Fluid Mechanics/Formulas

This section serves as a general review section.

Table of Useful Formulas

Name	Equation	Notes	subject
Acceleration of a fluid particle	${\vec {a}}={D{\vec {V}} \over Dt}={\partial {\vec {V}} \over \partial t}+u{\partial {\vec {V}} \over \partial x}+v{\partial {\vec {V}} \over \partial y}+w{\partial {\vec {V}} \over \partial z}$		Fluid Mechanics/Kinematics
Ideal Gas Law	${p \over \rho }=RT$ , $R={{\bar {R}} \over M}$ , ${\bar {R}}{=}8.314$ 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	$y_{R}{=}{I_{s}c \over y_{c}A}+y_{c}$ $x_{R}{=}{I_{s}c \over x_{c}A}+x_{c}$		Fluid Statics
Navier-Stokes Vector form	$\rho {D{\vec {V}} \over Dt}=-{\vec {\nabla }}p+\rho {\vec {g}}+\mu \nabla ^{2}{\vec {V}}$		Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in x	$\rho ({\partial u \over \partial t}+u{\partial u \over \partial x}+v{\partial u \over \partial y}+w{\partial u \over \partial z})=-{\partial p \over \partial x}+\rho g_{x}+\mu ({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}})$		Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in y	$\rho ({\partial v \over \partial t}+u{\partial v \over \partial x}+v{\partial v \over \partial y}+w{\partial v \over \partial z})=-{\partial p \over \partial y}+\rho g_{y}+\mu ({\partial ^{2}v \over \partial x^{2}}+{\partial ^{2}v \over \partial y^{2}}+{\partial ^{2}v \over \partial z^{2}})$		Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in z	$\rho ({\partial w \over \partial t}+u{\partial w \over \partial x}+v{\partial w \over \partial y}+w{\partial w \over \partial z})=-{\partial p \over \partial z}+\rho g_{z}+\mu ({\partial ^{2}w \over \partial x^{2}}+{\partial ^{2}w \over \partial y^{2}}+{\partial ^{2}w \over \partial z^{2}})$		Fluid Mechanics/Differential Analysis of Fluid Flow
Shear Stress	$\tau =\mu {du \over dy}$	${\frac {N}{m^{2}}}$	Fluid Mechanics/Analysis Methods
Stream Function	$u={\partial \psi \over \partial y}$ and $v=-{\partial \psi \over \partial x}$		Kinematics
Conservation of Mass, Steady in compressible	$\nabla {\vec {u}}={\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}=0$		Fluid Mechanics/Differential Analysis of Fluid Flow
Fluid Rotation	$\underbrace {{\frac {1}{2}}({\partial w \over \partial y}-{\partial v \over \partial z})} _{\omega _{x}}+\underbrace {{\frac {1}{2}}({\partial u \over \partial z}-{\partial w \over \partial x})} _{\omega _{y}}+\underbrace {{\frac {1}{2}}({\partial v \over \partial x}-{\partial u \over \partial y})} _{\omega _{z}}=\omega$	=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	${\frac {u}{v}}={dx \over dy}$		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	${{\vec {\nabla }}{\dot {\vec {V}}}}$	0 for incompressible	Fluid Mechanics/Differential Analysis of Fluid Flow
Vorticity	${\vec {\zeta }}=2{\vec {\omega }}={{\vec {\nabla }}\times {\vec {V}}}$		Fluid Mechanics/Differential Analysis of Fluid Flow
Specific Weight	$\gamma =\rho g$	${\frac {kg}{m^{2}s^{2}}}$	Fluid Mechanics/Analysis Methods
Surface Tension	$\delta p{=}{2\sigma \over R}$	of droplet	Fluid Mechanics/Analysis Methods
Capillary Rise in Tube	$h={2\sigma \cos {\theta } \over \gamma R}$		Fluid Mechanics/Analysis Methods
Torque	$dT=r\tau dA$	Nm	Other
Streamline Coordinates	${\vec {V}}{=}V{\hat {S}}$	V always tan to ${\hat {S}}$	Fluid Mechanics/Analysis Methods
Control volume 1st law of thermodynamics	${\partial \over \partial t}\int _{cv}e\rho dVol+\int _{cs}e\rho \mathbf {V} \cdot \mathbf {\hat {n}} dA{=}({\dot {Q_{netin}}}+{\dot {W_{netin}}})_{cv}$ || || Fluid Mechanics/Control Volume Analysis

Common Symbols, Terms and meanings

Symbol	Meaning	Units (SI)	Notes	Subject
$\tau$ (tau)	Shear Stress	${\frac {N}{m^{2}}}$	$=\mu {du \over dy}$	Fluid Mechanics/Analysis Methods
$\nu$ (nu)	Kinematic Viscosity	${m^{2} \over s}$	${\frac {\mu }{\rho }}$
$\gamma$ (gamma)	Specific Weight	${\frac {kg}{m^{2}s^{2}}}$	${=}\rho gH_{2}O4^{o}9.807kN/m^{3}$	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