Newton's Viscosity Law Calculator
Enter force, area, velocity and gap to compute viscosity and shear stress via τ = η·(dv/dx).
Input Data
Results
At a glance:Newton's law of viscosity (Newton, 1687) states that for fluid layers sliding past one another, the shear stress τ is proportional to the velocity gradient dv/dx: τ = η·(dv/dx). The proportionality constant η is the dynamic viscosity (Pa·s), measuring internal friction. Equivalently η = F·d/(A·v), where F is shear force, d the gap, A the area and v the velocity. Newtonian fluids (water, oil, air) have constant η; non-Newtonian fluids (blood, paint, polymers) have η that varies with shear rate. Gas viscosity rises with temperature; liquid viscosity falls. 1 Pa·s = 10 poise.
Formula
Shear stress: τ = F/A
Velocity gradient: dv/dx = v/d
Viscosity: η = τ/(dv/dx) = F·d/(A·v)
Newtonian fluid: τ = η·(dv/dx)
Reynolds number: Re = ρ·v·L/η
$$\tau = \eta \frac{dv}{dx}, \quad \eta = \frac{F \cdot d}{A \cdot v}$$How to Use
- Enter shear force F (N), area A (m²), velocity v (m/s) and gap d (m).
- The calculator returns viscosity, shear stress and velocity gradient.
Case Studies
Lubricating oil and engine protection
SAE 5W-30: η ≈ 0.05 Pa·s cold, ≈ 0.01 Pa·s hot. Multi-grade keeps lubrication at start and a stable film when hot.
Bearing gap d ≈ 0.01 mm, v ≈ 10 m/s → dv/dx ≈ 1e6 1/s, τ ≈ 10000 Pa, film intact (elastohydrodynamic).
Synthetic oil has higher viscosity index for extreme temperatures.
Blood rheology and cardiovascular health
Whole-blood η ≈ 3–4 mPa·s (non-Newtonian, higher at low shear). Hyperviscosity raises thrombosis risk.
Plasma exchange treats high-viscosity syndrome by removing abnormal proteins.
FAQ
What is the difference between dynamic and kinematic viscosity?
Dynamic viscosity η (Pa·s) = shear stress / velocity gradient. Kinematic viscosity ν = η/ρ (m²/s), dividing by density. Pipe design often uses ν (Reynolds number Re = vL/ν). 1 cSt = 1e-6 m²/s.
Why does liquid viscosity drop with temperature?
Liquid viscosity comes from intermolecular attraction; higher temperature increases kinetic energy and weakens cohesion, so layers slide more easily and η falls (Arrhenius: η = A·exp(E/RT)). Water drops ~6× from 0°C to 100°C. Gases behave oppositely.
What is a non-Newtonian fluid?
A fluid whose η changes with shear rate. Shear-thinning (ketchup, paint) lowers η under shear; shear-thickening (cornstarch) raises it; Bingham plastics (toothpaste) need a yield stress. Blood is shear-thinning.
Does viscosity depend on density?
No direct relation. Honey η ≈ 10 Pa·s, density 1.4 g/cm³; mercury η ≈ 1.5 mPa·s, density 13.6 g/cm³. Viscosity is internal friction, not weight.
How to reduce fluid viscosity?
Heat it (most common), add diluent, apply shear (for shear-thinning fluids), or ultrasonication. Heating gases instead raises viscosity.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.