# Fluid Mechanics/Formulas

This section serves as a general review section.

## Table of Useful Formulas

Name Equation Notes subject
Acceleration of a fluid particle ${\displaystyle {\vec {a}}={D{\vec {V}} \over Dt}={\partial {\vec {V}} \over \partial t}+u{\partial {\vec {V}} \over \partial x}+v{\partial {\vec {V}} \over \partial y}+w{\partial {\vec {V}} \over \partial z}}$  Fluid Mechanics/Kinematics
Ideal Gas Law ${\displaystyle {p \over \rho }=RT}$  , ${\displaystyle R={{\bar {R}} \over M}}$  , ${\displaystyle {\bar {R}}{=}8.314}$  kJ/(kmol*K) Fluid Mechanics/Compressible Flow
Buoyancy force ${\displaystyle F_{B}=\gamma V}$  V=Volume Fluid Statics
Pressure variation in motionless incompressible fluid ${\displaystyle p_{1}{=}\gamma h+p_{2}}$  Fluid Statics
Hydrostatic Force on plane surface ${\displaystyle F_{R}{=}\gamma h_{c}A}$  ${\displaystyle h_{c}}$  =vertical dist centroid of area Fluid Statics
Hydrostatic Force on curved surface ${\displaystyle y_{R}{=}{I_{s}c \over y_{c}A}+y_{c}}$
${\displaystyle x_{R}{=}{I_{s}c \over x_{c}A}+x_{c}}$
Fluid Statics
Navier-Stokes Vector form ${\displaystyle \rho {D{\vec {V}} \over Dt}=-{\vec {\nabla }}p+\rho {\vec {g}}+\mu \nabla ^{2}{\vec {V}}}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in x ${\displaystyle \rho ({\partial u \over \partial t}+u{\partial u \over \partial x}+v{\partial u \over \partial y}+w{\partial u \over \partial z})=-{\partial p \over \partial x}+\rho g_{x}+\mu ({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}})}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in y ${\displaystyle \rho ({\partial v \over \partial t}+u{\partial v \over \partial x}+v{\partial v \over \partial y}+w{\partial v \over \partial z})=-{\partial p \over \partial y}+\rho g_{y}+\mu ({\partial ^{2}v \over \partial x^{2}}+{\partial ^{2}v \over \partial y^{2}}+{\partial ^{2}v \over \partial z^{2}})}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Navier-Stokes in z ${\displaystyle \rho ({\partial w \over \partial t}+u{\partial w \over \partial x}+v{\partial w \over \partial y}+w{\partial w \over \partial z})=-{\partial p \over \partial z}+\rho g_{z}+\mu ({\partial ^{2}w \over \partial x^{2}}+{\partial ^{2}w \over \partial y^{2}}+{\partial ^{2}w \over \partial z^{2}})}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Shear Stress ${\displaystyle \tau =\mu {du \over dy}}$  ${\displaystyle {\frac {N}{m^{2}}}}$  Fluid Mechanics/Analysis Methods
Stream Function ${\displaystyle u={\partial \psi \over \partial y}}$  and ${\displaystyle v=-{\partial \psi \over \partial x}}$  Kinematics
Conservation of Mass, Steady in compressible ${\displaystyle \nabla {\vec {u}}={\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}=0}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Fluid Rotation ${\displaystyle \underbrace {{\frac {1}{2}}({\partial w \over \partial y}-{\partial v \over \partial z})} _{\omega _{x}}+\underbrace {{\frac {1}{2}}({\partial u \over \partial z}-{\partial w \over \partial x})} _{\omega _{y}}+\underbrace {{\frac {1}{2}}({\partial v \over \partial x}-{\partial u \over \partial y})} _{\omega _{z}}=\omega }$  =0 if irrotational Fluid Mechanics/Differential Analysis of Fluid Flow
Streamline Flow ${\displaystyle Q=\psi _{B}-\psi _{A}}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Streamline ${\displaystyle {\frac {u}{v}}={dx \over dy}}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Streakline ${\displaystyle {\frac {dx}{dt}}=u,{\frac {dy}{dt}}=v,{\frac {dz}{dt}}=w}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Volumetric Dilation ${\displaystyle {{\vec {\nabla }}{\dot {\vec {V}}}}}$  0 for incompressible Fluid Mechanics/Differential Analysis of Fluid Flow
Vorticity ${\displaystyle {\vec {\zeta }}=2{\vec {\omega }}={{\vec {\nabla }}\times {\vec {V}}}}$  Fluid Mechanics/Differential Analysis of Fluid Flow
Specific Weight ${\displaystyle \gamma =\rho g}$  ${\displaystyle {\frac {kg}{m^{2}s^{2}}}}$  Fluid Mechanics/Analysis Methods
Surface Tension ${\displaystyle \delta p{=}{2\sigma \over R}}$  of droplet Fluid Mechanics/Analysis Methods
Capillary Rise in Tube ${\displaystyle h={2\sigma \cos {\theta } \over \gamma R}}$  Fluid Mechanics/Analysis Methods
Torque ${\displaystyle dT=r\tau dA}$  Nm Other
Streamline Coordinates ${\displaystyle {\vec {V}}{=}V{\hat {S}}}$  V always tan to ${\displaystyle {\hat {S}}}$  Fluid Mechanics/Analysis Methods
Control volume 1st law of thermodynamics

${\displaystyle {\partial \over \partial t}\int _{cv}e\rho dVol+\int _{cs}e\rho \mathbf {V} \cdot \mathbf {\hat {n}} dA{=}({\dot {Q_{netin}}}+{\dot {W_{netin}}})_{cv}}$  || || Fluid Mechanics/Control Volume Analysis

## Common Symbols, Terms and meanings

Symbol Meaning Units (SI) Notes Subject
${\displaystyle \tau }$  (tau) Shear Stress ${\displaystyle {\frac {N}{m^{2}}}}$  ${\displaystyle =\mu {du \over dy}}$  Fluid Mechanics/Analysis Methods
${\displaystyle \nu }$  (nu) Kinematic Viscosity ${\displaystyle {m^{2} \over s}}$  ${\displaystyle {\frac {\mu }{\rho }}}$
${\displaystyle \gamma }$  (gamma) Specific Weight ${\displaystyle {\frac {kg}{m^{2}s^{2}}}}$  ${\displaystyle {=}\rho gH_{2}O4^{o}9.807kN/m^{3}}$  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 ${\displaystyle \psi }$  =constant Fluid Mechanics/Kinematics
${\displaystyle \psi }$  (psi) Stream Function 1 Fluid Mechanics/Kinematics
${\displaystyle \mu }$  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
${\displaystyle {\check {u}}}$  internal energy per unit mass Fluid Mechanics/Control Volume Analysis
${\displaystyle {\dot {Q_{netin}}}}$  rate of heat transfer Fluid Mechanics/Control Volume Analysis
${\displaystyle {\dot {W_{netin}}}}$  rate of work transfer Fluid Mechanics/Control Volume Analysis

## Common Physical Properties

gamma water 62.4 lb/ft^3