# 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 |