Non-Newtonian Classification, the Kelvin Creep Model, and the Pseudoplastic Flow Curve: 22-Agric-A6 Physical Properties of Biological Materials and Food Products, May 2016, Q6

22-Agric-A6 Physical Properties of Biological Materials and Food Products · May 2016

Question 6: Non-Newtonian Classification, the Kelvin Creep Model, and the Pseudoplastic Flow Curve (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Check: the question is read literally as the local pressure gradient $\Delta P/L$ plotted against distance travelled down the pipe, $L$, at a fixed flow rate — not against flow rate itself. For any time-independent fluid, fully developed laminar flow at constant $Q$ gives a constant $\Delta P/L$ all along the pipe, so a gradient that changes with distance is only possible for a time-dependent fluid whose viscosity is itself evolving with the shear history each fluid element has accumulated since entering the pipe.

(a) Classification from the pressure-gradient trend along the pipe. As a food material moves down the pipeline at constant flow rate, every fluid element accumulates shear history, and for a time-dependent material that history changes the local apparent viscosity, and hence the local pressure gradient, as a function of position.

(b) Exponentially increasing $\Delta P/L$ along the pipe indicates a rheopectic (time-hardening) material: continued shear progressively builds internal structure (e.g. shear-induced particle/floc alignment or bond formation), so viscosity — and the pressure gradient needed to sustain the same flow rate — rises the further the material has travelled, approaching a fully structured asymptote.
(c) Exponentially decaying $\Delta P/L$ indicates a thixotropic (time-thinning) material: continued shear breaks down internal structure at a rate proportional to the structure remaining (first-order kinetics), so apparent viscosity, and the pressure gradient, decays smoothly toward a fully broken-down asymptote as material moves downstream.
(a) Linearly decreasing $\Delta P/L$ indicates a time-thinning material whose structural breakdown proceeds at a bounded, roughly constant rate (a zero-order rather than first-order breakdown kinetics) until the structure is exhausted, rather than the ever-slowing, asymptotic decay of a first-order thixotropic breakdown.

In every case, a time-independent fluid (Newtonian, or a simple power-law pseudoplastic/dilatant with a shear history that does not itself alter the material) gives a flat $\Delta P/L$ versus $L$ once flow is fully developed, which is the reference case shown in the figure below.

Local pressure gradient versus distance down the pipe at constant flow rate, for the three time-dependent trends named in the question, against a flat time-independent reference.

(b) Kelvin (Kelvin–Voigt) model for creep. The Kelvin model represents a viscoelastic solid as a Hookean spring (modulus $E$) and a Newtonian dashpot (viscosity $\eta$) connected in parallel, so both elements always carry the same strain and the total stress is the sum of the spring's and the dashpot's contributions, $\sigma = E\epsilon + \eta\,d\epsilon/dt$. Under a suddenly applied and then held constant stress $\sigma_0$ (a creep test), solving this first-order equation gives $$\epsilon(t) = \frac{\sigma_0}{E}\left(1-e^{-t/\tau}\right),\qquad \tau = \frac{\eta}{E},$$ which starts at zero strain (the dashpot prevents the spring from extending instantaneously) and rises smoothly toward the equilibrium elastic strain $\sigma_0/E$ — a delayed (retarded) elastic response, never overshooting the equilibrium value. On removal of the stress, the strain relaxes back to zero along the mirror-image decay $\epsilon(t)=\epsilon_{\max}e^{-t/\tau}$, so all of the deformation is eventually recovered given enough time. Fitting a measured creep curve to this equation — reading off the equilibrium strain to get $E$, and the time to reach $(1-1/e)$ of that strain to get the retardation time $\tau$, hence $\eta=E\tau$ — is the standard way the Kelvin model is used to reduce food creep data to two material constants.

(c) Pseudoplastic flow curve across the full shear-rate range. Real structured (pseudoplastic) food fluids do not follow the power law $\tau=K\dot\gamma^n$ indefinitely; over a wide enough shear-rate range the curve shows three regions, sketched below on log-log axes. At very low ("creeping-flow") shear rates the imposed shear is too gentle to disturb the fluid's resting structure any further, so the apparent viscosity saturates at a constant lower Newtonian plateau $\mu_0$ and $\tau$ is again proportional to $\dot\gamma$. At intermediate shear rates the structure progressively breaks down with increasing shear rate, giving the familiar power-law, shear-thinning region ($n<1$, concave on log-log axes). At very high shear rates the structure is essentially fully broken down, so further increases in shear rate can no longer reduce viscosity, and the curve again becomes linear at an upper Newtonian plateau $\mu_\infty < \mu_0$. Newtonian-like (locally linear) behaviour is therefore expected at both extremes of shear rate, because at each extreme the fluid's structural state is no longer changing with further changes in shear rate, so the local relationship between stress and shear rate reverts to a constant proportionality — even though the fluid is globally non-Newtonian across the full range.

Shear stress versus shear rate for a pseudoplastic fluid from creeping flow to very high shear rate: Newtonian-like plateaus bracket the power-law shear-thinning region.

Concept & Theory

Topic: Time-dependent (thixotropic/rheopectic) versus time-independent non-Newtonian classification; the Kelvin–Voigt viscoelastic model; the structural (three-region) flow curve.

Key relations: Kelvin creep, $\epsilon(t)=(\sigma_0/E)(1-e^{-t/\tau})$, $\tau=\eta/E$; structural flow curve bounded by $\mu_0$ (low-shear) and $\mu_\infty$ (high-shear) Newtonian plateaus with a power-law region between.

Why this works: a time-dependent fluid's viscosity is a function of shear history, not just the instantaneous shear rate, so a spatially varying shear history (as accumulates along a pipeline at constant $Q$) produces a spatially varying viscosity and pressure gradient even though the flow rate itself never changes; a time-independent fluid has no such memory and therefore gives a constant gradient once flow is fully developed. The Kelvin model's dashpot-in-parallel topology is specifically what prevents an instantaneous strain jump on loading, which is the experimental signature (delayed onset of strain) that distinguishes creep in a viscoelastic solid from the Maxwell (spring-and-dashpot-in-series) model used for stress relaxation.

Common pitfall: using the Kelvin model (parallel spring-dashpot, correct for creep/retardation) and the Maxwell model (series spring-dashpot, correct for stress relaxation) interchangeably — each reproduces only one of the two experiments correctly, because a Maxwell element strains instantaneously under load (no delay) while a Kelvin element cannot. Assuming a pseudoplastic fluid's power-law index $n$ is a universal constant rather than a local description valid only over the shear-rate range actually tested is the second common pitfall in part (c).

Source: Steffe, Rheological Methods in Food Process Engineering, viscoelasticity and time-dependent flow chapters; Rao et al., Engineering Properties of Foods.

Question 6 of 9: Non-Newtonian Classification, the Kelvin Creep Model, and the Pseudoplastic Flow Curve

Question 6: Non-Newtonian Classification, the Kelvin Creep Model, and the Pseudoplastic Flow Curve (20 marks)