Hydrostatic Equilibrium and Its States

Key Takeaways

• The branch of study that focuses on the pressure on a fluid and the pressure exerted by a fluid on an immersed object is called hydrostatics.

• In the air, a condition arises where the pressure gradient force balances the gravitational force, called the hydrostatic equilibrium state. 

• When the center of buoyancy and center of gravity are at the same point, the state of equilibrium is neutral, provided the weight and buoyant forces are equal and opposite.

Pressure increases with depth

Shear stress and pressure force are two significant parameters in fluid applications. Hydrostatics is the study of the pressure in a fluid or the pressure exerted by a fluid, and is essential to understanding the characteristics of fluids at rest. In hydrostatics, hydrostatic equilibrium is the condition in which there is a balance between external forces like gravitational force and pressure force. In this article, we will explore the vertical pressure gradient and hydrostatic equilibrium.

Vertical Pressure Gradient

Pressure is the force per unit. In a fluid, pressure at the bottom is equal to the weight of the column of fluid above. The fluid pressure decreases with height. Consider a column of air and a thin horizontal slice from the column. The pressure exerted against the bottom is greater than the pressure from the top. There is a pressure gradient experienced in the column of air as we go vertically upwards. The net force acting on the slice of air column upwards is due to the pressure gradient and can be given by the equation:

A is the horizontal cross-sectional area of the air column and P is the difference in pressures at the bottom and top.

Hydrostatic Equilibrium

As aforementioned, the force acting on the slice of air column can be calculated using equation 1.

Also, the force is given by the equation below:

G is the acceleration due to gravity and m is the mass.

The mass of the air column can be given by:

is the air density, A is the area of the slice of air,  and z is the slice thickness.

In the air, a condition rises where the pressure gradient force balances the gravitational force; such a condition is called the hydrostatic equilibrium state. Using equations 1, 3, and 4, the hydrostatic equilibrium can be mathematically represented as: 

The hydrostatic equilibrium describes the balance between the upward pressure and downward gravity in a fluid. The force balance in the fluid under hydrostatic conditions is stationary. Usually, a negative sign is incorporated in the mathematical expression to indicate a drop in pressure with height. 

Hydrostatic Equilibrium and Buoyancy

The stability of a floating or submerged body in a fluid is based on hydrostatic equilibrium conditions. When an object is fully or partially submerged in a liquid, hydrostatic pressure is exerted on all the surfaces of the object in contact with the liquid. The forces on the sides of the body are equal and opposite, typically canceling each other out. As the pressure increases with depth, the hydrostatic forces at the bottom surface of the submerged object are greater than the hydrostatic forces at the top. The hydrostatic forces that are not canceled push the body down. The unequal force leads to a resultant upward hydrostatic force acting on the body, called buoyancy.

In hydrostatic equilibrium conditions, only two forces act on the body:

1. Weight due to gravity

2. Buoyant force

In hydrostatic equilibrium, the weight and buoyant force are equal and opposite.

The Three States of Equilibrium in Submerged Bodies

For the equilibrium condition of a submerged body, the weight of the body should be equal to the buoyant force. Depending on the relative position of the center of gravity and the center of buoyancy, the equilibrium state differs between:

• Stable equilibrium: The weight and buoyant force are equal and opposite. Under this condition, if the center of buoyancy lies above the center of gravity, it is in a stable state of equilibrium.

• Unstable equilibrium: The conditions for unstable equilibrium are:

  1. The center of gravity should be above the center of buoyancy.

  2. The weight and buoyant forces should be equal and opposite.

• Neutral equilibrium: When the center of buoyancy and center of gravity are at the same point, the state of equilibrium is neutral, provided the weight and buoyant forces are equal and opposite.

Hydrostatic equilibrium is important when dealing with fluids and objects that are either immersed or floating in fluid. Design challenges can be overcome or balanced by analyzing the hydrostatic forces and pressure acting on the fluid. Cadence’s suite of CFD tools can assist you with analyzing the interaction of fluids with a submerged body.

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