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Bernoulli’s Equation
Introduction
A fluid in motion is subjected to various forces which results in variation of the
acceleration and the energies involved in the flow phenomenon of the fluid.
The study and analysis of the fluid flow by considering forces and energies involved is
known as fluid dynamics.
Similar to mechanics of solids, the dynamics of fluid is also governed by Newton’s
second law of motion which states that “the rate of change of momentum of a body is
directly proportional to the resultant external force and it takes place in the direction of
the net applied force on it.” Mathematically,
F = ma
Where
F = Net external force
m = Mass of the fluid element on which force acts
a = total acceleration
Forces acting on the fluid in Motion
In the study of the fluid motion, forces per unit volume of the fluid are considered.
A fluid may subjected to various forces during its course of motion such as
1. Gravity
2. Pressure force
3. Viscous force
4. Turbulence
4. Surface tension
5. Elastic force or compressibility
The gravity force (Fg) is due to the weight of the fluid and is equal to mg. The gravity
force per unit volume is equal to ρg.
The pressure force (Fp) is exerted on the fluid element due to the pressure gradient
between two points in the direction of flow.
The viscous force (Fv) is due to the viscosity of the flowing fluid.
The turbulent force (Ft) is due to the turbulence of the flow i.e. in the turbulent flow the
fluid particles move from one layer to other and there is continuous momentum
transfer between adjacent layers.
The surface tension force (Fs) is due to the cohesive property of the fluid mass. This
force is considered only when the depth of flow is extremely small.
The compressibility force (Fe) is due to the elastic property of the fluid and it is
important for compressible fluids or in the fluids in which the elastic properties are
considerable.
According to Newton‟s second law
Fg + Fp + Fv +Ft + Fs +Fe = Fi
Eqn (1)
Where Fi is inertia force equal to ma.
In most of the fluids flow, surface tension force and the compressibility force is not
significant. Therefore, Fs and
Fe = 0
So Eqn 1 becomes
Fg + Fp + Fv +Ft = Fi
If flow is viscous i.e. laminar flow , then Ft =
Eqn (2)
(Reynold‟s eqn)
0
Eqn 2 becomes
Fg + Fp + Fv = Fi
Eqn (3) (Navier stokes eqn)
When fluid is non- viscous i.e. ideal or the viscosity of fluid is negligible, then FV =
Hence Eqn 3 becomes
Fg + Fp = Fi
Eqn (4) (Euler‟s eqn)
(Eqn (4) is essential in deriving bernoulli‟s equation)
0
Note : 1. Reynold’s equation is used in the analysis of the turbulent flows.
2. Navier stokes equation is employed in the analysis of the viscous flows.
Assumptions (Validity domain) of Bernoulli’s Equation
1. The flow must be steady.
2. Incompressible flow
3. Non-viscous or ideal fluid
4. Irrotational flow (i.e. no tangential force) along whole section.
5. Rotational flow only along the stream line (not on entire section).
Bernoulli’s Equation
Consider a fluid element is flowing along a stream tube of crossectional area dA
between two arbitrarily chosen sections 1-1 and 2-2. Suppose in time interval dt, the
fluid element moves through a short distance ds.
Let Z1 and Z2 be the distance of a point on sections 1-1 and 2-2 from datum.
Suppose P1 and P2 be the pressure and V1 and V2 are the velocities at sections 1-1 and 2-2
respectively.
From euler‟s equation (4)
Fp + Fg = Fi
PdA – (P + dP) dA – mg Cosθ = m a
dv dv
Where a = v. ds + dt
Convective
acceleration
Temporal
acceleration
dv
for steady flow, dt = 0
therefore,
dv
PdA – (P + dP) dA – mg Cosθ = m v. ds
but m= ρ.ds.dA
dv
-dP.dA – ρgdA.dsCosθ = (ρds.dA).v. ds
From figure , Z2 - Z1 = dZ = dsCosθ
dv
-dP.dA – ρgdA.dZ = (ρds.dA).v. ds
-dP – ρg.dZ = ρ.v.dv
-dP
ρ - g.dz = v.dv
dP
∫ ρ +∫v.dv + ∫g.dz = 0
Since flow is incompressible, mass density (ρ) of the fluid will be independent of
pressure. Hence
P2 – P1 V22- V12
+
+ g(Z2- Z1) = 0
ρ
2
P1 V12
P2 V22
ρ + 2 + gZ1 = ρ + 2 + gZ2
Eqn (5)
P1 V12
P2 V22
+
+
Z
=
1
ω 2g
ω + 2g + Z2
Eqn (6)
where ω is specific weight or weight density of fluid
Eqn (6) can be applied between any two sections of a pipe or channel or any other
passage through which ideal fluid is flowing.
P V2
ω + 2g + Z = Constant
Eqn (7)
Equation (5) (6) and (7) is known as Bernoulli‟s equation for steady incompressible flow
of an ideal (non-viscous fluid).
Note :
1. For Irrotational flow the value of constant „C‟ is equal for all stream lines so
Bernoulli‟s equation is applicable at any section but for rotational flow the value of
P V2
constant ( ω + 2g + Z) is different for different sections. Therefore, Bernoulli‟s equation
is valid for rotational flow only along the stream line as along a stream line constant „C‟
remains same.
2. Equations (5) (6) and (7) are the energy equations, since each term in these equations
represents the energy possessed by the flowing fluid.
Each term in Equation (5) represents energy per unit mass of the flowing fluid and each
term in Equation (6) and (7) denotes energy per unit weight of the flowing fluid which
is expressed as N.m/N.
P
V2
3. In equation (7) the term ω is known as pressure head or static head ; the term 2g is
known as velocity head or kinetic head and Z is known as potential head or datum
head.
4. The sum of the pressure head, the velocity head and the potential head is known as
the total head or the total energy per unit weight of the fluid.
5. Thus, the Bernoulli’s equation (7) states that “In a steady, irrotational flow of an
incompressible fluid the total energy at any point is constant”.
P
6. The sum of the pressure head and the potential head i.e. (ω +Z) is known as
piezometric head.
P2
7. If V2 is greater than V1 then the piezometric head ( ω +Z2) must be less than the
P1
piezometric head ( ω +Z1). However if the two points considered lie along the same
horizontal plane then Z1 = Z2, in which case the changes in velocity cause
corresponding change in pressure.
Modified Bernoulli’s Equation
The equation (6) has been derived for an ideal fluid (non-viscous) for which there is no
loss of energy. But in actual practice, during the flow of real fluids there is always some
loss of energy in the form of heat due to the viscosity of the fluid and turbulent motion
of the fluid particles.
Hence Equation (6) can be written as
P1 V12
P2 V22
ω + 2g + Z1 = ω + 2g + Z2 + hL
Eqn (9)
where hL is the loss of energy (or head) between sections under consideration.
Eqn (9) can be applied between any two sections of a pipe or channel or any other
passage through which real fluid is flowing.
Sample Question
1. A free jet of water emerging from a nozzle of diameter 75 mm attached to a pipe of
225 mm diameter as shown in figure. The velocity of water at point A is 18 m/sec.
Neglect friction in the pipe and nozzle. Nozzle tip velocity is Vc then find
(a) Vc
(b) Pressure at B in kpa.
Solution: Assume
1. Flow is steady
2. Flow is incompressible
3. Irrotational flow
Apply Bernoulli‟s equation (6) at points A and tip of nozzle C
Assume datum at nozzle tip C.
PA 182
PC VC2
+
+
21
=
ω 2g
ω + 2g + 0
Since PA = PC
VC= 27.12 m/sec.
(b) Apply Continuity equation between point B and point C
ρb AbVb = ρc Ac Vc
Since
ρ b = ρc
VB = 3.01 m/sec
Now apply Bernoulli‟s equation between B and C, assuming point as datum.
PB 3.012
PC 27.122
+
+
0
=
ω
2g
ω + 2g + 0.5
Pc= Patm , PB = Patm + ( PB)gauge
(PB)gauge = 368.3 KPa
References
1. Hydraulics and Fluid Mechanics by Modi and Seth
2. Fluid Mechanics by R K Bansal