Unit 8 · 8.4 — Fluids and Conservation Laws

14 learning items · ~3% exam weight (unit share)

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Concept: The Continuity Equation is a direct algebraic statement of the conservation of mass; ideal incompressible fluids must speed up their flow velocity when the cross-sectional area of a pipe narrows.
Concept: Bernoulli's Equation is a direct algebraic statement of the conservation of mechanical energy per unit volume for a flowing, ideal fluid along a streamline.
Concept: The Venturi effect states that where a fluid's velocity increases, its internal static pressure must decrease as a consequence of Bernoulli's principle.
Concept: Torricelli's Theorem (governing the exit speed of fluid from a puncture) is derived from Bernoulli's equation with identical pressure terms canceled and surface velocity assumed to be zero.
Equation: $A 1 v 1 = A 2 v 2 (Continuity Equation / Conservation of Volume Flow Rate)$
Equation: $P 1 + ρ g y 1 + \frac{1}{2} ρ v 1^{2} = P 2 + ρ g y 2 + \frac{1}{2} ρ v 2^{2} (Bernoulli’s Equation)$
Equation: $v = 2 g Δ y (Torricelli’s Theorem shortcut for exit speed)$
Vocabulary: Volume Flow Rate (Q) — The geometric volume of fluid passing through a given cross-sectional plane per second (measured in m^3/s).
Skill: Mathematical Routines — Algebraically manipulating Bernoulli's Equation by strategically setting velocity to zero at extremely large fluid surfaces (v1 approx 0) to solve for puncture exit speeds.
Skill: QQT FRQ — Explain conceptually why fluid droplets from a narrower fountain nozzle reach a greater maximum height, then verify the claim by deriving the relationship using Continuity and Bernoulli principles.
Concept: Fluid flow is driven by energy differences, where variations in pressure or potential energy cause matter to move between locations.
Equation: $C o n t in u i t y E q u a t i o n : \frac{V}{t} = A \cdot v, s t a t in g t ha tt h e v o l u m e f l o w r a t er e main sco n s t an tt h r o ug h o u t a c l ose d p i p e .$
Concept: Bernoulli's principle describes how changes in a fluid's elevation result in corresponding changes to its internal pressure and velocity.
Equation: $T or r i ce l l i^{'} s T h eor e m : v = 2 \cdot g \cdot h, w hi c h r e l a t es t h es p ee d o f esc a p in g f l u i d t o i t s d e pt hb e l o w t h es u r f a ce .$

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