Bernoulli's Principle And Continuity

Two principles act together for an ideal fluid in a horizontal pipe. The equation of continuity, $A_1{v}_1=A_2{v}_2$, expresses conservation of mass for an incompressible fluid and shows that where the cross-section narrows the speed must rise. Bernoulli's theorem then expresses conservation of energy along a streamline, $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$; with no height change the height term drops out, so an increase in kinetic energy per unit volume must be offset by a fall in pressure. Hence in the constriction the water moves faster and its pressure is lower. The option 'speed decreases and pressure increases' reverses the continuity result. The option 'both increase' violates energy conservation, since faster flow cannot also carry higher pressure here. The option 'both decrease' contradicts continuity for the speed. This combined reasoning is the foundation of the NCERT treatment of streamlined flow and underlies devices such as the venturimeter and atomiser. As a plausibility check, the familiar drop in pressure where a fluid speeds up explains lift on aerofoils, confirming that the narrow, fast region indeed has the lower pressure.