An unknown liquid flows smoothly through a 6.0-mm-diameter horizontal tube where the pressure gradient is 600 Pa/m. Then the tube diameter gradually shrinks to 3.0 mm. What is the pressure gradient in this narrower portion of the tube?

Verified step by step guidance

1. Understand that this problem involves the principle of fluid dynamics, specifically the Hagen-Poiseuille equation, which relates the pressure gradient, flow rate, and tube dimensions for laminar flow. The equation is: $Q=\frac{\pi }{\mathrm{\Delta P}}\cdot \frac{r^4}{\cdot }$.
2. Recognize that the flow rate $Q$ is constant throughout the tube because the liquid is incompressible. This means the pressure gradient in the narrower section must adjust to maintain the same flow rate.
3. Write the Hagen-Poiseuille equation for both sections of the tube. For the wider section: $Q=\frac{\pi }{\mathrm{\Delta P}}\cdot \frac{r^4}{\cdot }$. For the narrower section: $Q=\frac{\pi }{\mathrm{\Delta P}}\cdot \frac{r^4}{\cdot }$.
4. Set the flow rates equal to each other and solve for the pressure gradient in the narrower section. The relationship will be: $\frac{\mathrm{\Delta P}}{\mathrm{\Delta P}}=\frac{r^4}{r^4}$.
5. Substitute the radii of the two sections (3.0 mm and 6.0 mm) into the equation to find the ratio of the pressure gradients. Multiply the given pressure gradient in the wider section (600 Pa/m) by this ratio to find the pressure gradient in the narrower section.

Key Concepts

Continuity Equation

The Continuity Equation states that for an incompressible fluid flowing through a pipe, the mass flow rate must remain constant. This means that the product of the cross-sectional area and the fluid velocity is constant along the flow. As the diameter of the tube decreases, the velocity of the fluid must increase to maintain this constant flow rate.

Bernoulli's Principle

Bernoulli's Principle relates the pressure, velocity, and height of a fluid in steady flow. It states that an increase in the fluid's velocity occurs simultaneously with a decrease in pressure or potential energy. In the context of the tube, as the diameter decreases and the fluid velocity increases, the pressure in the narrower section will drop, which is crucial for calculating the new pressure gradient.

Pressure Gradient

The pressure gradient is defined as the change in pressure per unit length of a fluid flow. It is a driving force for fluid movement, indicating how pressure varies along the flow direction. In this scenario, understanding how the pressure gradient changes when the tube diameter decreases is essential for determining the new pressure conditions in the narrower section of the tube.