Last modified on 22 October 2013, at 22:15

Fluid Mechanics/Formulas

This section serves as a general review section.

Table of Useful FormulasEdit

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 $y_R {{=}} {I_sc \over y_c A } + y_c$
$x_R {{=}} {I_sc \over x_c A } + x_c$
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 ${\partial u \over \partial x }+{\partial v \over \partial y} + {\partial w \over \partial z} =0$ 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
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

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

Common Symbols, Terms and meaningsEdit

Symbol Meaning Units (SI) Notes Subject
$Stress$ (tau) Shear Stress $\frac{N}{m^2}$ $= \mu {du \over dy}$ Fluid Mechanics/Analysis Methods
$Kinematic Viscosity$ (nu) Kinematic Viscosity ${m^2 \over s}$ $\frac{\mu}{\rho}$
$Specific Weight$ (gamma) Specific Weight $\frac{kg}{m^2 s^2}$ ${{=}} \rho g H_2O 4^o 9.807 kN/m^3$ 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 $Stream Function$ =constant Fluid Mechanics/Kinematics
$Stream Function$ (psi) Stream Function 1 [Fluid Mechanics/Kinematics]]
$viscosity$ 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
$\dot{Q_{net in}}$ rate of heat transfer Fluid Mechanics/Control Volume Analysis
$\dot{W_{net in}}$ rate of work transfer Fluid Mechanics/Control Volume Analysis

Common Physical PropertiesEdit

gamma water 62.4 lb/ft^3