I have a two-dimensional velocity field $$\mathbf{V}=\begin{pmatrix} u \\ v \end{pmatrix}=\begin{pmatrix} 0.5+0.8x \\ 1.5+2.5\sin(\omega t)-0.8y \end{pmatrix}$$ from Example 4-5 in "Fluid Mechanics: Fundamentals and Applications" 2nd Edition.
I have correctly derived the streamlines from $$\frac{dy}{dx}=\frac{v}{u}$$ at $t=2$ giving $$y=\frac{C}{0.8(0.5+0.8x)}+1.875$$ in agreement with the text.
Using a Runge-Kutta numerical integration method I approximated the pathlines from $0<t<2$ s using these equations. $$u=\frac{dx}{dt}=0.5+0.8x$$ $$v=\frac{dy}{dt}=1.5+2.5\sin(\omega t)-0.8y$$ Plotting the equations agrees with the figure in the text. My results are on the left
- red: starting from $(0.5,0.5)$
- blue: starting from $(0.5,2.5)$
- green: starting from $(0.5,4.5)$
and the light pink curves in the right figure are from the text.
These questions (here, and here) deal with methods to find the streaklines. This link, however, has been the most helpful in understanding how to derive the necessary equations. But my results do not agree with the text. The text defines the method for plotting the streaklines this way.
Finally, the streaklines are simulated by following the paths of many fluid tracer particles released at the given three locations at times between $t=0$ s and $t=2$ s, and connecting the locus of their positions at $t=2$ s.
I want to try and avoid solving the ODEs by hand since $y(t)$ will involve an integrating factor, but more importantly, to arrive at equations for $x=x(x',t,t')$ and $y=y(y',t,t')$ without going through lots of error-prone algebra.
From the third link above $(x',y')$ are the future positions of the particle at time $t'$ and $t$ is the current time.
To close out the question, I am looking for a way to use my approximate values for the pathlines to plot my streaklines from $0<t<2$ s.
EDIT:
It has just occurred to me that in the first of the previous links, an ODE that defines a streakline is mentioned. The ODE is from Wikipedia and is given here. $$\left\{\begin{matrix} \frac{d\vec{x}_{str}}{dt}=\vec{u}_P(\vec{x}_{str},t) \\ \vec{x}_{str}(t=\tau_P)=\vec{x}_{P0} \end{matrix}\right.$$ From Wikipedia...
$\vec{u}_P(\vec{x},t)$ is the velocity of a particle $P$ at location $\vec{x}$ and time $t$. The parameter $\tau_P$, parameterizes the streakline $\vec{x}_{str}(t,\tau_P)$ and $t_0\le\tau_P\le t$, where $t$ is a time of interest.
The second link from above describes this parameterization, but I am struggling to understand how to work it out in general, not just for this velocity field. But if I can make that work then the ODE should be able to be solved numerically with a similar Runge-Kutta method. However, this link suggests that there are errors in the Wikipedia article.
The parameterized ODEs from $0<\tau<t$ should be the following.$$\frac{dx_\tau}{d\theta}=0.5+0.8x_\tau(\theta)$$ $$\frac{dy_\tau}{d\theta}=1.5+2.5\sin\left(\omega(\tau+\theta)\right)-0.8y_\tau(\theta)$$
Getting some help setting up the numerical integration is where I'm stuck. I'm unsure how to deal with the $\tau$ and $\theta$ terms; i.e, which term is marching forward and what values to put in for $\tau$.