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Fluid Dynamics

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Bernoulli's Equation

Describes the conservation of energy in a flowing fluid, represented as

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

where

P

is the fluid pressure,

\rho

the density,

v

the flow velocity, and

h

the elevation.

Navier-Stokes Equations

A set of nonlinear partial differential equations that describe the motion of fluid substances, represented as

\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla P + \mu \nabla^2 \mathbf{v} + \mathbf{f}

where

\mathbf{v}

is the fluid velocity,

P

the pressure,

\mu

the dynamic viscosity, and

\mathbf{f}

represents body forces.

Reynolds Number

Dimensionless quantity used to predict flow patterns in different fluid flow situations, represented as

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

where

v

is the flow velocity,

L

a characteristic length,

\rho

the density, and

\mu

the dynamic viscosity of the fluid.

Continuity Equation

Expresses the conservation of mass in fluid dynamics, often simplified for incompressible flow as

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

or for steady, incompressible flow as

A_1v_1 = A_2v_2

where

A

is the cross-sectional area and

v

is the flow velocity.

Laminar Flow

Type of fluid motion in which all the particles in the fluid move along smooth, regular paths in layers or lamina, with minimal mixing and disruption between the layers.

Turbulent Flow

Type of fluid motion characterized by chaotic changes in pressure and flow velocity. It is associated with high momentum convection and eddy formation.

Boundary Layer

The layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity are significant, generating a velocity gradient perpendicular to the flow direction.

Darcy-Weisbach Equation

An empirical equation that describes the pressure losses due to friction along a given length of pipe with a constant diameter, represented as

\Delta P = f \frac{L}{D} \frac{1}{2} \rho v^2

where

\Delta P

is the pressure loss,

f

the Darcy friction factor,

L

the pipe length,

D

the pipe diameter,

v

the flow velocity, and

\rho

the fluid density.

Mach Number

A dimensionless quantity representing the ratio of the speed of an object moving through a fluid to the speed of sound in that fluid, defined as

Ma = \frac{v}{c}

where

v

is object's velocity and

c

is the speed of sound in the fluid.

Stream Function

A mathematical function used to simplify the description of a two-dimensional, incompressible flow, where the function's value remains constant along streamlines and its partial derivatives are related to the velocity components.

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