Viscosity and Poiseuille Flow (4B) - MCAT Chemical and Physical Foundations of Biological Systems

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

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

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

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

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.

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.

$Q$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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