How to draw arrow flow lines around 2D circle?

Question

I am drawing a tikz diagram of a circle with equally spaced field flow lines flowing around it, starting from below, up, around, then up. I was not able to find a post that was helpful for this one. I've tried using \draw..controls which allows curved paths using control points, for Bezier curves. My difficulty is that it only allows two control points, so flowing around something like a cirlce seems impossible. Too much distance between points and it goes thru it. Too little, then it can't go all the way around. I'd love to figure out a way to do this. Any ideas to draw equally spaced flow lines around a circle? Thanks. An minimum working example code is below, although I'm not sure where this should go. Thx again.

\begin{tikzpicture}
\filldraw [black,fill=gray] (3,0) circle (1);
\draw[->] (2.5, -2.0) .. controls (2.5,-1.2) and (2.25,-1.0) .. (0.8,0);
\draw[->] (2.3, -2.0) .. controls (2.3,-1.2) and (2.05,-1.0) .. (0.6,0);
\end{tikzpicture}  

Per the suggestion, I've added an image...a pretty bad one, but hopefully it still helps to show what I would like to do. Thx again!

Answer 1

Here is a proposal that looks somewhat similar to what you draw by hand.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{intersections}
\begin{document}

\begin{tikzpicture}
\filldraw [black,fill=gray] (0,0) circle (1);
\path (0,-3) coordinate (low) (0,3) coordinate (high);
\begin{scope}[overlay]
\foreach \X in {250,290}
{\path[name path global=\X-ray] (0,0) -- (\X:3);}
\end{scope}
\foreach \Z [count=\Y]in {0.3,0.6,0.9}
{\path[name path=circle-\Y] (0,0) circle(1+\Z);
\foreach \X in {250,290}
{\path[name intersections={of=\X-ray and circle-\Y,by=P-\Y-\X}];}
\draw[-latex] ([xshift=2mm]P-\Y-250 |-low) -- 
([xshift=2mm,yshift=-2mm]P-\Y-250) to[out=90,in=-20] (P-\Y-250)
arc(-110:-250:1+\Z) to[out=20,in=-90] ++(0.2,0.2) -- ([xshift=2mm]P-\Y-250 |-high);
\draw[-latex] ([xshift=-2mm]P-\Y-290 |-low) -- 
([xshift=-2mm,yshift=-2mm]P-\Y-290) to[out=90,in=-160] (P-\Y-290)
arc(-70:70:1+\Z) to[out=160,in=-90] ++(-0.2,0.2) -- ([xshift=-2mm]P-\Y-290 |-high);
}
\end{tikzpicture}  

\end{document}

Answer 2

It is a very late answer, but it is worth to show the "physical" solution, as the nice picture provided by @marmot is poorly related to the real problem.

In the same spirit as my answer to this question: field-lines-of-a-coplanar-waveguide, but with much less mathematics: Assuming an Eulerian fluid (incompressible and no viscosity) the lines of flow are the gradient lines of a "velocity potential" V(x,y) which is an harmonic function (Laplacian=0) outside the disk. Using the boundary conditions at both infinity and disk peripheral, one easily get:

where r and \theta are the polar coordinates, assuming $R=1$ for simplicity. This function is the imaginary part of the holomorphic function $z\mapsto z-1/z$.

It is then simple to get the flow lines by calling gnuplot to draw the contour lines of the real part of this function.

By this way we get the following MWE:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\begin{document}

\def\gnuplotscript{
   ii={0,1};
   ZZ(x,y)=(x+ii*y)-1/(x+ii*y);
   set sample 100;
   unset surface;
   unset key;
   set contour;
   set view map;
   set cntrparam cubicspline;
   set size square;
   set cntrparam levels 12;
   set isosample 20;
}

\begin{tikzpicture}
\begin{axis}[no markers, axis on top, 
    tick label style={font=\small},
    xlabel={$x$}, ylabel={$y$},
    xmin=-2.5, xmax=2.5, ymin=-2.5, ymax=2.5,
    width=12cm, height=12cm
]    
\addplot[contour prepared,raw gnuplot, thick, 
    contour/draw color=red,contour/labels=false] 
    gnuplot {%
    \gnuplotscript
    splot[-2.5:2.5][-2.5:2.5] (x**2+y**2)<1?0:abs(real(ZZ(x,y)));                  
    };
\draw[color=red,thick] (axis cs: 0,2.5) -- (axis cs: 0,1) (axis cs: 0,-2.5) -- (axis cs: 0,-1);
\draw [ultra thick, draw=black]  (axis cs: 0,0) circle(1);
\end{axis}
\end{tikzpicture}

\end{document}

and the result:

Answer 3

Run with xelatex (needs some time)

\documentclass[pstricks]{standalone}
\usepackage{pst-func}

\begin{document}
\begin{pspicture*}(-5,-2.2)(5.5,3.5)
 \pscircle(0,0){1}%
 \psaxes{->}(0,0)(-5,-2)(5.2,3)%
 \multido{\rA=0.01+0.2}{5}{%
    \psplotImp[linewidth=1pt,linecolor=blue,polarplot,
    stepFactor=0.2](-6,-6)(5,2.4){%
        r dup mul 1.0 r div sub phi sin dup mul mul \rA\space sub }}%
 \uput*[45](0,2){$f(r,\phi)=\left(r^2-\frac{1}{r}\right)\cdot\sin^2\phi=0$}
\end{pspicture*}
\end{document}