Filling between two lines

As mentioned in the comments you can create a path from the arcs you already defined for filling. I just slightly redefined your arca :

\documentclass[11pt, a4paper]{article}
\usepackage{textgreek}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tikz}

\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}
%Outer lining of the tube.
\draw {[rounded corners=0.2cm] (6.3,1.0) -- (6.3,-1.0) -- (8.3,-1.0) -- (8.3,-4.0)};
\draw {[rounded corners=0.2cm] (6.0,1.0) -- (6.0,-1.3) -- (8.0,-1.3) -- (8.0,-4.0)};
\draw {[rounded corners=0.2cm] (9.3,-4.0) -- (9.3,-1.0) -- (11.6,-1.0) -- (11.6,0.0)};
\draw {[rounded corners=0.2cm] (9.6,-4.0) -- (9.6,-1.3) -- (11.9,-1.3) -- (11.9,0.0)};

%Round bend which also needs to be filled with a color.
\def\arca {(9.3,-4.0) arc (180:0:-0.5 and -0.5)}
\def\arcb {(8.0,-4.0) arc (180:0:0.8 and -0.8)}
% constructing, drawing and filling path
\path[draw,fill=red] (8,-2.5) -- (8,-4) -- \arcb -- (9.6,-4) -- (9.6,-3.5) -- (9.3,-3.5) -- (9.3,-4)  -- \arca  -- (8.3,-4) -- (8.3,-2.5) -- cycle;

%Lines that are indicating a level. Space under these lines needs to be colored.
\draw [dashed] (8.3,-2.5) -- (10.3,-2.5);
\draw [dashed] (9.6,-3.5) -- (10.3,-3.5);
\draw [thick, <->] (10.2,-2.5) -- (10.2,-3.5);
\draw (10.6,-3.0) node {\begin{small}$\Delta P$\end{small}};
\end{tikzpicture}
\end{figure}
\end{document}

An alternative with double lines. double distance defines pipe width and double=... filling color which by default is white. This means that over a non white background the pipe is not transparent.

\documentclass[tikz,border=2mm]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[pipe/.style={draw, double, double distance=3mm, rounded corners=2mm},
    pipefull/.style={pipe, double=red}]

%Outer lining of the tube.
\draw[pipe] (6,1) |- ++(2,-2) -- ++(0,-3) coordinate (aux1) arc(-180:-0:0.6) coordinate (aux2) |-++(2,3)--++(0,2);

\draw[pipefull] ([yshift=2cm]aux1) coordinate(top) -- (aux1) arc(-180:-0:0.6) -- ([yshift=1cm]aux2) coordinate(bottom) ;

\draw[dashed] (top)--++(0:20mm) coordinate (aux3);
\draw[dashed] (bottom) -- (aux3|-bottom) coordinate (aux4);
\draw[<->,thick] ([xshift=-2mm]aux3)-- node[right, font=\small]{$\Delta P$} ([xshift=-2mm]aux4);

\end{tikzpicture}
\end{document}