Resume of Equilibrium of Accelerated Fluid Masses

Lecture 5:Monday 24thJanuary2011

Resume of equilibrium of accelerated fluid masses

- translation

- rotation

Description of fluids in motion

- definitions

- Continuity principle

Examples

CIVIL, ENVIRONMENTAL and GEOMATIC ENGINEERING DEPARTMENT

1st YEAR FLUIDSCEGE1009 MECHANISMS

3.Fluids in motion

3.1Definitions:

Rather than describing a flow pattern by the motion of its individual constituent fluid particles at different times (the "Lagrangian" approach), this section will concentrate on describing the flow by means of the three orthogonal components of velocity (u, v, w) at any point (x, y, z) at a given time t. This is known as the Eulerian description. It is characterised by the use of streamlines to visualise the flow pattern. Streamlines are lines whose tangents are at any instant parallel to the local velocity vector at all points along their lengths. From this definition it is clear that no flow can take place across a streamline.

A streamtube is an imaginary tube within a flow whose walls consist of streamlines. No flow can enter or leave a streamtube except at its ends.

Incidentally, if one adopts a Lagrangian description of a flow, it will be characterised by pathlines - which are the trajectories of individual fluid particles.

Steady flow is one in which all three velocity components remain independent of time at all points in the flow field. Expressed mathematically:

u = f1 (x, y, z) but NOT a function of t

v = f2 (x, y, z) but NOT a function of t,

w = f3 (x, y, z) but NOT a function of t.

Conversely, an unsteady flow DOES vary with time.

Two-dimensional flow is one in which the flow pattern is independent of one space coordinate - in other words it is the same in any plane orthogonal to that coordinate direction. An example might be the flow over a weir or past an aircraft wing.

One-dimensional flow is one in which the flow is independent of two space coordinates - in other words velocities vary only with one coordinate. Flow along pipes is often characterised in this way.

For convenience when deriving solutions, it is often necessary to define a Control Volume. This is a region within the flow pattern enclosing a number of streamlines and bounded by a Control Surface (often chosen so that no flow takes place across parts of it).

3.2Continuity principle:

The law of conservation of mass, when applied to an identifiable collection of matter, states that the total mass of the system remains unaltered whatever change of conditions it might experience. However, this is not the most convenient definition when dealing with the flow of a fluid.

Instead, it is easier to consider the flow into and out of a fixed control volume. Here, the continuity principle is usually expressed: "the rate of mass flow into the control volume - the mass flow out of the control volume = the rate of increase of mass in the control volume".

For the particular case of steady flow, this reduces to the statement that the rate of mass flow into the control volume equals the mass outflow ("what goes in ... must come out").

Consider the mass flow (or mass flux) entering and leaving the control volume ABCD. First looking at the mass flow through a small element A of the control surface: if the streamline passing through the element makes an angle  with a line drawn normal to the surface, then the mass flux, , into the Control Volume through the element is given by:

where q is the magnitude of the local velocity vector.

The total mass fluxinto the control volume is then:

If the flow is incompressible and uniform across any cross section of area A, we then have:

=  q A

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For example, consider the flow of an incompressible fluid along a diverging duct:

Inflow In =  q1 A1

Outflow Out = q2 A2

Hence: q1 A1 = q2 A2, and

q2 = q1 A1 / A2,

The volume flow rate, often referred to as the discharge, is normally defined:

If the flow is incompressible but not uniform, it is still possible to find a simple relationship for the mass flux by defining a mean velocity for the flow across any cross-section:

Consider flow between two plates with a developing parabolic velocity distribution between:

Hence mean velocities act as one-dimensional representations of the true velocity distribution.

Example:

If the velocity distribution between the two plates is described by:

u = k (d2 - y2), then the total mass flux can be found by integrating  u between the plates.

This can then be equated to the mass flux in terms of the mean velocity U to give:

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