Fluid Mechanics Formulas

Complete collection of fluid mechanics formulas with detailed explanations.

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Fluid Properties

Density

\rho = \frac{m}{V}

Density equals mass divided by volume.

Notation:

ρ:Density

m:Mass

V:Volume

Units:

ρ:kg/m³

m:kg

V:m³

Applications:

•Fluid characterization
•Buoyancy calculations
•Flow measurements

Limitations:

Homogeneous fluid

Pressure

P = \frac{F}{A}

Pressure equals force divided by area.

Notation:

P:Pressure

F:Force

A:Area

Units:

P:Pa (N/m²)

F:N

A:m²

Applications:

•Hydraulic systems
•Atmospheric pressure
•Fluid flow

Limitations:

Normal force only

Viscosity

\tau = \mu\frac{du}{dy}

Shear stress equals dynamic viscosity times velocity gradient.

Notation:

τ:Shear stress

μ:Dynamic viscosity

du/dy:Velocity gradient

Units:

τ:Pa

μ:Pa·s

du/dy:s⁻¹

Applications:

•Fluid flow resistance
•Lubrication
•Drag calculations

Limitations:

Newtonian fluids

Kinematic Viscosity

\nu = \frac{\mu}{\rho}

Kinematic viscosity equals dynamic viscosity divided by density.

Notation:

ν:Kinematic viscosity

μ:Dynamic viscosity

ρ:Density

Units:

ν:m²/s

μ:Pa·s

ρ:kg/m³

Applications:

•Reynolds number
•Flow similarity
•Fluid dynamics

Limitations:

Incompressible fluid

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Hydrostatics

Hydrostatic Pressure

P = P_0 + \rho gh

Pressure at depth equals atmospheric pressure plus density times gravitational acceleration times depth.

Notation:

P:Pressure at depth

P₀:Atmospheric pressure

ρ:Density

g:Gravitational acceleration

h:Depth

Units:

P, P₀:Pa

ρ:kg/m³

g:9.81 m/s²

h:m

Applications:

•Underwater pressure
•Dam design
•Submarine operations

Limitations:

Incompressible fluid

Buoyant Force

F_b = \rho_f V_f g

Buoyant force equals fluid density times displaced volume times gravitational acceleration.

Notation:

F_b:Buoyant force

ρ_f:Fluid density

V_f:Displaced volume

g:Gravitational acceleration

Units:

F_b:N

ρ_f:kg/m³

V_f:m³

g:9.81 m/s²

Applications:

•Floating objects
•Ship design
•Submarine buoyancy

Limitations:

Static equilibrium

Archimedes' Principle

F_b = W_{displaced}

Buoyant force equals weight of displaced fluid.

Notation:

F_b:Buoyant force

W_displaced:Weight of displaced fluid

Units:

F_b, W_displaced:N

Applications:

•Floating bodies
•Density measurements
•Buoyancy calculations

Limitations:

Static conditions

Pascal's Principle

\frac{F_1}{A_1} = \frac{F_2}{A_2}

Pressure is transmitted equally in all directions in a confined fluid.

Notation:

F₁, F₂:Forces

A₁, A₂:Areas

Units:

F₁, F₂:N

A₁, A₂:m²

Applications:

•Hydraulic lifts
•Brake systems
•Pressure transmission

Limitations:

Incompressible fluid

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Fluid Dynamics

Continuity Equation

A_1v_1 = A_2v_2

Volume flow rate is constant in a pipe with varying cross-sectional area.

Notation:

A₁, A₂:Cross-sectional areas

v₁, v₂:Velocities

Units:

A₁, A₂:m²

v₁, v₂:m/s

Applications:

•Pipe flow
•Nozzle design
•Flow rate calculations

Limitations:

Incompressible flow

Bernoulli's Equation

P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2

Total energy per unit volume is constant along a streamline.

Notation:

P₁, P₂:Pressures

ρ:Density

v₁, v₂:Velocities

g:Gravitational acceleration

h₁, h₂:Heights

Units:

P₁, P₂:Pa

ρ:kg/m³

v₁, v₂:m/s

g:9.81 m/s²

h₁, h₂:m

Applications:

•Aerodynamics
•Venturi meters
•Flow measurement

Limitations:

Ideal fluid, steady flow

Reynolds Number

Re = \frac{\rho vD}{\mu}

Reynolds number equals density times velocity times diameter divided by dynamic viscosity.

Notation:

Re:Reynolds number

ρ:Density

v:Velocity

D:Characteristic length

μ:Dynamic viscosity

Units:

Re:dimensionless

ρ:kg/m³

v:m/s

D:m

μ:Pa·s

Applications:

•Flow regime determination
•Turbulence prediction
•Similarity analysis

Limitations:

Characteristic length scale

Poiseuille's Law

Q = \frac{\pi r^4\Delta P}{8\mu L}

Volume flow rate equals π times radius to fourth power times pressure difference divided by 8 times viscosity times length.

Notation:

Q:Volume flow rate

r:Radius

ΔP:Pressure difference

μ:Dynamic viscosity

L:Length

Units:

Q:m³/s

r:m

ΔP:Pa

μ:Pa·s

L:m

Applications:

•Blood flow
•Pipe flow
•Capillary flow

Limitations:

Laminar flow, circular pipe

⚡

Advanced Fluid Mechanics

Drag Force

F_D = \frac{1}{2}C_D\rho Av^2

Drag force equals half drag coefficient times density times area times velocity squared.

Notation:

F_D:Drag force

C_D:Drag coefficient

ρ:Density

A:Cross-sectional area

v:Velocity

Units:

F_D:N

C_D:dimensionless

ρ:kg/m³

A:m²

v:m/s

Applications:

•Aerodynamics
•Vehicle design
•Wind resistance

Limitations:

Steady flow

Stokes' Law

F_D = 6\pi\mu rv

Drag force on a sphere equals 6π times viscosity times radius times velocity.

Notation:

F_D:Drag force

μ:Dynamic viscosity

r:Radius

v:Velocity

Units:

F_D:N

μ:Pa·s

r:m

v:m/s

Applications:

•Particle settling
•Sedimentation
•Microfluidics

Limitations:

Small Reynolds number

Surface Tension

\gamma = \frac{F}{L}

Surface tension equals force divided by length.

Notation:

γ:Surface tension

F:Force

L:Length

Units:

γ:N/m

F:N

L:m

Applications:

•Capillary action
•Bubble formation
•Wetting phenomena

Limitations:

Liquid-gas interface

Capillary Rise

h = \frac{2\gamma\cos\theta}{\rho gr}

Capillary rise equals twice surface tension times cosine of contact angle divided by density times gravitational acceleration times radius.

Notation:

h:Capillary rise

γ:Surface tension

θ:Contact angle

ρ:Density

g:Gravitational acceleration

r:Radius

Units:

h:m

γ:N/m

θ:radians

ρ:kg/m³

g:9.81 m/s²

r:m

Applications:

•Soil moisture
•Paper chromatography
•Microfluidics

Limitations:

Small diameter tubes

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Bernoulli Principle Hydrostatic Pressure Continuity Equation Hydraulic Press Venturi Meter Pitot Tube Stokes' Law Reynolds Number Buoyancy