Calculation of Dynamic Viscosity from Shear Stress
Reference ID: MET-98F4 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The calculation of dynamic viscosity from measured shear stress is a fundamental
procedure in process engineering, especially when characterising liquids in
rheometers, viscometers, or plate‑gap flow cells. Viscosity governs how a fluid
resists deformation under shear and directly influences pump sizing, pipe
network design, heat‑transfer performance, and product quality. By relating the
applied force on a known area to the resulting shear rate, engineers obtain a
quantitative measure of the fluid’s internal friction that can be reported in
SI units (pascal‑seconds) or industry‑standard centipoise.
Methodology & Formulas
The calculation proceeds through a series of conversions and algebraic relationships
derived from basic mechanics and fluid‑flow theory.
Unit‑conversion constants (defined once for reuse):
cP_to_Pa_s – conversion factor from centipoise to pascal‑seconds.
bar_to_Pa – conversion factor from bar to pascals.
K_offset – offset to convert Celsius to Kelvin.
Input quantities (expressed in practical engineering units):
Force F (newtons)
Plate area A (square metres)
Plate gap (distance) ℓ (metres)
Plate velocity V (metres per second)
Temperature T_C (°C)
Pressure P_bar (bar)
Fluid density ρ (kilograms per cubic metre)
Convert temperature and pressure to absolute units (if required by downstream
calculations):
Absolute temperature: T_K = T_C + K_offset
Absolute pressure: P_Pa = P_bar × bar_to_Pa
Shear stress (τ):
τ = F / A
Shear rate (γ̇):
γ̇ = V / ℓ
Dynamic viscosity (μ):
From the definition μ = τ / γ̇, which can also be written as
μ = (F × ℓ) / (A × V).
Report dynamic viscosity in centipoise:
μ_cP = μ / cP_to_Pa_s
Kinematic viscosity (ν):
ν = μ / ρ
Convert kinematic viscosity to Stokes (where 1 St = 1 × 10⁻⁴ m²·s⁻¹):
ν_St = ν / 1e-4
The resulting values—shear stress, shear rate, dynamic viscosity (both in
pascal‑seconds and centipoise), and kinematic viscosity (in square metres per
second and Stokes)—provide a complete rheological description of the fluid under
the test conditions. These quantities can be directly used in design equations,
simulation models, or quality‑control specifications.
Dynamic viscosity (μ) is obtained by dividing the shear stress (τ) by the shear rate (γ̇):
μ = τ / γ̇
• Shear stress (τ) – typically measured in Pa (N·m⁻²) using a torque transducer or pressure sensor.
• Shear rate (γ̇) – the velocity gradient, expressed in s⁻¹. For a simple Couette flow, γ̇ = (ω·R)/h, where ω is the rotational speed, R the radius of the inner cylinder, and h the gap width.
Ensure consistent units (Pa·s) and correct temperature compensation, as viscosity is temperature‑dependent.
For non‑Newtonian fluids, the apparent viscosity varies with shear rate. Use the appropriate rheological model (e.g., Power‑law, Bingham, Herschel‑Bulkley) to relate τ and γ̇.
Example – Power‑law model: τ = K·γ̇ⁿ → μ_app = τ/γ̇ = K·γ̇ⁿ⁻¹.
Determine K (consistency index) and n (flow behavior index) from a flow curve (τ vs. γ̇) and apply the model at the operating shear rate of your process.
Viscosity changes exponentially with temperature. Apply the Arrhenius‑type or Vogel‑Fulcher‑Tammann (VFT) correlation:
μ(T) = μ₀·exp[ E_a / (R·T) ] (Arrhenius)
or log μ = A + B/(T‑C) (VFT)
Measure μ at a reference temperature (T₀), determine the constants (E_a, A, B, C) from calibration data, then calculate μ at the process temperature (T). Many process control systems embed these correlations for real‑time compensation.
Instrument drift – torque transducers can lose calibration over time; perform regular checks.
Gap size tolerance – small errors in the gap (h) of a Couette or parallel‑plate viscometer cause large shear‑rate errors.
Temperature gradients – uneven heating leads to local viscosity variations.
Wall slip – especially with low‑viscosity or highly shear‑thinning fluids; use roughened surfaces or correction factors.
Inertial effects – at high rotational speeds, centrifugal forces add apparent stress; stay within the instrument’s Reynolds number limits.
Mitigating these factors improves the reliability of the calculated dynamic viscosity.
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Worked Example: Determining Dynamic and Kinematic Viscosity from Measured Shear Stress
Scenario
A process engineer at a chemical plant is evaluating the fluid handling characteristics of a high‑density oil flowing through a narrow gap viscometer. The engineer measures a shear stress of 15 Pa on the fluid while the top plate moves at 0.05 m s⁻¹ across a 1 mm gap. The fluid density is known to be 870 kg m⁻³. The engineer must calculate the fluid’s dynamic viscosity (µ) and kinematic viscosity (ν).
