1. Chapter 4 Description of a Fluid in Motion

2. Main Topics Introduction Fundamental Physical Laws Fluid Flow Fields
Steady and Unsteady Flows
Visualizing the velocity field: Flow Lines
Streamline
Pathline
Steakline
Timeline
Systems and Control Volumes
A relation between System and the Control Volume Approaches
One and Two Dimensional Flows

3. 4.0 Introduction
The development of an analytical description of fluid flow is based upon the expression of the physical laws related to fluid flow in a suitable mathematical form. Accordingly, we shall present the pertinent physical laws and discuss the methods used to describe a fluid in motion.
In this chapter vector notation will be used occasionally, primarily for the purpose of abbreviating otherwise lengthy expressions.
In many ways this chapter is one of the most important chapter in the course, for it paves the way for everything that follows. The student who masters its contents before proceeding will be rewarded for his/her efforts. Before taking up the main business of the chapter, we pause briefly to make a few comments regarding three kinds of time derivatives used in this chapter. We might illustrate them with a homely example -- namely reporting the concentration (population) of fish in the Singapore River. Because the fish are moving, the fish concentration c will be a function of position (x,y,z) and time (t).

4. 4.0 Introduction Partial Time Derivative, ∂c/∂t
Total Time Derivative, dc/dt
Substantial Time Derivative, Dc/Dt

5. 4.1 Fundamental Physical Laws
The law of Conservation of mass
Continuity equation
Newton’s Second Law of Motion
Momentum equation
First Law of Thermodynamics
Energy equation

6. 4.2 Fluid Flow Fields Lagrangian Representation v(x(t), y(t), z(t), t)
Eulerian Representation
v(x1, y1, z1, t)
The difference between these approaches lies in the manner in which the position in the field is identified.
For Eulerian approach, we express at a fixed position in space, the variables of a continuous ‘string’ of fluid particles moving by this position. On the other hand, for Lagrangian approach, we try to study a particular fluid particle in the flow, we must ‘follow the particle’.

7. 4.2 Fluid Flow Fields
An amusing illustration of the Eulerian and Lagrangian descriptions of fluid motion as applied to traffic on a highway has been suggested by White.
Engineers who design the highway are concerned with the number of cars that must pass over the road, the movement of traffic as a whole, and the number of cars that may be expected to enter or leave the highway at each ramp. An Eulerian description is perfectly suited because the highway designer has no interest in any particular car and its characteristics- whether it is red or blue, manual or automatic transmission, and so on. A police officer patrolling the highway, however, is interested in identifying those particular cars that are breaking the law and in giving the drivers tickets. Only a Lagrangian approach would stand up in court!

8. 4.3 Steady and Unsteady Flows
If the flow at every point in the fluid is independent of time, the flow is termed steady. If the flow at a point varies with time, the flow is termed unsteady.
It is possible in certain cases to transform an unsteady flow problem to a steady flow problem by changing the frame of reference.
Fig and 4.3.2

9. 4.4 Visualizing the Velocity Field: Flow Lines
Streamline – an imaginary line that is everywhere tangent to the fluid velocity vector
Pathline - the curve marked out by the trajectory of a particular fluid particle as it moves through the flow field
Streakline - a fluid line (that is, a line composed of fluid particles) made up of all particles that have passed a certain point
Timeline - a line of fluid particles that have been marked at a particular instant of time
Visualization

10. 4.5 Systems and Control Volumes
In employing the fundamental and subsidiary laws, either one of the following modes of application may be adopted.
1) The activities of each and every given mass must be such as to satisfy the basic laws and the pertinent subsidiary laws.
2) The activities in each and every volume in space must be such that the basic laws and the pertinent subsidiary laws are satisfied.
In the first instance, the laws are applied to an identified quantity of matter called system. A system may change shape, position and thermal condition, but must always entail the same matter.
For the second case, a definite volume is designated in space, and the boundary of this volume is known as the control surface. The amount and identity of this matter in the control volume may change with time., but the shape of the control volume is most of time fixed.
The activities of each and every given mass must be such as to satisfy the basic laws and the pertinent subsidiary laws – System Approach
The activities in each and every volume in space must be such that the basic laws and the pertinent subsidiary laws are satisfied – Control Volume Approach

11. 4.6 A Relation between the System and the Control Volume Approaches
In thermodynamics, one usually makes a distinction between those properties of a substance whose measure depends on the mass of the substance present and those properties whose measure is independent of the mass of the substance present. The former are called extensive properties while the latter are called intensive properties. For each extensive variable, one can introduce by distributive measurements, the corresponding intensive property, simply divided by mass, such quantity is termed specific. It is with extensive properties that we shall now relate the system approach with the control volume approach.
Fig.4.6.1
Derivation of eq.4.6.5

12. 4.7 One and Two Dimensional Flows
One dimensional flow is a simplification, where all properties and flow characteristics are assumed to be expressible as function of one space coordinate and time. The position is usually the location along some path or conduit. For instance, a one dimensional flow in a pipe would require that the velocity, pressure etc. be constant over any given cross section at any given time. In reality, flow in pipes and conduits is never truly one dimensional, since the velocity will vary over the cross section.
If the departure is not too great or if average effects at a cross section are of interest, one dimensional flow may be assumed to exist. For instance, in pipe and duct, this assumption is often acceptable, where
1. Variation of cross section of the container is not too excessive
2. Curvature of the streamlines is not excessive
3. Velocity profile is known not to change appreciably along the duct
Two dimensional flow is distinguished by the condition that all properties and flow characteristics are functions of two coordinates, say x, y and time and hence do not change along the z direction at a given instant. All planes normal to the z direction will, at a given instant, have the same stream line pattern.
Fig.4.7.1

13. Points to remember
Partial Time Derivative, ∂c / ∂t observes how the concentration changes with time at a fixed position in space. Hence by ∂c / ∂t we mean the "changing of c with respect to t, holding x, y, z constant.“
Total Time Derivative, dc / dt observes how concentration changes with respect to time.
Substantial Time Derivative, Dc / Dt observes how concentration changes with respect to time along the flow.
Eulerian approach observes the property at a particular fixed coordinates, whereas, Lagrangian approach will follow a specific fluid particle to see how the property will change along the way.

14. Points to remember
A system may change shape, position and thermal condition, but must always entail the same matter. A control volume is a definite volume in space, the amount and identity of this matter in the control volume may change with time, but the shape of the control volume is most of time fixed.
Reynolds transport equation has the following form,
The velocity vector is observed from the a reference frame xyz and the control volume is also observed from the same reference frame, as such, we can say that the velocity vector
is observed from the control volume.