Final answer:
The relationship between pressure and velocity in a horizontal pipe is described by Bernoulli's equation, which indicates that an increase in fluid velocity leads to a decrease in pressure, and the flow rate is directly proportional to the pressure differential.
Step-by-step explanation:
For a horizontal pipe, the relationship between pressure and velocity can be understood through Bernoulli's equation which states that for an incompressible, frictionless fluid, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline. This means that if the velocity of a fluid increases within a pipe, the pressure in the pipe must decrease assuming the height remains constant (since we are considering a horizontal pipe, potential energy changes due to height differences can be ignored). The flow rate (№) is given by the equation № = Av, where № is the flow rate, A is the cross-sectional area of the pipe, and v is the average velocity of the fluid flow. Therefore, when the pipe's cross-sectional area decreases, the velocity must increase to maintain a constant flow rate, leading to a decrease in pressure according to Bernoulli's principle.
Additionally, a pressure difference causes fluid to flow from a region of high pressure to a region of low pressure, with the flow rate being directly proportional to this pressure differential. This is why the flow rate increases when the pressure difference between two points increases. These principles are essential for understanding how fluids behave in various systems and are applicable to many fields such as engineering, meteorology, and medicine.