Consider a horizontal constricted tube.
Let A_{1 }and A_{2} be the cross-sectional areas at points P_{1} and P_{2}, respectively.
Let v_{1} and v_{2} be the corresponding flow speeds.
ρis the density of the fluid in the pipeline.
By the equation of continuity,
A_{1}v_{1 }= A_{2}v_{2 }... (1)
v_{2}/ v_{1} =A_{1}/A_{2 }> 1 (…A_{1}>A_{2})
Therefore, the speed of the liquid increases as it passes through the constriction. Since the meter is assumed to be horizontal, from Bernoulli’s equation we get,
\(p_1+\frac{1}{2}ρv_1^2\) =\(p_2+\frac{1}{2}ρv_2^2\)
\(p_1+\frac{1}{2}ρv_1^2\) =\(p_2+\frac{1}{2}ρv_1^2(\frac{A_1}{A_2})^2\)
\(p_1-p_2=\frac{1}{2}ρv_1^2\left[(\frac{A_1}{A_2})^2-1\right]\)
Again, since A_{1}>A_{2 }the bracketed term is positive so that p_{1} > p_{2}. Thus, as the fluid passes through the constriction or throat, the higher speed results in lower pressure at the throat.