Fluid Statics, Fluid Dynamics, Bernoulli, Energy Losses, Flow Measurement, and Size Reduction (Notes)

Fluid Statics

• Fluid statics is a branch of fluid mechanics that focuses on fluids at rest, in a state of stable equilibrium. Key areas of study include the behavior of liquids and gases when static, the pressure they exert on surfaces, and how this pressure changes with depth, particularly under gravity.

• Governing principles essential for understanding fluid statics:

  • (1) Isotropy of pressure: At any point within a static fluid, the pressure is exerted equally in all directions. This means that if you imagine a tiny fluid element at a point, the force per unit area on any infinitesimal surface passing through that point will be the same, regardless of the orientation of the surface. This principle is fundamental for deriving hydrostatic equations.

  • (2) Constant pressure at horizontal planes: In a continuous, static fluid, all points lying on the same horizontal plane (i.e., at the same depth) experience the same pressure. This is a direct consequence of the balance of forces in a fluid at rest; if there were a pressure difference, the fluid would flow.

• Pressure definitions:

  • Pressure is fundamentally defined as the normal force (F) applied per unit area (A) on a surface: P = \frac{F}{A}. Its SI unit is the Pascal (Pa), equivalent to Newtons per square meter (N/m^2).

  • Absolute pressure: This is the actual pressure measured with reference to a perfect vacuum, which is zero pressure. All pressure values are positive in absolute terms.

  • Gauge pressure: This is the pressure measured relative to the local atmospheric pressure. It indicates how much the measured pressure is above or below the ambient atmospheric pressure. A vacuum can be expressed as a negative gauge pressure. Standard atmospheric pressure at sea level is approximately 760 mm Hg, 1.013 × 10^5 Pa, or 14.7 psi.

• Hydrostatic pressure in a stationary column of fluid:

  • For a column of fluid with a uniform cross-sectional area S (constant along its height) and subjected to a surface pressure P_0 (e.g., atmospheric pressure), the pressure at a depth h below the surface is given by the hydrostatic equation:

    P = P_0 + \rho g h

    where P is the pressure at depth h, \rho (rho) is the density of the fluid (assumed constant for incompressible fluids), g is the acceleration due to gravity (\approx 9.81 \text{ m/s}^2), and h is the vertical depth from the surface. This formula accounts for the weight of the fluid column above the point of measurement.

  • For any two points at different heights within the same stationary fluid column, the pressure difference (\Delta P) between levels separated by a vertical height h is solely dependent on the fluid's density and gravity:

    \boxed{\Delta P = \rho g h}

    This means pressure increases linearly with depth in an incompressible fluid.

• Pressure as “head”:

  • In liquids, particularly in hydraulics and fluid engineering, pressure differences are frequently expressed as a