Continuity Equation and Bernoulli Equation

Bernoulli's Equation

Bernoulli's equation relates the pressure, speed, and height of an ideal fluid (incompressible, non-viscous, steady flow) at any two points along a streamline.

$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$

The total mechanical energy per unit volume is conserved along a streamline.

$P$=Static pressure(Pa)

$\frac{1}{2}\rho v^2$=Dynamic pressure (kinetic energy per unit volume)(Pa)

$\rho gh$=Hydrostatic pressure (potential energy per unit volume)(Pa)

Horizontal pipe ($h_1 = h_2$)

→

$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$ — height terms cancel

Static fluid ($v = 0$)

→

Reduces to $P + \rho gh = \text{constant}$ — the hydrostatic equation

Three Assumptions: The fluid must be (1) incompressible, (2) non-viscous, and (3) in steady (laminar) flow.

Energy Conservation: Derived from the work-energy theorem — net work done by pressure forces equals the change in KE + PE of the fluid element.

Dimensional Check: Each term has dimensions of $[M][L]^{-1}[T]^{-2}$ (pressure), confirming dimensional consistency.

Core Insight: Where the speed is high, the pressure is low — this inverse relationship drives all Bernoulli applications.

Torricelli's Theorem

Torricelli's Theorem states that the speed of efflux from an orifice is equivalent to the velocity gained by an object falling freely from the free surface to the hole's depth.

$$v = \sqrt{2gh}$$

The exit velocity depends solely on the vertical depth of the fluid above the hole.

$v$=Efflux speed(m/s)

$g$=Acceleration due to gravity(m/s²)

$h$=Depth of orifice below surface(m)

Pressurized tank ($P_{top} > P_{atm}$)

→

$v = \sqrt{2gh + \frac{2(P_{top} - P_{atm})}{\rho}}$ — gauge pressure adds to velocity

Orifice pointed upward

→

The jet rises back to the original water surface level (KE reconverts to PE)

Density Independence: The exit speed is independent of the liquid's density — $\rho$ cancels from both sides of Bernoulli's equation.

Atmospheric Cancellation: Both the top surface and the exit hole are at atmospheric pressure $P_0$, so $P_0$ cancels out.

Free-Fall Equivalence: The efflux speed exactly matches $v = \sqrt{2gh}$ for a ball falling height $h$ — pure PE → KE conversion.

Maximum Range: For a jet from height $y$ above ground in a tank of total height $H$, horizontal range $R = 2\sqrt{y(H - y)}$, maximized when $y = H/2$.

Speed-Pressure Relationship

A direct consequence of Bernoulli's equation for horizontal flow: where the speed of the fluid is high, the static pressure is low, and vice versa.

$$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$$

For a horizontal pipe, the height terms cancel, leaving an inverse trade-off between pressure and speed.

$P_1, P_2$=Static pressures at two points(Pa)

$v_1, v_2$=Flow speeds at the same points(m/s)

$v_1 = 0$ (stagnation)

→

$P_1 - P_2 = \frac{1}{2}\rho v_2^2$ — all dynamic pressure converts to static

Streamline Crowding: In narrower regions, streamlines bunch together — fluid moves faster, pressure drops.

Danger Near Trains: A person near a fast train feels pulled toward it because the fast-moving air between them and the train has lower pressure than the still air behind.

Two Boats Pulled Together: Parallel boats moving in the same direction create fast-flowing water between them, lowering pressure and pushing them toward each other.

The Venturi Effect & Meter

The Venturi effect is the pressure drop that occurs when fluid flows through a constriction. A Venturi meter uses this effect to measure flow rate by reading the pressure difference via a manometer.

$$P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2)$$

The pressure drop between the wide section and the narrow throat equals the gain in dynamic pressure.

$P_1$=Pressure in wide section(Pa)

$P_2$=Pressure in throat(Pa)

$v_1, v_2$=Flow speeds in wide and narrow sections(m/s)

$A_2 \ll A_1$

→

$v_1 \approx 0$ and $P_1 - P_2 \approx \frac{1}{2}\rho v_2^2$ — the simplified Venturi relation

Continuity Link: The constriction forces the same volume through a smaller area per second, so $v$ must increase ($A_1 v_1 = A_2 v_2$).

Manometer Reading: The U-tube manometer between wide and narrow sections shows the pressure head difference directly.

Carburetor Application: In a car engine, air rushing through a Venturi constriction creates low pressure that draws petrol vapour into the air stream.

Filter Pump / Aspirator: A fast water jet through a constriction creates low pressure that draws air in from a side tube.

Dynamic Lift & Magnus Effect

Dynamic lift on an aerofoil arises because the wing shape forces air to move faster over the top (lower pressure) than the bottom (higher pressure). The Magnus effect creates lateral force on spinning objects.

$$F_{lift} = \frac{1}{2}\rho(v_{top}^2 - v_{bottom}^2) \times A$$

The net upward force equals the pressure difference between bottom and top surfaces multiplied by the wing area.

$\Delta P$=Pressure difference between bottom and top(Pa)

$A$=Wing (or surface) area(m²)

$v_{top}, v_{bottom}$=Air speeds above and below the wing(m/s)

Camber Design: The wing's curved upper surface forces air to travel a longer path in the same time, increasing $v_{top}$ and lowering pressure above.

Magnus Effect: A spinning cricket ball drags air with it — speed adds on one side, subtracts on the other, creating a lateral pressure difference that produces 'swing'.

Chimney Effect: Wind blowing across a chimney top creates low pressure above, enhancing the upward draft of smoke. Taller chimneys are more effective.

Atomizer / Sprayer: Air blown over the top of a tube dipped in liquid creates low pressure, lifting liquid up and atomizing it (used in perfume bottles and paint sprayers).

A Pitot tube measures fluid velocity by comparing stagnation point pressure (where flow stops) to static pressure in the undisturbed stream.

$$v = \sqrt{\frac{2(P_{stag} - P_{static})}{\rho}}$$

Converts measured dynamic pressure into flow velocity.

$P_{stag}$=Stagnation (total) pressure at the tube opening(Pa)

$P_{static}$=Static pressure from a side tap(Pa)

$\rho$=Fluid density(kg/m³)

Stagnation Point: The point directly facing the flow where fluid velocity drops to zero and all KE converts to pressure energy.

Aircraft Use: Pitot tubes mounted on airplane fuselages measure airspeed by sensing the ram pressure of incoming air.

Dimensional Check: $\sqrt{Pa / (kg/m^3)} = \sqrt{(kg \cdot m^{-1} \cdot s^{-2}) / (kg \cdot m^{-3})} = \sqrt{m^2/s^2} = m/s$ ✓

Blood Pressure & Biological Flow

Blood pressure measurement relies on Bernoulli's principle — when an external cuff pressure equals the systolic pressure, flow resumes through a constricted artery at high speed, producing turbulence detected by a stethoscope.

Systolic vs Diastolic: Systolic (~120 torr) is peak pressure during a heartbeat; diastolic (~75–80 torr) is the minimum between beats. 1 torr = 133.3 N/m².

Sphygmomanometer: The inflatable cuff compresses the artery until flow stops. As pressure is released, the first sounds (Korotkoff sounds) mark systolic pressure; their disappearance marks diastolic.

Continuity in Capillaries: Blood from the aorta (radius ~1 cm, $v \approx 30$ cm/s) flows through millions of capillaries with combined cross-section ~2000 cm², so capillary speed drops to ~0.05 cm/s.

Posture Effect: Blood pressure in the feet is higher than in the head by $\rho g \Delta h$ — lying horizontal minimizes this variation.