Intro to Fluid Dynamics and Bernoulli’s Balance

Final Questions and Homework Assignment

  • A homework assignment will be given today, designated as Homework Number Four.

Introduction to Bernoulli's Equation

  • Bernoulli's equation is a foundational concept in fluid dynamics.
  • It is essential for understanding the balance of energy within fluid systems.

Equations Reviewed in Class

  • Reminder of previously discussed equations in this course:
    • Pascal's Principle
    • Bernoulli's Equation (focus of today's class)
    • Three other equations related to fluid dynamics.

Energy Balance in Bernoulli’s Equation

  • Bernoulli’s equation conserves energy among three key components:
    1. Pressure Energy: The internal energy of the fluid exerted as pressure in the system.
    2. Kinetic Energy: Often referred to as dynamic energy, associated with the fluid’s movement.
    3. Potential Energy: Energy related to the fluid's height above a reference point.
  • The components must balance as fluids flow from one point to another in a system.
  • If energy is lost during flow due to friction or leakage, this imbalance indicates a problem in the system.

Conditions for Applying Bernoulli's Equation

  • Assumptions necessary for the application of Bernoulli’s principle include:
    • Steady Flow: No accumulation of flow in the system.
    • Streamlined Flow: The fluid flows in layers with no turbulence.
    • Isothermal Conditions: The fluid density remains constant.
    • Incompressibility: The fluid density does not change under pressure.

Terminology in Fluid Flow

  • Vocabulary used in textbooks may include:
    • Streamlines: Visual depiction of fluid flow direction.
    • Pathlines: Tracing the trajectory of individual fluid particles over time.
    • Streaklines: Lines obtained by tracing the points of fluid particles flowing through a fixed point in time.
  • For this course, all three can often be treated as the same under streamlined conditions.

Fluid Dynamics Diagramming

  • Importance of drawing schematics to illustrate flow direction and identify key points.
  • Choose points in the system where measurements will be made (e.g., inlet and outlet).

Assumptions Revisited

  • Key assumptions for using Bernoulli's equation include:
    • No frictional losses.
    • Fluid is incompressible.
    • Constant temperature.
    • No leaks in the system.

Example Problem: Fluid Flow from a Tank

  • Consider a cylindrical tank filled with fluid that drains from a small opening at the bottom:
    • Characteristics of the tank:
    • Height (h) and diameter at the top.
    • Diameter of the outlet (d).
  • The objective is to derive an expression for the exit velocity of the fluid when opened, assuming height remains constant.

Key Assumptions for Example Problem

  • Pressure in the tank and at the outlet is atmospheric (cancelling this in the equation).
  • Velocity at point one (at the top of the tank) is negligible or approximately zero compared to the velocity at point two.

Derivation using Bernoulli’s Equation

  1. Cancel terms based on assumptions (both pressures being atmospheric, negligible initial velocity).
  2. Using potential energy difference, the resultant velocity can be expressed as: v2=extsqrt(2gh)v_2 = ext{sqrt}(2gh) where:
    • $g$ = acceleration due to gravity
    • $h$ = height of the fluid column above the outlet.
  3. Volumetric flow rate (Q): For a steady flow, calculate the volumetric flow rate at various points in the system. Q=AvQ = A v
    • Where:
    • $A$ is the area of the cross-section of the pipe and $v$ is velocity.

Mass Conservation Principle

  • The principle of mass conservation states:
    • Mass flow rate into a section = mass flow rate out of that section.
    • This applies to fluid dynamics and affects velocity as the diameter of the pipe changes.

Effects of Constriction in Pipe

  • If the diameter decreases, the velocity at that point must increase due to the conservation of mass (continuity equation).
  • This is governed by: A<em>1v</em>1=A<em>2v</em>2A<em>1 v</em>1 = A<em>2 v</em>2
    • Where $A1$ and $A2$ are cross-sectional areas at points 1 and 2 respectively.

Calculation Example and Pressure Relation

  • The pressure difference ($ ext{ΔP}$) across two points can be calculated using hydrostatic pressure relation: extΔP=hoghext{ΔP} = ho g h
    • Where $h$ is the height difference between the two points.

Conclusion of Class Discussion

  • Emphasis on the significance of assumptions and correct application of Bernoulli’s equation.
  • Importance of diagrams and clearly defining parameters in problem-solving for fluid dynamics.