Introduction to Fluid Mechanics Chapter 2 Fundamental Concepts

Main Topics Fluid as a Continuum Velocity Field Stress Field Viscosity Surface Tension Description and Classification of Fluid Motions

Fluid as a Continuum   Fluid as a Continuum – disregard the behavior of individual molecules. (A less rigid assumption than that in solid mechanics – in some cases such as rarefied gas flow, we have to abandon this concept) As a consequence of the continuum assumption, each fluid property is assumed to have a definite value at every point in space   (The concept of a continuum breaks down whenever the mean free path of the molecules becomes the same order of magnitude as the smallest significant characteristic dimension of the problem – rarefied gas in the upper reaches of the atmosphere).

Velocity Field

Velocity Field Consider also Steady and Unsteady Flows 1D, 2D, and 3D Flows Timelines, Pathlines, and Streaklines

One-, Two-, and Three-Dimensional Flows Everything is inherently three-dimensional – to simplify the problem – fewer dimensions are frequently considered.

 Fundamental Concepts Timeline -- the flow lines (structures, or fields) at a given instant.   Pathline – the path or trajectory traced out by a moving fluid particle (using dye or smoke). Streakline – The line joining the fluid particles, which had, at some time, passed through the given location in space. Streamline – lines drawn in the flow field so that at a given instant they are tangent to the direction of flow at every point in the flow field. (Since the streamlines are tangent to the velocity vector at every point in the flow field, there can be no flow across a streamline.) In a steady flow, pathlines, streaklines, and streamlines are identical lines in the flow field (Example 2.1)

The Use of Streamlines (Google images)

Stress Conditions on a Surface Stress condition on a surface passing through a given point under a coordinate system

Stress Conditions at a Point (Stress Field) Since there could be numerous surfaces passing through a point, the stress condition on a single surface may be inadequate to describe the stress condition at the point. However, it has been proven that the stress conditions on any three mutually perpendicular surfaces passing through the given point could be adequate to describe the stress condition at that point. Or the stress conditions on the surfaces of an infinitesimal rectangular cuboid at that point may be used to represent the stress conditions at that point.

Stress condition at a given point The first subscript is related to the direction of the surface, while the second is related to the direction of the stress. Every stress is marked in the positive direction.

Viscosity Newtonian Fluids Most of the common fluids (water, air, oil, etc.) “Linear” fluids Although any shear stress could produce a fluid deformation, a higher deformation rate requires a larger shear stress (or shear force). Therefore, a fluid still has the resistive characteristics against an applied shear stress in terms of deformation rate. This resistive characteristics may be considered as a property of a given fluid and can be measured experimentally in term of the concept of viscosity.

 Viscosity With different forces applied, the deformation rate is different. The velocity difference between the top and bottom plates is du

 Newtonian Fluids For Newtonian fluids (most common fluids, such as water and air), experiments have found that the shear stress as applied is directly proportional to the rate of deformation:

Viscosity Non-Newtonian Fluids Special fluids (e.g., most biological fluids, toothpaste, some paints, etc.) “Non-linear” fluids

Surface Tension

Description and Classification of Fluid Motions

 Laminar Flow

Turbulent flows   Fluid particles rapidly mix as they move along due to random three-dimensional velocity fluctuations. Semi-empirical theories in conjunction with experimental data are the common approach for a turbulent flow. Computational solutions are also available through the use of some empirical parameters, however. http://www.google.com/images

Turbulent flows   http://www.google.com/images