Fluid Mechanics Fundamentals for the FE Exam
Fluid mechanics is a topic that spans nearly every engineering discipline. Whether you are taking the Civil, Mechanical, Chemical, or Environmental FE exam, you will encounter fluid mechanics questions. This review covers the core concepts that appear most frequently.
Fluid Properties
- Density (rho): Mass per unit volume. For water at standard conditions, rho = 1,000 kg/m^3 (62.4 lb/ft^3).
- Specific weight (gamma): Weight per unit volume. gamma = rho * g. For water, gamma = 9,810 N/m^3.
- Specific gravity (SG): The ratio of a fluid's density to water's density. SG = rho_fluid / rho_water.
- Dynamic viscosity (mu): A measure of a fluid's resistance to shearing motion. Units: Pa*s or lb*s/ft^2.
- Kinematic viscosity (nu): nu = mu / rho. Units: m^2/s or ft^2/s.
- Surface tension: The cohesive force at the fluid surface that causes the surface to behave like an elastic membrane.
- Vapor pressure: The pressure at which a liquid begins to boil at a given temperature. Important for cavitation analysis.
Fluid Statics (Hydrostatics)
Pressure Variation with Depth
In a static fluid, pressure increases linearly with depth:
P = P_0 + gamma * h
Where P_0 is the pressure at the surface and h is the depth below the surface. This relationship is fundamental to hydrostatic problems.
Manometers
Manometers measure pressure differences using columns of liquid. To analyze a manometer, start at one end and work through the fluid columns, adding pressure when going down and subtracting when going up:
P_A + gamma_1 * h_1 - gamma_2 * h_2 - gamma_3 * h_3 = P_B
Hydrostatic Forces on Surfaces
For a submerged flat surface:
- Horizontal surfaces: F = P * A = gamma * h * A, where h is the depth to the centroid of the surface.
- Inclined surfaces: F = gamma * h_c * A, where h_c is the vertical depth to the centroid. The force acts at the center of pressure, which is always below the centroid.
- Center of pressure depth: y_cp = y_c + I_c / (y_c * A), where I_c is the second moment of area about the centroidal axis and y_c is the distance from the surface along the inclined plane to the centroid.
Buoyancy
Archimedes' principle: The buoyant force on a submerged or floating body equals the weight of the fluid displaced.
F_b = gamma_fluid * V_displaced
For a floating body, the buoyant force equals the weight of the body. This determines how deep the body sits in the fluid.
Fluid Dynamics
Conservation of Mass (Continuity Equation)
For steady, incompressible flow through a pipe or duct:
A_1 * V_1 = A_2 * V_2 = Q (constant flow rate)
As the cross-sectional area decreases, velocity increases, and vice versa.
Bernoulli's Equation
For steady, incompressible, inviscid flow along a streamline:
P_1 / gamma + V_1^2 / (2*g) + z_1 = P_2 / gamma + V_2^2 / (2*g) + z_2
Each term represents a type of head (energy per unit weight): pressure head, velocity head, and elevation head. The sum is constant along a streamline.
Assumptions and limitations of Bernoulli's equation:
- Steady flow
- Incompressible fluid
- No friction (inviscid) or heat transfer
- Applied along a streamline
When friction is present, add head loss terms to account for energy dissipation.
Reynolds Number
The Reynolds number is the ratio of inertial forces to viscous forces:
Re = rho * V * D / mu = V * D / nu
It determines the flow regime:
- Re less than 2,100: Laminar flow (smooth, orderly)
- 2,100 less than Re less than 4,000: Transition region
- Re greater than 4,000: Turbulent flow (chaotic, mixing)
Pipe Flow and Head Loss
The Darcy-Weisbach equation gives the major (friction) head loss in a pipe:
h_f = f * (L/D) * V^2 / (2*g)
Where f is the Darcy friction factor. For laminar flow, f = 64 / Re. For turbulent flow, use the Moody diagram or the Colebrook equation.
Minor losses occur at fittings, valves, entrances, and exits:
h_m = K * V^2 / (2*g)
Common K values are tabulated in the reference handbook.
Dimensional Analysis
Dimensional analysis uses the Buckingham Pi theorem to determine dimensionless groups that govern physical phenomena. Key dimensionless numbers:
- Reynolds number: Inertial to viscous forces (flow regime)
- Froude number: Inertial to gravitational forces (open channel flow)
- Mach number: Flow speed to sound speed (compressible flow)
- Weber number: Inertial to surface tension forces
Practice Tips
- Master Bernoulli's equation; it appears in many variations
- Practice reading the Moody diagram quickly
- Know the continuity equation and how to apply it
- Practice manometer problems and hydrostatic force calculations
- Understand when Bernoulli's equation applies and when it does not
Fluid mechanics builds logically from statics (hydrostatics) to dynamics (flow analysis). Master the fundamentals in order, and the more advanced topics will follow naturally.