Viscosity, Fluid Drag and Terminal Velocity
Ideal vs. Real Fluids and Types of Flow
Fluid flow is categorized as Streamline (laminar) or Turbulent based on the stability of particle paths and velocity consistency.
$$R_e = \frac{\rho v D}{\eta}$$
The Reynolds Number predicts flow regime transitions.
$R_e$=Reynolds Number(dimensionless)
$\rho$=Fluid density(kg/m³)
$v$=Flow velocity(m/s)
$D$=Characteristic length (diameter)(m)
$\eta$=Dynamic viscosity(Pa·s)
$R_e < 2000$
→Laminar flow — smooth, predictable streamlines
$2000 < R_e < 3000$
→Transition region — flow is unstable
$R_e > 3000$
→Turbulent flow — chaotic eddies and mixing
Steady Flow: Particles passing a point follow identical paths, ensuring constant velocity over time at that location. Streamlines never cross in steady flow.
Turbulence: Occurs at high velocities or low viscosities, characterized by chaotic eddies and vortices. Exact particle paths become unpredictable.
Ideal Fluid: Assumed to be Incompressible, non-viscous, and irrotational for simplified modeling. Real fluids always have some viscosity.
Dimensional Check: $R_e = \frac{[\text{kg/m}^3][\text{m/s}][\text{m}]}{[\text{Pa·s}]}$ — all units cancel, confirming it is dimensionless.
The Equation of Continuity
The Equation of Continuity expresses the Conservation of Mass for an Incompressible fluid moving through a channel of varying width.
$$A_1 v_1 = A_2 v_2$$
The volume flow rate remains constant throughout a closed system.
$A$=Cross-sectional area(m²)
$v$=Fluid velocity(m/s)
$Av$=Volume flow rate(m³/s)
$A_2 = A_1/2$
→$v_2 = 2v_1$ — halving the area doubles the speed
$A_2 = A_1/4$
→$v_2 = 4v_1$ — quartering area quadruples speed (diameter halved)
Flow Rate: The product $Av$ represents the volume of fluid passing a section per unit time (m³/s). This remains constant at every cross-section.
Incompressibility: Assumes density $\rho$ is uniform, making mass conservation ($\rho_1 A_1 v_1 = \rho_2 A_2 v_2$) simplify to volume conservation.
Falling Water Stream: Water from a tap narrows as it falls — gravity increases its speed $v$, so continuity requires the cross-sectional area $A$ to decrease.
Bernoulli's Principle
Bernoulli's Principle relates pressure, speed, and height, fundamentally stating that an increase in fluid speed coincides with a decrease in Static Pressure. It applies to ideal (non-viscous, incompressible, steady) fluid flow.
$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
An expression of energy conservation for flowing fluids — each term has units of pressure (Pa).
$P$=Static Pressure(Pa)
$\frac{1}{2}\rho v^2$=Dynamic Pressure (KE per unit volume)(Pa)
$\rho gh$=Hydrostatic Pressure (PE per unit volume)(Pa)
$h_1 = h_2$ (horizontal flow)
→$P + \frac{1}{2}\rho v^2 = \text{const}$ — pressure and speed trade off directly
$v_1 = v_2$ (uniform pipe)
→$P + \rho gh = \text{const}$ — reduces to hydrostatic pressure equation
Venturi Effect: Fluid pressure drops in a constricted section of a pipe where velocity is higher. Used in Venturi meters, carburetors, and atomizers.
Energy Components: Each term represents energy per unit volume — static pressure is internal energy, $\frac{1}{2}\rho v^2$ is kinetic, $\rho gh$ is gravitational potential.
Dimensional Check: All three terms have dimensions of $[\text{Pa}] = [\text{kg·m}^{-1}\text{·s}^{-2}]$, confirming dimensional consistency.
Viscosity and Stokes' Law
Viscosity ($\eta$) represents internal friction between adjacent fluid layers. It measures how much force is needed to slide one layer of fluid over another.
High Viscosity: Thick fluids like honey and glycerin ($\eta \approx 630 \times 10^{-3}$ Pa·s) resist flow strongly.
Low Viscosity: Water ($\eta \approx 0.8 \times 10^{-3}$ Pa·s) and air ($\eta \approx 0.019 \times 10^{-3}$ Pa·s) flow easily.
Temperature Dependence: Viscosity decreases with temperature in liquids (molecules move apart) but increases in gases (more molecular collisions).
Units: The SI unit of $\eta$ is Pa·s (= kg·m⁻¹·s⁻¹ = N·s·m⁻²). All are equivalent.
Stokes' Law quantifies the Drag Force on a sphere moving slowly through a viscosity-dominated medium under laminar conditions.
$$F_d = 6\pi \eta r v$$
Calculates the retarding force on a small sphere in laminar (low Reynolds number) conditions.
$F_d$=Viscous Drag Force(N)
$\eta$=Coefficient of dynamic viscosity(Pa·s)
$r$=Radius of sphere(m)
$v$=Velocity of sphere relative to fluid(m/s)
$v \to 0$
→$F_d \to 0$ — no drag when stationary
High speed ($R_e \gg 1$)
→Stokes' Law breaks down — drag becomes proportional to $v^2$
Linear Dependence: Unlike air resistance at high speeds ($\propto v^2$), Stokes' drag is proportional to $v$ — valid only for slow, laminar flow around the sphere.
Dimensional Check: $[F_d] = [\text{Pa·s}][\text{m}][\text{m/s}] = [\text{kg·m}^{-1}\text{·s}^{-1}][\text{m}][\text{m·s}^{-1}] = [\text{kg·m·s}^{-2}] = [\text{N}]$ ✓
Terminal Velocity
Terminal Velocity is the constant speed a falling object reaches when the net downward force becomes zero — the weight is exactly balanced by the upward drag force (and buoyancy in a fluid).
$$v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$$
Derived by setting $mg = F_d + F_b$ (weight = Stokes' drag + buoyancy) and solving for $v$.
$v_t$=Terminal velocity(m/s)
$r$=Radius of sphere(m)
$\rho_s$=Density of sphere(kg/m³)
$\rho_f$=Density of fluid(kg/m³)
$g$=Gravitational acceleration(m/s²)
$\eta$=Fluid viscosity(Pa·s)
$\rho_s = \rho_f$ (neutral buoyancy)
→$v_t = 0$ — object floats, no net downward force
$\rho_f \approx 0$ (falling through air)
→Simplifies to $v_t = \frac{2r^2 \rho_s g}{9\eta}$ (buoyancy negligible)
$\eta \to 0$ (no viscosity)
→$v_t \to \infty$ — object accelerates indefinitely (free fall)
Derivation: At terminal velocity: $\frac{4}{3}\pi r^3 \rho_s g = 6\pi \eta r v_t + \frac{4}{3}\pi r^3 \rho_f g$. Rearranging gives the formula above.
Approach to Terminal Velocity: A dropped sphere initially accelerates at $g$, then acceleration decreases exponentially as drag builds up, asymptotically reaching $v_t$. It never truly reaches $v_t$ but gets arbitrarily close.