Viscosity Calculator (Falling Sphere Method)

Easily calculate dynamic and kinematic viscosity using our Viscosity Calculator based on Stokes’ Law for a falling sphere. Enter the sphere and fluid properties to get results.

Viscosity Calculator

Typical Viscosity Values

Fluid	Temperature (°C)	Dynamic Viscosity (mPa·s)	Kinematic Viscosity (cSt or mm²/s)
Water	20	1.002	1.004
Water	100	0.282	0.294
Air	20	0.018	15.06
Glycerol	20	1410	1184
Honey	20	2000 – 10000	~1400 – 7000
Engine Oil (SAE 10W-30)	20	~200	~220
Engine Oil (SAE 10W-30)	100	~10	~11
Mercury	20	1.526	0.113

Approximate viscosity values for common fluids at specified temperatures.

What is Viscosity?

Viscosity is a fundamental property of fluids (liquids and gases) that measures their resistance to flow or deformation when subjected to shear stress or tensile stress. In simpler terms, it describes how “thick” or “sticky” a fluid is. For example, honey has a much higher viscosity than water, meaning it flows more slowly and resists movement more strongly. The Viscosity Calculator helps quantify this property.

Viscosity arises from the internal friction between adjacent layers of fluid that are moving at different velocities. The stronger the intermolecular forces within the fluid, the greater the resistance to flow, and thus the higher the viscosity.

Who Should Use a Viscosity Calculator?

A Viscosity Calculator is useful for:

• Engineers (Chemical, Mechanical, Civil): For designing pipelines, pumps, and mixing processes, and in lubrication applications.
• Scientists (Physicists, Chemists, Biologists): In fluid dynamics research, material science, and understanding biological fluids.
• Food Industry Professionals: To control the texture and flow properties of products like sauces, syrups, and dairy.
• Pharmaceutical Industry: For formulation of liquid medicines and creams.
• Paint and Coatings Industry: To ensure proper application properties.
• Oil and Gas Industry: For characterizing crude oils and drilling fluids.

Common Misconceptions

One common misconception is confusing viscosity with density. Density is the mass per unit volume of a substance, while viscosity is the resistance to flow. A fluid can be very dense but have low viscosity (like mercury), or have low density but high viscosity (like some gels). The Viscosity Calculator focuses solely on the flow resistance aspect.

Viscosity Formula and Mathematical Explanation

This Viscosity Calculator uses Stokes’ Law, which describes the force resisting the motion of a small sphere moving through a viscous fluid at low Reynolds numbers. When the sphere reaches terminal velocity (constant speed), the drag force due to viscosity balances the net force of gravity and buoyancy.

The formula for dynamic viscosity (η) derived from Stokes’ Law is:

η = (2 * g * r² * (ρ_s – ρ_f)) / (9 * v)

Where:

• η (eta) is the dynamic viscosity.
• g is the acceleration due to gravity (9.80665 m/s²).
• r is the radius of the sphere.
• ρ_s (rho_s) is the density of the sphere.
• ρ_f (rho_f) is the density of the fluid.
• v is the terminal velocity of the sphere.

Once dynamic viscosity (η) is known, kinematic viscosity (ν, nu) can be calculated as:

ν = η / ρ_f

Kinematic viscosity is the ratio of dynamic viscosity to fluid density.

Variables Table

Variable	Meaning	Unit (SI)	Typical Range (for calculator)
r	Radius of the sphere	m (meters)	0.0005 – 0.005 m (0.5 – 5 mm)
ρs	Density of the sphere	kg/m³	1000 – 20000 kg/m³
ρf	Density of the fluid	kg/m³	1 – 10000 kg/m³
v	Terminal velocity	m/s	0.0001 – 0.1 m/s (0.1 – 100 mm/s)
g	Acceleration due to gravity	m/s²	9.80665 m/s² (constant)
η	Dynamic viscosity	Pa·s (Pascal-seconds)	Varies widely
ν	Kinematic viscosity	m²/s	Varies widely

Practical Examples (Real-World Use Cases)

Example 1: Measuring Glycerol Viscosity

A small steel sphere (radius 1 mm, density 7850 kg/m³) is dropped into glycerol (density 1260 kg/m³). It reaches a terminal velocity of 2.1 mm/s.

• Sphere Radius (r): 1 mm = 0.001 m
• Sphere Density (ρ_s): 7850 kg/m³
• Fluid Density (ρ_f): 1260 kg/m³
• Terminal Velocity (v): 2.1 mm/s = 0.0021 m/s

Using the Viscosity Calculator (or formula):

η = (2 * 9.81 * (0.001)² * (7850 – 1260)) / (9 * 0.0021) ≈ 1.41 Pa·s = 1410 mPa·s

ν = 1.41 / 1260 ≈ 0.00112 m²/s = 1120 cSt

This result is close to the known viscosity of glycerol around 20°C.

Example 2: Estimating Light Oil Viscosity

A glass sphere (radius 2 mm, density 2500 kg/m³) falls through a light oil (density 850 kg/m³) with a terminal velocity of 50 mm/s.

• Sphere Radius (r): 2 mm = 0.002 m
• Sphere Density (ρ_s): 2500 kg/m³
• Fluid Density (ρ_f): 850 kg/m³
• Terminal Velocity (v): 50 mm/s = 0.05 m/s

The Viscosity Calculator would show:

η = (2 * 9.81 * (0.002)² * (2500 – 850)) / (9 * 0.05) ≈ 0.288 Pa·s = 288 mPa·s

ν = 0.288 / 850 ≈ 0.000339 m²/s = 339 cSt

This suggests a moderately viscous oil.

How to Use This Viscosity Calculator

1. Enter Sphere Radius: Input the radius of the sphere used in the experiment in millimeters (mm).
2. Enter Sphere Density: Input the density of the material the sphere is made of in kilograms per cubic meter (kg/m³).
3. Enter Fluid Density: Input the density of the fluid whose viscosity you are measuring in kg/m³.
4. Enter Terminal Velocity: Input the constant velocity the sphere reaches as it falls through the fluid in millimeters per second (mm/s). This is typically measured by timing the sphere’s fall over a known distance after it has reached constant speed.
5. Calculate: Click the “Calculate” button or simply change input values.
6. Read Results: The calculator will display:
   • Dynamic Viscosity (η): The primary result, shown in Pascal-seconds (Pa·s) and milliPascal-seconds (mPa·s).
   • Kinematic Viscosity (ν): Shown in square meters per second (m²/s) and centiStokes (cSt).
   • Reynolds Number (Re): An indicator of the flow regime. Stokes’ Law is most accurate for Re < 1.
   • Drag Force: At terminal velocity, this equals the net gravitational force.
7. Reset: Use the “Reset” button to clear inputs to default values.
8. Copy: Use the “Copy Results” button to copy the inputs and results to your clipboard.

Ensure all input values are positive and realistic. The Viscosity Calculator will provide error messages for invalid inputs.

Key Factors That Affect Viscosity Results

1. Temperature: This is the most significant factor. For liquids, viscosity generally decreases sharply as temperature increases. For gases, viscosity increases with temperature. Accurate temperature control and measurement are crucial for reliable viscosity measurements.
2. Pressure: For most liquids, viscosity increases slightly with increasing pressure, but the effect is usually small at moderate pressures. For gases, viscosity is largely independent of pressure at low to moderate pressures but increases at very high pressures.
3. Fluid Composition: The chemical nature of the fluid and the presence of dissolved or suspended substances strongly influence viscosity. Impurities or changes in concentration can alter viscosity significantly.
4. Shear Rate (for Non-Newtonian fluids): The calculator assumes a Newtonian fluid, where viscosity is independent of shear rate (how fast the fluid is being deformed). However, many fluids (like ketchup, paint, blood) are non-Newtonian, and their viscosity changes with shear rate. Stokes’ Law and this calculator are less suitable for strongly non-Newtonian fluids without modification. You might explore rheology for more.
5. Accuracy of Measurements: The precision of the sphere’s radius, densities, and especially the terminal velocity measurement directly impacts the accuracy of the calculated viscosity. Small errors in radius are magnified because it is squared in the formula.
6. Sphere and Container Size: If the sphere is too large compared to the container holding the fluid, wall effects can influence the terminal velocity and thus the calculated viscosity. The container diameter should be much larger than the sphere diameter. Our fluid dynamics basics page discusses this.
7. Terminal Velocity Achievement: Ensure the sphere has truly reached terminal velocity before measurements are taken. It needs sufficient distance to fall and accelerate to a constant speed.

Understanding these factors is crucial when using the Viscosity Calculator and interpreting its results, especially when comparing with standard values or literature data obtained under specific conditions. You might also find our temperature conversion tool useful.

Frequently Asked Questions (FAQ)

What is the difference between dynamic and kinematic viscosity?

Dynamic viscosity (or absolute viscosity, η) measures the fluid’s internal resistance to flow under shear stress. Kinematic viscosity (ν) is the ratio of dynamic viscosity to the fluid’s density (ν = η/ρ). Kinematic viscosity is often more convenient in fluid dynamics calculations involving the Reynolds number.

What units are used for viscosity?

The SI unit for dynamic viscosity is Pascal-second (Pa·s) or kg/(m·s). Common non-SI units include Poise (P, 1 P = 0.1 Pa·s) and centiPoise (cP, 1 cP = 1 mPa·s). The SI unit for kinematic viscosity is m²/s. Common non-SI units include Stokes (St, 1 St = 10⁻⁴ m²/s) and centiStokes (cSt, 1 cSt = 1 mm²/s = 10⁻⁶ m²/s).

Why does the calculator mention Reynolds number (Re)?

Stokes’ Law, used by this Viscosity Calculator, is derived assuming laminar flow around the sphere, which typically occurs at low Reynolds numbers (Re < 1). As Re increases, the flow becomes more turbulent, and Stokes' Law becomes less accurate. The calculator warns you if Re is significantly above 1.

How does temperature affect viscosity?

For liquids, viscosity generally decreases exponentially with increasing temperature because the increased kinetic energy of molecules overcomes intermolecular forces. For gases, viscosity increases with temperature due to more frequent molecular collisions. This calculator does not explicitly account for temperature; you must know the fluid density at the measurement temperature.

What is a Newtonian fluid?

A Newtonian fluid is one where the viscosity is constant regardless of the shear rate or agitation applied to it (at a constant temperature and pressure). Water, air, and many oils behave as Newtonian fluids under many conditions. This calculator assumes the fluid is Newtonian.

What about non-Newtonian fluids?

Non-Newtonian fluids (e.g., ketchup, paint, blood, polymer solutions) have viscosities that change with the applied shear rate. Their behavior is more complex and cannot be fully described by a single viscosity value using this simple falling sphere method without further considerations.

Can I use this calculator for gases?

While the principle applies, the falling sphere method is practically very difficult for gases due to their low viscosity and density, leading to very slow terminal velocities or requiring very small, light spheres and extremely precise measurements. Other methods are usually preferred for gases.

What are typical sphere materials?

Steel, glass, or ceramic balls are often used due to their well-defined densities and spherical shape. The material should be denser than the fluid and chemically inert with it.