Fluid Mechanics Concepts

Newton's Law

• Describes the relationship between shear stress and the velocity gradient in fluids.
• Shear stress (τ) is related to the dynamic viscosity (µ) and the velocity gradient (du/dy).

Capillary Elevation

• Capillary elevation occurs when a liquid rises or falls in a narrow tube due to surface tension.
• The height (h) of the liquid elevation or depression is related to the surface tension (σ), contact angle (θ), and the tube radius (r).
• The formula is h = (2σ cosθ)/(ρgr) where σ = surface tension, θ = contact angle, ρ = density of liquid, g = acceleration due to gravity, and r = radius of the tube.

Streamlines

• Streamlines are curves in a flow field that are everywhere tangent to the velocity vector at each point in the fluid.
• Streamlines help visualize the flow pattern and show flow direction.

Prandtl Tube

• Used for measuring fluid velocity.
• The formula for the velocity (v) of a fluid using a Prandtl tube is given as v = 2√[(p2−p1)/ρ] where p2 and p1 pressures, and ρ is density.

Moody Diagram

• Shows the relationship between the friction factor (λ) and the Reynolds number (Re) for various pipe roughness.
• The Moody diagram is used to determine the friction factor for turbulent flow in pipes.
• Friction factor depends on Re and roughness.

Buckingham Pi Theorem

• Used to reduce the number of variables in complex physical problems.
• The Buckingham Pi theorem is a technique for finding dimensionless groups from a set of variables in order to simplify the problem.
• Based on dimensional analysis
• Reduces the number of variables via dimensionless groups

Euler's Equation

• Describes the conservation of linear momentum in a fluid flow.
• The relationship between pressure gradient and fluid acceleration is given as a function of the fluid velocity.
• The equation considers the spatial variations of pressure, momentum, and the fluid acceleration.

Reynolds Number

• A dimensionless number describing the ratio of inertial forces to viscous forces in a fluid.
• Used to determine the flow regime (laminar or turbulent) in a pipe or channel.
• A critical Reynolds number separates laminar from turbulent flows, and it's denoted as Re.

Flow in a tube

• The fluid velocity, pressure, and density of the fluid are important factors that must be considered when studying flow in a tube.
• Laminar flow has smooth layers, turbulent flow is chaotic.

Power of a fluid

• Power is calculated by taking the product between the mass flow rate, gravity constant, elevation difference and the density of the fluid, as a function of the area of the pipe or channel.
• Power of the fluid is a function of the mass flow rate, elevation difference, and cross-sectional area of the pipe, or channel.

Laval Nozzle

• A Laval nozzle is a converging-diverging nozzle used to accelerate a fluid to supersonic speeds.
• The fluid expands and accelerates in the diverging section to achieve supersonic speeds.

Efficiency of a nozzle

• Efficiency of a nozzle is a measure of how effectively the nozzle accelerates the fluid.
• It is a measure of how much of the input energy is converted into kinetic energy of the fluid.

Hydrostatic Pressure

• Pressure exerted by a fluid at rest due to the force of gravity.
• It depends only on the height and density of the fluid.
• Formula: P = ρgh.

Hydrostatic Paradox

• The equal hydrostatic pressure forces at every point in a fluid at the same height, regardless of the shape or size of the container.

Bernoulli Equation

• Describes the conservation of energy in steady, incompressible, and inviscid flow.
• Expresses the relationship between static and dynamic pressure, kinetic energy and pressure.

Torricelli's law

• Formula : V= √(2gh)
• Equation that predicts the speed of a fluid emerging from an opening in container.

Dimensional Analysis

• Method for deriving and analyzing physical relationships or models.

Nozzle Efficiency

• A calculation of how effectively energy is transferred using the Laval Nozzle.

Kapilární Elevation

• Equation is h=2σ cosθ/ ρgr
• Height of elevation in a narrow tube due to surface tension.

Viscosity

• Fluid resistance to flow.
• Liquid resistance, or internal friction.

Laminar Flow

• Characterized by smooth, parallel layers of fluid moving past each other.

Turbulent Flow

• Chaotic flow marked by fluctuations in velocity and pressure.

Reynolds transport theorem

• A fundamental concept in fluid mechanics, which describes how to analyze the conservation laws of mass, momentum, and energy for open and closed systems.

Velocity

• The speed and direction at a given point in a fluid flow.

Pressure

• Force per unit area exerted by a fluid.

Density

• Mass per unit volume of a fluid.

Specific Weight

• Weight per unit volume of a fluid.

Surface Tension

• Force that tends to minimize the surface area of a liquid.

Contact Angle

• Angle between the tangent at the intersection of the liquid and solid surfaces and the solid surface.

More Notes

• Includes numerous other related concepts like momentum, energy, and conservation laws.