How to plot this the streamlines xy=C by latex?

Question

How to plot this the streamlines xy=C by latex? I attached my code but it does not generate the wanted figure.

this is my code

\documentclass[tikz,border=10pt,multi]{standalone}
\usetikzlibrary{shapes.symbols}

\makeatletter

\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\makeatother



    \begin{document}

    \begin{tikzpicture}
    \begin{axis}[
        axis lines = left,
        xlabel = $x$,
        ylabel = {$f(x)$},
    ]
    %Below the red is defined
    \addplot [
        domain=-2:2, 
       samples=20, 
        color=red,
    ]
    { 1/x};
    \addlegendentry{$xy=-1$}
    %Here  is defined
    \addplot [
        domain=-2:2, 
        samples=20, 
        color=green,
        ]
        {-3/x};
    \addlegendentry{$xy=3$}
    \addplot [
        domain=-2:2, 
        samples=20, 
        color=green,
        ]
        {5/x};
    \addlegendentry{$xy=-5$}

    \end{axis}
    \end{tikzpicture}

    \end{document}

Answer 1

Here I present a solution using non-linear spacing that also works for quite low values for C or "high" values for Max (see code). For example test with values 0.1, 0.2 and 0.3 respectively or a Max value of 10. They work perfectly fine using only 22 samples per (full) line.

For details on how the solution works, please have a look at the comments in the code.

% used PGFPlots v1.16
\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
    % load library to use a "cool" `cycle list'
    \usetikzlibrary{
        pgfplots.colorbrewer,
    }
    \pgfplotsset{
        % use this `compat' level or higher to use the advanced positioning
        % for the axis labels
        compat=1.3,
        /pgf/declare function={
            % declare a variable to store the min and max value for the function
            Max=5;
            % declare the function to use
            f(\C,\x) = \C/\x;
            %
            % -----------------------------------------------------------------
            % calculate the lower and upper boundaries (the domain values)
            lb(\C) = sqrt(\C);
            ub = Max;
            %
            %%% non-linear spacing: <https://stackoverflow.com/a/39140096/5776000>
            % "non-linearity factor"
            a = 1.0;
            % function to use for the non-linear spacing
            Y(\x) = exp(a*\x);
            % rescale to former limits
            X(\C,\x)  = (Y(\x) - Y(lb(\C)))/(Y(ub) - Y(lb(\C))) * (ub - lb(\C)) + lb(\C);
            % -----------------------------------------------------------------
        },
        % load a "cool" cycle list
        cycle list/Blues-5,
    }
\begin{document}
\begin{tikzpicture}[
    % define a style for the plot labels
    Label/.style={
        node font=\tiny,
        black,
        pos=#1,
    },
]
    \begin{axis}[
        % axis lines should be centered
        axis lines=center,
        % set axis labels
        xlabel={$x$},
        ylabel={$f(x)$},
        % set them right and above the axis lines, respectively
        xlabel style={
            at={(xticklabel* cs:1)},
            anchor=west,
        },
        ylabel style={
            at={(yticklabel* cs:1)},
            anchor=south,
        },
        % axis unit ratio should be the same for both axis lines
        axis equal image=true,
        % don't show any ticks (and tick labels)
        ticks=none,
        % the lines should be `smooth'
        smooth,
        % when using the non-linear spacing approach you don't need that much samples
        samples=11,
%        % ---------------------------------------------------------------------
%        % for debugging purposes only
%        % ---------------------------------------------------------------------
%        % add small marks to the lines
%        every axis plot post/.append style={
%            mark=*,
%            mark size=0.3pt,
%        },
%        % ---------------------------------------------------------------------
    ]

        % first draw the lines for $C = 0$
        \pgfplotsinvokeforeach {0} {
            % the horizontal line
            % used `forget plot' so the `cycle list index' isn't increased
            \addplot+ [domain=-Max:Max,forget plot,>->] {#1}
                node [Label=0.4,above] {$C = #1$}
                node [Label=0.6,above] {$C = #1$}
            ;
            % the vertical line
            \addplot+ [domain=-Max:Max,>->] (#1,x)
                node [Label=0.25,left] {$#1$}
                node [Label=0.75,left] {$#1$}
            ;
        }

            % define a length for the shift of the plot labels
            % (using "above left" and similar causes the labels to be
            %  "too far away" from the plot they belong to and are almost
            %  *in* the next plot)
            \pgfmathsetlengthmacro{\Shift}{3pt}

            % define a factor to enlarge the domain to get the parabolic effect
            % of the end points of the different "C" plots
            % (for linear spacing use the value 0.25)
            \pgfmathsetmacro{\Factor}{0.01}

        % now draw plots for $C > 0$
        \pgfplotsinvokeforeach {1,2,3} {
            % to do so we split each line into two part
            % this is advantage because of the symmetry of the lines and
            % thus we can avoid the need for a lot of "samples" in the steep
            % parts of the curve.
            %
            % -----------------------------------------------------------------
            % linear spacing approach
            % -----------------------------------------------------------------
%            % quadrant I
%            \addplot+ [domain=+sqrt(#1):{Max+\Factor*#1},forget plot,->]  {f(#1,x)}
%                node [Label=0,shift={(\Shift,\Shift)}] {$+#1$};
%            % the steep parts can be drawn using a parametric plot
%            \addplot+ [domain=+sqrt(#1):{Max+\Factor*#1},forget plot,-<]  ({f(#1,x)},x);
%
%            % quadrant II
%            \addplot+ [domain={-Max-\Factor*#1}:-sqrt(#1),forget plot,>-] {-f(#1,x)}
%                node [Label=1,shift={(-\Shift,\Shift)}] {$-#1$};
%            \addplot+ [domain=+sqrt(#1):{Max+\Factor*#1},forget plot,->]  ({-f(#1,x)},x);
%
%            % quadrant III
%            \addplot+ [domain={-Max-\Factor*#1}:-sqrt(#1),forget plot,>-] {f(#1,x)}
%                node [Label=1,shift={(-\Shift,-\Shift)}] {$+#1$};
%            \addplot+ [domain={-Max-\Factor*#1}:-sqrt(#1),forget plot,<-] ({f(#1,x)},x);
%
%            % quadrant IV
%            \addplot+ [domain=+sqrt(#1):{Max+\Factor*#1},forget plot,->]  {-f(#1,x)}
%                node [Label=0,shift={(\Shift,-\Shift)}] {$-#1$};
%            \addplot+ [domain={-Max-\Factor*#1}:-sqrt(#1),>-]             ({-f(#1,x)},x);

            % -----------------------------------------------------------------
            % non-linear spacing approach
            % -----------------------------------------------------------------
            % quadrant I
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,->]  ({X(#1,x)},{f(#1,X(#1,x))})
                node [Label=0,shift={(\Shift,\Shift)}] {$+#1$};
            % the steep parts can be drawn using a parametric plot
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,-<]  ({f(#1,X(#1,x))},{X(#1,x)});

            % quadrant II
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,-<] ({-X(#1,x)},{f(#1,X(#1,x))})
                node [Label=0,shift={(-\Shift,\Shift)}] {$-#1$};
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,->]  ({-f(#1,X(#1,x))},{X(#1,x)});

            % quadrant III
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,-<] ({-X(#1,x)},{-f(#1,X(#1,x))})
                node [Label=0,shift={(-\Shift,-\Shift)}] {$+#1$};
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,->] ({-f(#1,X(#1,x))},-{X(#1,x)});

            % quadrant IV
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,forget plot,->]  ({X(#1,x)},{-f(#1,X(#1,x))})
                node [Label=0,shift={(\Shift,-\Shift)}] {$-#1$};
            \addplot+ [domain=+sqrt(#1):Max+\Factor*(#1)^2,-<]             ({f(#1,X(#1,x))},{-X(#1,x)});
        }

    \end{axis}
\end{tikzpicture}
\end{document}

Answer 2

@Barry -- There's a lot going on in your stagnation point flow, but here's a start, with lots more work on labels and arrowheads needed.

\documentclass[tikz,border=10pt,multi]{standalone}
\usetikzlibrary{shapes.symbols}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
    axis x line=center,
    axis y line=center,
    xlabel = $x$,
    ylabel = {$y$},
    ymax=3,
    ymin=-3,
    ticks=none,
]
\node[right] at (-2,2) {$C=-2$};
\node[right] at (-0.7,0.2) {$C=0$};

%Below the red is defined
\addplot [
    domain=-2:-0.1, 
   samples=50, 
    color=red,
]
{1/x};
\addplot [
    domain=0.1:2, 
   samples=50, 
    color=red,
]
{(1/x)};  
\addplot [
    domain=-2:-0.1, 
   samples=50, 
    color=red,
]
{ -1/x};
\addplot [
    domain=0.1:2, 
   samples=100, 
    color=red,
]
{(-1/x)};

% Now the teal
   \addplot [
    domain=-2.:-0.1, 
   samples=50, 
    color=teal,
]
{(0.5/x)};
   \addplot [
    domain=0.1:2, 
   samples=50, 
    color=teal,
]
{(0.5/x)};
\addplot [
    domain=-2:-0.1, 
   samples=100, 
    color=teal,
]
{ -0.5/x};   
\addplot [
    domain=0.1:2, 
   samples=50, 
    color=teal,
]
{(-0.5/x)};

%Here green  is defined
    \addplot [
    domain=-2:-0.1, 
    samples=50, 
    color=green,
    ]
    {-2/x};
\addplot [
    domain=-2:-0.1, 
    samples=50, 
    color=green,
    ]
    {2/x};
\addplot [
    domain=0.1:2, 
    samples=50, 
    color=green,
    ]
    {-2/x};
\addplot [
    domain=0.1:2, 
    samples=50, 
    color=green,
    ]
    {-2/x};

 \end{axis}
\end{tikzpicture}

\end{document}