
CBSE Important Questions›

CBSE Previous Year Question Papers›
 CBSE Previous Year Question Papers
 CBSE Previous Year Question Papers Class 12
 CBSE Previous Year Question Papers Class 10

CBSE Revision Notes›

CBSE Syllabus›

CBSE Extra Questions›

CBSE Sample Papers›
 CBSE Sample Papers
 CBSE Sample Question Papers For Class 5
 CBSE Sample Question Papers For Class 4
 CBSE Sample Question Papers For Class 3
 CBSE Sample Question Papers For Class 2
 CBSE Sample Question Papers For Class 1
 CBSE Sample Question Papers For Class 12
 CBSE Sample Question Papers For Class 11
 CBSE Sample Question Papers For Class 10
 CBSE Sample Question Papers For Class 9
 CBSE Sample Question Papers For Class 8
 CBSE Sample Question Papers For Class 7
 CBSE Sample Question Papers For Class 6

ISC & ICSE Syllabus›

ICSE Question Paper›
 ICSE Question Paper
 ISC Class 12 Question Paper
 ICSE Class 10 Question Paper

ICSE Sample Question Papers›
 ICSE Sample Question Papers
 ISC Sample Question Papers For Class 12
 ISC Sample Question Papers For Class 11
 ICSE Sample Question Papers For Class 10
 ICSE Sample Question Papers For Class 9
 ICSE Sample Question Papers For Class 8
 ICSE Sample Question Papers For Class 7
 ICSE Sample Question Papers For Class 6

ICSE Revision Notes›
 ICSE Revision Notes
 ICSE Class 9 Revision Notes
 ICSE Class 10 Revision Notes

ICSE Important Questions›

Maharashtra board›

RajasthanBoard›
 RajasthanBoard

Andhrapradesh Board›
 Andhrapradesh Board
 AP Board Sample Question Paper
 AP Board syllabus
 AP Board Previous Year Question Paper

Telangana Board›

Tamilnadu Board›

NCERT Solutions Class 12›
 NCERT Solutions Class 12
 NCERT Solutions Class 12 Economics
 NCERT Solutions Class 12 English
 NCERT Solutions Class 12 Hindi
 NCERT Solutions Class 12 Maths
 NCERT Solutions Class 12 Physics
 NCERT Solutions Class 12 Accountancy
 NCERT Solutions Class 12 Biology
 NCERT Solutions Class 12 Chemistry
 NCERT Solutions Class 12 Commerce

NCERT Solutions Class 10›

NCERT Solutions Class 11›
 NCERT Solutions Class 11
 NCERT Solutions Class 11 Statistics
 NCERT Solutions Class 11 Accountancy
 NCERT Solutions Class 11 Biology
 NCERT Solutions Class 11 Chemistry
 NCERT Solutions Class 11 Commerce
 NCERT Solutions Class 11 English
 NCERT Solutions Class 11 Hindi
 NCERT Solutions Class 11 Maths
 NCERT Solutions Class 11 Physics

NCERT Solutions Class 9›

NCERT Solutions Class 8›

NCERT Solutions Class 7›

NCERT Solutions Class 6›

NCERT Solutions Class 5›
 NCERT Solutions Class 5
 NCERT Solutions Class 5 EVS
 NCERT Solutions Class 5 English
 NCERT Solutions Class 5 Maths

NCERT Solutions Class 4›

NCERT Solutions Class 3›

NCERT Solutions Class 2›
 NCERT Solutions Class 2
 NCERT Solutions Class 2 Hindi
 NCERT Solutions Class 2 Maths
 NCERT Solutions Class 2 English

NCERT Solutions Class 1›
 NCERT Solutions Class 1
 NCERT Solutions Class 1 English
 NCERT Solutions Class 1 Hindi
 NCERT Solutions Class 1 Maths

JEE Main Question Papers›

JEE Main Syllabus›
 JEE Main Syllabus
 JEE Main Chemistry Syllabus
 JEE Main Maths Syllabus
 JEE Main Physics Syllabus

JEE Main Questions›
 JEE Main Questions
 JEE Main Maths Questions
 JEE Main Physics Questions
 JEE Main Chemistry Questions

JEE Main Mock Test›
 JEE Main Mock Test

JEE Main Revision Notes›
 JEE Main Revision Notes

JEE Main Sample Papers›
 JEE Main Sample Papers

JEE Advanced Question Papers›

JEE Advanced Syllabus›
 JEE Advanced Syllabus

JEE Advanced Mock Test›
 JEE Advanced Mock Test

JEE Advanced Questions›
 JEE Advanced Questions
 JEE Advanced Chemistry Questions
 JEE Advanced Maths Questions
 JEE Advanced Physics Questions

JEE Advanced Sample Papers›
 JEE Advanced Sample Papers

NEET Eligibility Criteria›
 NEET Eligibility Criteria

NEET Question Papers›

NEET Sample Papers›
 NEET Sample Papers

NEET Syllabus›

NEET Mock Test›
 NEET Mock Test

NCERT Books Class 9›
 NCERT Books Class 9

NCERT Books Class 8›
 NCERT Books Class 8

NCERT Books Class 7›
 NCERT Books Class 7

NCERT Books Class 6›
 NCERT Books Class 6

NCERT Books Class 5›
 NCERT Books Class 5

NCERT Books Class 4›
 NCERT Books Class 4

NCERT Books Class 3›
 NCERT Books Class 3

NCERT Books Class 2›
 NCERT Books Class 2

NCERT Books Class 1›
 NCERT Books Class 1

NCERT Books Class 12›
 NCERT Books Class 12

NCERT Books Class 11›
 NCERT Books Class 11

NCERT Books Class 10›
 NCERT Books Class 10

Chemistry Full Forms›
 Chemistry Full Forms

Biology Full Forms›
 Biology Full Forms

Physics Full Forms›
 Physics Full Forms

Educational Full Form›
 Educational Full Form

Examination Full Forms›
 Examination Full Forms

Algebra Formulas›
 Algebra Formulas

Chemistry Formulas›
 Chemistry Formulas

Geometry Formulas›
 Geometry Formulas

Math Formulas›
 Math Formulas

Physics Formulas›
 Physics Formulas

Trigonometry Formulas›
 Trigonometry Formulas

CUET Admit Card›
 CUET Admit Card

CUET Application Form›
 CUET Application Form

CUET Counselling›
 CUET Counselling

CUET Cutoff›
 CUET Cutoff

CUET Previous Year Question Papers›
 CUET Previous Year Question Papers

CUET Results›
 CUET Results

CUET Sample Papers›
 CUET Sample Papers

CUET Syllabus›
 CUET Syllabus

CUET Eligibility Criteria›
 CUET Eligibility Criteria

CUET Exam Centers›
 CUET Exam Centers

CUET Exam Dates›
 CUET Exam Dates

CUET Exam Pattern›
 CUET Exam Pattern
Bernoullis Equation Formula
The Bernoulli Principle, formulated by the Swiss mathematician Daniel Bernoulli, is a fundamental concept in fluid dynamics that describes the behavior of moving fluids. It states that in a steady, incompressible flow of a fluid with negligible viscosity, an increase in the fluid’s speed occurs simultaneously with a decrease in its pressure or potential energy. The Bernoulli equation mathematically represents this principle and can be expressed as: P+1/ρv^{2}+ρgh=constant. where P is the fluid pressure, ρ is the fluid density, v is the fluid velocity, g is the acceleration due to gravity, and h is the height above a reference point. This equation highlights the tradeoff between pressure, kinetic energy, and potential energy in a flowing fluid. Learn more about Bernoulli’s equation principle, formula, and examples based on it.
Quick Links
ToggleWhat is Bernoulli’s Equation Principle?
The Bernoulli Principle, named after the Swiss mathematician Daniel Bernoulli, is a key principle in fluid dynamics that describes the conservation of energy in a flowing fluid. According to this principle, for an incompressible, nonviscous fluid undergoing steady flow, the sum of the pressure energy, kinetic energy, and potential energy per unit volume remains constant along any streamline.
Bernoulli’s principle, applies to liquids in ideal conditions, and therefore, pressure and density are inversely proportional to each other, which means that a slowmoving fluid exerts more pressure than a fastmoving fluid. Fluids, in this case, refer to gases as well as liquids. This principle underlies many applications. Some very common examples are when an aeroplane is trying to stay aloft, or even the most common, mundane things like shower curtains curling inward.
A similar phenomenon occurs in the case of rivers with varying widths. Water speeds are slower in larger areas and faster in smaller areas. Students must think the liquid pressure will be higher. However, contrary to the explanation above, the liquid pressure decreases in the narrow part of the flow and increases in the wide part of the flow.
Bernoulli’s Equation Formula
Mathematically, the Bernoulli Equation is expressed as:
\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]
where:
\( P \) is the fluid pressure,
\( \rho \) is the fluid density,
\( v \) is the fluid velocity,
\( g \) is the acceleration due to gravity,
\( h \) is the height above a reference point.
Explanation of the Bernoulli Equation Formula
Pressure Energy: \( P \) – Represents the energy due to the fluid’s pressure. Higher pressure means higher energy.
Kinetic Energy: \( \frac{1}{2} \rho v^2 \) – Represents the energy due to the fluid’s motion. Faster moving fluid has more kinetic energy.
Potential Energy: \( \rho gh \) – Represents the energy due to the fluid’s elevation in a gravitational field. Higher elevation means more potential energy.
Key Points
Incompressible Flow: The fluid density (\( \rho \)) remains constant.
NonViscous Fluid: The fluid has no internal friction (viscosity), which means no energy is lost due to internal resistance.
Steady Flow: The velocity of the fluid at any given point does not change over time.
Along a Streamline: The principle applies to a single streamline, which is a path traced by a fluid particle under steady flow.
Applications of the Bernoulli Principle
Aerodynamics: Explains how airplane wings generate lift. The faster airflow over the curved upper surface of the wing results in lower pressure compared to the slower airflow beneath the wing, creating lift.
Venturi Effect: In a constricted section of a pipe, fluid speed increases and pressure decreases, which is utilized in devices like the Venturi meter for measuring fluid flow rate.
Hydraulic Engineering: Helps design efficient systems for water distribution and sewage systems, ensuring proper flow and pressure management.
Medical Devices: Explains the functioning of devices like the atomizer, which uses fluid pressure differences to create a fine spray.
Bernoulli Equation Formula Solved Examples
Example 1: Water flows through a horizontal pipe. The speed of water at point A is 2 m/s, and the pressure is 200,000 Pa. At point B, the speed of water increases to 3 m/s. What is the pressure at point B?
Solution:
Given:
\( v_A = 2 \text{ m/s} \)
\( P_A = 200,000 \text{ Pa} \)
\( v_B = 3 \text{ m/s} \)
Since the pipe is horizontal, the height \( h \) is the same at both points, so \( h_A = h_B \)
Using the Bernoulli Equation:
\[ P_A + \frac{1}{2} \rho v_A^2 = P_B + \frac{1}{2} \rho v_B^2 \]
Rearrange to solve for \( P_B \):
\[ P_B = P_A + \frac{1}{2} \rho (v_A^2 – v_B^2) \]
Assuming the density of water \( \rho \) is 1000 kg/m³:
\[ P_B = 200,000 + \frac{1}{2} \times 1000 \times (2^2 – 3^2) \]
\[ P_B = 200,000 + 500 \times (4 – 9) \]
\[ P_B = 200,000 + 500 \times (5) \]
\[ P_B = 200,000 – 2500 \]
\[ P_B = 197,500 \text{ Pa} \]
Example 2: The speed of air over the top surface of an airplane wing is 80 m/s, and the speed below the wing is 60 m/s. If the pressure below the wing is 101,325 Pa, what is the pressure above the wing? Assume the density of air is 1.225 kg/m³.
Solution:
Given:
\( v_{\text{top}} = 80 \text{ m/s} \)
\( v_{\text{bottom}} = 60 \text{ m/s} \)
\( P_{\text{bottom}} = 101,325 \text{ Pa} \)
Density of air \( \rho = 1.225 \text{ kg/m}^3 \)
Using Bernoulli’s Equation:
\[ P_{\text{top}} + \frac{1}{2} \rho v_{\text{top}}^2 = P_{\text{bottom}} + \frac{1}{2} \rho v_{\text{bottom}}^2 \]
Rearrange to solve for \( P_{\text{top}} \):
\[ P_{\text{top}} = P_{\text{bottom}} + \frac{1}{2} \rho (v_{\text{bottom}}^2 – v_{\text{top}}^2) \]
Substitute the values:
\[ P_{\text{top}} = 101,325 + \frac{1}{2} \times 1.225 \times (60^2 – 80^2) \]
\[ P_{\text{top}} = 101,325 + 0.6125 \times (3600 – 6400) \]
\[ P_{\text{top}} = 101,325 + 0.6125 \times (2800) \]
\[ P_{\text{top}} = 101,325 – 1715 \]
\[ P_{\text{top}} = 99,610 \text{ Pa} \]
Example 3: Water flows from a large tank through a small hole at the bottom. The surface of the water in the tank is 5 meters above the hole. What is the speed of water exiting the hole? Assume the water surface area is much larger than the hole area.
Solution:
Given:
Height \( h = 5 \text{ m} \)
Pressure at the top of the water surface and at the exit of the hole is atmospheric pressure, so they cancel out in the Bernoulli equation.
Assuming the velocity at the top surface \( v_{\text{top}} \approx 0 \text{ m/s} \)
Using Bernoulli’s Equation:
\[ P_{\text{top}} + \frac{1}{2} \rho v_{\text{top}}^2 + \rho gh_{\text{top}} = P_{\text{bottom}} + \frac{1}{2} \rho v_{\text{bottom}}^2 + \rho gh_{\text{bottom}} \]
Since \( P_{\text{top}} = P_{\text{bottom}} \) and \( v_{\text{top}} \approx 0 \):
\[ \rho gh = \frac{1}{2} \rho v_{\text{bottom}}^2 \]
Simplify to solve for \( v_{\text{bottom}} \):
\[ gh = \frac{1}{2} v_{\text{bottom}}^2 \]
\[ v_{\text{bottom}}^2 = 2gh \]
\[ v_{\text{bottom}} = \sqrt{2gh} \]
Substitute \( g = 9.81 \text{ m/s}^2 \) and \( h = 5 \text{ m} \):
\[ v_{\text{bottom}} = \sqrt{2 \times 9.81 \times 5} \]
\[ v_{\text{bottom}} = \sqrt{98.1} \]
\[ v_{\text{bottom}} \approx 9.9 \text{ m/s} \]
These examples illustrate how the Bernoulli Equation can be applied to different scenarios involving fluid flow, demonstrating its versatility in solving practical problems.
FAQs (Frequently Asked Questions)
1. What is the Bernoulli Equation?
The Bernoulli Equation is a principle in fluid dynamics that describes the conservation of energy in a flowing fluid. It states that for an incompressible, nonviscous fluid undergoing steady flow, the sum of the fluid’s pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline. The equation is expressed as: P+1/2ρv^{2}+ρgh=constant
2. What do the terms in the Bernoulli Equation represent?
In the Bernoulli Equation:
 $P$ represents the fluid pressure.
 $ρ$ is the fluid density.
 $v$ is the fluid velocity.
 $g$ is the acceleration due to gravity.
 $h$ is the height above a reference point.
These terms account for the pressure energy (
$P$P), kinetic energy (
$\frac{1}{2} \rho v^2$1/2ρv^{2}), and potential energy ($ρgh$) per unit volume of the fluid.
3. What assumptions are made in the Bernoulli Equation?
The Bernoulli Equation assumes:
 The fluid is incompressible (constant density).
 The fluid is nonviscous (no internal friction).
 The flow is steady (fluid properties at a point do not change over time).
 The flow occurs along a streamline.
4. What is the significance of the Bernoulli Equation in fluid dynamics?
The Bernoulli Equation is significant because it explains the relationship between pressure, velocity, and elevation in a fluid flow. It is used to analyze various fluid flow scenarios, predict fluid behavior, and design systems such as pipelines, airfoils, and ventilation systems.
5. Can the Bernoulli Equation be applied to all types of fluid flow?
No, the Bernoulli Equation is not applicable to all types of fluid flow. It is specifically valid for incompressible, nonviscous fluids in steady flow along a streamline. It does not apply to compressible flows (like gases at high speeds), flows with significant viscosity (like oil), or turbulent flows.