Fluid Flow, Continuity, and Bernoulli’s Equation (4B) - MCAT Chemical and Physical Foundations of Biological Systems

$P_2<P_1$. Bernoulli’s principle shows that increased speed in horizontal flow reduces pressure to conserve energy.

$P_1-P_2=\frac{3}{2}\rho v_1^2$. From simplified Bernoulli’s equation, pressure drop equals the difference in dynamic pressures for $v_2=2v_1$.

$P_2-P_1=-\rho g\Delta h$. With equal speeds, Bernoulli’s equation shows pressure difference opposes gravitational potential change.

A curve everywhere tangent to the local fluid velocity. A streamline traces the path where its direction matches the instantaneous velocity vector of the fluid particles.

Smooth, layered flow with minimal mixing between layers. Laminar flow features parallel layers sliding past each other with viscous forces dominating, preventing mixing.

Chaotic flow with eddies and significant mixing. Turbulent flow involves inertial forces dominating, leading to irregular motion and vortex formation.

$\text{Re}=\frac{\rho vD}{\mu}$. Reynolds number Re quantifies the ratio of inertial to viscous forces using density, speed, diameter, and viscosity.

$\text{Re}\gtrsim 2000$. In pipe flow, Re exceeding approximately 2000 indicates inertial forces overpower viscous damping, promoting turbulence.

$Q=\frac{\pi r^4\Delta P}{8\mu L}$. Poiseuille’s law derives from viscous flow assumptions, showing flow rate proportional to pressure gradient and radius to the fourth power.

$Q=\frac{dV}{dt}$. Volumetric flow rate $Q$ measures the volume of fluid $dV$ passing through a cross-section per unit time $dt$.

$Q=Av$. For steady incompressible flow, volumetric flow rate $Q$ equals cross-sectional area $A$ times average fluid speed $v$.

$A_1v_1=A_2v_2$. Continuity for incompressible fluids ensures constant flow rate, so the product of area and speed remains equal at different points.

$\text{m}^3!!/\text{s}$. In SI units, volumetric flow rate $Q$ is expressed as cubic meters per second, reflecting volume over time.

$\dot{m}=\rho Q$. Mass flow rate $\dot{m}$ is the product of fluid density $\rho$ and volumetric flow rate $Q$, giving mass per unit time.

$P_{abs}=P_g+P_{atm}$. Absolute pressure $P_{abs}$ includes atmospheric pressure $P_{atm}$ added to gauge pressure $P_g$, accounting for total pressure.

$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$. Bernoulli’s equation conserves energy per unit volume along a streamline, summing static, dynamic, and gravitational terms.

$P_1+\frac{1}{2}\rho v_1^2+\rho gh_1=P_2+\frac{1}{2}\rho v_2^2+\rho gh_2$. Bernoulli’s equation equates the sum of pressures and energies at two points on a streamline for conservation in ideal flow.

$\frac{1}{2}\rho v^2$. Dynamic pressure represents the kinetic energy per unit volume of the moving fluid in Bernoulli’s equation.

$\rho gh$. Hydrostatic pressure arises from gravitational potential energy per unit volume at height $h$ in a fluid of density $\rho$.

Steady, incompressible flow. The continuity equation $A_1v_1=A_2v_2$ assumes steady flow with constant density, ensuring mass conservation.

Steady, incompressible, nonviscous flow on a streamline. Bernoulli’s equation in simplest form requires no viscosity, constant density, steady flow, and evaluation along a streamline.

$v=\frac{Q}{A}$. Average fluid speed $v$ derives from rearranging the flow rate equation $Q=Av$ for a given cross-section.

$P_1+\frac{1}{2}\rho v_1^2=P_2+\frac{1}{2}\rho v_2^2$. In horizontal flow, gravitational terms cancel in Bernoulli’s equation, leaving static and dynamic pressure balance.

$v_2=\frac{1}{3}v_1$. Continuity dictates that speed decreases inversely with area increase to conserve volume flow rate.