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