# 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

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