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Qwestrum Engineering360 · Chemical Engineering · Momentum Transfer (Fluid Mechanics)

Non Newtonian Flow

Non-Newtonian fluids have a shear-rate-dependent viscosity; the power-law model τ = K(dV/dy)^n covers shear-thinning (n < 1) and shear-thickening (n > 1) behaviour, and the Metzner-Reed Reynolds number extends pipe-flow friction correlations to them.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • n < 1: shear-thinning (pseudoplastic); n > 1: shear-thickening (dilatant)
  • Binghamplastic:τ=τy+μp(dVdy)abovetheyieldstressτyBingham plastic: \tau = \tau_{y} + \mu_{p} (\frac{dV}{dy}) above the yield stress \tau_{y}
  • Apparentviscosityμa=K(dVdy)(n−1)forapower−lawfluidApparent viscosity \mu_{a} = K (\frac{dV}{dy})^(n-1) for a power-law fluid

Topic details

Introduction

Many process fluids — polymers, slurries, foods — are non-Newtonian, so this topic generalises pipe-flow analysis. You classify a fluid from its flow curve (shear stress versus shear rate), fit the power-law or Bingham-plastic model, compute an apparent viscosity, and use the Metzner-Reed Reynolds number to pick a friction factor exactly as you would for a Newtonian fluid.

Key relations & formulas

τ=μ(dVdy)\tau = \mu (\frac{dV}{dy})
(Newtonian reference)
τ=K(dVdy)n\tau = K (\frac{dV}{dy})^n
(power-law / Ostwald-de Waele fluid)
ReMR=ρV(2−n)Dn/(K8(n−1))Re_{MR} = \rho V^(2-n) D^n / (K 8^(n-1))
(Metzner-Reed Reynolds number)

Notation and sign conventions

Relation 1 —
τ=μ\tau = \mu
τ=μ(dVdy)\tau = \mu (\frac{dV}{dy})
(Newtonian reference)
Write this relation with symbols exactly as in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
τ=K\tau = K
τ=K(dVdy)n\tau = K (\frac{dV}{dy})^n
(power-law / Ostwald-de Waele fluid)
Write this relation with symbols exactly as in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
Re_{MR} = \rho V^
ReMR=ρV(2−n)Dn/(K8(n−1))Re_{MR} = \rho V^(2-n) D^n / (K 8^(n-1))
(Metzner-Reed Reynolds number)
Write this relation with symbols exactly as in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Concept in depth

A Newtonian fluid has a constant viscosity, but many industrial fluids thin under shear (pseudoplastic, like paint that flows when brushed) or thicken (dilatant, like a starch suspension). The power-law model captures both with a consistency index K and a flow index n; the resulting apparent viscosity changes with shear rate, so there is no single viscosity to plug into the ordinary Reynolds number. The Metzner-Reed number restores a Newtonian-like framework so that the familiar friction-factor chart still applies. Bingham plastics add a yield stress that must be exceeded before any flow starts, relevant to toothpaste, drilling muds and sludges.

Assumptions and validity limits

State assumptions explicitly before using any relation for non newtonian flow — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Momentum Transfer viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Momentum Transfer papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to non newtonian flow.
4. Use equation 1:
τ=μ\tau = \mu
.
5. Use equation 2:
τ=K\tau = K
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Non Newtonian Flow appears in pipes, packed beds, and pumps. In Indian chemical curricula this topic is tested because it connects theory to fluid flow in process equipment.
GATE and semester exams often combine non newtonian flow with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use non newtonian flow?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students use the Newtonian Reynolds number for power-law fluids, forget that apparent viscosity depends on shear rate (and hence on velocity), and misclassify shear-thinning versus shear-thickening from the sign of (n − 1). Neglecting the yield stress in a Bingham problem is another error.

Quick revision checklist

Before attempting non newtonian flow problems, confirm you can:
1. n < 1: shear-thinning (pseudoplastic); n > 1: shear-thickening (dilatant)
2.
Binghamplastic:τ=τy+μp(dVdy)abovetheyieldstressτyBingham plastic: \tau = \tau_{y} + \mu_{p} (\frac{dV}{dy}) above the yield stress \tau_{y}

3.
Apparentviscosityμa=K(dVdy)(n−1)forapower−lawfluidApparent viscosity \mu_{a} = K (\frac{dV}{dy})^(n-1) for a power-law fluid
Revise the solved examples in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Fluid classification from flow index

Problem

A power-law fluid has n = 0.6. Is it shear-thinning or shear-thickening, and how does apparent viscosity change with shear rate?

Solution

Since n < 1, the fluid is shear-thinning (pseudoplastic). Apparent viscosity μ_a = K(dV/dy)^(n−1) = K(dV/dy)^(−0.4) decreases as shear rate rises.

Conceptual check — Non Newtonian Flow

Problem

In a Momentum Transfer semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of non newtonian flow." What should a complete answer include?

Exams & GATE

Use Metzner-Reed Re for the friction factor in power-law pipe flow.

📖 Standard books (India)

  • Unit Operations of Chemical Engineering — McCabe, Smith & Harriott

    Read: Syllabus unit

    Momentum, heat, and mass transfer operations

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