Fluid Flow, Continuity, and Bernoulli’s Equation (4B) - Chem & Physics

1. State the relationship between gauge pressure $P_g$ and absolute pressure $P_{abs}$.

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

2. What is the definition of volumetric flow rate $Q$ for a fluid?

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

3. For a horizontal pipe, what does Bernoulli's equation reduce to between points $1$ and $2$?

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

4. Identify the correct expression for average speed $v$ in a tube given $Q$ and cross-sectional area $A$.

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

5. Which condition must hold for Bernoulli's equation to apply in its simplest form?

   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.

6. What is the definition of hydrostatic (gravitational) pressure term in Bernoulli's equation?

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

7. Using continuity, if $A_2=3A_1$, what is $v_2$ in terms of $v_1$ for incompressible flow?

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

8. What is the definition of a streamline in fluid flow?

   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.

9. Using continuity, if a pipe narrows so that $A_2=\frac{1}{2}A_1$, what is $v_2$ in terms of $v_1$?

   $v_2=2v_1$. Continuity requires speed to double when area halves to maintain constant flow rate in incompressible flow.

10. For horizontal flow, if speed increases from $v_1$ to $v_2$ with $v_2>v_1$, how does pressure change?

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

11. State the Reynolds number formula for flow in a cylindrical tube.

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

12. State Poiseuille's law for laminar flow rate $Q$ through a cylindrical tube of radius $r$ and length $L$.

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

13. In a vertical pipe with equal speeds ($v_1=v_2$), what is $P_2-P_1$ if $h_2-h_1=\Delta h$?

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

14. What is the continuity relationship between two points in an incompressible fluid ($1$ and $2$)?

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

15. State Bernoulli's equation for steady, incompressible, nonviscous flow along a streamline.

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

16. State the continuity equation relating flow rate, area, and speed for steady incompressible flow.

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

17. What is the definition of dynamic pressure in Bernoulli's equation?

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

18. Identify the correct unit for volumetric flow rate $Q$ in SI units.

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

19. For pipe flow, which inequality indicates a strong tendency toward turbulence using Reynolds number?

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

20. What is the definition of mass flow rate $\dot{m}$ in terms of density and $Q$?

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

21. In a horizontal pipe, if $v_2=2v_1$, what is $P_1-P_2$ in terms of $\rho$ and $v_1$?

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

22. Write Bernoulli's equation explicitly between two points ($1$ and $2$) on a streamline.

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

23. What is the definition of turbulent flow in a pipe?

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

24. Which condition must hold for the continuity form $A_1v_1=A_2v_2$ to be valid?

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

25. What is the definition of laminar flow in a pipe?

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

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