Why does the speed of a liquid increase and its pressure decrease when a liquid passes through a constriction in a horizontal pipe?

#### Solution

**Consider a horizontal constricted tube.**

Let A_{1} and A_{2} be the cross-sectional areas at points 1 and 2, respectively. Let v_{1} and v_{2} be the corresponding flow speeds. ρ is the density of the fluid in the pipeline. By the equation of continuity,

v_{1}A_{1} = v_{2}A_{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 = 1/2rho"v"_1^2 = "p"_2 + 1/2rho_2^2`

∴ `"p"_1 + 1/2rho"v"_1^2 = "p"_2 + 1/2rho"v"_1^2 ("A"_1/"A"_2)^2` .....[from eq.(1)]

∴ `"p"_1 - "p"_2 = 1/2"pv"_1^2 [("A"_1/"A"_2)^2 - 1]` ...(2)

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.