Viscosity & Shear Stress Calculator

Calculate shear stress, viscosity, and flow velocity using Newton's law of viscosity.

Water=0.001, Honey=2-10, Air=1.8e-5
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Viscosity Relates Shear Stress to Velocity Gradient

For a Newtonian fluid, shear stress is proportional to the rate at which velocity changes across adjacent fluid layers. Newton’s law of viscosity is τ=μdu/dy. Dynamic viscosity μ measures resistance to shearing deformation, while kinematic viscosity ν=μ/ρ divides that resistance by density. Gases and liquids can have very different temperature dependence.

The velocity gradient is not the same as velocity. A fluid moving as a rigid plug with no gradient has no Newtonian viscous shear in its interior, while strong gradients near a stationary wall create shear stress. Non-Newtonian fluids do not have one constant μ independent of shear rate.

τ=μdu/dy,   ν=μ/ρ
SymbolMeaningWhy it appears / units
τShear stressPa=N/m².
μDynamic viscosityPa·s.
du/dyVelocity gradients⁻¹.
νKinematic viscositym²/s.

Large viscosity means more shear stress is required for the same velocity gradient. Reynolds number uses viscosity to compare inertial and viscous effects, so viscosity strongly influences whether a flow tends to remain laminar or become turbulent.

Viscosity units provide a strong consistency check. In Re=ρvD/μ, using dynamic viscosity μ requires density ρ. If kinematic viscosity ν is used instead, Re=vD/ν. Mixing those two forms double-counts or omits density and can shift the Reynolds number by orders of magnitude.

Worked Examples

Example 1: Water flowing between plates: η=0.001, dv/dy=100s⁻¹
τ=0.001×100
Result: τ=0.1 Pa — small shear stress in water
Water is a low-viscosity Newtonian fluid
Example 2: Kinematic viscosity of water: η=0.001, ρ=1000
ν=0.001/1000
Result: ν=10⁻⁶ m²/s = 1 cSt
Standard reference value
Example 3: Simple shear
μ=0.10Pa·s, du/dy=20s−1
Result: τ=2.0Pa
Stress scales directly with gradient for a Newtonian fluid.
Example 4: Kinematic viscosity
μ=1.0mPa·s, ρ=1000kg/m³
Result: ν=1.0×10−6m²/s
This is the familiar order of magnitude for water near room conditions.

Common Mistakes

⚠️
Confusing dynamic and kinematic viscosity

Dynamic viscosity μ has units Pa·s; kinematic viscosity ν=μ/ρ has units m²/s.

⚠️
Using velocity instead of velocity gradient

Viscous shear depends on how velocity changes with position, not on absolute bulk speed alone.

⚠️
Assuming every fluid is Newtonian

Paints, blood, polymer solutions, suspensions, and other fluids can have viscosity that depends on shear rate or time.

Frequently Asked Questions

Newtonian vs non-Newtonian fluids?
Newtonian: viscosity constant regardless of shear rate (water, air, oil). Non-Newtonian: shear-thinning (ketchup, blood, paint — gets easier to flow under stress), shear-thickening (cornstarch in water).
Temperature effect on viscosity?
Liquids: viscosity DECREASES with temperature (hydrogen bonds break). Gases: viscosity INCREASES with temperature (more molecular collisions). Oil viscosity at -20°C vs 100°C varies by 100× — critical for engine design.
Why is viscosity important near solid walls?
The no-slip condition makes fluid velocity match the wall, producing velocity gradients near the surface. Those gradients create viscous shear stress and drag.
How does temperature affect viscosity?
Liquid viscosity generally decreases strongly as temperature rises, while gas viscosity generally increases with temperature over ordinary ranges.
What is a Newtonian fluid?
It is a fluid for which shear stress is linearly proportional to shear rate with a viscosity that does not depend on that shear rate under the stated conditions.
How is viscosity related to Reynolds number?
Re=ρvL/μ. Larger viscosity lowers Reynolds number for the same density, speed, and length scale, making viscous effects more dominant relative to inertia.
How are dynamic and kinematic viscosity related?
Kinematic viscosity is dynamic viscosity divided by density: ν=μ/ρ. Dynamic viscosity measures resistance to shear in Pa·s, while kinematic viscosity has units m2/s and incorporates density. The two should not be interchanged in Reynolds-number formulas without the corresponding density factor.

Formula Explorer connections

Interpretation: This relationship connects pressure, velocity, density, viscosity, geometry or transport in a fluid system. Assumption: Check whether flow is steady, incompressible, laminar, fully developed or one-dimensional. Reynolds and Mach regimes determine whether simplified formulas are valid.

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