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Fluid Mechanics Fundamentals

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Continuity Equation

The Continuity Equation in fluid mechanics is represented by

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0

and it states that the mass of fluid remains constant as it flows through a control volume. It is used to relate the velocity and cross-sectional area of flow in ducts and pipes.

Reynolds Number

The Reynolds Number, expressed as

Re = \frac{\rho v L}{\mu}

, dictates the flow regime of a fluid in a pipe or around a body. It's used to predict the transition from laminar flow to turbulent flow. A higher Reynolds number indicates turbulent flow.

Navier-Stokes Equations

The Navier-Stokes Equations, represented as a set of nonlinear partial differential equations, describe the motion of fluid substances. These equations are fundamental in predicting the flow characteristics of fluids in engineering and physics.

Bernoulli's Equation

Bernoulli's Equation, given by

P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}

, describes the conservation of energy in a flowing fluid. It applies to incompressible, non-viscous flows and indicates that the sum of pressure potential, kinetic energy per unit volume, and gravitational potential energy per unit volume must remain constant along a streamline.

Pascal's Law

Pascal's Law states that a change in pressure at any point in an enclosed incompressible fluid is transmitted undiminished to all points in the fluid, as well as to the enclosing walls. This principle is the operating principle behind hydraulic systems.

Darcy's Law

Darcy's Law provides an equation that describes the flow of fluid through a porous medium. The law is given by

Q = -\frac{kA}{\mu L}(P_2 - P_1)

where

Q

is the volumetric flow rate,

k

is the permeability of the medium,

A

is the area,

\mu

is dynamic viscosity,

L

is the length, and

P_1

and

P_2

are the pressures at two points.

Viscosity

Viscosity, symbolized by $\mu$ , quantifies a fluid's resistance to shear or flow. It is used in calculating Reynolds number and affects the flow characteristics in fluid dynamics.

Archimedes' Principle

Archimedes' Principle states that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid displaced by the body. This principle is foundational for the calculation of buoyancy and explains why ships float.

Specific Gravity

Specific Gravity, denoted as SG, is a dimensionless quantity that describes the ratio of the density of a fluid to the density of a reference substance (usually water). It is given by

SG = \frac{\rho_{fluid}}{\rho_{reference}}

and is commonly used for fluid identification and buoyancy calculations.

Capillary Action

Capillary Action is a phenomenon where liquid flows through a porous medium or narrow spaces against gravity, due to adhesive and cohesive forces, which can be represented mathematically by the Jurin's law. It's prevalent in biological systems and thin tube measurements.

Mach Number

Mach Number, $M$ , is a dimensionless quantity representing the ratio of the speed of a fluid to the speed of sound in that fluid.

M = \frac{v}{c}

where

v

is the fluid velocity and

c

is the speed of sound. It is crucial for designs involving supersonic and hypersonic flows, such as in aerodynamics.

Surface Tension

Surface Tension is a force per unit length, or energy per unit area, that acts at the interface between two fluids or a fluid and a solid. It can be measured and is significant in the study of capillarity and droplet formation.

Young-Laplace Equation

The Young-Laplace Equation describes the pressure difference across the interface of two fluids due to surface tension, and is given by

\Delta P = \gamma (\frac{1}{r_1} + \frac{1}{r_2})

where

\gamma

is the surface tension and

r_1

and

r_2

are the principal radii of curvature of the interface. This equation is applied in phenomena like droplet formation.

Laplace Equation

The Laplace Equation,

\nabla^2 \Phi = 0

, is a second-order partial differential equation describing the behavior of electric, gravitational, and fluid potential fields in a region where there are no sources or sinks. It's essential in potential flow theory and electrostatics.

Stokes' Law

Stokes' Law is used to calculate the drag force acting on spherical objects in a viscous fluid, which is given by

F_d = 6\pi \mu r v

where

F_d

is the drag force,

\mu

is the fluid viscosity,

r

is the radius of the sphere, and

v

is the velocity of the sphere in the fluid. This law is useful in determining terminal velocities of particles.

Buoyancy

Buoyancy is a force exerted by a fluid to support the weight of partially or fully submerged objects, governed by Archimedes' Principle. It is dependent on the density of the fluid and the volume of the displaced fluid.

Diffusion Coefficient

The Diffusion Coefficient, often denoted as $D$ , quantifies the rate at which particles diffuse in a medium. It is a key parameter in Fick's laws of diffusion and is important for manipulating mass transfer processes.

Fick's Laws of Diffusion

Fick's First Law relates the diffusive flux to the concentration gradient of the particle being diffused, and is given by

J = -D \nabla c

where

J

is the diffusion flux,

D

is the diffusion coefficient, and

c

is the particle concentration. Fick's Second Law predicts how diffusion causes the concentration to change with time.

Friction Factor

The Friction Factor in fluid dynamics is a dimensionless quantity used in the Darcy-Weisbach equation to estimate the pressure drop or head loss due to friction along a pipe or duct. In laminar flow, it is inversely proportional to Reynolds number.

Cavitation

Cavitation occurs when local fluid pressure drops below vapor pressure, leading to the formation of vapor bubbles within the liquid, which collapse as they move to higher pressure regions. This phenomenon can cause damage to equipment and is a concern in pump and turbine operation.

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