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Introduction to Fluid Mechanics

Chapter 2

Fundamental Concepts

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Main Topics

Fluid as a ContinuumVelocity FieldStress FieldViscositySurface TensionDescription 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).

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Velocity Field

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Velocity Field

Consider alsoSteady and Unsteady Flows1D, 2D, and 3D FlowsTimelines, Pathlines, and

Streaklines

One-, Two-, and Three-Dimensional Flows

Everything is inherently three-dimensional – to simplify the problem – fewer dimensions are frequently considered.

Fundamental ConceptsTimeline -- 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.

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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

The velocity difference between the top and bottom plates is du

With different forces applied, the deformation rate is different.

tan/ yl

ytu

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:

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ViscosityNon-Newtonian Fluids

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

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Surface Tension

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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.

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Turbulent flows

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