Gonit Vigyan

🌊 Fluid Mechanics

Kinematics of Moving Fluids, Equation of Continuity, Euler's Equation, Bernoulli's Theorem

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1. Kinematics of Moving Fluids

1.1 Types of Fluid Flow

Type	Description	Example
Steady Flow	Velocity at a point does not change with time ($\partial \vec{v}/\partial t = 0$)	Smooth flow in a pipe
Unsteady Flow	Velocity changes with time	Pulsating flow
Uniform Flow	Velocity does not change with position ($\partial \vec{v}/\partial s = 0$)	Flow in a straight pipe of constant cross-section
Non-uniform Flow	Velocity changes with position	Flow in a tapered pipe
Laminar Flow	Fluid particles move in smooth, parallel layers	Low velocity, high viscosity
Turbulent Flow	Irregular, chaotic motion with eddies	High velocity, low viscosity
Rotational Flow	Fluid elements rotate about their own axis ($\nabla \times \vec{v} \neq 0$)	Flow near boundaries
Irrotational Flow	Fluid elements do not rotate ($\nabla \times \vec{v} = 0$)	Ideal flow away from boundaries
Compressible Flow	Density changes ($\rho \neq \text{constant}$)	High-speed gas flow
Incompressible Flow	Density constant ($\rho = \text{constant}$)	Most liquid flows

1.2 Streamlines, Pathlines, and Streaklines

• Streamline: A curve whose tangent at any point gives the direction of velocity at that point. For steady flow, streamlines are fixed in space.
• Pathline: The actual path traced by a fluid particle over time.
• Streakline: The locus of all particles that have passed through a particular point at some earlier time.

For steady flow: Streamlines, pathlines, and streaklines coincide.

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2. Equation of Continuity

2.1 Statement

The equation of continuity is a mathematical expression of the law of conservation of mass in fluid dynamics.

2.2 General Form (Integral Form)

The rate of increase of mass inside a fixed volume equals the net rate of mass inflow through its surface:

$$\frac{\partial}{\partial t} \int_V \rho \, dV = -\oint_S \rho \vec{v} \cdot d\vec{S}$$

2.3 Differential Form (Continuity Equation)

Using the divergence theorem:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$$

Where:

• $\rho$ = density of fluid
• $\vec{v}$ = velocity vector
• $\nabla \cdot (\rho \vec{v})$ = divergence of mass flux

2.4 Special Cases

For Steady Flow ($\partial \rho/\partial t = 0$):

$$\nabla \cdot (\rho \vec{v}) = 0$$

For Incompressible Flow ($\rho = \text{constant}$):

$$\nabla \cdot \vec{v} = 0$$

For One-Dimensional Flow (Pipe Flow):

$$\rho_1 A_1 v_1 = \rho_2 A_2 v_2 = \text{constant (mass flow rate)}$$

For incompressible flow:

$$A_1 v_1 = A_2 v_2 = Q \quad \text{(volume flow rate)}$$

Where:

• $A$ = cross-sectional area
• $v$ = flow velocity
• $Q$ = volume flow rate (discharge)

2.5 Physical Interpretation

• If area decreases, velocity increases (and vice versa).
• In incompressible flow, fluid speeds up in constrictions.
• This explains why a river flows faster in narrow sections.

Exam Connection (2025 Q39):

"As water flows from a faucet, stream of water becomes narrower as it descends. The guiding principle for this observation is:"

1. Bernoulli's equation
2. Pascal's law
3. Continuity equation
4. Archimedes' principle

Answer: (3) Continuity equation

Reasoning: As water falls, it accelerates due to gravity. By continuity equation $A_1 v_1 = A_2 v_2$, if velocity increases, area must decrease to maintain constant flow rate.

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3. Euler's Equation of Motion

3.1 Statement

Euler's equation describes the motion of an inviscid (frictionless) fluid. It is essentially Newton's second law applied to a fluid element.

3.2 Derivation (Conceptual)

Consider a fluid element. The forces acting on it are:

1. Pressure forces (from surrounding fluid)
2. Body forces (like gravity)

For an inviscid fluid, shear stresses are zero. Applying Newton's second law to a fluid element gives:

3.3 Euler's Equation (Vector Form)

$$\rho \frac{D\vec{v}}{Dt} = -\nabla P + \vec{f}$$

Where:

• $\frac{D}{Dt} = \frac{\partial}{\partial t} + (\vec{v} \cdot \nabla)$ is the material derivative (also called convective derivative)
• $\rho$ = density
• $\vec{v}$ = velocity field
• $P$ = pressure
• $\vec{f}$ = body force per unit volume (e.g., $\rho \vec{g}$ for gravity)

3.4 Component Form (in Cartesian coordinates)

For the x-component:

$$\rho \left( \frac{\partial v_x}{\partial t} + v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z} \right) = -\frac{\partial P}{\partial x} + f_x$$

3.5 Physical Meaning

The material derivative $\frac{D\vec{v}}{Dt}$ represents the acceleration of a fluid particle as it moves through space. It has two parts:

• Local acceleration $\partial \vec{v}/\partial t$: change due to time variation at a fixed point
• Convective acceleration $(\vec{v} \cdot \nabla)\vec{v}$: change due to motion to regions of different velocity

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4. Bernoulli's Theorem

4.1 Statement

Bernoulli's theorem states that for an ideal (inviscid, incompressible), steady flow along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant.

4.2 Mathematical Form

Along a streamline:

$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$

Where:

• $P$ = static pressure
• $\frac{1}{2}\rho v^2$ = dynamic pressure (kinetic energy per unit volume)
• $\rho gh$ = hydrostatic pressure (potential energy per unit volume)
• The constant is called the total pressure or stagnation pressure

4.3 Alternative Forms

Per unit mass:

$$\frac{P}{\rho} + \frac{1}{2}v^2 + gh = \text{constant}$$

Per unit weight (head form):

$$\frac{P}{\rho g} + \frac{v^2}{2g} + h = \text{constant} = H$$

Where $H$ is the total head.

4.4 Assumptions (Important for Exams)

Bernoulli's equation is valid only when:

1. Steady flow ($\partial/\partial t = 0$)
2. Incompressible flow ($\rho = \text{constant}$)
3. Inviscid flow (no viscosity, frictionless)
4. Along a streamline (not across streamlines)
5. No shaft work (no pumps or turbines between sections)

4.5 Applications of Bernoulli's Theorem

Application	Explanation
Venturi meter	Measures flow rate using pressure difference in constriction
Pitot tube	Measures flow velocity by comparing static and stagnation pressure
Airplane wing lift	Faster air over curved top surface creates lower pressure, generating lift
Atomizer / Sprayer	High-speed air creates low pressure, drawing liquid up
Flow from a tank (Torricelli's theorem)	$v = \sqrt{2gh}$ for efflux velocity

4.6 Torricelli's Theorem (Special Case)

For a tank with a small hole at depth $h$ below the free surface:

$$v = \sqrt{2gh}$$

This is obtained by applying Bernoulli between the free surface (where $v \approx 0$, $P = P_{\text{atm}}$) and the hole (where $P = P_{\text{atm}}$).

4.7 Limitations

• Cannot be used in regions of separated flow or boundary layers.
• Frictionless assumption fails for real fluids over long distances.
• For gases, incompressible assumption fails at high velocities (Mach number > 0.3).

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5. Important Formulas Summary

Concept	Formula	Notes
Continuity (general)	$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$	Mass conservation
Continuity (incompressible)	$\nabla \cdot \vec{v} = 0$	$\rho = \text{constant}$
Continuity (1D, incompressible)	$A_1 v_1 = A_2 v_2$	Volume flow rate constant
Continuity (1D, compressible)	$\rho_1 A_1 v_1 = \rho_2 A_2 v_2$	Mass flow rate constant
Euler's equation	$\rho \frac{D\vec{v}}{Dt} = -\nabla P + \vec{f}$	Inviscid flow
Bernoulli (along streamline)	$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$	Under assumptions
Torricelli's theorem	$v = \sqrt{2gh}$	Efflux velocity

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6. Exam-Specific Problem Types

Type A: Continuity Equation (2025 Q39)

Problem: Water flows through a pipe of diameter 10 cm at 2 m/s. If the pipe narrows to 5 cm diameter, what is the velocity in the narrow section?

Solution:

$$A_1 v_1 = A_2 v_2 \implies \pi r_1^2 v_1 = \pi r_2^2 v_2$$ $$(5^2) \times 2 = (2.5^2) \times v_2 \implies 25 \times 2 = 6.25 \times v_2$$ $$v_2 = \frac{50}{6.25} = 8 \, \text{m/s}$$

Type B: Bernoulli Application

Problem: Water flows through a horizontal pipe with a constriction. At a point where the diameter is 10 cm, the pressure is 200 kPa and velocity is 2 m/s. At a constriction where diameter is 5 cm, find the pressure. (Density of water = 1000 kg/m³)

Solution:

1. Find velocity at constriction using continuity:

$$A_1 v_1 = A_2 v_2 \implies (\pi \times 5^2) \times 2 = (\pi \times 2.5^2) \times v_2$$ $$25 \times 2 = 6.25 \times v_2 \implies v_2 = 8 \, \text{m/s}$$

2. Apply Bernoulli (horizontal pipe, so $h$ cancels):

$$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$$ $$200 \times 10^3 + \frac{1}{2}(1000)(2^2) = P_2 + \frac{1}{2}(1000)(8^2)$$ $$200,000 + 2000 = P_2 + 32,000$$ $$P_2 = 202,000 - 32,000 = 170,000 \, \text{Pa} = 170 \, \text{kPa}$$

Type C: Torricelli's Theorem

Problem: A large tank filled with water has a small hole 5 m below the free surface. Find the velocity of efflux. (Take $g = 10 \, \text{m/s}^2$)

Solution:

$$v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5} = \sqrt{100} = 10 \, \text{m/s}$$

Type D: Venturi Meter

Problem: In a Venturi meter, the diameter at inlet is 20 cm and at throat is 10 cm. If the pressure difference between inlet and throat is 30 kPa, find the flow rate. (Density of fluid = 1000 kg/m³)

Solution:

Using Bernoulli and continuity:

$$Q = A_1 A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho(A_1^2 - A_2^2)}}$$

But simpler approach:

1. Let $v_1$ and $v_2$ be velocities at inlet and throat.
2. From continuity: $A_1 v_1 = A_2 v_2 = Q$, so $v_2 = (A_1/A_2) v_1$
3. From Bernoulli: $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$
4. Substitute $v_2$: $$P_1 - P_2 = \frac{1}{2}\rho (v_2^2 - v_1^2) = \frac{1}{2}\rho v_1^2 \left( \left( \frac{A_1}{A_2} \right)^2 - 1 \right)$$
5. Solve for $v_1$, then $Q = A_1 v_1$.

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7. Common Misconceptions to Avoid

Misconception	Truth
"Bernoulli applies everywhere in a fluid."	False: Only along a streamline, under specific assumptions.
"If velocity increases, pressure always decreases."	True only along a streamline in inviscid flow. Across streamlines, pressure may vary differently.
"Continuity equation applies only to incompressible flow."	False: General continuity applies to all flows; incompressible is a special case.
"Euler's equation includes viscous forces."	False: Euler's equation is for inviscid flow. Navier-Stokes includes viscosity.

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8. Predicted Question Types for CUET PG

1. Continuity Equation (High Probability): Qualitative (faucet narrowing) or numerical (pipe diameter change). (2025 Q39)
2. Bernoulli's Theorem (High Probability): Numerical application (find pressure or velocity in constriction).
3. Torricelli's Theorem (Medium Probability): Velocity of efflux from a tank.
4. Assumptions of Bernoulli (Medium Probability): Which conditions must hold for Bernoulli to be valid?
5. Euler's Equation (Low Probability): Recognition questions about the material derivative or forces considered.
6. Venturi/Pitot Tube (Low-Medium Probability): How they work and basic calculations.

Key Takeaways for Exam Day

• Continuity: $A_1 v_1 = A_2 v_2$ (incompressible)
• Bernoulli: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
• Torricelli: $v = \sqrt{2gh}$
• Assumptions for Bernoulli: Steady, incompressible, inviscid, along streamline
• Euler's equation: Inviscid flow analog of Newton's second law

By mastering these concepts and practicing the problem types above, you'll be well-prepared for fluid mechanics questions on the CUET PG exam.