Types of Fluid Flow

Steady and unsteady flow

Steady flow: flow in which fluid properties are not changing w.r.t. time at given cross-
section.

Unsteady flow: flow in which fluid properties are changing w.r.t. time at given cross
section.

Uniform and Non uniform flow

Uniform flow: Fluid is said to be in uniform flow if the velocity is not changing w.r.t. cross
section at a given interval of time.

Non- uniform flow: Fluid is said to be in non uniform flow if the velocity is changing w.r.t.
cross section at a given interval of time.
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Laminar and Turbulent flow

Laminar flow: A laminar flow is one in which fluid flow is in the form of layers, and there
is no intermixing of fluid particles or molecular momentum transfer.

Turbulent flow: A turbulent flow is one in which there is a high order of intermixing of
fluid particles.

Rotational and irrotational flow

Rotational flow: If the fluid particles rotate about their axis or center of mass.

Irrotational flow: If the fluid particles are not rotated about their axis or center of mass.

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 Equation of Continuity in Fluid Mechanics
The product of cross sectional area of the pipe and the fluid
speed at any point along the pipe is constant.

This product is equal to volume flow per second or simply flow

rate
A*V = constant

The rate a fluid enters a pipe must equal the

rate the fluid leaves the pipe.

i.e. There can be no sources or sinks of fluid.

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 Equation of Continuity Derivation
Consider a fluid flowing through a pipe of non-uniform size.
The particles in the fluid move along the same lines in a steady flow.

If we consider the flow for a short interval of time

he fluid at the lower end of the pipe cover a distance
with the velocity V1 , then

Distance covered by the fluid is ,

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 Equation of Continuity in Fluid Mechanics
Distance covered by the fluid is , 𝟏 𝟏
Let, A1 be the area of cross section of the lower end then ,
Volume of the fluid that flows into the pipe at the lower end

V= A1 𝟏 A1 𝟏

If is the density of the fluid, then the mass of the

fluid contained in the shaded region of lower end
of the pipe is :
𝟏
A1 𝟏 -------------(1)
𝟏 𝟏
now , the mass flux defined as the mass of fluid per unit time passing through any
cross section at lower end is :
∆𝒎𝟏
𝟏 A1 𝟏 -------------(2)
∆𝒕
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 Equation of Continuity in Fluid Mechanics
 If the fluid moves with velocity V2 through the upper end of pipe having across sectional area A2 in time t
then the mass flux at the upper end is given by :
∆
∆
A2 2

2 A2 -------------(3)
 Since the flow is steady , so the density of the fluid
between the lower and upper end of the pipe does
not change with time.

 Thus the mass flux at the lower end must be equal to the mass flux at upper end so:

A1 2 A2 -------------(4)
 In a more general form we can write:
A

This relation describes the law of conservation of mass in fluid dynamics.

If the fluid is incompressible , then density is constant for steady flow of incompressible fluid so
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 Equation of Continuity in Fluid Mechanics
1 2
Now equation (4) can be written as:
A1 A2

 In general :

A
A , is often called the “ volume Flow Rate ” equation to volume/time, with SI
units of m3/s

 As A , V
A1 A2
 As Velocity increases, streamlines get closer

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 Bernoulli’s equation
“Bernoulli’s theorem state that in a steady flow of an incompressible
fluid, the total energy at any point of the fluid is constant”
𝑤ℎ𝑒𝑟𝑒, 𝑃: pressure
𝜌: 𝑑𝑒𝑛𝑠𝑖𝑡𝑦,
V: velocity
h: elevation
Pressure g: gravitational acceleration
Energy P.E. per
K.E. per
unit volume
unit volume

Velocity of fluid at a narrow passage is faster is faster than at a broad passage

As the velocity increases, the pressure decreases (Bernoulli's Principle)
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 Applications Bernoulli’s Principle
Torricelli: In Torricelli theorem Bernoulli’s principle is used to
determine the speed of fluid coming from a spigot on an open tank.
𝑉1= 2𝑔(𝑦2 − 𝑦1)

Airplanes: A sailboat can move against the wind, using the pressure
differences on each side of the sail, and using the keel to keep from
going sideways.

Baseballs: A ball’s path will curve due to its spin,

which results in the air speeds on the two sides of the
ball not being equal.

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 Applications Bernoulli’s equation
Venturi meter: device used for the measurement of flow in a pipeline
(a) converging cone
(b) Short Cylindrical throat
(c) Diverging cone

Blood Flow : A person with constricted arteries

will find that they may experience a temporary
lack of blood to the brain as blood speeds up to
get past the constriction, thereby reducing the
pressure.

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 Bernoulli’s equation Derivation
 Consider a pipe with varying diameter and height
through which an incompressible fluid is flowing.
 The relationship between the areas of cross-sections
A, the flow speed v, height from the ground y, and
pressure p at two different points 1 and 2 are given in
the figure.
Assumptions:
 The density of the incompressible fluid remains constant at both points.
 The energy of the fluid is conserved as there are no viscous forces in the fluid.
 Therefore, the work done on the fluid is given as:
dW = F1dx1 – F2dx2

dW = p1A1dx1 – p2A2dx2

dW = p1dv – p2dv = (p1 – p2)dv

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 Bernoulli’s equation Derivation
We know that the work done on the fluid was due to
the conservation of change in gravitational potential
energy and change in kinetic energy.
The change in kinetic energy of the fluid is given as:

-
The change in potential energy is given as:

dU = m2gy2 – m1gy1 = dvg(y2 – y1)

Therefore, the energy equation is given as:

dW = dK + dU

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(p1-p2) dv = dv - dvg

(p1-p2) = - g
Rearranging the above equation, we get

p1 + g p2 + g
This is Bernoulli’s equation.

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