Viscosity Calculator

Newton's Law of Viscosity and Fundamental Definition

Dynamic viscosity (\eta) is defined through Newton's law of viscosity:

\tau = \eta(dv)/(dy)

where: - \tau = (F)/(A) = shear stress (Pa) - \eta = dynamic viscosity (Pa·s) - (dv)/(dy) = velocity gradient or shear rate (s⁻¹)

**Rearranging for Viscosity**

\eta = (\tau)/(dv/dy) = (F/A)/(dv/dy)

**Physical Interpretation** - Viscosity measures **resistance to flow** - Higher viscosity → more resistance → slower flow - Viscosity relates **stress to strain rate** - Fundamental property of fluid behavior

**Units** - **SI**: Pa·s (Pascal-seconds) or N·s/m² - **CGS**: Poise (P) or centipoise (cP) - 1 P = 0.1 Pa·s - 1 cP = 0.001 Pa·s = 1 mPa·s - **Imperial**: lb·s/ft²

**Newtonian vs Non-Newtonian** - **Newtonian fluids**: Viscosity is constant (water, air, most simple fluids) - **Non-Newtonian fluids**: Viscosity depends on shear rate (paint, blood, ketchup)

**Physical Meaning** Viscosity represents the "thickness" or "stickiness" of a fluid: - **High viscosity**: Thick, flows slowly (honey, molasses) - **Low viscosity**: Thin, flows easily (water, air) - **Zero viscosity**: Ideal fluid (no internal friction)

\eta = (\tau)/(dv/dy)

Molecular Origin of Viscosity

Viscosity arises from intermolecular forces and momentum transfer:

**Mechanism** - Molecules in adjacent layers move at different speeds - **Momentum transfer** between layers - **Intermolecular forces** resist relative motion - Creates **internal friction**

**For Gases** - Viscosity arises from **molecular collisions** - Faster molecules transfer momentum to slower layers - Increases with temperature (more collisions) - Independent of pressure (at moderate pressures)

**For Liquids** - Viscosity arises from **intermolecular forces** - Molecules must overcome attractive forces to flow - Decreases with temperature (weaker forces) - Strongly dependent on molecular structure

**Temperature Dependence**

**Gases**: \eta(T) ≈ \eta_0√(\fracT)T_0

Viscosity **increases** with temperature.

**Liquids**: \eta(T) = \eta_0 e^B/T

Viscosity **decreases** with temperature (exponential).

**Pressure Dependence** - **Gases**: Weak dependence (at moderate pressures) - **Liquids**: Weak dependence (except at very high pressures) - **High pressures**: Significant increases possible

Measurement Methods

Various methods measure viscosity:

**1. Capillary Viscometers** - Measure flow time through capillary - Poiseuille's law relates to viscosity - Simple and accurate - Common: Ubbelohde, Ostwald

**2. Rotational Viscometers** - Rotating cylinder or cone - Measure torque required - Works for non-Newtonian fluids - Can measure at different shear rates

**3. Falling Ball Viscometers** - Measure terminal velocity of sphere - Stokes' law relates to viscosity - Simple but less accurate

**4. Vibrating Viscometers** - Measure damping of vibrations - Fast measurement - Good for process control

**5. Couette Flow** - Parallel plates with relative motion - Direct measurement of stress/strain - Very accurate - Research applications

Applications in Engineering and Science

Viscosity is crucial in many applications:

**1. Fluid Flow** - Determines flow resistance - Affects pressure drop - Influences pump requirements - Critical for pipe design

**2. Lubrication** - Prevents wear - Reduces friction - Carries away heat - Protects surfaces

**3. Heat Transfer** - Affects convective heat transfer - Influences boundary layers - Determines heat exchanger performance

**4. Mixing and Processing** - Affects mixing efficiency - Determines processing conditions - Influences product quality

**5. Biological Systems** - Blood flow - Cell mechanics - Drug delivery - Physiological processes

Kinematic Viscosity Relationship

Kinematic viscosity relates to dynamic viscosity:

\nu = (\eta)/(\rho)

where \nu is kinematic viscosity and \rho is density.

**Physical Significance** - Kinematic viscosity: Momentum diffusivity - Dynamic viscosity: Resistance to flow - Both important in different contexts

**Reynolds Number** Uses kinematic viscosity:

Re = (\rho UL)/(\eta) = (UL)/(\nu)

**Applications** - Flow characterization - Boundary layer analysis - Natural convection - Sedimentation

Worked Example

Inputs

• shearStress: 10
• velocityGradient: 1000

Result: Dynamic Viscosity: 0.01 Pa·s (10 cP)

Explanation

For a shear stress \tau = 10 Pa and velocity gradient dv/dy = 1000 s⁻¹:

Calculate dynamic viscosity: \eta = (\tau)/(dv/dy) \eta = (10)/(1000) \eta = 0.01 Pa·s = 10 cP

This is typical for a light oil at room temperature.

Second Scenario

Inputs

• shearStress: 7.5
• velocityGradient: 1000

Result: Dynamic Viscosity: 0.01 Pa·s (10 cP)

Explanation

This scenario uses different inputs (shearStress = 7.5, velocityGradient = 1000) to show how changing one variable affects the viscosity result. Run the calculator above with these values to get the exact updated output with step-by-step work.