Pressure in Fluids

Thrust And Pressure

When dealing with forces exerted by fluids or on surfaces, it's important to distinguish between the total force acting perpendicularly on a surface and the pressure, which is the force per unit area.

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Thrust

Thrust is the total force acting perpendicular to a surface. It is a vector quantity, measured in Newtons (N).

When a fluid is in contact with a surface, it exerts forces on that surface. The thrust is the magnitude of the component of the total force exerted by the fluid that is perpendicular to the surface. For a fluid at rest, the force exerted by the fluid on any surface in contact with it is always perpendicular to the surface.

Example: When a block is placed on a table, the weight of the block exerts a downward force. The thrust on the table is the magnitude of this weight, acting perpendicularly to the table's surface (if the table is horizontal). When a boat floats on water, the buoyant force exerted by the water on the boat's bottom surface is a form of upward thrust.

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Pressure

Pressure ($P$) is defined as the thrust acting per unit area of the surface. It is a measure of how concentrated the force is on a given area.

$ \text{Pressure} = \frac{\text{Thrust}}{\text{Area}} = \frac{F_{perp}}{A} $

where $F_{perp}$ is the magnitude of the force acting perpendicularly to the surface, and $A$ is the area of the surface.

Pressure is a scalar quantity. Although it is related to a force vector, pressure at a point in a fluid at rest acts equally in all directions. The SI unit of pressure is Newton per square metre (N/m$^2$), which is also called Pascal (Pa). $1 \text{ Pa} = 1 \text{ N/m}^2$.

Other common units of pressure include:

• Bar: $1 \text{ bar} = 10^5 \text{ Pa} \approx 1$ atmosphere.
• Atmosphere (atm): Standard atmospheric pressure at sea level, $1 \text{ atm} = 101325 \text{ Pa} \approx 1.01 \times 10^5 \text{ Pa}$.
• Torr: $1 \text{ Torr} = 133.322$ Pa (named after Torricelli, related to mmHg).
• Millimetres of mercury (mmHg): Pressure that supports a column of mercury 1 mm high. $760 \text{ mmHg} = 1 \text{ atm}$.

The concept of pressure is especially important in fluids (liquids and gases) because they cannot withstand tangential or shear stress when at rest. Forces exerted by fluids at rest are always perpendicular to the surfaces they are in contact with.

Example: A sharp knife cuts better than a blunt knife. If the same force is applied, the sharp knife has a much smaller edge area, resulting in a much higher pressure, which helps in cutting. The pressure exerted by a block on a table is its weight divided by the area of contact with the table. A camel can walk on sand easily because its broad feet distribute its weight over a larger area, resulting in lower pressure on the sand.

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Pressure In Fluids

Pressure is a particularly important concept when studying fluids (liquids and gases). Fluids are substances that can flow and take the shape of their container. Unlike solids, fluids at rest cannot sustain shear stress.

In a fluid at rest, the force exerted by the fluid on any surface is always perpendicular to the surface. This is true whether the surface is the wall of the container or an imaginary surface within the fluid itself.

One of the key properties of pressure in a fluid at rest is that it is transmitted equally in all directions. At any point within a fluid, the pressure is the same regardless of the orientation of the imaginary surface passing through that point.

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Pressure at a Point in a Fluid

Consider an infinitesimal volume element within a fluid at rest. For this element to be in equilibrium, the net force on it must be zero. By considering the forces exerted by the surrounding fluid on this element, it can be shown that the pressure at a point in a fluid at rest is independent of direction. Imagine a small imaginary prism within the fluid. By analysing the forces on the faces of the prism, and letting the size of the prism shrink to zero, one can demonstrate that the pressure is the same in all directions at that point.

This property, that pressure at a point in a fluid at rest is the same in all directions, is sometimes referred to as Pascal's principle of pressure transmission at a point.

This principle is fundamental to understanding how pressure behaves within fluids and forms the basis for Pascal's Law concerning the transmission of pressure throughout a fluid.

Pressure

Pressure is a measure of the force exerted perpendicularly per unit area. In fluids, pressure is particularly significant due to the fluid's inability to withstand shear stress at rest and its ability to transmit pressure.

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Pascal’s Law

Pascal's Law, formulated by Blaise Pascal, describes how pressure is transmitted within a fluid.

The law states:

Pressure applied to an enclosed fluid at rest is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

This means that if you apply extra pressure at one point in a confined fluid, that same amount of extra pressure is added to the pressure at every other point in the fluid and to the pressure on the walls of the container.

Imagine a sealed container filled with a fluid. If you push down on a piston in the container, increasing the pressure at the piston's surface, this increase in pressure is felt uniformly throughout the entire fluid volume and on all the inner surfaces of the container.

Pascal's Law is a consequence of the fact that fluids at rest cannot sustain shear stress. Any attempt to apply pressure that is not uniform would create unbalanced internal forces that would cause the fluid to flow until the pressure is uniform.

This principle is the basis for the operation of many hydraulic systems.

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Variation Of Pressure With Depth ($ P = P_0 + \rho gh $)

In a fluid column (like a tank of water or the Earth's atmosphere), the pressure increases with depth. This is due to the weight of the fluid above the point of measurement.

Consider a point at depth $h$ below the surface of a liquid open to the atmosphere. Let the pressure at the surface be $P_0$ (this is typically atmospheric pressure). Consider a cylindrical column of the liquid with cross-sectional area $A$, extending from the surface down to the point at depth $h$. The volume of this liquid column is $V = Ah$. If the density of the liquid is $\rho$, the mass of this column is $m = \rho V = \rho A h$. The weight of this liquid column is $W = mg = \rho A h g$.

This weight $W$ is the force pushing down on the liquid at depth $h$ due to the column of liquid above it. In addition, there is the pressure $P_0$ acting on the surface of the liquid from the atmosphere above it, contributing a downward force $F_0 = P_0 A$.

The total downward force acting on the area $A$ at depth $h$ is $F_{total} = W + F_0 = \rho A h g + P_0 A$.

The pressure $P$ at depth $h$ is the total downward force per unit area:

$ P = \frac{F_{total}}{A} = \frac{\rho A h g + P_0 A}{A} = \frac{A (\rho h g + P_0)}{A} $

$ P = P_0 + \rho h g $

This formula gives the total pressure (absolute pressure) at a depth $h$ in a liquid of uniform density $\rho$, where $P_0$ is the pressure at the surface. The term $\rho h g$ is the pressure due to the column of liquid above the point, often called the hydrostatic pressure.

Key points about pressure in a fluid column:

• Pressure increases linearly with depth $h$.
• Pressure is the same at all points at the same horizontal level within the fluid, regardless of the shape of the container (this is why water finds its own level).
• Pressure does not depend on the area of the surface, only on the depth, density, and surface pressure.
• This formula is approximate if the fluid density $\rho$ is not constant with depth (e.g., in the atmosphere or very deep oceans, where density changes with pressure and temperature).

(Image Placeholder: A container filled with liquid. Show the surface with atmospheric pressure P0. Indicate a point at depth h below the surface. Show a column of liquid above this point with height h and area A. Label the pressure P at depth h.)

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Atmospheric Pressure And Gauge Pressure

Atmospheric Pressure ($P_{atm}$): The Earth is surrounded by a layer of air called the atmosphere. The air has weight, and this weight exerts pressure on the Earth's surface and everything on it. This pressure is called atmospheric pressure. It is the pressure exerted by the column of air from the top of the atmosphere down to the point of measurement. Atmospheric pressure varies with altitude (lower at higher altitudes) and weather conditions.

Standard atmospheric pressure at sea level is $1 \text{ atm} \approx 1.013 \times 10^5$ Pa.

Atmospheric pressure is measured using a barometer. A simple mercury barometer consists of a glass tube closed at one end, filled with mercury, and inverted into a trough of mercury. The height of the mercury column supported by the atmospheric pressure is measured. Standard atmospheric pressure supports a column of mercury 76 cm (or 760 mm) high.

Gauge Pressure ($P_{gauge}$): Many pressure measuring instruments (like tyre pressure gauges or blood pressure monitors) measure the pressure relative to the local atmospheric pressure. This is called gauge pressure.

Absolute Pressure ($P_{absolute}$): The total pressure at a point is called absolute pressure. It is the sum of the gauge pressure and the local atmospheric pressure.

$ P_{absolute} = P_{gauge} + P_{atm} $

In the formula $P = P_0 + \rho h g$, if $P_0$ is the atmospheric pressure on the surface of the liquid, then $P$ is the absolute pressure at depth $h$. The hydrostatic pressure $\rho h g$ is the gauge pressure at that depth, relative to the pressure at the surface.

$ P_{absolute} = P_{atm} + \rho h g $

$ P_{gauge} = P_{absolute} - P_{atm} = \rho h g $

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Hydraulic Machines

Pascal's Law is the fundamental principle behind the operation of hydraulic machines such as hydraulic presses, hydraulic lifts, and hydraulic brakes. These machines use the incompressibility of liquids and the transmission of pressure to multiply force.

Consider a simple hydraulic lift consisting of two cylinders of different cross-sectional areas ($A_1$ and $A_2$, with $A_2 > A_1$), connected by a horizontal tube, and filled with an incompressible liquid (like oil). Pistons are fitted in both cylinders.

(Image Placeholder: A diagram showing two interconnected cylinders with pistons. The left cylinder is narrow with area A1 and piston force F1. The right cylinder is wider with area A2 and piston force F2. Both are filled with liquid.)

If a small force $F_1$ is applied to the piston in the narrower cylinder (area $A_1$), it creates an increase in pressure in the liquid at that point:

$ \Delta P = \frac{F_1}{A_1} $

According to Pascal's Law, this increase in pressure $\Delta P$ is transmitted undiminished throughout the liquid to the piston in the wider cylinder (area $A_2$). The liquid exerts an upward force $F_2$ on this piston due to this pressure increase:

$ F_2 = \Delta P \times A_2 $

Substitute $\Delta P = F_1/A_1$:

$ F_2 = \left(\frac{F_1}{A_1}\right) A_2 $

$ F_2 = F_1 \left(\frac{A_2}{A_1}\right) $

Since $A_2 > A_1$, the ratio $\frac{A_2}{A_1}$ is greater than 1. This means that the force $F_2$ exerted on the larger piston is greater than the force $F_1$ applied to the smaller piston. The force is multiplied by the ratio of the areas. This multiplication of force is the principle behind hydraulic lifts and presses.

Note that while the force is multiplied, the work done is conserved (neglecting friction). If the small piston moves down by a distance $d_1$, the volume of liquid displaced is $V_1 = A_1 d_1$. Since the liquid is incompressible, the same volume must be displaced in the larger cylinder, causing the large piston to move up by a distance $d_2$, such that $V_2 = A_2 d_2 = V_1$. So, $A_2 d_2 = A_1 d_1$, or $d_2 = d_1 (A_1/A_2)$. Since $A_1/A_2 < 1$, the large piston moves a smaller distance than the small piston. The work done on the small piston is $W_1 = F_1 d_1$. The work done by the large piston is $W_2 = F_2 d_2 = \left(F_1 \frac{A_2}{A_1}\right) \left(d_1 \frac{A_1}{A_2}\right) = F_1 d_1$. Thus, $W_1 = W_2$, and energy is conserved.

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