Lesson 5.3: Fluid Kinematics (Streamline, Continuity, Potential Flow)

Fluid Kinematics deals with motion of fluids without considering the forces causing the motion. GATE and PSU exams often test flow patterns, velocity relationships, and basic fluid motion principles.

🔹 1. Introduction

• Definition: Kinematics of fluid studies velocity, acceleration, and flow patterns of fluids.

• Applications: Pipe flow, open channels, pumps, turbines

• Key Concepts:

  1. Streamline: Path traced by a fluid particle in steady flow

  2. Pathline: Actual path followed by a particle

  3. Streakline: Locus of all particles passing through a point

• Steady vs Unsteady Flow:

  • Steady: velocity at a point does not change with time

  • Unsteady: velocity varies with time

🔹 2. Continuity Equation

• Principle: Mass conservation → mass flow rate remains constant along a streamline

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

• Incompressible Fluid: ρ1=ρ2\rho_1 = \rho_2ρ1​=ρ2​ →

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

• Example: Pipe reduces from 0.3 m² to 0.1 m² cross-section, inlet velocity 2 m/s → outlet velocity?

V2=A1V1A2=0.3∗20.1=6m/sV_2 = \frac{A_1 V_1}{A_2} = \frac{0.3*2}{0.1} = 6 m/sV2​=A2​A1​V1​​=0.10.3∗2​=6m/s

🔹 3. Potential Flow

• Definition: Flow that is irrotational and can be described by a velocity potential function (φ)

$$V=\nabla \varphi$$

• Stream Function (ψ): Represents flow along streamlines, constant along a streamline

• Relation: ∂ϕ∂x=u,∂ϕ∂y=v\frac{\partial \phi}{\partial x} = u, \frac{\partial \phi}{\partial y} = v∂x∂ϕ​=u,∂y∂ϕ​=v

• Applications: Ideal flow analysis, pump and turbine design, aerodynamics

🔹 4. Flow Patterns

• Laminar Flow: Smooth, parallel layers (Re < 2000)

• Turbulent Flow: Irregular, chaotic (Re > 4000)

• Reynolds Number (Re):

Re=ρVDμRe = \frac{\rho V D}{\mu}Re=μρVD​

• Example: Water flow in 0.1 m diameter pipe at 1 m/s, ν = 1e-6 m²/s → Re = ?

Re=VDν=1∗0.11e−6=1e5⇒TurbulentRe = \frac{VD}{ν} = \frac{1*0.1}{1e-6} = 1e5 \Rightarrow \text{Turbulent}Re=νVD​=1e−61∗0.1​=1e5⇒Turbulent

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE ME 2017):
Water flows through a pipe reducing from 0.25 m² to 0.1 m², inlet velocity 1.5 m/s. Find outlet velocity using continuity.

Example 2 (PSU Exam):
Calculate Reynolds number for water in a 0.05 m diameter pipe, velocity 2 m/s, ν = 1.004e-6 m²/s → classify flow.

🔹 6. Practice Exercises

1. Determine outlet velocity for given pipe area reduction.

2. Compute Reynolds number for different fluids and pipe diameters.

3. Sketch streamline, pathline, and streakline for steady/unsteady flow.

4. Solve basic potential flow problems using φ and ψ functions.

5. Explain laminar vs turbulent flow with examples.

🔹 7. Summary

• Streamline, Pathline, Streakline: Describe particle motion

• Continuity Equation: Mass conservation in flow

• Potential Flow: Irrotational, velocity potential and stream function

• Reynolds Number: Predicts laminar vs turbulent flow

• Applications: Pumps, turbines, pipe flow, hydraulics