Let us consider the following hypotheses for fluid flow:
- Incompressible flow
- Steady flow
That being stated, let us consider a horizontal pipe with constant diameter, thus constant cross-sectional area. According to conservation of energy, we have
\begin{equation} \frac{P_1}{\rho g} + \frac{V_1^2}{2g} = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + h_L \tag{1} \label{1} \end{equation}
where the term $h_L$ relates all losses between point 1 and 2. Applying Eq. \ref{1} to calculate $V_2$, it is possible to define the following equation
\begin{equation} \frac{V_2^2}{2g} = \frac{P_1 - P_2}{\rho g} + \frac{V_1^2}{2 g} - h_L \tag{2} \label{2} \end{equation}
It is also possible to calculate $V_2$ by applying conservation of mass
\begin{equation} V_2 A_2 = V_1 A_1 \tag{3} \label{3} \end{equation}
As the pipe has the same cross-sectional area, Eq. \ref{3} can be written as \begin{equation} V_2 = V_1 \tag{4} \label{4} \end{equation}
Comparing Eq. \ref{2} and \ref{4}, the unique way to Eq. \ref{2} be true is if \begin{equation} \frac{P_1 - P_2}{\rho g} - h_L = 0 \tag{5} \label{5} \end{equation}
Is it Eq. \ref{5} valid, exclusively, due to hypothesis 2? As the flow is steady, it shouldn't be accelerating. Is it correct?