# ENGR2000 Lecture Notes - Lecture 3: Trac, Reynolds Transport Theorem, Curtin University

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2nd-Year Fluid Mechanics, Faculty of Science & Engineering, Curtin University

ENGR2000: FLUID MECHANICS

For Second-year Chemical, Petroleum, Civil & Mechanical Engineering

FLUID MECHANICS LECTURE NOTES

CHAPTER 3: MASS CONSERVATION, MOMENTUM AND ENERGY

3.1 Introduction

In order to introduce the ideas of conservation, it is often necessary to isolate

a part of the system and consider what is going on within that part of the

ﬂow ﬁeld. This is achieved by using a control volume (CV). We enclose a

part of the whole system by a notional boundary and perform our analysis

using the CV – what is happening within it and what is happening at its

boundaries. Examples of CV’s are seen in Fig. 3.1.

FIGURE 3.1: Examples of control volumes

Chapter 3 −Page 1

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2nd-Year Fluid Mechanics, Faculty of Science & Engineering, Curtin University

Note that a CV can include regions where there is no ﬂuid. The CV surface

- called the Control Surface (CS) - should not cross solid boundaries nor,

strictly, include solid components; however, we can ‘isolate’ such elements by

putting a separate CS around them. Often (see lower example in Fig. 3.1)

we do not bother to draw this isolating boundary in since it does not aﬀect

the analysis.

The CV’s illustrated above are stationary; we can use moving CV’s – would

these be of greater use? In mathematical terminology should we use an

Eulerian (ﬁxed frame) or Lagrangian (moving frame) approach to under-

standing ﬂow behaviour? Consider the following analogy to ﬂuid ﬂow: Traﬃc

is travelling along a road and we wish to characterise it. The Eulerian would

set up a check-point at a particular location and measure the number of cars

per hour passing that point. In contrast, the Lagrangian would follow partic-

ular vehicles and determine their destinations. Each is valid form of analysis

meeting diﬀerent interests.

In Fluid Mechanics, we are not generally interested in the fate of a particular

‘lump’ of ﬂuid and so we adopt a Eulerian approach. In Solid Mechanics the

Lagrangian approach is used because we are concerned with the motion (eg.

locii and orbits) of particular parts of the structure.

3.2 Mass conservation

The principle of mass conservation is based upon the physical recognition that

mass is neither created nor destroyed (certainly in the Newtonian world!).

In Fluid Mechanics, this is interpreted in terms of mass ﬂow. For a control

volume in which the mean ﬂow ﬁeld is time independent, we can say that

the rate of mass ﬂow into a CV must be equal to the rate at which it leaves.

To illustrate this idea, consider Fig. 3.X1.

Chapter 3 −Page 2

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2nd-Year Fluid Mechanics, Faculty of Science & Engineering, Curtin University

FIGURE 3.X1: Fluid mass ﬂow through a simple pipe system

Denoting a mass ﬂow-rate as ˙m(the over-dot means ’rate with time’, i.e. the

time derivative) with units kg/s, then it is obvious that

˙m1= ˙m2+ ˙m3(3.X1)

to show that this is a conservation principle, re-write it as

0 = ˙m1−˙m2−˙m3

|{z }

Net mass ﬂow-rate into CV

(3.X2)

Thus the net mass ﬂow rate into (or out of) the CV is zero so no ﬂuid mass

is ‘being created’ inside the CV.

With the above ideas in mind, we now undertake a general analysis of mass

conservation that can be applied to any ﬂow ﬁeld.

For simplicity we shall consider a two-dimensional ﬂow. We set up a CV in

the ﬂow as shown in Fig. 3.2. To make life easy we choose a square CV with

sides in the x- and y-directions. The choice of CV can be very important in

facilitating the analysis.

Chapter 3 −Page 3

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