Fluid Dynamics: Equation of Continuity, Bernoulli's Theorem & Torricelli's Law

Fluid Dynamics (Hydrodynamics) treats fluids in motion. To analyze complex flow patterns rigorously for engineering and entrance examinations, we model fluid behavior using an ideal fluid: an idealized substance that is (1) incompressible (ρ = const), (2) non-viscous (η = 0), (3) in steady laminar flow, and (4) irrotational.

1. Streamline Flow & The Equation of Continuity

In steady (streamline) flow, every particle passing through a particular point follows the identical path and possesses the same velocity as the particle preceding it. A tangent drawn at any point along a streamline yields the direction of fluid velocity at that point. Streamlines never intersect; if they did, a particle at the junction would have two ambiguous velocities simultaneously.

               End 1 (Inlet)                                   End 2 (Constriction)
               Area A1, Velocity v1                          Area A2, Velocity v2
               +-----------------                            /----------------+
               |                                            /                 |
               |  -----> v1       _/       -----> v2 |
               |                    [ Narrow Neck: A2 < A1 ]                 |
               +-----------------/                          -----------------+
                                        v2 > v1
                        Mass Conservation: A1 * v1 = A2 * v2
      

By conservation of mass, the mass of fluid entering a tube of flow in time Δ t must equal the mass exiting in the same time interval:

For an incompressible fluid (ρ₁ = ρ₂ = ρ):

2. Derivation of Bernoulli's Theorem

Bernoulli's Principle (1738) represents the fundamental statement of the work-energy theorem applied to non-viscous, steady fluid flow along a streamline:

Consider an incompressible fluid flowing through a non-uniform pipe from height h₁ to height h₂. In a time interval Δ t:

• Work done by pressure force at inlet: W₁ = F₁ Δ x₁ = P₁ A₁ (v₁ Δ t) = P₁ Δ V.
• Work done by pressure force at outlet: W₂ = -F₂ Δ x₂ = -P₂ A₂ (v₂ Δ t) = -P₂ Δ V.
• Net work done by pressure forces: W_net = (P₁ - P₂)Δ V.
• Change in kinetic energy: Δ K = 1/2Δ m (v₂² - v₁²) = 1/2ρ Δ V (v₂² - v₁²).
• Change in gravitational potential energy: Δ U = Δ m g (h₂ - h₁) = ρ Δ V g (h₂ - h₁).

Applying the Work-Energy Theorem (W_net = Δ K + Δ U):

Rearranging terms:

Alternative "Head" Formulation

Dividing the entire equation by ρ g gives all terms in units of length (meters of fluid column):

• P/(ρ g) = Pressure Head
• v²/2g = Velocity (Dynamic) Head
• h = Elevation (Datum) Head

3. Torricelli's Law of Efflux & Tank Emptying Time

Consider a large open storage tank filled with liquid of density ρ to a total depthH, possessing a small orifice of areaalocated at a depthhbelow the open free surface (a ≪ A, whereA is the cross-sectional area of the tank):

       Free Surface: Level 1 (Area A, P1 = P0, v1 approx 0)
       ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
       |                                                   |
       |                   |                               |
       |                   | h                             |
       |                   v                               |
       |                Orifice: Level 2 (Area a, P2 = P0) =====> Speed of Efflux v2
       |                                                   |
       |                                      (H - h)      |
       +---------------------------------------------------+
       |<----------------- Horizontal Range R ------------>|
      

Applying Bernoulli's equation between the open free surface (1) and the discharge orifice (2):

By continuity, v₁ = (a/A)v₂ ≈ 0 since a ≪ A:

Kinematics of the Efflux Stream

• Time of Flight to Ground: Fluid leaves horizontally from elevation (H - h):
  t = √((2(H - h))/g)
• Horizontal Range (R):
  R = v · t = √(2gh) × √((2(H - h))/g) = 2√(h(H − h))
• Maximum Range Condition: By differentiating R with respect to h, the range is maximized when the hole is drilled precisely at mid-depth (h = H/2):
  R_max = 2√((H/2)(H − H/2)) = H
• Complementary Depths: Two holes drilled at depths h and (H - h) produce identical horizontal ranges (R₁ = R₂).
• Time to Empty Tank Completely:
  T_empty = (A/a)√(2H/g)

4. Applications: Venturimeter, Aerofoils & Magnus Effect

The Venturimeter

A Venturimeter measures the volumetric discharge rate Q of liquid flowing through a pipeline. It consists of a wide tube of area A₁ connected to a narrow throat of area A₂ (A₂ < A₁):

From continuity: v₂ = (A₁ / A₂)v₁. Because A₂ < A₁, v₂ > v₁. By Bernoulli's equation, this acceleration causes the pressure at the throat to drop (P₂ < P₁):

The rate of flow Q is given by:

Dynamic Aerodynamic Lift (Aerofoil)

An airplane wing (aerofoil) is sculpted with a convex upper curvature and a flat lower surface. As the wing moves forward through the air, streamlines crowd together above the curved top, forcing air to travel faster (v_top > v_bottom). By Bernoulli's principle, the static pressure on top drops below that beneath the wing (P_top < P_bottom). The resulting upward pressure difference produces aerodynamic lift:

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