Fluid Properties and Hydrostatics (4B) - Chem & Physics

1. State the continuity equation for steady, incompressible flow in a pipe.

   $A_1 v_1=A_2 v_2$. The continuity equation conserves volume flow rate for incompressible fluids, relating area and velocity inversely.

2. If tube radius doubles, by what factor does $Q$ change in Poiseuille flow (all else constant)?

   $Q$ increases by a factor of $16$. In Poiseuille flow, volumetric rate scales with the fourth power of radius, so doubling radius multiplies flow by 16.

3. State the Laplace pressure for a spherical soap bubble with two surfaces.

   $\Delta P=\frac{4\gamma}{r}$. Laplace pressure in soap bubbles accounts for two curved surfaces, doubling the pressure difference compared to single interfaces.

4. State the hydrostatic pressure formula for a fluid at depth $h$ below the surface.

   $P=P_0+\rho g h$. Hydrostatic pressure at depth includes atmospheric pressure plus the weight of the overlying fluid column per unit area.

5. State the Laplace pressure for a spherical liquid droplet with one surface.

   $\Delta P=\frac{2\gamma}{r}$. Young-Laplace equation gives the pressure jump across a single curved liquid-gas interface due to surface tension.

6. What is the definition of pressure in terms of force and area?

   $P=\frac{F}{A}$. Pressure measures the perpendicular force distributed over a surface area, crucial for understanding fluid statics and dynamics.

7. What is the definition of density in terms of mass and volume?

   $\rho=\frac{m}{V}$. Density quantifies mass concentration within a given volume, fundamental to material properties and buoyancy calculations.

8. State Poiseuille's law for laminar flow of a Newtonian fluid in a cylindrical tube.

   $Q=\frac{\pi r^4\Delta P}{8\eta L}$. Poiseuille's law describes laminar viscous flow, with rate proportional to pressure gradient and fourth power of radius.

9. Which quantity is equal at the same depth in a connected, static fluid: pressure or volume?

   Pressure (same depth implies same $P$ in static connected fluid). In hydrostatic equilibrium, pressure equalizes at identical depths in connected static fluids due to Pascal's law.

10. If pipe area decreases to $\frac{1}{3}$ of its original value, what happens to speed for incompressible flow?

    Speed increases by a factor of $3$. Continuity for incompressible flow requires velocity to inversely scale with area, tripling speed when area thirds.

11. What is the buoyant force on a submerged object in terms of displaced fluid?

    $F_B=\rho_{\text{fluid}} g V_{\text{disp}}$. Archimedes' principle equates buoyant force to the gravitational weight of the fluid volume displaced by the object.

12. State the hydraulic press relation between forces and piston areas.

    $\frac{F_1}{A_1}=\frac{F_2}{A_2}$. In hydraulic presses, equal pressures result in forces proportional to piston areas, allowing force amplification.

13. What is the shear stress in a Newtonian fluid in terms of viscosity and velocity gradient?

    $\tau=\eta\frac{dv}{dy}$. Shear stress in Newtonian fluids is proportional to the velocity gradient, with viscosity as the constant of proportionality.

14. What is the gauge pressure at depth $h$ in a fluid of density $\rho$?

    $P_{\text{gauge}}=\rho g h$. Gauge pressure represents the excess pressure due to the fluid column's weight, excluding atmospheric contributions.

15. What is the volumetric flow rate $Q$ in terms of cross-sectional area and speed?

    $Q=Av$. Volumetric flow rate represents the volume passing per unit time, derived from cross-sectional area and fluid speed.

16. What is the surface tension force magnitude along a contact line of length $L$?

    $F=\gamma L$. Surface tension generates a tangential force proportional to the length of the interface, minimizing surface area.

17. What is the condition for an object to float in a fluid, stated using densities?

    $\rho_{\text{object}}<\rho_{\text{fluid}}$. Floating occurs when an object's density is less than the fluid's, ensuring buoyant force balances weight.

18. What is the fraction of an object's volume submerged when it floats at rest?

    $\frac{V_{\text{sub}}}{V}=\frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}$. Equilibrium in floating objects sets the submerged fraction equal to the density ratio, balancing buoyancy and weight.

19. State Pascal's principle for an enclosed, incompressible fluid.

    $\Delta P$ applied to a confined fluid is transmitted undiminished. Pascal's principle ensures uniform pressure transmission in confined incompressible fluids, enabling hydraulic systems.

20. What is the fluidic resistance $R$ of a cylindrical tube in laminar flow?

    $R=\frac{8\eta L}{\pi r^4}$. Fluidic resistance in tubes arises from viscosity, inversely scaling with the fourth power of radius for laminar flow.

21. State the Reynolds number for flow in a tube using density, speed, diameter, and viscosity.

    $\mathrm{Re}=\frac{\rho v D}{\eta}$. Reynolds number quantifies the ratio of inertial to viscous forces, predicting flow regime transitions in fluids.

22. Identify the pressure difference between two points separated vertically by $\Delta h$ in a static fluid.

    $\Delta P=\rho g\Delta h$. Hydrostatic pressure varies linearly with vertical depth due to the cumulative weight of the fluid column.

23. Identify the typical flow regime when $\mathrm{Re}<2000$ in a cylindrical pipe.

    Laminar flow. Low Reynolds numbers indicate dominance of viscous forces, resulting in smooth, layered fluid motion without turbulence.

24. State Bernoulli's equation along a streamline for an ideal fluid.

    $P+\frac{1}{2}\rho v^2+\rho g h=\text{constant}$. Bernoulli's equation conserves mechanical energy per unit volume along streamlines in ideal fluid flow.

25. State the capillary rise height $h$ in a tube of radius $r$ with contact angle $\theta$.

    $h=\frac{2\gamma\cos\theta}{\rho g r}$. Capillary action height balances surface tension adhesion against gravity, depending on tube radius and wetting angle.

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