Viscosity

Viscosity is a fundamental physical property of fluids (both liquids and gases) that quantifies their internal resistance to flow. It acts as an internal friction between adjacent layers of fluid that are in relative motion, converting kinetic energy into thermal energy.

Microscopic Origin of Viscosity

When a fluid flows over a fixed solid horizontal surface, the fluid layer in direct contact with the solid surface adheres to it due to strong adhesive forces and remains stationary (zero velocity). As the vertical distance from the solid surface increases, the velocity of the subsequent fluid layers increases progressively, reaching a maximum at the top layer. This sequential variation creates a velocity gradient. Viscosity arises due to strong inter-molecular cohesive forces in liquids, and molecular momentum transfer due to collisions in gases, resisting the relative sliding motion between these parallel layers.

Newton’s Law of Viscosity

For a streamlined, laminar flow of a fluid, the tangential viscous force (F) acting between two adjacent layers is directly proportional to the surface area (A) of the layers and the velocity gradient (dv/dx) perpendicular to the direction of flow.

Where:

• F represents the internal viscous dragging force.
• A represents the cross-sectional area of the fluid layers in contact.
• dv/dx represents the velocity gradient (change in velocity dv per unit distance dx).
• η (eta) is the constant of proportionality known as the Coefficient of Viscosity.
• The negative sign indicates that the viscous force acts in a direction opposite to the fluid flow.

Units and Dimensions of Coefficient of Viscosity (η)

• SI Unit: Pascal-second (Pa⋯) or Kilogram per meter-second (kg·m^-1· s^-1), also termed as Poiseuille (Pl).
• CGS Unit: Poise (P).
• Conversion Factor: 1 Pascal-second = 10 Poise.
• Dimensional Formula: [M^1L-1T^-1].

Fluid Classification Based on Viscous Behavior

Fluids are categorized into two primary groups based on how their viscosity responds to the applied shear stress or rate of deformation.

Newtonian Fluids

Fluids that strictly obey Newton’s law of viscosity, where the coefficient of viscosity (η) remains constant regardless of the speed or stress applied during mixing or pumping.

• Examples: Water, alcohol, mineral oils, gasoline, and true solutions.

Non-Newtonian Fluids

Fluids whose viscosity changes dynamically depending on the shear stress applied to them. They do not maintain a constant coefficient of viscosity.

• Thixotropic / Shear-Thinning Fluids: Viscosity decreases as stress increases (e.g., Blood, modern paints, tomato ketchup, and quicksand).
• Dilatant / Shear-Thickening Fluids: Viscosity increases rapidly as stress increases (e.g., Cornstarch mixed with water, also known as Oobleck).

Physical Variables Modifying Viscosity

Impact of Temperature Variation

• In Liquids: Viscosity is inversely proportional to temperature. As temperature rises, the thermal kinetic energy of the liquid molecules increases, weakening the inter-molecular cohesive forces holding the layers together. Consequently, the liquid flows more easily (e.g., honey flows smoothly when heated).
• In Gases: Viscosity is directly proportional to the square root of absolute temperature (η ∝ √(T)). The viscosity of gases does not depend on cohesive forces but on molecular momentum transfer. Heating a gas causes the molecules to move faster and collide more frequently, increasing the resistance to structural flow.

Impact of Pressure Variation

• In Liquids: Generally, an increase in pressure forces liquid molecules closer together, strengthening cohesive bonds and increasing viscosity. (Exception: Water exhibits a anomalous decrease in viscosity under light initial pressure increases).
• In Gases: The viscosity of gases is practically independent of pressure variations under normal operational limits, as gas density changes compensate for mean free path variations.

Critical Velocity and Reynolds Number (R_e)

The nature of fluid flow through a conduit changes drastically depending on its velocity, transitioning from structured alignment to chaotic turbulence.

Types of Fluid Flow

• Streamline / Laminar Flow: A smooth, orderly flow where every fluid particle passing a particular point moves along exactly the same path as the preceding particles.
• Turbulent Flow: A chaotic, irregular flow characterized by eddies, swirls, and rapid mixing when the fluid exceeds a threshold velocity.

Critical Velocity (v_c)

The maximum velocity up to which a fluid flow remains streamlined. If the fluid velocity exceeds this critical value, the flow instantly transforms into turbulent flow.

Reynolds Number (R_e)

A dimensionless parameter used by engineers to predict the flow regime (laminar or turbulent) of a fluid moving inside a pipe. It represents the ratio of inertial forces to viscous forces within the fluid:

Where:

• ρ is the density of the fluid.
• v is the velocity of the fluid flow.
• d is the internal diameter of the pipe.
• η is the coefficient of viscosity.

Flow Regime Indicators

• If R_e < 1000, the fluid flow is strictly laminar/streamlined. Viscous forces dominate.
• If 1000 < R_e < 2000, the flow is unsteady and transitioning between states.
• If R_e > 2000, the fluid flow becomes completely turbulent. Inertial forces dominate.

Stokes’ Law and Terminal Velocity

When a solid spherical body moves through a viscous fluid medium, it experiences a retarding force due to fluid friction.

Statement of Stokes’ Law

According to George Gabriel Stokes, the backward viscous drag force (F) acting on a perfectly smooth spherical body of radius r moving with a velocity v through an infinite, homogeneous fluid of viscosity η is given by:

Terminal Velocity (v_t)

When an object falls through a viscous fluid, it initially accelerates due to gravity. As its velocity increases, the upward viscous drag force grows proportionally (per Stokes’ Law). Eventually, the upward forces (viscous drag + buoyant force) exactly balance the downward gravitational weight of the object. At this equilibrium point, the net force acting on the body becomes zero, the acceleration ceases, and the object continues to fall at a constant maximum velocity called Terminal Velocity.

Mathematical Expression for Terminal Velocity

Where:

• r is the radius of the falling spherical body.
• ρ_o is the absolute density of the solid object.
• ρ_f is the absolute density of the fluid medium.
• g is the acceleration due to gravity.
• η is the coefficient of viscosity of the fluid.

Crucial Inference

Terminal velocity is directly proportional to the square of the object’s radius (v_t ∝ r²). Thus, larger particles fall significantly faster through a fluid medium than smaller particles of identical material.

UPSC High-Yield Scientific Trivia

Mechanics of Parachute Descents

When a skydiver jumps from an aircraft, gravity causes them to accelerate rapidly. When they deploy a parachute, the vast surface area dramatically increases the air resistance (viscous drag of air). Because the upward drag force quickly matches the downward weight, the skydiver achieves a safe, slow terminal velocity of only a few meters per second, permitting a secure landing.

Formation of Raindrops and Cloud Mechanics

Clouds consist of millions of microscopic water droplets. Due to their minute radius (r), their calculated terminal velocity according to Stokes’ Law (v_t ∝ r²) is extremely small — often less than a few centimeters per second. Because this falling speed is so low, normal updrafts of wind are sufficient to keep cloud droplets suspended in the sky. They only fall as rain when droplets coalesce into larger, heavier drops with a high terminal velocity.

Engine Oil Viscosity Ratings (SAE Nomenclature)

Motor oils used in automobiles utilize standardized alphanumeric classification tags such as “SAE 10W-40”. The “10W” (Winter) rating indicates the fluid’s low-temperature viscosity capability, ensuring the oil remains thin enough to circulate quickly during cold engine startups. The “40” rating indicates the high-temperature viscosity stability, ensuring the oil does not become dangerously thin under intense engine heat, thereby maintaining a protective lubricating film over moving metal components.

Volcanic Lava Dynamics

The morphology of volcanic eruptions depends directly on the viscosity of the magma. Mafic/Basaltic lava is rich in iron and magnesium but low in silica, giving it a very low viscosity. It flows rapidly over great distances, forming broad, gently sloping shield volcanoes (e.g., Hawaiian volcanoes). In contrast, Felsic/Rhyolitic lava is rich in silica, making it highly viscous. It resists flow and traps volcanic gases, leading to explosive, violent eruptions that build steep composite volcanoes (e.g., Mount Vesuvius).

Blood Viscosity and Clinical Diagnostics

Human blood is roughly 3 to 4 times more viscous than water, primarily due to suspended Red Blood Cells (RBCs). An abnormal increase in blood viscosity (Hyperviscosity syndrome), often caused by polycythemia or high plasma protein levels, drastically increases the internal friction against artery walls. This forces the heart to pump harder to maintain systemic circulation, leading to high blood pressure (hypertension) and elevated risks of cardiovascular strokes.

Last Modified: May 27, 2026