FLUID FLOW

1.0 INTRODUCTION

The motion a fluid is usually extremely complex, but can be described fully by an expression
describing the location of a fluid particle in space at different times thus enabling determination
of the magnitude and direction of velocity and acceleration in the flow field at any instant of
time.

1.1 FLUID KINEMATICS

This is a branch of fluid mechanics that deals with the study of velocity and acceleration of
particles of fluids in motion and their distribution in space without considering any force or
energy involved.
The motion of fluid particles may be described by the following methods:
1. Langrangian method
2. Eulerian method
1. Langrangian Method
In this method, the observer concentrates on the movement of a single particle. The path
taken by the particle and the changes in its velocity and acceleration are studied.
2. Eulerian Method
In eulerian method, the observer concentrates on a point in the fluid system. Velocity,
acceleration and other characteristics of the fluids at that particular point are studied.
This method is almost exclusively used in fluid mechanics, especially because of its
mathematical simplicity. In fluid mechanics, we are not concerned with the motion of
each particle, but we study the general state of motion at various points in the fluid
system.

1.2 TYPES OF FLUID FLOW

Fluids may be classified as follows:
1. Steady and unsteady flows
2. Uniform and non-uniform flows
3. 0ne, two and three dimensional flows
4. Rotational and irrotational flows
5. Laminar and turbulent flows
6. Compressible and incompressible flows
1. Steady and Un steady Flows
Steady flow: This is a type of flow in which fluid characteristics like velocity, pressure,
density etc. at a point do not change with time. Mathematically, we have
𝜕𝑢 𝜕𝑣 𝜕𝑤
( 𝜕𝑡 ) = 0; ( 𝜕𝑡 ) = 0; ( 𝜕𝑡 ) =0
𝑥0 𝑦0 𝑧0 𝑥0 𝑦0 𝑧0 𝑥0 𝑦0 𝑧0
𝜕𝑝 𝜕𝜌
( 𝜕𝑡 ) = 0; ( 𝜕𝑡 ) = 0; and so on
𝑥0 𝑦0 𝑧0 𝑥0 𝑦0 𝑧0
 where (𝑥0 𝑦0 𝑧0 ) is a fixed point in a fluid field where these variables are being measured
with respect to (w.r.t.) time.

Unsteady flow: It is a type of flow in which the velocity, pressure, density etc. at a point
change with time. Mathematically, we have
𝜕𝑢 𝜕𝑣 𝜕𝑤
( 𝜕𝑡 ) ≠ 0; ( 𝜕𝑡 ) ≠ 0; ( 𝜕𝑡 ) ≠0
𝑥0 𝑦0 𝑧0 𝑥0 𝑦0 𝑧0 𝑥0 𝑦0 𝑧0
𝜕𝑝 𝜕𝜌
( 𝜕𝑡 ) ≠ 0; ( 𝜕𝑡 ) ≠ 0; and so on
𝑥0 𝑦0 𝑧0 𝑥0 𝑦0 𝑧0

2. Uniform and Non-uniform Flows

Uniform flow: The type of flow, in which the velocity at any given time does not change
with respect to space is called uniform flow. Mathematically, we have
Steady flow: This is a type of flow in which fluid characteristics like velocity, pressure,
density etc. at a point do not change with time. Mathematically, we have
𝜕𝑉
( ) =0
𝜕𝑠 𝑡 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
where, 𝜕𝑉 = change in velocity, and
𝜕𝑠 = displacement in any direction.
Non-uniform flow: It is that type of flow in which the velocity at any given time changes
with respect to space. Mathematically, we have
𝜕𝑉
( ) ≠0
𝜕𝑠 𝑡 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
3. One, Two and Three Dimensional Flows
One dimensional flow: It is that type of flow in which the flow parameter such as
velocity is a function of time and one space co-ordinate only. Mathematically,
𝑢 = 𝑓(𝑥), 𝑣 = 0 𝑎𝑛𝑑 𝑤 = 0
where u, v, and w are velocity components in x, y and z directions respectively.
Two dimensional flow: It is that type of flow in which the flow parameter such as
velocity is a function of time and two space co-ordinates. Mathematically,
𝑢 = 𝑓1 (𝑥, 𝑦)
𝑣 = 𝑓2 (𝑥, 𝑦)
𝑤=0
Three dimensional flow: It is that type of flow in which the flow parameter such as
velocity is a function of time and three space co-ordinates. Mathematically,
𝑢 = 𝑓1 (𝑥, 𝑦, 𝑧)
𝑣 = 𝑓2 (𝑥, 𝑦, 𝑧)
𝑤 = 𝑓3 (𝑥, 𝑦, 𝑧)
 4. Rotational and Irrotational Flows
Rotational flow: A flow is said to be rotational if the fluid particles while moving in the
direction of flow rotate about their mass centres. Flow near the solid boundaries is
rotational.
Irrotational flow: A flow is said to be irrotational if the fluid particles while moving in
the direction of flow do not rotate about their mass centres. Flow outside the boundary
layer is general considered irrotational.
Note: If the flow is irritational as well as steady, it is known as Potential flow.
5. Laminar and Turbulent Flows
Laminar flow: A laminar flow is one in which paths taken by the individual particles of
the fluid do not cross one another and moving in an orderly manner along a well defined
paths. This type of flow is also called stream-line flow or viscous flow.
Turbulent flow: A turbulent flow is that flow in which the fluid particles move in an
disorderly manner.
6. Compressible and Incompressible Flows
Compressible flow: It is that type of flow in which the density (ρ) of the fluid changes
from point to point (or in other words density is not constant for this flow).
Mathematically, ρ ≠ constant.
Incompressible flow: It is that flow in which density is constant for the fluid flow.
Liquids are generally considered flowing incompressibly. Mathematically, ρ = constant.

1.3 TYPES OF FLOW LINES

Whenever a fluid is in motion, its innumerable particles move along certain lines depending
upon the conditions of flow. Flow lines are of several types.
1. Pathline
A pathline is the path followed by a fluid particle in motion. A path line shows the
direction of particular particle as it moves ahead.
2. Streamline
A streamline may be defined on as an imaginary line within the flow so that the tangent
at any point on it indicates the velocity at that point.
3. Streakline
The streakline is a curve which gives an instantaneous picture of the location of the fluid
particles, which have passed through a given point.

1.4 DISCHARGE OR RATE OF FLOW

The total quantity of fluid flowing in unit time past any particular cross-section of a stream is
called the discharge or rate of flow at that section. It can be measured either in terms of mass,
in which case it is referred to as the mass rate of flow 𝑚̇ and measured in units such as
kilograms per second (kg/s), or in terms of volume, which it is known as the volume rate of
 flow Q, measured in units such as cubic metres per second (m3/s). Let us consider a liquid
flowing through a pipe.
Let, A = Area of cross-section of the pipe, m2 and
V = Average velocity of the liquid, m/s.
Discharge, Q = Area x average velocity = A.V (1)

1.5 CONTINUITY OF FLOW

Matter is neither created nor destroyed. This principle of conservation of mass can be applied
to a flowing fluid. It can be stated as follows:
“If no fluid is added or removed from the pipe in any length then the mass passing across
different sections shall be same.”
2
1

Fig. 1 Fluid flow through a pipe

Considering two cross-sections of a pipe as shown Fig. 1

A1 = Area of the pipe at section 1,
V1 = Velocity of the fluid at section 1,
ρ1 = Density of the fluid at section 1,
and. A1, V1, ρ1 are corresponding values at section 2.
The total quantity of fluid passing through section 1 = ρ1 A1 V1
and, the total quantity of fluid passing through section 2 = ρ2 A2 V2

From the law of conservation of matter (theorem of continuity), we have

𝝆𝟏 𝑨𝟏 𝑽𝟏 = 𝝆𝟐 𝑨𝟐 𝑽𝟐 (2)

Equation (2) is applicable to the compressible as well as incompressible fluids and is called
Continuity Equation. In case of incompressible fluids, ρ1 = ρ2 and the continuity equation
(2) becomes
𝑨𝟏 𝑽𝟏 = 𝑨𝟐 𝑽𝟐 (3)

The continuity of flow equation is one of the major tools of fluid mechanics, providing a
means of calculating velocities at different points in a system.
The continuity equation can also be applied to determine the relation between the flows into
and out of a junction. In Fig. 2, for steady conditions,
 Total inflow to junction = Total outflow from junction,
𝝆𝟏 𝑸𝟏 = 𝝆𝟐 𝑸𝟐 + 𝝆𝟑 𝑸𝟑

For an incompressible fluid, 𝜌1 = 𝜌2 = 𝜌3 so that

𝑸𝟏 = 𝑸𝟐 + 𝑸𝟑 or 𝑨𝟏 𝑽𝟏 = 𝑨𝟐 𝑽𝟐 + 𝑨𝟑 𝑽𝟑 (𝟒)

1.6 CONTINUITY EQUATION IN CARTESIAN CO-ORDINATES

Consider a fluid element (control volume) – parallelepiped with sides dx, dy and dz as shown
in Fig. 3.

C
G

B F
dz
D
H

E
A dy
dx

Fig. 3 Fluid element in three-dimensional flow

 Let, ρ = Mass density of the fluid at a particular instant;

u, v, w = Components of velocity of flow entering the three faces of the parallelepiped.

Rate of mass of fluid entering the face ABCD (i.e. fluid influx)

= ρ x velocity in X-direction x area of ABCD

= 𝜌𝑢𝑑𝑦𝑑𝑧 (𝑖)

In general case, both mass density ρ and velocity u will change in the X-direction and so,

Rate of mass of fluid leaving the face EFGH (i.e. fluid efflux)
𝜕 𝜕
= 𝜌𝑢𝑑𝑦𝑑𝑧 + (𝜌𝑢𝑑𝑦𝑑𝑧)𝑑𝑥 = = [ 𝜌𝑢 + (𝜌𝑢)𝑑𝑥] 𝑑𝑦𝑑𝑧 (𝑖𝑖)
𝜕𝑥 𝜕𝑥

Mass accumulated per unit time due to flow in X-direction

𝜕
= 𝜌𝑢𝑑𝑦𝑑𝑧 − [ 𝜌𝑢 + (𝜌𝑢)𝑑𝑥] 𝑑𝑦𝑑𝑧
𝜕𝑥

𝜕
= − (𝜌𝑢)𝑑𝑥𝑑𝑦𝑑𝑧 (𝑖𝑖𝑖)
𝜕𝑥

Similarly.

Mass accumulated per unit time due flow in Y-direction

𝜕
= − (𝜌𝑣)𝑑𝑥𝑑𝑦𝑑𝑧 (𝑖𝑣)
𝜕𝑦

Mass accumulated per unit time due flow in Z-direction

𝜕
= − (𝜌𝑤)𝑑𝑥𝑑𝑦𝑑𝑧 (𝑣)
𝜕𝑧
The total (or net) gain in fluid mass per unit time for fluid along the three co-
ordinates axes

𝜕 𝜕 𝜕
= −[ (𝜌𝑢) + (𝜌𝑣) + (𝜌𝑤)] 𝑑𝑥𝑑𝑦𝑑𝑧 (𝑣𝑖)
𝜕𝑥 𝜕𝑦 𝜕𝑧

Rate of change of mass of the parallelepiped (control volume)

 𝜕𝜌
= 𝑑𝑥𝑑𝑦𝑑𝑧 (𝑖𝑣)
𝜕𝑡
Total (or net) gain in fluid mass per unit time along XYZ-direction

= Rate of change of mass per unit time in control volume

𝜕 𝜕 𝜕 𝜕𝜌
= −[ (𝜌𝑢) + (𝜌𝑣) + (𝜌𝑤)] 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝑑𝑥𝑑𝑦𝑑𝑧
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡

or
𝝏 𝝏 𝝏 𝝏𝝆
(𝝆𝒖) + (𝝆𝒗) + (𝝆𝒘) + =𝟎 (𝟓)
𝝏𝒙 𝝏𝒚 𝝏𝒛 𝝏𝒕

This equation (5) is the general equation of continuity in three dimensions and is
applicable to any type of flow and for any fluid whether steady or unsteady, compressible
or incompressible.
𝝏𝝆
For steady flow ( 𝝏𝒕 = 𝟎), incompressible fluids (𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) the equation reduces to

𝝏𝒖 𝝏𝒗 𝝏𝒘
+ + =𝟎 (𝟔)
𝒅𝒙 𝒅𝒚 𝒅𝒛

For two dimensional flow, equation (6) reduces to

𝜕𝑢 𝜕𝑣
+ =0
𝑑𝑥 𝑑𝑦
therefore, (w = 0)

1.7 CONTINUITY EQUATION IN CYLINDRICAL CO-ORDINATES

The form of the continuity equation for a system of cylindrical co-ordinates r, θ and z, in
which r and θ are measured in a plane corresponding to the x-y plane for Cartesian co-
ordinates, can be found by using the relations between polar and Cartesian co-ordinate:

𝑥 2 + 𝑦 2 = 𝑟 2 , (𝑦/𝑥) = tan 𝜃,
𝑢 = 𝑣𝑟 𝑐𝑜𝑠𝜃 − 𝑣𝜃 𝑠𝑖𝑛𝜃, 𝑣 = 𝑣𝑟 𝑠𝑖𝑛𝜃 + 𝑣𝜃 𝑐𝑜𝑠𝜃,
𝜕 𝜕 𝜕𝑟 𝜕 𝜕𝜃 𝜕 𝜕 𝜕𝑟 𝜕 𝜕𝜃
= + , = + ,
𝜕𝑥 𝜕𝑟 𝜕𝑥 𝜕𝜃 𝜕𝑥 𝜕𝑦 𝜕𝑟 𝜕𝑦 𝜕𝜃 𝜕𝑦
This results in equation (6) becoming
𝟏 𝝏 𝟏 𝝏𝒗𝜽 𝝏𝒘
[ (𝒓𝒗𝒓 )] + + =𝟎 (𝟕)
𝒓 𝝏𝒓 𝒓 𝝏𝜽 𝝏𝒛

In the case of two-dimensional flow, this can be simplified further. Putting 𝜕𝑤⁄𝜕𝑧 = 0
and writing

𝝏 𝜕𝑣𝑟
(𝒓𝒗𝒓 ) = (𝑟 + 𝑣𝑟 ),
𝝏𝒓 𝜕𝑟
equation (7) becomes

𝒗𝒓 𝝏𝒗𝒓 𝟏 𝝏𝒗𝜽
+ + =𝟎 (𝟖)
𝒓 𝝏𝒓 𝒓 𝝏𝜽