Fluid Mechanics/Formulas

This section serves as a general review section.

Table of Useful Formulas

Name	Equation	Notes	subject
Acceleration of a fluid particle	$\overrightarrow{a}=\frac{D\overrightarrow{V}}{Dt}=\frac{\partial \overrightarrow{V}}{\partial t}+u\frac{\partial \overrightarrow{V}}{\partial x}+v\frac{\partial \overrightarrow{V}}{\partial y}+w\frac{\partial \overrightarrow{V}}{\partial z}$		Fluid Mechanics/Kinematics
Ideal Gas Law	$\frac{p}{\rho }=RT$ , $R=\frac{\overline{R}}{M}$ , $\overline{R}=8.314$ kJ/(kmol*K)		Fluid Mechanics/Compressible Flow
Buoyancy force	$F_B=\gamma V$	V=Volume	Fluid Statics
Pressure variation in motionless incompressible fluid	$p_1=\gamma h+p_2$		Fluid Statics
Hydrostatic Force on plane surface	$F_R=\gamma h_{c}A$	$h_c$ =vertical dist centroid of area	Fluid Statics
Hydrostatic Force on curved surface	$y_R=\frac{I_{s}c}{y_{c}A}+y_c$ $x_R=\frac{I_{s}c}{x_{c}A}+x_c$		Fluid Statics
Navier-Stokes Vector form	$\rho \frac{D\overrightarrow{V}}{Dt}=-\overrightarrow{\mathrm{\nabla }}p+\rho \overrightarrow{g}+\mu {\mathrm{\nabla }}^2\overrightarrow{V}$		Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in x	$\rho (\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z})=-\frac{\partial p}{\partial x}+\rho g_x+\mu (\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2})$		Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in y	$\rho (\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z})=-\frac{\partial p}{\partial y}+\rho g_y+\mu (\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}v}{\partial z^2})$		Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in z	$\rho (\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z})=-\frac{\partial p}{\partial z}+\rho g_z+\mu (\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2})$		Fluid Mechanics/Differential Analysis of Fluid Flow
Shear Stress	$\tau =\mu \frac{du}{dy}$	$\frac{N}{m^2}$	Fluid Mechanics/Analysis Methods
Stream Function	$u=\frac{\partial \psi }{\partial y}$ and $v=-\frac{\partial \psi }{\partial x}$		Kinematics
Conservation of Mass, Steady in compressible	$\mathrm{\nabla }\overrightarrow{u}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$		Fluid Mechanics/Differential Analysis of Fluid Flow
Fluid Rotation	$\underset{{\omega }_x}{\underbrace{\frac{1}{2}(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z})}}+\underset{{\omega }_y}{\underbrace{\frac{1}{2}(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x})}}+\underset{{\omega }_z}{\underbrace{\frac{1}{2}(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})}}=\omega$	=0 if irrotational	Fluid Mechanics/Differential Analysis of Fluid Flow
Streamline Flow	$Q={\psi }_B-{\psi }_A$		Fluid Mechanics/Differential Analysis of Fluid Flow
Streamline	$\frac{u}{v}=\frac{dx}{dy}$		Fluid Mechanics/Differential Analysis of Fluid Flow
Streakline	$\frac{dx}{dt}=u,\frac{dy}{dt}=v,\frac{dz}{dt}=w$		Fluid Mechanics/Differential Analysis of Fluid Flow
Volumetric Dilation	$\overrightarrow{\mathrm{\nabla }}\dot{\overrightarrow{V}}$	0 for incompressible	Fluid Mechanics/Differential Analysis of Fluid Flow
Vorticity	$\overrightarrow{\zeta }=2\overrightarrow{\omega }=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{V}$		Fluid Mechanics/Differential Analysis of Fluid Flow
Specific Weight	$\gamma =\rho g$	$\frac{kg}{m^2{s}^2}$	Fluid Mechanics/Analysis Methods
Surface Tension	$\delta p=\frac{2\sigma }{R}$	of droplet	Fluid Mechanics/Analysis Methods
Capillary Rise in Tube	$h=\frac{2\sigma \cos \theta }{\gamma R}$		Fluid Mechanics/Analysis Methods
Torque	$dT=r\tau dA$	Nm	Other
Streamline Coordinates	$\overrightarrow{V}=V\hat{S}$	V always tan to $\hat{S}$	Fluid Mechanics/Analysis Methods
Control volume 1st law of thermodynamics	$\frac{\partial }{\partial t}{\int }_{cv}e\rho dVol+{\int }_{cs}e\rho 𝐕\cdot \widehat{𝐧}dA=(\dot{Q_{netin}}+\dot{W_{netin}}{)}_{cv}$ || || Fluid Mechanics/Control Volume Analysis

Common Symbols, Terms and meanings

Symbol	Meaning	Units (SI)	Notes	Subject
$\tau$ (tau)	Shear Stress	$\frac{N}{m^2}$	$=\mu \frac{du}{dy}$	Fluid Mechanics/Analysis Methods
$\nu$ (nu)	Kinematic Viscosity	$\frac{m^2}{s}$	$\frac{\mu }{\rho }$
$\gamma$ (gamma)	Specific Weight	$\frac{kg}{m^2{s}^2}$	$=\rho g{H}_{2}O{4}^{o}9.807kN/m^3$	Fluid Mechanics/Analysis Methods
Lagrangian	With particle			Fluid Mechanics/Kinematics
Eulerian	Field perspective			Fluid Mechanics/Kinematics
Streakline	continually released markers			Fluid Mechanics/Kinematics
Pathline	path of one particle			Fluid Mechanics/Kinematics
Stream Lines	tangent to velocity		$\psi$ =constant	Fluid Mechanics/Kinematics
$\psi$ (psi)	Stream Function	1		Fluid Mechanics/Kinematics
$\mu$	viscosity	N/m^2s		Fluid Mechanics/Kinematics
e	total stores energy per unit mass for each particle in the system		Fluid Mechanics/Control Volume Analysis
$\check{u}$	internal energy per unit mass		Fluid Mechanics/Control Volume Analysis
$\dot{Q_{netin}}$	rate of heat transfer		Fluid Mechanics/Control Volume Analysis
$\dot{W_{netin}}$	rate of work transfer		Fluid Mechanics/Control Volume Analysis

Common Physical Properties

gamma water	62.4 lb/ft^3