Fluid Mechanics Course

Essential Foundation: Critical prerequisite for plasma physics (MHD), astrophysics, and engineering

Fluid Statics

The study of fluids at rest - pressure, buoyancy, and hydrostatic equilibrium

Introduction

Fluid statics (or hydrostatics) deals with fluids at rest. In static equilibrium, there are no shear stresses acting on fluid elements - only normal stresses (pressure). This simplification makes the analysis tractable while providing essential insights applicable to dams, submarines, atmospheric pressure, and more.

The fundamental principle is that pressure at a point in a static fluid acts equally in all directions (Pascal's principle) and varies only with depth in a gravitational field.

Pressure Fundamentals

Definition of Pressure

Pressure is the normal force per unit area exerted by a fluid on a surface:

$$p = \frac{F}{A} = \lim_{\Delta A \to 0} \frac{\Delta F_n}{\Delta A}$$

Units: Pa (Pascal) = N/mยฒ = kg/(mยทsยฒ). Also: 1 atm = 101,325 Pa = 760 mmHg = 14.7 psi

Pascal's Law

Pressure at a point in a static fluid is the same in all directions (isotropic):

$$p_x = p_y = p_z = p$$

This is proven by considering force equilibrium on a small fluid wedge. Shear stresses are zero in a static fluid, leaving only normal pressure forces.

Pressure Transmission

A pressure change applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the container. This principle is the basis for hydraulic systems:

$$\frac{F_1}{A_1} = \frac{F_2}{A_2} \implies F_2 = F_1 \cdot \frac{A_2}{A_1}$$

Hydraulic lifts, brakes, and presses exploit this mechanical advantage.

Hydrostatic Pressure Variation

Basic Hydrostatic Equation

For a fluid at rest in a gravitational field, the pressure increases with depth:

$$\frac{dp}{dz} = -\rho g$$

Where z is measured upward. For incompressible fluids (constant ฯ):

$$p_2 - p_1 = -\rho g (z_2 - z_1) = \rho g h$$

where h = zโ‚ - zโ‚‚ is the depth below point 1.

Pressure in the Atmosphere

For gases, density varies with pressure and temperature. For an isothermal atmosphere:

$$p = p_0 \exp\left(-\frac{gz}{RT}\right) = p_0 \exp\left(-\frac{z}{H}\right)$$

where H = RT/g โ‰ˆ 8.5 km is the scale height for Earth's atmosphere.

Gauge Pressure

Pressure relative to atmospheric: pgauge = pabs - patm

Absolute Pressure

Measured from absolute zero pressure (perfect vacuum)

Manometry

Manometers use the hydrostatic equation to measure pressure differences by observing liquid column heights.

Simple U-Tube Manometer

$$p_A - p_B = \rho_{man} g h$$

The height difference h of manometer fluid (mercury, water, oil) indicates pressure difference.

Inclined Manometer

For small pressure differences, an inclined tube amplifies the reading:

$$\Delta p = \rho g L \sin\theta$$

where L is the length along the incline and ฮธ is the angle from horizontal.

Multi-Fluid Manometer

For complex systems with multiple fluids, apply hydrostatic equation through each section:

$$p_1 + \sum_i \rho_i g h_i = p_2$$

Barometer

Measures absolute atmospheric pressure. Mercury barometer:

$$p_{atm} = \rho_{Hg} g h$$

At sea level: h โ‰ˆ 760 mm Hg = 29.92 in Hg

Forces on Submerged Surfaces

Plane Surfaces

The hydrostatic force on a submerged plane surface is:

$$F_R = \int_A p \, dA = \int_A (\rho g h) \, dA = \rho g \bar{h} A = p_c A$$

where pc is the pressure at the centroid and A is the surface area.

The center of pressure (where resultant acts) is below the centroid:

$$y_{cp} = \bar{y} + \frac{I_{xc}}{\bar{y} A}$$

where Ixc is the second moment of area about the centroidal axis.

Curved Surfaces

For curved surfaces, decompose into horizontal and vertical components:

Horizontal Component

FH = Force on vertical projection of curved surface

Vertical Component

FV = Weight of fluid above (or below) the curved surface

Buoyancy and Archimedes' Principle

Archimedes' Principle

A body immersed in a fluid experiences an upward buoyant force equal to the weight of displaced fluid:

$$F_B = \rho_{fluid} g V_{displaced}$$

The buoyant force acts through the centroid of the displaced volume (center of buoyancy).

Floating Bodies

For a floating body in equilibrium:

$$F_B = W \implies \rho_{fluid} g V_{submerged} = \rho_{body} g V_{body}$$
$$\frac{V_{submerged}}{V_{body}} = \frac{\rho_{body}}{\rho_{fluid}}$$

Stability of Floating Bodies

For rotational stability, the metacenter M must be above the center of gravity G:

$$GM = \frac{I_{waterline}}{V_{submerged}} - BG$$

where Iwaterline is the second moment of the waterline area about the tilt axis. GM > 0 indicates stable equilibrium.

Applications

Dam Design

Calculate hydrostatic forces on dam walls to ensure structural stability against overturning and sliding.

Submarines

Buoyancy control through ballast tanks, hull design for pressure at depth.

Hydraulic Systems

Pascal's law applied in lifts, brakes, presses, and actuators.

Atmospheric Science

Pressure variation with altitude, barometric formula, weather systems.

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