Final answer:
To determine the change in pressure for an incompressible fluid flowing through a pipe with a changing cross-sectional area, apply the conservation of mass and Bernoulli's principle, leading to an expression involving the density, cross-sectional areas, and velocities at two points.
Step-by-step explanation:
Understanding Fluid Flow and Pressure Change
Fluid flow through a pipe involves the conservation of mass and, for incompressible fluids, the principle of continuity. Considering an incompressible fluid with constant density ρ flowing through a pipe with a reduction in cross-sectional area, we can derive an equation for the change in pressure using these principles.
The conservation of mass for an incompressible fluid dictates that the mass flow rate must be the same at any two points along the pipe.
Therefore, ρA1v1 = ρA2v2, where A1 and A2 are the cross-sectional areas at points 1 and 2, and v1 and v2 are the fluid velocities at these points, respectively.
Bernoulli's principle is then applied to incompressible fluids at constant depth, stating that an increase in fluid velocity results in a decrease in pressure.
Hence, for the given constants ρ, A1, A2, and v1, we can express the change in pressure between two points as
ΔP = P1 - P2, which is inversely related to the square of the velocities.