MCAT Chemical and Physical Foundations of Biological Systems Flashcards: 4b Viscosity Poiseuille Flow

What this deck covers

This deck focuses on 4b Viscosity Poiseuille Flow, giving you a quick way to review the definitions, rules, and examples that matter most for MCAT Chemical and Physical Foundations of Biological Systems.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the shear stress magnitude at the tube wall in laminar flow with pressure drop $\Delta P$ over length $L$?

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

Flashcard 2: What is the Reynolds number formula for flow in a tube using density $\rho$, speed $v$, diameter $D$, and viscosity $\eta$?

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

Flashcard 3: Which flow regime is typically assumed for Poiseuille flow: laminar or turbulent?

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

Flashcard 4: Using Poiseuille flow, how does $Q$ scale with pressure difference $\Delta P$ (all else constant)?

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

Flashcard 5: What is the relationship between maximum and average speed in Poiseuille flow?

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

Flashcard 6: What boundary condition at the wall is assumed in Poiseuille flow (name the condition)?

Answer: 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.

Flashcard 7: Identify the approximate Reynolds number threshold above which pipe flow tends to become turbulent.

Answer: 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.

Flashcard 8: If tube radius doubles ($r\to 2r$), by what factor does $Q$ change (Poiseuille, constant $\Delta P$)?

Answer: $Q$ 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.

Flashcard 9: If viscosity triples ($\eta\to 3\eta$), by what factor does $Q$ change (Poiseuille, constant $\Delta P$)?

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

Flashcard 10: If tube length halves ($L\to \frac{L}{2}$), by what factor does $Q$ change (Poiseuille, constant $\Delta P$)?

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

Flashcard 11: If pressure difference doubles ($\Delta P\to 2\Delta P$), by what factor does $Q$ change (Poiseuille)?

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

Flashcard 12: Two identical tubes are placed in series; by what factor does total resistance $R$ change versus one tube?

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

Flashcard 13: What is the velocity profile $v(r)$ for laminar Poiseuille flow in a tube of radius $R$?

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

Flashcard 14: What is the definition of dynamic viscosity $\eta$ in terms of shear stress and velocity gradient?

Answer: $\eta=\frac{\tau}{\frac{dv}{dy}}$. 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.

Flashcard 15: What SI unit is used for dynamic viscosity $\eta$?

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

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

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

Flashcard 17: What SI unit is used for kinematic viscosity $\nu$?

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

Flashcard 18: What does it mean for a fluid to be Newtonian regarding the relation between $\tau$ and $\frac{dv}{dy}$?

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

Flashcard 19: State the volumetric flow rate (Poiseuille) equation for laminar flow in a cylindrical tube.

Answer: $Q=\frac{\pi r^4\Delta P}{8\eta L}$. 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.

Flashcard 20: State the hydraulic resistance $R$ for laminar flow in a cylindrical tube, defined by $\Delta P=QR$.

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

Flashcard 21: What is the relation between average flow speed $v_{\text{avg}}$, flow rate $Q$, and radius $r$?

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

Flashcard 22: Using Poiseuille flow, how does $Q$ scale with tube radius $r$ (all else constant)?

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

Flashcard 23: Using Poiseuille flow, how does $Q$ scale with viscosity $\eta$ (all else constant)?

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

Flashcard 24: Using Poiseuille flow, how does $Q$ scale with tube length $L$ (all else constant)?

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

Flashcard 25: Two identical tubes are placed in parallel; by what factor does total resistance $R$ change versus one tube?

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