Fluid Properties and Hydrostatics (4B) - MCAT Chemical and Physical Foundations of Biological Systems

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

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

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

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

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

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

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

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

$Q$ 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.

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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