Fluid Dynamics

Fluid dynamics studies fluids in motion. To simplify calculations, we examine an ideal fluid, which is non-viscous and incompressible (constant density $\rho$ ).

Volume Flow Rate and Velocity

Volume flow rate ( $Q$ ) measures the volume of fluid passing a cross-sectional area per unit time.

Q = \frac{dV}{dt} = \frac{d}{dt}(A x) = A \frac{dx}{dt} = A v

(1)

The SI unit for flow rate is $\text{m}^3/\text{s}$ . A common conversion is $1 \text{ L/min} = 1.67 \cdot 10^{-5} \text{ m}^3/\text{s}$ .

Continuity Equations

Mass conservation dictates that the mass of fluid entering a conduit must equal the mass exiting. Mass flow rate is given by $\frac{dm}{dt} = \rho A v$ . Equating the mass flow rate at two points yields the general continuity equation:

\rho_1 A_1 v_1 = \rho_2 A_2 v_2

(2)

For an incompressible fluid, density remains constant ( $\rho_1 = \rho_2$ ), reducing the expression to the incompressible continuity equation:

A_1 v_1 = A_2 v_2 \implies Q_1 = Q_2

(3)

Fluid speed varies inversely with cross-sectional area ( $v \propto \frac{1}{A}$ ). For a circular pipe of radius $r$ , speed varies inversely with the square of the radius ( $v \propto \frac{1}{r^2}$ ).

Example 1

Water flows through a hose of radius $r_1 = 0.900 \text{ cm}$ at a rate of $Q = 0.500 \text{ L/s}$ and emerges through a nozzle of diameter $d_2 = 0.500 \text{ cm}$ . Calculate the water speed in the hose ( $v_1$ ) and in the nozzle ( $v_2$ ).

Hint Click to unblur

Convert $Q$ to $\text{m}^3/\text{s}$ . Use Equation 1 for the hose and Equation 3 for the nozzle.

Solution Click to unblur

Converting flow rate yields $Q = 5.00 \cdot 10^{-4} \text{ m}^3/\text{s}$ .

For the hose ( $r_1 = 0.00900 \text{ m}$ ):

v_1 = \frac{Q}{\pi r_1^2} = \frac{5.00 \cdot 10^{-4} \text{ m}^3/\text{s}}{\pi (0.00900 \text{ m})^2} \approx 1.96 \text{ m/s}

For the nozzle ( $r_2 = 0.00250 \text{ m}$ ):

v_2 = \left(\frac{r_1}{r_2}\right)^2 v_1 = \left(\frac{0.900 \text{ cm}}{0.250 \text{ cm}}\right)^2 (1.96 \text{ m/s}) \approx 25.4 \text{ m/s}