Viscosity and Poiseuille Flow (4B) - Chem & Physics

1. If viscosity triples ($\eta\to 3\eta$), by what factor does $Q$ change (Poiseuille, constant $\Delta P$)?

   $Q$ decreases by a factor of $3$. Tripling viscosity reduces flow rate to one-third, as it inversely scales in Poiseuille flow under constant conditions.

2. What SI unit is used for kinematic viscosity $\nu$?

   $\frac{\text{m}^2}{\text{s}}$. Kinematic viscosity's SI unit is area per time, reflecting its role in momentum diffusion without density dependence.

3. What is the relationship between maximum and average speed in Poiseuille flow?

   $v_{\max}=2v_{\text{avg}}$. In parabolic laminar flow, the centerline maximum velocity is twice the cross-sectional average due to the velocity distribution.

4. What SI unit is used for dynamic viscosity $\eta$?

   $\text{Pa}\cdot\text{s}=\frac{\text{N}\cdot\text{s}}{\text{m}^2}=\frac{\text{kg}}{\text{m}\cdot\text{s}}$. The SI unit for dynamic viscosity derives from force per area times time, equivalent to pascal-seconds or its breakdowns in base units.

5. State the hydraulic resistance $R$ for laminar flow in a cylindrical tube, defined by $\Delta P=QR$.

   $R=\frac{8\eta L}{\pi r^4}$. Hydraulic resistance analogs to electrical resistance, where flow rate relates to pressure drop, incorporating geometric and viscous factors for cylindrical tubes.

6. Two identical tubes are placed in series; by what factor does total resistance $R$ change versus one tube?

   Total $R$ increases by a factor of $2$. Series connection adds resistances linearly, doubling total resistance for identical tubes analogous to electrical circuits.

7. What boundary condition at the wall is assumed in Poiseuille flow (name the condition)?

   No-slip condition: $v=0$ at the wall. The no-slip condition assumes fluid adheres to solid surfaces, leading to zero velocity at tube walls in viscous flows.

8. Which flow regime is typically assumed for Poiseuille flow: laminar or turbulent?

   Laminar flow. Poiseuille flow derives under assumptions of steady, incompressible, laminar conditions with fully developed velocity profiles.

9. What is the Reynolds number formula for flow in a tube using density $\rho$, speed $v$, diameter $D$, and viscosity $\eta$?

   $\text{Re}=\frac{\rho v D}{\eta}$. Reynolds number quantifies the ratio of inertial to viscous forces, using diameter for pipe flow characterization.

10. What does it mean for a fluid to be Newtonian regarding the relation between $\tau$ and $\frac{dv}{dy}$?

    $\tau\propto\frac{dv}{dy}$ with constant $\eta$. Newtonian fluids exhibit a linear relationship between shear stress and velocity gradient, maintaining constant viscosity independent of shear rate.

11. What is the velocity profile $v(r)$ for laminar Poiseuille flow in a tube of radius $R$?

    $v(r)=\frac{\Delta P}{4\eta L}\left(R^2-r^2\right)$. The parabolic profile results from viscous drag, with velocity maximum at the center and zero at walls due to no-slip.

12. If tube radius doubles ($r\to 2r$), by what factor does $Q$ change (Poiseuille, constant $\Delta P$)?

    $Q$ increases by a factor of $16$. Doubling radius raises flow rate by $2^4=16$ due to the strong geometric dependence in Poiseuille's equation.

13. If tube length halves ($L\to \frac{L}{2}$), by what factor does $Q$ change (Poiseuille, constant $\Delta P$)?

    $Q$ increases by a factor of $2$. Halving length halves resistance, doubling flow rate since flow is inversely proportional to length in Poiseuille's law.

14. What is the relation between average flow speed $v_{\text{avg}}$, flow rate $Q$, and radius $r$?

    $v_{\text{avg}}=\frac{Q}{\pi r^2}$. Average flow speed equals volumetric flow rate divided by cross-sectional area, assuming uniform density in incompressible fluids.

15. If pressure difference doubles ($\Delta P\to 2\Delta P$), by what factor does $Q$ change (Poiseuille)?

    $Q$ increases by a factor of $2$. Doubling pressure gradient linearly increases driving force, thus doubling volumetric flow rate in laminar flow.

16. What is the definition of kinematic viscosity $\nu$ in terms of $\eta$ and density $\rho$?

    $\nu=\frac{\eta}{\rho}$. Kinematic viscosity represents the ratio of dynamic viscosity to fluid density, indicating momentum diffusivity.

17. State the volumetric flow rate (Poiseuille) equation for laminar flow in a cylindrical tube.

    $Q=\frac{\pi r^4\Delta P}{8\eta L}$. Poiseuille's equation describes laminar flow rate driven by pressure difference, inversely affected by viscosity and length while strongly dependent on radius to the fourth power.

18. What is the definition of dynamic viscosity $\eta$ in terms of shear stress and velocity gradient?

    $\eta=\frac{\tau}{\frac{dv}{dy}}$. Dynamic viscosity quantifies a fluid's resistance to shear, defined as the shear stress divided by the rate of change of velocity perpendicular to the flow direction.

19. Two identical tubes are placed in parallel; by what factor does total resistance $R$ change versus one tube?

    Total $R$ decreases by a factor of $2$. Parallel connection halves effective resistance as reciprocals add, increasing overall flow capacity for identical tubes.

20. Using Poiseuille flow, how does $Q$ scale with viscosity $\eta$ (all else constant)?

    $Q\propto\frac{1}{\eta}$. Viscosity resists fluid motion, so flow rate decreases inversely with increasing viscosity under constant pressure and geometry.

21. What is the shear stress magnitude at the tube wall in laminar flow with pressure drop $\Delta P$ over length $L$?

    $\tau_w=\frac{\Delta P\,R}{2L}$. Wall shear stress balances the pressure force over the tube's length, derived from force equilibrium in cylindrical coordinates.

22. Identify the approximate Reynolds number threshold above which pipe flow tends to become turbulent.

    Turbulence tends to occur for $\text{Re}\gtrsim 2000$. The threshold indicates transition from laminar to turbulent regimes, where inertial forces dominate viscous damping in pipe flows.

23. Using Poiseuille flow, how does $Q$ scale with tube radius $r$ (all else constant)?

    $Q\propto r^4$. In Poiseuille flow, the fourth-power dependence on radius arises from the integration of the parabolic velocity profile over the tube's cross-section.

24. Using Poiseuille flow, how does $Q$ scale with pressure difference $\Delta P$ (all else constant)?

    $Q\propto\Delta P$. Pressure difference drives flow, resulting in direct proportionality with flow rate in laminar conditions per Poiseuille's law.

25. Using Poiseuille flow, how does $Q$ scale with tube length $L$ (all else constant)?

    $Q\propto\frac{1}{L}$. Longer tubes increase frictional losses, making flow rate inversely proportional to length for a given pressure drop.

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