Ideal Fluid Dynamics

To simplify the study of moving fluids we introduce the ideal fluid model:

• Incompressible ($\rho = \text{const}$): density does not vary with pressure. Real liquids approximate this condition well at moderate speeds.
• Non-viscous ($\eta = 0$): no internal friction between fluid layers. In reality all fluids have some viscosity (water, air), but for many problems its effect is negligible.

Steady flow: the fluid properties (velocity, pressure, density) at every point do not change with time: $\partial(\cdot)/\partial t = 0$. Streamlines are curves whose tangent at every point indicates the velocity direction. In steady flow, streamlines coincide with trajectories and never intersect.

Laminar vs turbulent flow:

• Laminar: the fluid flows in orderly, parallel layers. Typical at low speeds and in narrow conduits (e.g., dripping honey).
• Turbulent: the motion is chaotic with vortices and fluctuations. Typical at high speeds and wide conduits (e.g., water from a fire hydrant).

Real-world examples:

• Faucet: at moderate flow the water comes out laminar (clear jet); opening fully makes it turbulent (white jet due to air bubbles).
• Cigarette smoke: initially rises in laminar flow, then becomes turbulent.
• Blood flow: in healthy arteries flow is laminar; heart murmurs indicate turbulence.

The continuity equation expresses mass conservation. For an incompressible fluid, the volumetric flow rate $Q$ (volume passing through a cross-section per unit time) is constant: $$Q = A v = \text{const} \quad [\text{m}^3/\text{s}]$$ If the cross-section narrows, the velocity increases proportionally, and vice versa: $A_1 v_1 = A_2 v_2$.

For a compressible fluid, the mass flow rate is conserved: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2 = \dot{m} = \text{const}$.

Intuitive interpretation: imagine squeezing the end of a garden hose — the water comes out faster because the same amount of water must pass through a smaller opening in the same time.

Real-world applications and examples:

• Fire hose: squeezing the nozzle reduces the exit area, increasing water velocity and range.
• River: where the riverbed narrows the current accelerates; where it widens it slows down.
• Blood circulation: blood velocity decreases in capillaries (huge total cross-section) and increases in arteries (small cross-section).
• Ventilation ducts: grilles use narrowing to accelerate air.
• Venturi tubes: use constriction to measure flow rate.

Bernoulli's theorem follows from the conservation of mechanical energy along a streamline of an ideal fluid in steady flow: $$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$
Interpretation of the three terms (energy per unit volume):

• $P$ — static pressure: energy associated with pressure forces.
• $\frac{1}{2}\rho v^2$ — dynamic pressure: kinetic energy per unit volume.
• $\rho g h$ — gravity pressure: gravitational potential energy per unit volume.

Key consequence: at the same elevation, where velocity increases pressure decreases, and vice versa. This is the Bernoulli effect, underlying numerous phenomena and applications.

Real-world applications and examples:

• Aerodynamic lift: an airplane wing has an asymmetric profile. Air on the upper surface flows faster (longer path), creating a low-pressure zone above the wing. The pressure difference generates an upward force (lift).
• Carburetor: the narrowing of the air duct creates a depression that draws fuel from the reservoir.
• Atomizer (perfume): squeezing the bulb creates a fast air jet above a vertical tube; the low pressure draws the liquid, which is atomized.
• Curveball (Magnus effect): the ball's rotation creates an air speed difference on opposite sides, curving the trajectory.
• Roof torn off by wind: strong wind over a roof creates a low-pressure zone; the higher internal pressure can lift the roof.
• Chimney: wind at the chimney top creates a depression that enhances smoke extraction.

Torricelli's theorem — efflux from a tank: Consider a tank with a hole at depth $h$ below the free surface. Applying Bernoulli between the free surface ($v_1 \approx 0$, $P_1 = P_{atm}$) and the hole ($P_2 = P_{atm}$): $$P_{atm} + \rho g h = P_{atm} + \frac{1}{2}\rho v^2 \quad \Rightarrow \quad v = \sqrt{2gh}$$ The efflux velocity is identical to that of a body in free fall from height $h$. The flow rate through the hole is $Q = A_{hole} v = A_{hole} \sqrt{2gh}$.

Venturi tube — flow rate meter: A conduit with a central constriction. Applying Bernoulli and continuity between sections 1 (wide) and 2 (narrow), the flow rate is obtained from the pressure difference: $$\Delta P = P_1 - P_2 = \frac{1}{2}\rho\left(v_2^2 - v_1^2\right)$$ Combining with $A_1 v_1 = A_2 v_2$ yields $v_1$ and thus $Q = A_1 v_1$.

Aerodynamic lift in detail: The wing has an asymmetric profile: the upper surface is more curved than the lower surface. Streamlines bunch together above the wing, increasing velocity and decreasing pressure. The difference $P_{lower} - P_{upper}$ generates a net upward force: $$F_{lift} \approx \frac{1}{2} C_L \rho A v^2$$ where $C_L$ is the lift coefficient (dependent on wing shape and angle of attack), $\rho$ the air density, $A$ the wing area, and $v$ the flight speed.

Other applications:

• Siphon: uses pressure difference to transfer liquid from one level to another through a tube.
• Venturi meter: flow measurement in industrial pipes, oil pipelines, aqueducts.
• Drain time: integrating the flow rate yields the time needed to drain a tank.