A liquid moves through a horizontal pipe. The pressure is

Explanation

In an incompressible fluid (assuming the liquid is incompressible) flowing through a horizontal pipe, Bernoulli's principle applies. This principle states that for an incompressible fluid in steady flow, the sum of the pressure head, kinetic energy head, and potential energy head remains constant throughout the flow path.Pressure head: This refers to the pressure (P) divided by the density (ρ) of the fluid.Kinetic energy head: This is the kinetic energy (KE) of the fluid per unit weight, calculated as KE / (ρg), where g is the acceleration due to gravity.Potential energy head: This is the potential energy (PE) of the fluid per unit weight, calculated as PE / (ρg), where PE is typically the height of the fluid above a reference point.In a horizontal pipe, the potential energy head remains constant (since there's no change in height). Therefore, for constant flow rate (meaning the same amount of liquid passes through any point in the pipe per unit time), the following relationship holds:P1/ρ + KE1/(ρg) = P2/ρ + KE2/(ρg)where:P1 and P2 are the pressures at two different points in the pipeKE1 and KE2 are the kinetic energies at the corresponding pointsSince the pipe is horizontal (no change in height), the potential energy heads (PE/ρg) are equal at all points. This leaves us with:P1 + KE1/ρ = P2 + KE2/ρIf the diameter is larger at a specific point (point 2), the flow velocity (v) will be lower at that point compared to a narrower diameter section (point 1) due to the principle of continuity (mass flow rate remains constant). This relationship can be expressed as:A1 * v1 = A2 * v2where:A1 and A2 are the cross-sectional areas at points 1 and 2, respectivelySince the area (A) is proportional to the square of the radius (r), doubling the radius (doubling the diameter) results in a four-fold increase in the area (A). To maintain the same mass flow rate, the velocity (v) at the wider section (point 2) must decrease by a factor of 1/4 compared to the narrower section (point 1).Combining these relationships:Lower v2 leads to a lower kinetic energy head (KE2) at the wider section compared to the narrower section.Since the sum of pressure and kinetic energy head needs to be constant, the pressure (P2) at the wider section must be lower to compensate for the lower kinetic energy head.Therefore, the pressure is lower where the diameter is large in an incompressible fluid flowing through a horizontal pipe based on Bernoulli's principle and the principle of continuity.