Fluid Flow – Types and Continuity

Fluid mechanics ka ek fundamental branch hai fluid dynamics, jo fluids in motion ke behavior ko study karta hai. Fluid flow ke types, characteristics, and continuity principles ko samajhna engineering, hydraulics, aerodynamics, and process industries me essential hai.

1. Introduction to Fluid Flow

Fluid Flow:

Movement of fluid particles from one point to another under the influence of pressure, gravity, or other forces.

Importance of studying fluid flow:

1. Design of pipes, pumps, and turbines
2. Understanding natural flows like rivers, ocean currents, and atmospheric circulation
3. Design of aircraft, ships, and automobiles
4. Industrial applications: chemical reactors, water supply systems

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2. Types of Fluid Flow

Fluid flow can be classified based on nature of motion, velocity profile, and time-dependence.

2.1 Based on Flow Pattern

2.1.1 Laminar Flow

• Fluid flows in parallel layers with no mixing
• Velocity profile: parabolic for pipe flow
• Reynolds number (Re) < 2000 → laminar flow
• Examples:
  • Flow of oil in thin tubes
  • Slow-moving viscous fluids

2.1.2 Turbulent Flow

• Chaotic and irregular flow
• Mixing occurs between layers
• Reynolds number (Re) > 4000 → turbulent flow
• Examples:
  • River currents
  • Water in large pipes
  • Airflow over wings

2.1.3 Transitional Flow

• Re = 2000–4000 → flow switches between laminar and turbulent
• Sensitive to disturbances

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2.2 Based on Compressibility

2.2.1 Incompressible Flow

• Density of fluid considered constant
• Applies to liquids and low-speed gases (Mach < 0.3)

2.2.2 Compressible Flow

• Density varies significantly
• Important in high-speed gas dynamics, jets, rockets

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2.3 Based on Viscosity

2.3.1 Ideal Flow (Inviscid)

• Neglects viscous effects
• Governing equations: Euler’s equation

2.3.2 Real Flow

• Includes viscosity
• Governing equations: Navier-Stokes equations

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2.4 Based on Streamlines

2.4.1 Steady Flow

• Fluid properties at a point do not change with time

∂∂t=0\frac{\partial}{\partial t} = 0∂t∂​=0

• Example: Water through a constant-diameter pipe at constant speed

2.4.2 Unsteady Flow

• Properties change with time
• Example: Flow during pump start-up, tides, and flood waves

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2.5 Based on Flow Geometry

2.5.1 One-Dimensional Flow

• Velocity varies in one direction
• Example: Flow through a long straight pipe

2.5.2 Two-Dimensional Flow

• Velocity varies in two directions
• Example: Flow over an airfoil, open channel flow

2.5.3 Three-Dimensional Flow

• Velocity varies in all three directions
• Example: Ocean currents, complex industrial flows

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3. Reynolds Number and Flow Type

Reynolds number (Re):

Dimensionless number that predicts laminar or turbulent flow.

Re=ρVDμ=VDνRe = \frac{\rho V D}{\mu} = \frac{V D}{\nu}Re=μρVD​=νVD​

Where:

• ρ = fluid density
• V = flow velocity
• D = characteristic length (pipe diameter)
• μ = dynamic viscosity
• ν = kinematic viscosity

Flow regimes:

• Re < 2000 → Laminar
• 2000 < Re < 4000 → Transitional
• Re > 4000 → Turbulent

Applications:

• Pipe design, chemical reactors, aerodynamic studies

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4. Continuity Equation

Continuity principle:

Mass is conserved in fluid flow.

• For incompressible fluids, the mass flow rate is constant along the flow.

4.1 Mathematical Formulation

• Consider a pipe with varying cross-section: A₁ → A₂
• Velocity: V₁ → V₂

Mass flow rate: m˙=ρAV\dot{m} = \rho A Vm˙=ρAV

• For incompressible fluid (ρ constant):

A1V1=A2V2A_1 V_1 = A_2 V_2A1​V1​=A2​V2​

• This is the Continuity Equation:

Velocity increases when area decreases, and vice versa

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4.2 General Form (3D)

∂ρ∂t+∇⋅(ρV)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0∂t∂ρ​+∇⋅(ρV)=0

• For steady incompressible flow:

∇⋅V=0\nabla \cdot \mathbf{V} = 0∇⋅V=0

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4.3 Examples

Example 1: Pipe Constriction

• Pipe: Diameter 0.1 m → 0.05 m
• Velocity at inlet: 2 m/s
• Outlet velocity:

V2=V1(A1A2)=2(π(0.1)2/4π(0.05)2/4)=8 m/sV_2 = V_1 \left(\frac{A_1}{A_2}\right) = 2 \left(\frac{\pi (0.1)^2 /4}{\pi (0.05)^2 /4}\right) = 8 \, m/sV2​=V1​(A2​A1​​)=2(π(0.05)2/4π(0.1)2/4​)=8m/s

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Example 2: Flow Rate

• Flow rate (Q) = A × V
• Q constant along pipe for incompressible fluid

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5. Velocity and Streamlines

• Streamline: Path followed by fluid particle
• Steady flow → streamlines fixed
• Unsteady flow → streamlines change with time
• Streamtube: Bundle of streamlines → mass conservation applied

m˙=ρVA=constant along streamtube\dot{m} = \rho V A = \text{constant along streamtube}m˙=ρVA=constant along streamtube

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6. Practical Applications of Continuity

1. Pipe Networks
   • Velocity and diameter design using A₁V₁ = A₂V₂
2. Hydraulic Machines
   • Pump and turbine analysis
3. Aerodynamics
   • Wing design, wind tunnels
4. Open Channel Flow
   • Rivers, canals, and spillways
5. Industrial Process Engineering
   • Flow of liquids and gases in reactors, pipelines

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7. Flow Visualization

• Dye injection → visualize laminar and turbulent flow
• Particle image velocimetry (PIV) → measure velocity field
• Smoke tunnels → airflow visualization
• Laminar flow → smooth, parallel streamlines
• Turbulent flow → chaotic mixing

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8. Factors Affecting Flow Type

1. Velocity of fluid (V) → higher → turbulent
2. Fluid viscosity (μ) → higher → laminar
3. Pipe diameter (D) → larger → turbulent
4. Density (ρ) → affects Reynolds number

Combined in Reynolds number → predicts flow regime

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9. Continuity in Compressible Flow

• For gases, density varies → mass flow:

m˙=ρAV=constant\dot{m} = \rho A V = \text{constant}m˙=ρAV=constant

• Requires ρ variation to be considered:

ρ1A1V1=ρ2A2V2\rho_1 A_1 V_1 = \rho_2 A_2 V_2ρ1​A1​V1​=ρ2​A2​V2​

• Example: Nozzle design, supersonic flows

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10. Graphical Representation

1. Velocity vs Cross-section → inverse relation
2. Streamlines → laminar vs turbulent
3. Reynolds number vs flow type → transition chart

Re
|
|          Turbulent
|        *
|      *
|    *
|  *
|_*________________ Laminar

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11. Numerical Problems

Problem 1: Pipe Constriction

• A pipe narrows from 0.2 m² to 0.05 m²
• Velocity at inlet = 1 m/s
• Find outlet velocity:

V2=A1A2V1=0.20.05⋅1=4m/sV_2 = \frac{A_1}{A_2} V_1 = \frac{0.2}{0.05} \cdot 1 = 4 m/sV2​=A2​A1​​V1​=0.050.2​⋅1=4m/s

Problem 2: Flow Rate

• Water flow Q = 0.5 m³/s, pipe diameter 0.1 m
• Velocity:

V=QA=0.5π(0.1)2/4≈63.66m/sV = \frac{Q}{A} = \frac{0.5}{\pi (0.1)^2 /4} \approx 63.66 m/sV=AQ​=π(0.1)2/40.5​≈63.66m/s

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12. Summary Table

Concept	Formula / Relation	Notes
Mass flow rate	m˙=ρAV\dot{m} = \rho A Vm˙=ρAV	Constant along flow
Continuity equation	A1V1=A2V2A_1 V_1 = A_2 V_2A1​V1​=A2​V2​	Incompressible fluids
Reynolds number	Re=ρVDμRe = \frac{\rho V D}{\mu}Re=μρVD​	Predicts laminar/turbulent
Laminar flow	Re < 2000	Parallel layers, smooth
Turbulent flow	Re > 4000	Chaotic, mixing
Streamline	Path of particle	Tangent to velocity vector