces to the inertia force.

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