Viscosity — Revision Notes

⚡ 30-Second Revision

Viscosity ($\eta$) — Internal resistance to fluid flow.
Newton's Law — $\tau = \eta \frac{dv}{dy}$ (Shear stress = Viscosity $\times$ Velocity gradient).
Units of $\eta$ — Pa s (SI), Poise (CGS, $1,\text{Poise} = 0.1,\text{Pa s}$). Dimensional formula: $[ML^{-1}T^{-1}]$.
Temperature Effect — Liquids: $\eta \downarrow$ as $T \uparrow$. Gases: $\eta \uparrow$ as $T \uparrow$.
Stokes' Law — Viscous drag $F_v = 6\pi\eta r v$ for a sphere.
Terminal Velocity — $v_t = \frac{2r^2(\rho - \rho_f)g}{9\eta}$. (Sphere density $\rho$, fluid density $\rho_f$).

2-Minute Revision

Viscosity is the internal friction within a fluid, resisting its flow. It's quantified by the coefficient of dynamic viscosity ($\eta$), which relates shear stress (force per unit area) to the velocity gradient (rate of change of velocity with distance perpendicular to flow).

The SI unit is Pascal-second (Pa s), and its dimensions are $[ML^{-1}T^{-1}]$. A key concept is the contrasting effect of temperature: liquid viscosity decreases with temperature due to weakened intermolecular forces, while gas viscosity increases due to enhanced molecular momentum transfer.

Stokes' Law describes the viscous drag force ($F_v = 6\pi\eta r v$) on a small sphere moving through a viscous fluid. This law is crucial for understanding terminal velocity, the constant maximum speed an object reaches when falling through a fluid, where its weight is balanced by buoyant force and viscous drag.

The terminal velocity formula, $v_t = \frac{2r^2(\rho - \rho_f)g}{9\eta}$, is frequently tested.

5-Minute Revision

Viscosity is a fluid's inherent resistance to flow, analogous to friction in solids. It arises from internal forces between fluid layers moving at different speeds. Newton's Law of Viscosity, $\tau = \eta \frac{dv}{dy}$, defines the coefficient of dynamic viscosity ($\eta$) as the ratio of shear stress (tangential force per unit area) to the velocity gradient (rate of change of velocity across layers).

The SI unit for $\eta$ is Pa s (or N s/m$^2$), and its dimensional formula is $[ML^{-1}T^{-1}]$. Remember $1,\text{Poise} = 0.1,\text{Pa s}$.

Temperature significantly affects viscosity, but differently for liquids and gases. For liquids, viscosity decreases with increasing temperature because higher thermal energy weakens the cohesive intermolecular forces. For gases, viscosity increases with temperature due to more frequent and energetic molecular collisions, leading to greater momentum transfer between layers.

Stokes' Law is fundamental for objects moving in viscous fluids: $F_v = 6\pi\eta r v$, where $F_v$ is the viscous drag force, $r$ is the sphere's radius, and $v$ is its velocity. This law is crucial for understanding terminal velocity ($v_t$).

When an object falls through a viscous fluid, it accelerates until the sum of the upward buoyant force ($F_B$) and viscous drag ($F_v$) equals its downward weight ($W$). At this point, the net force is zero, and it moves at a constant terminal velocity.

For a sphere, $v_t = \frac{2r^2(\rho - \rho_f)g}{9\eta}$, where $\rho$ is the sphere's density and $\rho_f$ is the fluid's density. This formula shows that $v_t$ is proportional to $r^2$ and inversely proportional to $\eta$.

Practice problems involving these formulas and conceptual questions on temperature effects.

Prelims Revision Notes

1
Definition — Viscosity is the internal friction of a fluid, resisting flow. It's the 'thickness' of a fluid.
2
Newton's Law of Viscosity — Shear stress ($\tau$) is proportional to velocity gradient ($\frac{dv}{dy}$). $\tau = \eta \frac{dv}{dy}$.
3
Coefficient of Dynamic Viscosity ($\eta$) — Constant of proportionality. SI unit: Pa s or N s/m$^2$. CGS unit: Poise (P). $1,\text{Poise} = 0.1,\text{Pa s}$.
4
Dimensional Formula of $\eta$ — $[ML^{-1}T^{-1}]$. Derived from $\eta = \frac{F/A}{dv/dy}$.
5
Effect of Temperature

* Liquids: Viscosity decreases as temperature increases (intermolecular forces weaken). * Gases: Viscosity increases as temperature increases (more molecular collisions, greater momentum transfer).

1
Effect of Pressure — Generally negligible for liquids; for gases, largely independent over a range, but increases at very high pressures.
2
Stokes' Law — Viscous drag force ($F_v$) on a small sphere of radius $r$ moving with velocity $v$ in a fluid of viscosity $\eta$: $F_v = 6\pi\eta r v$. Valid for laminar flow, small spheres, low speeds.
3
Terminal Velocity ($v_t$) — Constant maximum velocity attained by an object falling in a viscous fluid when net force is zero ($W = F_B + F_v$).
4
Terminal Velocity Formula (for sphere) — $v_t = \frac{2r^2(\rho - \rho_f)g}{9\eta}$.

* $\rho$: density of sphere. * $\rho_f$: density of fluid. * $g$: acceleration due to gravity.

1
Proportionalities — $v_t \propto r^2$, $v_t \propto (\rho - \rho_f)$, $v_t \propto 1/\eta$.
2
No-slip condition — Fluid layer in contact with a solid surface has zero relative velocity.
3
Laminar Flow in Pipe — Parabolic velocity profile; max at center, zero at walls.

Vyyuha Quick Recall

To remember the temperature effect on viscosity: Liquids Lower (viscosity with temp), Gases Grow (viscosity with temp). Think 'LLGG' for 'Liquids Lower, Gases Grow'.