A venturi flowmeter in an oil pipeline has radius half that of the pipe. Oil flows in the unconstricted pipe at \(1.9 \mathrm{m} / \mathrm{s}\). If the pressure difference between unconstricted flow and venturi is \(16 \mathrm{kPa}\) what's the oil's density?

Bernoulli's Equation is a fundamental principle in fluid mechanics that describes the behavior of a fluid moving along a streamline. It relates various forms of energy in a fluid flow, stating that the total mechanical energy of the flowing fluid remains constant. In the context of our venturi flowmeter problem, the equation is employed to connect the pressures and the velocities in different sections of the pipe.

Considering potential energy changes are negligible, the equation simplifies to \( P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \), where:\

• \(P_1\) and \(P_2\) are the fluid pressures at two points along the flow,
• \(\rho\) is the fluid density, and
• \(v_1\) and \(v_2\) are the fluid velocities at these points.

By employing Bernoulli's Equation, determining the density of the oil becomes possible by examining the pressure difference and velocities in different parts of the system.

Fluid Dynamics is the branch of physics concerned with the motion of fluids (liquids and gases) and the forces acting on them. It involves understanding how fluids flow, the patterns they form, and the energy involved in moving the fluid particles. The discipline uses various equations, including Bernoulli's Equation, to predict fluid behavior under different conditions.

In the venturi flowmeter, fluid dynamics helps us comprehend how oil behaves when passing through a restricted section of a pipe. The pressure decreases as the oil accelerates through the venturi, an effect that can be measured to calculate the flow rate or properties of the fluid, like density. This demonstrates fluid dynamics in action, with the Venturi effect showcasing how a fluid's velocity and pressure are inversely related in areas of varying cross-sectional area.

The Continuity Equation is a crucial concept in fluid dynamics that expresses the idea of conservation of mass in a flow. For an incompressible fluid, the equation states that the mass flow rate must remain constant from one cross-section of a pipe to another, which implies that any change in the cross-sectional area leads to a corresponding change in the fluid's velocity to maintain the same flow rate.

This principle is encapsulated in the equation \( A_1v_1 = A_2v_2 \), where:\

• \(A_1\) and \(A_2\) are the cross-sectional areas,
• \(v_1\) and \(v_2\) are the fluid velocities at these sections.

In our problem, since the venturi has half the radius of the pipe, its cross-sectional area is one-fourth of the pipe's area. Thus, the velocity in the venturi must be four times greater (\(v_2 = 4v_1\)) than in the unconstricted pipe to satisfy the Continuity Equation, which is critical for calculating the change in pressure and hence, the density of the oil.

Density Computation involves calculating the mass per unit volume of a substance. For fluids, density can change with temperature and pressure, but for incompressible fluids like the oil in our problem, density remains constant throughout the flow. By arranging the information from Bernoulli's Equation and Continuity Equation, we can isolate and solve for the fluid's density.

In our exercise, once we've accounted for the changes in velocity due to the venturi and understood the pressure difference created by these changes, we use the formula \( \rho = -\frac{\Delta P}{\frac{3}{2} v_1^2} \) to compute the oil's density. This computation allows us to determine how much mass of the oil passes through a given volume in the pipeline, which is essential for various applications in the industry, such as transportation and quality control.