Streamlines

The direction of magnetic field lines are streamlines represented by the alignment of iron filings sprinkled on paper placed above a bar magnet

Potential-flow streamlines achieving the Kutta condition around a NACA airfoil with upper and lower streamtubes identified.

Streamlines are defined by^[4]

${\displaystyle {d{\vec {x}}_{S} \over ds}\times {\vec {u}}({\vec {x}}_{S})={\vec {0}},}$

where "${\displaystyle \times }$" denotes the vector cross product and ${\displaystyle {\vec {x}}_{S}(s)}$ is the parametric representation of just one streamline at one moment in time.

If the components of the velocity are written ${\displaystyle {\vec {u}}=(u,v,w),}$ and those of the streamline as ${\displaystyle {\vec {x}}_{S}=(x_{S},y_{S},z_{S}),}$ we deduce^[4]

${\displaystyle {dx_{S} \over u}={dy_{S} \over v}={dz_{S} \over w},}$

which shows that the curves are parallel to the velocity vector. Here ${\displaystyle s}$ is a variable which parametrizes the curve ${\displaystyle s\mapsto {\vec {x}}_{S}(s).}$ Streamlines are calculated instantaneously, meaning that at one instance of time they are calculated throughout the fluid from the instantaneous flow velocity field.

A streamtube consists of a bundle of streamlines, much like communication cable.

The equation of motion of a fluid on a streamline for a flow in a vertical plane is:^[5]

${\displaystyle {\frac {\partial c}{\partial t}}+c{\frac {\partial c}{\partial s}}=\nu {\frac {\partial ^{2}c}{\partial r^{2}}}-{\frac {1}{\rho }}{\frac {\partial p}{\partial s}}-g{\frac {\partial z}{\partial s}}}$

The flow velocity in the direction ${\displaystyle s}$ of the streamline is denoted by ${\displaystyle c}$. ${\displaystyle r}$ is the radius of curvature of the streamline. The density of the fluid is denoted by ${\displaystyle \rho }$ and the kinematic viscosity by ${\displaystyle \nu }$. ${\displaystyle {\frac {\partial p}{\partial s}}}$ is the pressure gradient and ${\displaystyle {\frac {\partial c}{\partial s}}}$ the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: ${\displaystyle {\frac {\partial c}{\partial t}}=0}$. ${\displaystyle g}$ denotes the gravitational acceleration.

Pathlines

A long-exposure photo of spark from a campfire shows the pathlines for the flow of hot air.

Pathlines are defined by

${\displaystyle {\begin{cases}\displaystyle {\frac {d{\vec {x}}_{P}}{dt}}(t)={\vec {u}}_{P}({\vec {x}}_{P}(t),t)\\[1.2ex]{\vec {x}}_{P}(t_{0})={\vec {x}}_{P0}\end{cases}}}$

The subscript ${\displaystyle P}$ indicates that we are following the motion of a fluid particle. Note that at point ${\displaystyle {\vec {x}}_{P}}$ the curve is parallel to the flow velocity vector ${\displaystyle {\vec {u}}}$, where the velocity vector is evaluated at the position of the particle ${\displaystyle {\vec {x}}_{P}}$ at that time ${\displaystyle t}$.

Streaklines

Example of a streakline used to visualize the flow around a car inside a wind tunnel.

Streaklines can be expressed as,

${\displaystyle {\begin{cases}\displaystyle {\frac {d{\vec {x}}_{str}}{dt}}={\vec {u}}_{P}({\vec {x}}_{str},t)\\[1.2ex]{\vec {x}}_{str}(t=\tau _{P})={\vec {x}}_{P0}\end{cases}}}$

where, ${\displaystyle {\vec {u}}_{P}({\vec {x}},t)}$ is the velocity of a particle ${\displaystyle P}$ at location ${\displaystyle {\vec {x}}}$ and time ${\displaystyle t}$. The parameter ${\displaystyle \tau _{P}}$, parametrizes the streakline ${\displaystyle {\vec {x}}_{str}(t,\tau _{P})}$ and ${\displaystyle t_{0}\leq \tau _{P}\leq t}$, where ${\displaystyle t}$ is a time of interest.

Notes

References

• Faber, T.E. (1995). Fluid Dynamics for Physicists. Cambridge University Press. ISBN 0-521-42969-2.