Question

The surface of contact of two fluids of different densities that are at rest and do not mix is horizontal. Prove this general result ( \(a\) ) from the fact that the potential energy of a system must be a minimum in stable equilibrium, \((b)\) from the fact that at any two points in a horizontal plane in either fluid the pressures are equal.

Short answer

The surface of contact of two fluids of different densities at rest is horizontal because this orientation minimizes the system's potential energy and ensures that the pressure at any two points in a horizontal plane in either fluid is equal.

Step-by-step solution

Analyzing component (a) - Potential Energy

Consider a small fluid element whose potential energy is given by \(_PE = mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is the height of the fluid element. The potential energy of the entire fluid system is the integral of the potential energy of all such fluid elements. In equilibrium, the potential energy of the system must be at a minimum. Thus, the height \(h\) of each element should be as small as possible, implying a flat, horizontal surface between two fluids.

Analyzing component (b) - Pressure Balance

Consider two points in the same horizontal plane in either fluid. Since both points are at the same height, according to Pascal's law, the pressure at these points should be equal. Thus, the pressure throughout a horizontal plane is the same. If the interface between the fluids was not horizontal, this would lead to pressure inconsistencies, contradicting Pascal's law. Therefore, the interface between two fluids at rest must be horizontal.

Conclusion

From the principles of Minimum Potential Energy and Pascal's law, it was demonstrated that the surface of contact of two immiscible fluids that are at rest must be horizontal. This is because the system's potential energy is minimized and the pressure in a horizontal plane in each fluid must be equal.

Key concepts

Potential Energy

In the context of fluid equilibrium, potential energy plays a significant role. When two fluids of different densities are in contact, they will arrange themselves such that the system's potential energy is minimized. This is because nature leans towards a state of minimum energy for stability.

Potential energy in a fluid system is determined by the position of its mass relative to the ground. For a small element of fluid, the potential energy can be calculated as \( PE = mgh \), where \( m \) denotes the mass, \( g \) is the gravitational acceleration, and \( h \) represents the height. In equilibrium, the system's total potential energy, which is the sum of all individual fluid elements, should be at its lowest possible value. To achieve this, the heights of the fluid elements are minimized, resulting in a horizontal surface between the fluids.

This principle ensures that when fluids are at rest and do not mix, their surface forms a horizontal interface to maintain a state of stable equilibrium.

Pascal's Law

Pascal's Law is crucial in understanding fluid equilibrium. It states that pressure is transmitted undiminished in an enclosed static fluid. This law applies to our scenario of two fluids at rest.

When two points lie on the same horizontal plane within a fluid, Pascal's Law tells us that the pressure at each point must be equal. If one were to assume otherwise, with one point having a higher pressure than the other, it would contradict the fundamental uniformity implied by Pascal's principle.

This concept ensures that in a stable fluid system, the pressures across a horizontal plane remain equal. Therefore, if the interface between the fluids were not horizontal, it would lead to inconsistencies in pressure distribution. By conforming to Pascal's Law, the surface must be flat, ensuring the equilibrium condition is upheld.

Pressure Balance

Pressure balance is a key concept in describing how fluids at rest achieve equilibrium. This balance is achieved by ensuring consistent pressure throughout a system at a particular level.

Imagine two points within the same horizontal plane of a fluid. According to the concept of pressure balance, the pressure at these two points must be identical. If they were different, the differing forces would induce flow, contradicting the condition of rest.

Thus, in a situation where two fluids do not mix and settle one over the other, the horizontal interface between the fluids aligns with the pressure balance principle. If the interface sloped, it would create a situation where identical vertical levels in the horizontal direction experienced unequal pressures, which isn't possible in equilibrium.

This pressure balance helps explain why the surface of contact between two immiscible fluids at rest is a flat, horizontal line.