A flask of conical shape (see figure) contains raw milk. The...

Name: Problem 96, Chapter 1: Problems and Solutions on Thermodynamics and Statistical Mechanics
Uploaded: 2021-09-04T22:18:11-08:00
Duration: 2 min 34 s
Channel: Mahendra K
Description: A flask of conical shape (see figure) contains raw milk. The pressure is measured inside the flask at the bottom. After a sufficiently long time, the cream rises to the top and the milk settles to the bottom. [You may assume that the total volume of liquid remains the same.] Does the pressure increase, decrease, or remain the same? Explain.

A flask of conical shape (see figure) contains raw milk. The pressure is measured inside the flask at the bottom. After a sufficiently long time, the cream rises to the top and the milk settles to the bottom. [You may assume that the total volume of liquid remains the same.] Does the pressure increase, decrease, or remain the same? Explain.

Key Concepts

Effect of Container Geometry

The geometry of the container, particularly when it does not have a uniform cross-sectional area, plays a crucial role in pressure measurements. In a conical vessel, for instance, the area changes with height, which means that the relationship between the volume of fluid and the fluid height is nonlinear. As a result, a redistribution of fluids with different densities can lead to changes in local pressure readings even though the total fluid weight remains the same, due to differences in how the weight is ‘focused’ over the smaller or larger areas at different depths.

Conservation of Mass and Volume

This concept emphasizes that if no fluid is added or removed, the total mass—and therefore the total weight—of the liquid column remains unchanged. Even when separation occurs, the conservation of volume ensures that the overall amount of fluid stays the same. However, because the volumes occupied by the individual layers depend on their densities, the vertical distribution of mass may change, affecting the pressure computed at a specific point in a non?uniform container.

Hydrostatic Pressure

This concept concerns the pressure in a static fluid being due to the weight of the fluid above a given point. In any container, the pressure at a depth is obtained by summing the contributions of all fluid layers above it; that is, by integrating the product of local density and gravitational acceleration over the vertical distance. This principle applies regardless of the specific shapes of the vessels or the fluids, provided the system is in equilibrium.

Fluid Stratification and Density Distribution

When a mixture separates into layers with different densities (stratification), the pressure at any point is determined by the sum of the pressures contributed by each distinct layer. In a layered, or stratified, system the pressure at the bottom is computed as the sum of the products of each layer’s density, gravitational acceleration, and its vertical thickness. This stratification can alter the local pressure even though the total mass of fluid remains constant.

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