03 Dinamika Fluida Persamaan Bernoulli Part1 MEKFLU

Introduction to Fluid Dynamics and Bernoulli's Principle

Overview of Chapter 3

• The discussion focuses on fluid dynamics, specifically the concept of Bernoulli's principle. This chapter contrasts with Chapter 2, which dealt with static fluids.

• Key topics include inviscid flow, application of Newton's second law, physical interpretation of Bernoulli's equation, and various types of pressure (static, dynamic, total).

Understanding Inviscid Flow

• Inviscid flow refers to fluid motion without viscosity; in reality, all fluids have some viscosity. This idealization allows for neglecting friction effects during fluid movement.

• The absence of viscosity means shear stress can be ignored in the analysis of inviscid flows. This concept is foundational for applying Newton’s second law in fluid dynamics.

Application of Newton's Second Law in Fluid Dynamics

Forces Acting on Fluid Particles

• Newton’s second law will be applied to analyze forces acting on a fluid particle: pressure force and body force (gravity). The resultant force is crucial for understanding fluid motion.

• Inertia plays a significant role as it relates to the mass and acceleration of the fluid particles being analyzed. Each particle can be treated as having a specific mass despite being infinitesimal in size.

Streamlines and Particle Motion

• A streamline represents an imaginary line along which a fluid particle moves; its velocity vector is tangent at every point along this line. Understanding streamlines helps visualize how fluids flow through different paths.

• Examples illustrate how particles follow these streamlines while maintaining their velocity vectors tangentially aligned with them throughout their journey within the flow field.

Steady Flow Conditions and Acceleration

Characteristics of Steady Flow

• Even under steady conditions where parameters do not change over time, velocities at different points (e.g., point A vs point B) may differ due to varying cross-sectional areas or other factors affecting flow speed.

• The application of Newton’s second law will consider both parallel and perpendicular components relative to streamlines when analyzing changes in velocity across different sections of flow paths.

Types of Flow Patterns

• Various scenarios are presented regarding acceleration along streamlines:

• Divergent flows lead to negative acceleration as mass decreases while moving from smaller to larger areas.

• Convergent flows result in positive acceleration as mass increases when moving from larger areas into smaller ones.

Newton's Second Law Along Streamlines

Infinitesimal Analysis

• An infinitesimal element representing a small section along a streamline will be analyzed using Newton’s laws; this involves considering forces acting parallel and perpendicular to the streamline direction for accurate modeling purposes.

Fluid Dynamics and Pressure Changes

Understanding Fluid Thickness and Pressure Variation

• The concept of fluid thickness is introduced, with Delta Ester oil assumed to have a specific thickness that interacts with the screen.

• The pressure at different points (A and B) can be analyzed using Taylor series to understand how pressure changes from a midpoint (P) based on differences in height.

• It is noted that for an inviscid fluid, shear stress is zero, and gravitational acceleration acts downward, influencing the pressure dynamics.

Geometric Considerations in Fluid Flow

• A triangle representation is used to illustrate relationships between Delta Z (height), Delta es (length), and angle theta; sine and cosine functions are applied for calculations.

• Newton's second law is applied to a free body diagram of the fluid particle, indicating that net forces acting on it will equal its inertia.

Forces Acting on Fluid Particles

• The force acting on a fluid particle along the streamline involves mass times acceleration; mass can be expressed as density multiplied by volume.

• Acceleration along the streamline can be broken down into differential components related to velocity changes over time.

Gravitational Effects on Pressure

• Gravitational force acts downward, represented as Delta W, which relates to volume and density of the fluid; this helps calculate gravitational effects along streamlines.

• The change in pressure due to gravity can be derived using geometric relationships involving angles and densities.

Calculating Net Forces in Fluid Dynamics

• Using Taylor series expansion allows for approximating pressure differences between various points within the fluid flow system.

• The net force calculation incorporates both left-side (A - ΔPS) and right-side pressures (B + ΔPS), leading to expressions for calculating resultant forces based on area considerations.

Finalizing Equations of Motion for Fluids

• By equating forces from both sides of a control volume, we derive expressions relating pressure changes directly tied to volumetric considerations.

• Substituting known values into Newton's second law leads us toward understanding how gravitational forces interact with pressure-induced forces within fluids.

This structured approach provides clarity regarding complex interactions within fluid dynamics while maintaining focus on key principles such as pressure variation, geometric influences, and fundamental laws governing motion.

Pressure Changes and Bernoulli's Equation

Understanding Pressure Changes

• The total pressure change is equal to the sum of partial pressure changes in both the S and n directions, leading to a conclusion that if the total pressure change (DP) is zero, then the individual components must also be zero.

• The approximation of total pressure can be simplified using only the partial pressures along a streamline, indicating that as long as certain conditions are met, we can treat these pressures uniformly.

Deriving Key Equations

• By manipulating equations, we arrive at a new equation (Equation 3.5), which states that DP plus half of velocity squared plus gamma times dz equals zero; this holds true along a streamline.

• Integrating under constant gamma leads us to Equation 3.6, which expresses DP in terms of velocity and height changes along a streamline.

Application of Bernoulli's Principle

• When integrating with constant density assumptions from ideal gas laws, we derive Bernoulli’s equation (Equation 3.7), which relates pressure, kinetic energy per unit volume, and potential energy per unit volume along streamlines.

• The application of Bernoulli’s equation requires specific conditions: neglecting viscous effects, steady flow conditions, incompressibility, and constant density throughout the flow path.

Example Scenario: Bicycle Rider

• An example involving a cyclist illustrates how to calculate pressure differences between two points based on varying speeds around them; it emphasizes understanding coordinate systems relative to moving objects for accurate analysis.

• In applying Bernoulli’s principle between two points on the cyclist's path—one where speed is maximum and another at rest—the resulting difference in pressure can be calculated using derived equations related to fluid dynamics principles.