Hydrostatic Pressure and Fluid Dynamics: Key Principles
Pascal’s Principle
Pascal stated that there is a fundamental difference between solids and liquids. Solids transmit forces only in the direction they are applied. In contrast, both fluids transmit pressures in all directions. This property has been proven experimentally since antiquity and is known as Pascal’s Principle. It can be phrased as: “Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.”
Hydrostatics
Hydrostatics applies to fluids at rest. A fluid is a substance that has no defined shape but takes the form of the container that holds it.
Pressure
Pressure is the normal force per unit area or surface.
- Density (D): The ratio between mass and volume.
- Relative Density (Drel): The ratio between the density of a substance and the density of distilled water. It is a dimensionless quantity.
- Specific Weight (Pspec): The ratio between the weight of a substance and its volume.
- Relative Specific Weight (Pspec,rel): The ratio between the specific weight of a substance and the specific weight of distilled water.
Hydrostatic Pressure
The pressure exerted depends on the height or depth of the liquid and the specific gravity of the liquid. If the container is open to the atmosphere, then atmospheric pressure acts on its surface. In this case, the absolute pressure at the bottom will be the sum of the atmospheric pressure and the pressure due to the liquid’s weight.
Fundamental Hydrostatic Law
In a liquid in equilibrium, two points located at the same level are subject to the same pressure (Pascal’s Principle). Two points at different levels have different pressures. The pressure difference between two points in a liquid in equilibrium is the product of the specific weight and the difference in level between these points.
Flow Rate
Flow rate is the product of the cross-sectional area and the fluid velocity at that point.
Bernoulli’s Equation
Fluids can move vertically, up or down, like a river descending from the mountains or smoke rising from a fireplace. Considering a uniform density fluid flowing through a tube, according to the law of conservation of mechanical energy, the work done on the fluid should be equal to the change in its mechanical energy.
Flotation of Bodies
When a body is immersed in a liquid, it is subject to two opposing forces: its weight, which tends downward, and the upthrust, which tends upward. Depending on the relationship between these two forces, the body can be in one of three situations:
- The body sinks to the bottom.
- The body remains suspended in the liquid (floating between two layers).
- The body floats on the surface of the liquid.
Flotation Condition
Any body floating on the surface of a liquid displaces a weight of liquid equal to its own weight.
Atmospheric Pressure Measurement
The Earth’s surface is surrounded by a layer of air. Due to the action of gravitational force, the weight of this gaseous mass exerts pressure on the Earth’s surface. This is called atmospheric pressure.
Torricelli’s Experiment
Torricelli used a graduated glass tube, one meter long and closed at one end. He completely filled it with mercury, then inverted it, plugging the free end with a finger, and placed it inside a container also containing mercury. The experiment showed that the mercury column in the tube descended until it stabilized at a certain height.
Archimedes’ Principle
When a body enters a liquid, the liquid exerts a vertical upward force called buoyant force. This force is numerically equivalent to the difference between the body’s weight in air and its apparent weight when immersed in the liquid. If the liquid is in a graduated cylinder, an increase in its volume is observed, corresponding to the volume of the submerged body. The buoyant force is equivalent to the weight of the fluid displaced by the body. This is known as Archimedes’ Principle, which states: “A body wholly or partially immersed in a fluid apparently loses a part of its weight equal to the weight of the fluid it displaces.”
Continuity Equation
Since no fluid can enter or leave through the walls of a tube, the volume of fluid flow passing through any cross-section must be constant.
Fluid Dynamics
Fluid dynamics is the study of fluids in motion. It is one of the most complex branches of mechanics. Considering ordinary fluid motion, such as a river overflowing, it would be impossible to describe the motion of each individual water droplet. However, when certain conditions are met, the fluid motion can be of a relatively simple type, known as steady or stationary flow.