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Mechanics

Bernoulli's Theorem and its Applications

Bernoulli's theorem is a statement of the law of conservation of energy for flowing fluids, relating pressure, velocity, and elevation. It applies to streamline, steady, non-viscous, and incompressible fluid flows.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Understanding the equation of continuity, conservation of energy basis, definition of static, dynamic, and hydrostatic pressure, Venturi meter, and Torricelli's law.

PGT extension

Rigorous work-energy derivation for fluid elements, generalizations for non-steady and compressible flows, Euler's equation of motion, and Magnus effect.

Prerequisites

Equation of continuity for fluids
Work-energy theorem
Hydrostatic pressure and Pascal's law

Conservation of Energy Basis

Bernoulli's theorem is derived directly from the work-energy theorem applied to a continuous volume of fluid.
The total mechanical energy per unit volume (or mass) remains constant along any streamline in the absence of dissipative viscous forces.
It involves three energy terms per unit volume: pressure energy, kinetic energy, and potential energy.

Pressure Heads in Fluid Flow

Static pressure head represents the potential energy stored in the fluid due to compressive forces per unit weight.
Velocity head or dynamic pressure represents the kinetic energy per unit weight of the flowing fluid.
Elevation head represents the potential energy per unit weight due to the gravitational field.

Venturi Meter Operation

A Venturi meter measures the rate of flow of an incompressible fluid through a pipe using Bernoulli's principle.
At the constriction, the cross-sectional area decreases, causing the fluid velocity to increase and the static pressure to drop.
The pressure difference between the main section and the throat is measured using a differential manometer.

Torricelli's Law of Efflux

Torricelli's theorem states that the velocity of efflux of a liquid through an orifice at a depth h below the free surface is equal to the velocity of a body falling freely through height h.
Derived by applying Bernoulli's equation between the open top surface of the liquid and the small orifice.
Assumes the cross-sectional area of the container is much larger than the area of the orifice.

Derivations

Derivation of Bernoulli's Equation via Work-Energy Theorem

Conditions: Ideal fluid, steady flow, streamline motion, incompressibility, non-viscous.

Consider a fluid element of mass m entering a pipe of varying cross-section and height between cross-section A₁ (at height z₁, pressure P₁, speed v₁) and A₂ (at height z₂, pressure P₂, speed v₂) in time dt.
Work done by the pressure force at the inlet is W₁ = P₁ A₁ v₁ d$t = P_{1} V$, where V is the volume of fluid passing per second.
Work done by the pressure force at the outlet is W₂ = -P₂ A₂ v₂ d$t = -P_{2} V$.
Net work done by pressure forces in time dt is $W_{net} = (P_{1} - P_{2}) V$.
Change in potential energy of the fluid element is Delta $U = mg(z_{2} - z_{1}) = \rho V g (z_{2} - z_{1})$.
Change in kinetic energy of the fluid element is Delta $K = (1/2) m v_{2}^2 - (1/2) m v_{1}^2 = (1/2) \rho V (v_{2}^2 - v_{1}^2)$.
Equating net work done to the total change in mechanical energy (W_net = Delta U + Delta K): (P₁ - P₂)$V = \rho V g (z_{2} - z_{1}) + (1/2) \rho V (v_{2}^2 - v_{1}^2)$.
Dividing by volume V and rearranging yields the standard Bernoulli form.

Result: P_{1} + \frac{1}{2}\rho v_{1}^{2} + \rho g $z_{1} = P_{2} + \frac{1}{2}\rho v_{2}^{2} + \rho g z_{2} = \text{constant}$

Derivation of Venturi Meter Flow Rate Formula

Conditions: Horizontal pipe (z₁ = z₂), ideal fluid flow.

Apply the equation of continuity between the inlet (area A₁, speed v₁) and the throat (area A₂, speed v₂): A₁ v₁ = A_{2} v₂, which gives v₁ = (A₂ / A₁) v₂.
Apply Bernoulli's equation for horizontal flow (z₁ = z₂): P₁ + (1/2) \rho v₁^2 = P_{2} + (1/2) \rho v_{2}^2$.
Express the pressure difference Delta $P = P_{1} - P_{2}$ as Delta $P = (1/2) \rho (v_{2}^2 - v_{1}^2)$.
Substitute v₁ into the pressure difference equation: Delta $P = (1/2) \rho v_{2}^2 [1 - (A_{2} / A_{1})^2]$.
Solve for velocity at the throat v₂ and compute volume flow rate $Q = A_{2} v_{2}$.

Result: $Q = A_{1} A_{2} \sqrt{\frac{2(P_{1} - P_{2})}{\rho(A_{1}^{2} - A_{2}^{2})}}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Bernoulli's Equation (Pressure Form)	$$P + \frac{1}{2}\rho v^{2} + \rho g z = \text{Constant}$$	P = static pressure, \rho = density, v = flow speed, g = acceleration due to gravity, z = vertical height Pa (N/m²)	Incompressible, non-viscous, steady streamline flow
Bernoulli's Equation (Head Form)	$$\frac{P}{\rho g} + \frac{v^{2}}{2g} + z = \text{Constant}$$	P/(\rho g) = pressure head, v²/(2g) = velocity head, z = elevation head meters (m)	Constant density fluid in a gravitational field
Equation of Continuity	$$A_{1}v_{1} = A_{2}v_{2} = \text{Constant}$$	A₁, A₂ = cross-sectional areas, v₁, v₂ = flow velocities m³/s	Incompressible fluid
Torricelli's Velocity of Efflux	$$v = \sqrt{2gh}$$	v = velocity of efflux, g = acceleration due to gravity, h = depth of orifice below free surface m/s	Large container cross-section compared to orifice
Venturi Meter Pressure Difference	$$P_{1} - P_{2} = \rho g h_{m} \left(\frac{\rho_{m}}{\rho} - 1\right)$$	h_m = manometer reading, \rho_m = manometer fluid density, \rho = main fluid density Pa	U-tube differential manometer connected across constriction
Euler's Equation of Motion for Fluids	$$\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla P + \mathbf{g}$$	v = velocity vector, P = pressure, \rho = density, g = gravity vector m/s²	Non-viscous fluid, foundation of Bernoulli's equation
Dynamic Pressure	$$q = \frac{1}{2}\rho v^{2}$$	q = dynamic pressure, \rho = fluid density, v = flow speed Pa	Incompressible flow
Horizontal Range of Efflux	$$R = 2sqrt{h(H - h)}$$	h = depth of hole from top, H = total height of liquid column m	Orifice discharging into open air at ground level H-h from base

Graphs and experiments

Venturi Tube Pressure Variation Graph

Draw: A plot of pipe diameter and static pressure P along the horizontal coordinate x of a converging-diverging pipe.

Interpret: Static pressure drops to a minimum value at the narrowest throat section where velocity is maximum, recovering partially as the pipe widens again.

Exam use: Directly tested in conceptual questions involving pressure-velocity trade-offs in conduits.

Prandtl-Pitot Tube Experiment

Draw: Schematic of an L-shaped tube pointing upstream into a flow with stagnation point at the tip and static holes on the side.

Interpret: Measures total pressure (stagnation pressure) and static pressure simultaneously, allowing calculation of flow velocity via $v = \sqrt(2(P_{0} - P)/\rho)$.

Exam use: Used to frame numerical and theoretical questions on fluid speed measurement.

Efflux Velocity vs. Depth (Torricelli's Law)

Draw: A parabolic or square-root curve showing efflux velocity v increasing with depth h.

Interpret: Demonstrates that velocity is proportional to the square root of the head h, mirroring free fall kinematics.

Exam use: Tested via graphical interpretation questions in PGT examinations.

Solved numericals

TGT core

Water flows through a horizontal pipe whose internal diameter is 4 cm at one end and 2 cm at the other end. If the speed of water at the wider end is 2 m/s, find the speed of water at the narrower end.

Use the equation of continuity: A₁ v₁ = A₂ v₂.
Since area is proportional to the square of diameter ($A = \pi d^2 / 4)$, (d₁^2) v₁ = (d₂^2) v₂.
Substitute values: (4 cm)^2 * (2 m/s) = (2 cm)^2 * v₂.
16 * $2 = 4 * v_{2} => v_{2} = 32 / 4 = 8 m/s$.

Answer: 8 m/s

TGT core

A cylindrical tank of height 5 m is filled with water. A small hole is punched at a depth of 1.25 m from the free surface. Find the velocity of efflux of water.

Apply Torricelli's law: v = sqrt{2gh}.
Given depth h = 1.25 m and $g = 9.8 m/s^{2}$.
$v = \sqrt{2 * 9.8 * 1.25} = \sqrt{24.5} = 4.95 m/s$.

Answer: 4.95 m/s

PGT extension

Water flows through a horizontal Venturi meter with inlet and throat cross-sectional areas of 20 cm² and 10 cm² respectively. The pressure difference between the inlet and throat is 3 kPa. Calculate the volume flow rate of water (density of wate$r = 1000 kg/m^{3})$.

Given A₁ = 20 cm² = 20 * 10^-4 m², A₂ = 10 cm² = 10 * 10^-4 m².
Pressure difference Delta P = 3000 Pa, rh$o = 1000 kg/m^{3}$.
Use Venturi flow rate formula: $Q = A_{1} A_{2} \sqrt{\frac{2 \delta P}{\rho (A_1^2 - A_2^2)}}$.
Calculate A₁ A₂ = 20 * 10 * 10^-8 = 2 * 10^-6 m⁴.
Calculate A₁² - A₂² = (400 - 100) * 10^-8 = 300 * 10^-8 m².
Substitute into Q formula: $Q = (2 * 10^{-6}) \sqrt{\frac{2 * 3000}{1000 * 300 * 10^{-8}}} = (2 * 10^{-6}) \sqrt{\frac{6000}{3 * 10^{-5}}} = (2 * 10^{-6}) \sqrt{2 * 10^{8}} = (2 * 10^{-6}) * 1.414 * 10^4 = 2.828 * 10^{-2} m^{3}/s$.

Answer: 2.83 x 10^{-2} m^3/s (or 28.3 liters/s)

Assumptions and limitations

Assumes fluid is completely incompressible (\rho = constant), which fails for high-speed gases where compressibility effects dominate.
Assumes zero viscosity (non-viscous fluid); real fluids experience internal friction and viscous dissipation of mechanical energy into heat.
Restricted to steady flow where fluid properties at any point do not change with time.
Valid along a single streamline; applying it across different streamlines requires irrotational flow conditions.

Common misconceptions

Common error: Bernoulli's theorem implies that fluid velocity causes pressure reduction directly without energy conservation.

Correct: Pressure drop and velocity increase are alternate manifestations of mechanical energy conservation along a streamline.

Work done by pressure forces accelerates the fluid element, changing its kinetic and potential energy.

Common error: Equal transit time fallacy: air particles taking upper and lower paths around an aerofoil must meet at the trailing edge.

Correct: Air passing over the curved upper surface of an aerofoil moves significantly faster than required by the equal transit time hypothesis.

Experimental observations show air over the top surface arrives at the trailing edge much earlier than air from the bottom surface.

Common error: Bernoulli's equation can be applied freely between any two arbitrary points in a turbulent or swirling viscous fluid.

Correct: It holds strictly only along a streamline for steady, non-viscous, incompressible, irrotational flow.

Viscous shear stresses and turbulence cause mechanical energy to dissipate, violating the constant total energy condition.

Quick revision

Bernoulli's theorem is the conservation of mechanical energy applied to ideal fluid flow along a streamline.
Bernoulli's Equation (Pressure Form): P + \frac{1}{2}\rho v^{2} + \rho g z = \text{Constant}
Bernoulli's Equation (Head Form): \frac{P}{\rho g} + \frac{v^{2}}{2g} + z = \text{Constant}

Must remember

Bernoulli's theorem is a direct consequence of the work-energy theorem applied to an ideal, non-viscous, incompressible fluid in steady streamline flow.
The total mechanical energy per unit volume along any streamline remains constant: $P + \frac{1}{2}\rho v^2 + \rho gz = \text{Constant}$.
Torricelli's Law states that the velocity of efflux through a small orifice at depth $h$ is $v = \sqrt{2gh}$, identical to the speed of a freely falling body from height $h$.
In a Venturi meter, an increase in fluid velocity at the constriction leads to a decrease in static pressure, which is measured by a differential manometer.
The equation of continuity $A_1 v_1 = A_2 v_2 = \text{Constant}$ represents the conservation of mass for an incompressible fluid.

Glossary

Static Pressure HeadThe potential energy stored in the fluid due to compressive forces per unit weight, represented as $P/(\rho g)$.

Velocity HeadThe kinetic energy per unit weight of the flowing fluid, represented as $v^2/(2g)$.

Dynamic PressureThe kinetic energy per unit volume of the fluid, given by $q = \frac{1}{2}\rho v^2$.

Elevation HeadThe potential energy per unit weight due to the gravitational field, represented as $z$ or $h$.

Euler's EquationThe equation of motion for an inviscid fluid: $\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla P + \mathbf{g}$.

Pressure Terms in Bernoulli's Equation

Parameter	Static Pressure ($P$)	Dynamic Pressure ($\frac{1}{2}\rho v^2$)	Hydrostatic Pressure ($\rho gz$)
Origin	Compressive forces / Molecular collisions	Kinetic energy of fluid motion	Gravitational potential energy
Dependency	Independent of local velocity magnitude	Proportional to the square of flow speed ($v^2$)	Proportional to vertical height ($z$)

Practice Quiz

Choose an option to reveal the explanation.

1. Bernoulli's theorem is fundamentally based on which of the following conservation laws?

Explanation: Bernoulli's theorem is derived directly from the work-energy theorem applied to a continuous volume of fluid, representing the conservation of mechanical energy along a streamline.

2. Which of the following conditions is NOT required for the strict validity of Bernoulli's equation?

Explanation: Bernoulli's equation assumes the absence of dissipative viscous forces (non-viscous fluid). Viscosity causes energy dissipation, violating the constant total mechanical energy condition.

3. In the head form of Bernoulli's equation $\frac{P}{\rho g} + \frac{v^2}{2g} + z = \text{Constant}$, what does the term $\frac{v^2}{2g}$ represent?

Explanation: The term $\frac{v^2}{2g}$ represents the kinetic energy per unit weight of the flowing fluid, commonly referred to as the velocity head.

4. An incompressible fluid flows through a pipe of varying cross-sectional area. If the area at a section is halved, what happens to the flow velocity?

Explanation: By the equation of continuity ($A_1 v_1 = A_2 v_2$), if the area $A$ is halved, the velocity $v$ must be doubled to keep the volume flow rate constant.

5. Torricelli's law states that the velocity of efflux of a liquid through an orifice at depth $h$ below the free surface is given by:

Explanation: Torricelli's theorem equates the potential energy drop of a liquid element to its kinetic energy gain, yielding $v = \sqrt{2gh}$.

6. A Venturi meter is used to measure:

Explanation: A Venturi meter utilizes Bernoulli's principle and a constriction in a pipe to measure the rate of flow of an incompressible fluid.

7. What is the expression for dynamic pressure $q$ of a fluid of density $\rho$ moving with speed $v$?

Explanation: Dynamic pressure represents the kinetic energy per unit volume of the fluid and is given by $q = \frac{1}{2}\rho v^2$.

8. In a horizontal pipe where fluid is flowing, if the cross-sectional area decreases, the static pressure at that constriction:

Explanation: As area decreases, velocity increases (continuity). By Bernoulli's theorem, an increase in kinetic energy (velocity) results in a drop in static pressure at the constriction.

9. Euler's equation of motion for an ideal fluid relates acceleration to pressure gradient and gravity. What is its mathematical form?

Explanation: Euler's equation expresses Newton's second law for a non-viscous fluid element, serving as the foundational differential equation for deriving Bernoulli's theorem.

10. What is the horizontal range $R$ of liquid efflux from a small hole at depth $h$ from the top of a liquid column of total height $H$, discharging into open air at ground level?

Explanation: Using kinematic equations with efflux velocity $v = \sqrt{2gh}$ and fall time $t = \sqrt{2(H-h)/g}$, the horizontal range is $R = v \cdot t = 2\sqrt{h(H-h)}$.

11. Which common misconception regarding aerofoils is incorrect?

Explanation: The equal transit time fallacy is incorrect; experimental data proves that air flowing over the upper surface reaches the trailing edge much earlier than air from the lower surface.

12. Units of pressure head $\frac{P}{\rho g}$ in the head form of Bernoulli's equation are:

Explanation: Pressure head represents energy per unit weight, which dimensionally reduces to length, measured in meters (m).

13. What happens to the total mechanical energy per unit volume along a streamline in a steady, non-viscous fluid flow?

Explanation: In the absence of dissipative viscous forces, the total mechanical energy per unit volume remains constant along any streamline according to Bernoulli's theorem.

14. In a U-tube differential manometer connected across a Venturi meter, the pressure difference $P_1 - P_2$ is given by:

Explanation: The pressure difference measured by a differential manometer across a Venturi meter accounting for manometric fluid density $\rho_m$ and main fluid density $\rho$ is $P_1 - P_2 = \rho g h_m \left(\frac{\rho_m}{\rho} - 1\right)$.

15. Why can Bernoulli's equation NOT be freely applied in turbulent or swirling viscous fluids?

Explanation: Viscous shear stresses and turbulence convert mechanical energy into thermal energy, violating the assumption of constant total mechanical energy along a streamline.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. Water flows through a horizontal pipe whose internal diameter changes from $4\text{ cm}$ to $2\text{ cm}$ at a constriction. If the speed of water at the larger section is $2\text{ m/s}$, the speed at the constriction is:

Explanation: By continuity equation $A_1 v_1 = A_2 v_2$, we have $(\pi d_1^2 / 4) v_1 = (\pi d_2^2 / 4) v_2$. Thus $v_2 = v_1 (d_1 / d_2)^2 = 2 \times (4/2)^2 = 2 \times 4 = 8\text{ m/s}$بيه.

2. A cylindrical tank of height $H$ is filled with water to the brim. A small hole is punched at a depth $h$ from the free surface. What depth $h$ maximizes the horizontal range of the efflux?

Explanation: The range is $R = 2\sqrt{h(H-h)}$. To maximize $R$, differentiate $h(H-h) = hH - h^2$ with respect to $h$ and set to zero: $H - 2h = 0 \implies h = H/2$.

3. The pressure difference between the main pipe and the throat of a Venturi meter is $4000\text{ Pa}$. If the density of water is $1000\text{ kg/m}^3$, what is the velocity of water at the throat assuming zero velocity at the main section?

Explanation: Using Bernoulli's equation with $v_1 \approx 0$, $P_1 - P_2 = \frac{1}{2}\rho v_2^2$. Thus $4000 = \frac{1}{2}(1000)v_2^2 \implies v_2^2 = 8 \implies v_2 = \sqrt{8} \approx 2.83\text{ m/s}$.

4. In a horizontal Venturi meter, the area of cross-section at the main pipe is $A_1$ and at the throat is $A_2$. If $A_1 = 2A_2$, the ratio of flow speeds $v_2/v_1$ is:

Explanation: By the equation of continuity $A_1 v_1 = A_2 v_2$, the ratio $v_2 / v_1 = A_1 / A_2 = 2/1 = 2:1$.

5. A small orifice is punched at the bottom of a water tank of height $5\text{ m}$. Taking $g = 10\text{ m/s}^2$, the velocity of efflux is:

Explanation: Using Torricelli's law $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5} = \sqrt{100} = 10\text{ m/s}$.

6. Which of the following terms in Bernoulli's pressure equation represents potential energy per unit volume?

Explanation: In Bernoulli's equation $P + \frac{1}{2}\rho v^2 + \rho gz = \text{Constant}$, $P$ is pressure energy per unit volume, $\frac{1}{2}\rho v^2$ is kinetic energy per unit volume, and $\rho gz$ is potential energy per unit volume due to gravity.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Bernoulli's theorem for fluid flow is conservation of:

Explanation: Bernoulli's theorem expresses the conservation of mechanical energy for an ideal fluid in streamline flow.

2. If the velocity of fluid flow at a point is doubled, the dynamic pressure at that point increases by a factor of:

Explanation: Dynamic pressure is given by $q = \frac{1}{2}\rho v^2$. Since $q \propto v^2$, doubling $v$ increases $q$ by a factor of $2^2 = 4$.

3. What is the physical significance of the constant in Bernoulli's equation?

Explanation: The constant is the total mechanical energy per unit volume (sum of static pressure, kinetic energy density, and potential energy density) along any streamline.

4. Torricelli's law is a special case of:

Explanation: Torricelli's law is derived directly by applying Bernoulli's equation between the open top surface of a container and a small orifice.

5. In a Venturi meter, the reading of the differential manometer shows a height difference $h_m$. If the density of the manometric liquid is doubled, the pressure difference measured will:

Explanation: From $P_1 - P_2 = \rho g h_m (\rho_m / \rho - 1)$, if $\rho_m$ is large compared to $\rho$, $P_1 - P_2 \propto \rho_m$, so doubling the manometer fluid density doubles the pressure difference for a given height reading.

6. Which of the following flow characteristics violates Bernoulli's equation assumption?

Explanation: Turbulence and internal friction (viscosity) introduce non-conservative shear forces that dissipate mechanical energy, violating the assumptions of Bernoulli's theorem.

7. The dimensions of the velocity head $\frac{v^2}{2g}$ are same as:

Explanation: Velocity head is measured in meters (m), which represents length or height dimensions.

8. Water is flowing through a tube of non-uniform cross-section. At a point where the cross-section is $20\text{ cm}^2$, the velocity is $2\text{ m/s}$. What is the velocity at a point where cross-section is $5\text{ cm}^2$?

Explanation: By continuity equation $A_1 v_1 = A_2 v_2$, we have $20 \times 2 = 5 \times v_2 \implies v_2 = 40 / 5 = 8\text{ m/s}$.

9. What happens to the static pressure of air above the wings of an airplane during flight compared to below the wings?

Explanation: Air flows faster over the curved upper surface of an aerofoil than underneath, resulting in lower static pressure above the wing and higher pressure below, creating net upward lift.

10. Euler's equation of motion reduces directly to Bernoulli's equation when integrated along:

Explanation: Integrating Euler's equation along a streamline for steady, inviscid, incompressible flow yields Bernoulli's equation.

Bernoulli's Theorem and its Applications

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Conservation of Energy Basis

Pressure Heads in Fluid Flow

Venturi Meter Operation

Torricelli's Law of Efflux

Derivations

Derivation of Bernoulli's Equation via Work-Energy Theorem

Derivation of Venturi Meter Flow Rate Formula

Formula sheet

Graphs and experiments

Venturi Tube Pressure Variation Graph

Prandtl-Pitot Tube Experiment

Efflux Velocity vs. Depth (Torricelli's Law)

Solved numericals

Water flows through a horizontal pipe whose internal diameter is 4 cm at one end and 2 cm at the other end. If the speed of water at the wider end is 2 m/s, find the speed of water at the narrower end.

A cylindrical tank of height 5 m is filled with water. A small hole is punched at a depth of 1.25 m from the free surface. Find the velocity of efflux of water.

Water flows through a horizontal Venturi meter with inlet and throat cross-sectional areas of 20 cm2 and 10 cm2 respectively. The pressure difference between the inlet and throat is 3 kPa. Calculate the volume flow rate of water (density of wate$r = 1000 kg/m^{3})$.

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Pressure Terms in Bernoulli's Equation

Practice Quiz

1. Bernoulli's theorem is fundamentally based on which of the following conservation laws?

2. Which of the following conditions is NOT required for the strict validity of Bernoulli's equation?

3. In the head form of Bernoulli's equation $\frac{P}{\rho g} + \frac{v^2}{2g} + z = \text{Constant}$, what does the term $\frac{v^2}{2g}$ represent?

4. An incompressible fluid flows through a pipe of varying cross-sectional area. If the area at a section is halved, what happens to the flow velocity?

5. Torricelli's law states that the velocity of efflux of a liquid through an orifice at depth $h$ below the free surface is given by:

6. A Venturi meter is used to measure:

7. What is the expression for dynamic pressure $q$ of a fluid of density $\rho$ moving with speed $v$?

8. In a horizontal pipe where fluid is flowing, if the cross-sectional area decreases, the static pressure at that constriction:

9. Euler's equation of motion for an ideal fluid relates acceleration to pressure gradient and gravity. What is its mathematical form?

10. What is the horizontal range $R$ of liquid efflux from a small hole at depth $h$ from the top of a liquid column of total height $H$, discharging into open air at ground level?

11. Which common misconception regarding aerofoils is incorrect?

12. Units of pressure head $\frac{P}{\rho g}$ in the head form of Bernoulli's equation are:

13. What happens to the total mechanical energy per unit volume along a streamline in a steady, non-viscous fluid flow?

14. In a U-tube differential manometer connected across a Venturi meter, the pressure difference $P_1 - P_2$ is given by:

15. Why can Bernoulli's equation NOT be freely applied in turbulent or swirling viscous fluids?

PYQ Patterns

1. Water flows through a horizontal pipe whose internal diameter changes from $4\text{ cm}$ to $2\text{ cm}$ at a constriction. If the speed of water at the larger section is $2\text{ m/s}$, the speed at the constriction is:

2. A cylindrical tank of height $H$ is filled with water to the brim. A small hole is punched at a depth $h$ from the free surface. What depth $h$ maximizes the horizontal range of the efflux?

3. The pressure difference between the main pipe and the throat of a Venturi meter is $4000\text{ Pa}$. If the density of water is $1000\text{ kg/m}^3$, what is the velocity of water at the throat assuming zero velocity at the main section?

4. In a horizontal Venturi meter, the area of cross-section at the main pipe is $A_1$ and at the throat is $A_2$. If $A_1 = 2A_2$, the ratio of flow speeds $v_2/v_1$ is:

5. A small orifice is punched at the bottom of a water tank of height $5\text{ m}$. Taking $g = 10\text{ m/s}^2$, the velocity of efflux is:

6. Which of the following terms in Bernoulli's pressure equation represents potential energy per unit volume?

Timed Topic Test

1. Bernoulli's theorem for fluid flow is conservation of:

2. If the velocity of fluid flow at a point is doubled, the dynamic pressure at that point increases by a factor of:

3. What is the physical significance of the constant in Bernoulli's equation?

4. Torricelli's law is a special case of:

5. In a Venturi meter, the reading of the differential manometer shows a height difference $h_m$. If the density of the manometric liquid is doubled, the pressure difference measured will:

6. Which of the following flow characteristics violates Bernoulli's equation assumption?

7. The dimensions of the velocity head $\frac{v^2}{2g}$ are same as:

8. Water is flowing through a tube of non-uniform cross-section. At a point where the cross-section is $20\text{ cm}^2$, the velocity is $2\text{ m/s}$. What is the velocity at a point where cross-section is $5\text{ cm}^2$?

9. What happens to the static pressure of air above the wings of an airplane during flight compared to below the wings?

10. Euler's equation of motion reduces directly to Bernoulli's equation when integrated along:

Water flows through a horizontal Venturi meter with inlet and throat cross-sectional areas of 20 cm² and 10 cm² respectively. The pressure difference between the inlet and throat is 3 kPa. Calculate the volume flow rate of water (density of wate$r = 1000 kg/m^{3})$.