Momentum Transport, Shell Balances and Differential Navier-Stokes

Momentum transport arises because moving fluid elements interact through stress. In a Newtonian liquid, viscosity produces stress proportional to strain rate. This is why shear-driven flow cannot be interpreted as a direct proportionality between stress and absolute velocity. Add a constant velocity to the entire apparatus and relative shear need not change. Pressure is different: it generates normal traction, and gradients in pressure can drive fluid even when no boundary moves. Conservation of momentum combines both processes with convective momentum carriage.

Two simple geometries reveal the distinction. A fixed bottom plate and a steadily moving top plate create plane Couette flow with an approximately linear velocity profile. In pressure-driven flow between stationary plates, the pressure gradient must be balanced by a changing viscous shear, producing a parabolic profile. Superpose the motions only because the governing simplified differential equation is linear for a constant-viscosity Newtonian fluid. The same superposition cannot be assumed when viscosity varies with temperature or shear rate.

A circular pipe is another useful analytical benchmark. In fully developed laminar flow, velocity vanishes at the wall and is greatest on the axis. Integrating velocity over circular annuli produces the fourth-power dependence of flow on radius. This extraordinary sensitivity makes laboratory diameter measurements important: a small uncertainty in radius leads to a larger relative uncertainty in flow. It also explains why fouling can sharply reduce flow even before a pipe is visibly obstructed. Nevertheless, the fourth-power law applies to pressure-controlled fully developed laminar flow, not an arbitrary pump connected to a turbulent network.

Reynolds number marks the relationship between inertial and viscous effects. It does not switch fluid equations at one universal value; the commonly quoted threshold belongs to straight-pipe flow under particular disturbance conditions. Transition is uncertain and can depend on inlet roughness and disturbances. The Darcy friction factor is especially easy to confuse with the Fanning factor, producing a fourfold calculation error. Every engineering formula needs its convention labeled, including whether mean or centerline velocity appears in Reynolds number.

To diagnose a pipe model, check velocity continuity, pressure-loss trends and power dissipation. At fixed mean velocity, longer pipe should generally require more pressure drop; at fixed geometry, larger viscosity increases laminar resistance. The hydraulic power required to overcome the modeled loss is ΔpQ. If pressure loss is positive but power appears negative, inspect the energy sign convention. A carefully chosen set of checks makes a model more reliable than a large number of visually similar simulation runs.

Unit 2, Lesson 1: Viscosity and Newtonian stress

Learning objective

Distinguish Newtonian viscosity from shear-dependent rheology using measured slope.

Conceptual model

Shear stress is proportional to rate of deformation for a Newtonian liquid under simple shear, not directly to velocity. A solid-body translation can have finite speed but zero velocity gradient and therefore zero viscous shear in this model.

In a Cartesian shear flow, τ_xy=μ du/dy. Dynamic viscosity has units Pa·s; kinematic viscosity ν=μ/ρ has units m²/s. These are not interchangeable in Reynolds number formulas.

Real polymer melts, suspensions and blood analogues may be shear-thinning, yielding or viscoelastic. A Newtonian calibration curve should be tested by varying shear rate and plotting stress against gradient.

Governing relationship

τ_xy = μ ∂u/∂y

Engineering interpretation

The relevant phenomena are shear gradient; stress tensor; viscosity units. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 1.800 m. Increase the driving gradient by 20% and evaluate τ_xy = μ ∂u/∂y. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 1.930 m. Increase the driving gradient by 30% and evaluate τ_xy = μ ∂u/∂y. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.060 m. Increase the driving gradient by 40% and evaluate τ_xy = μ ∂u/∂y. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 2.190 m. Increase the driving gradient by 50% and evaluate τ_xy = μ ∂u/∂y. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. A Newtonian liquid has dynamic viscosity 0.080 Pa·s between layers whose velocity changes by 0.24 m/s over 0.020 m. Estimate the viscous shear stress.

Solution with units. The velocity gradient is Δu/Δy=12 s⁻¹, so τ=μ du/dy=0.080×12=0.96 Pa. The sign depends on the wall normal; this number is the magnitude. If the fluid density is 1000 kg/m³, its kinematic viscosity is ν=8×10⁻⁵ m²/s.

Model audit. A shear-rate sweep is required to check that viscosity is actually independent of gradient.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in viscosity and newtonian stress, identifying dimensions for every quantity in τ_xy = μ ∂u/∂y.
2. What observable field or flux would you measure to check the claimed relationship for viscosity and newtonian stress? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for viscosity and newtonian stress simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 2, Lesson 2: Plane Couette and Poiseuille flow

Learning objective

Compare wall-driven motion with the parabolic pressure-driven component.

Conceptual model

The Navier–Stokes equation simplifies dramatically for fully developed planar flow. In pressure-driven flow between stationary plates, the shear-stress gradient balances the pressure gradient; integrating twice produces a parabola after enforcing no-slip at both walls.

Couette motion instead uses one moving wall; its profile is linear when pressure gradient vanishes. In a combined Couette–Poiseuille flow, add the wall-induced linear profile to the pressure-driven contribution, because the governing simplified equation is linear in velocity.

The velocity maximum need not be at the center if the upper plate moves. Check whether your predicted wall shears have the signs expected from momentum transfer before trusting the profile.

Governing relationship

u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y)

Engineering interpretation

The relevant phenomena are moving wall; stationary wall; pressure-driven layers. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 1.870 m. Increase the driving gradient by 20% and evaluate u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.000 m. Increase the driving gradient by 30% and evaluate u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.130 m. Increase the driving gradient by 40% and evaluate u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 2.260 m. Increase the driving gradient by 50% and evaluate u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. A plate moves at 0.5 m/s above a fixed plate separated by a 10 mm fluid gap. Find the fluid speed 4 mm above the lower plate with no pressure gradient.

Solution with units. For laminar plane Couette flow u(y)=U y/h. Here u(0.004)=0.5×0.004/0.010=0.20 m/s. The no-slip boundary values are u(0)=0 and u(h)=0.5 m/s. The gradient is 50 s⁻¹; shear magnitude follows after multiplying by viscosity.

Model audit. An applied axial pressure gradient superposes a curved Poiseuille component, so the simple straight profile would no longer hold.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in plane couette and poiseuille flow, identifying dimensions for every quantity in u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y).
2. What observable field or flux would you measure to check the claimed relationship for plane couette and poiseuille flow? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for plane couette and poiseuille flow simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 2, Lesson 3: Laminar pipe flow

Learning objective

Calculate diameter sensitivity, then explain why the fourth-power law fails in turbulence.

Conceptual model

For a Newtonian fluid in a long circular pipe, the fully developed axial momentum equation reduces to a radial balance between the pressure gradient and wall shear. Integration with finite centerline velocity and no slip gives a parabola in r.

Area-weighted integration yields Q=πR⁴Δp/(8μL) and average velocity Q/(πR²), half of the centerline velocity. Radius changes are exceptionally important: a 10% radius increase yields roughly 46% more laminar flow at fixed pressure drop.

Do not extrapolate this relation to high Reynolds numbers, rough pipes, entrances or pulsatile flow without a new model. The profile depends on viscosity and pressure conditions, not on an arbitrary visualization scale.

Governing relationship

Q = πR⁴Δp/(8μL)

Engineering interpretation

The relevant phenomena are fully developed axisymmetric flow; no-slip wall. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 1.940 m. Increase the driving gradient by 20% and evaluate Q = πR⁴Δp/(8μL). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.070 m. Increase the driving gradient by 30% and evaluate Q = πR⁴Δp/(8μL). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.200 m. Increase the driving gradient by 40% and evaluate Q = πR⁴Δp/(8μL). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 2.330 m. Increase the driving gradient by 50% and evaluate Q = πR⁴Δp/(8μL). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. An incompressible Newtonian liquid with μ=0.1 Pa·s travels through a long tube of radius 0.010 m and length 10 m under 4000 Pa pressure drop. Estimate laminar Q.

Solution with units. Poiseuille gives Q=π(0.010)⁴×4000/[8(0.1)(10)]=1.571×10⁻⁵ m³/s. The average speed is Q/(πR²)=0.0500 m/s and the centerline speed is 0.100 m/s. For ρ=1000 kg/m³, Re=ρ(2R)V/μ=10, safely laminar.

Model audit. A ten-fold viscosity decrease at the same pressure difference raises Q ten-fold in this model, but may also change the Reynolds regime.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in laminar pipe flow, identifying dimensions for every quantity in Q = πR⁴Δp/(8μL).
2. What observable field or flux would you measure to check the claimed relationship for laminar pipe flow? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for laminar pipe flow simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 2, Lesson 4: Reynolds number and friction

Learning objective

Use the Darcy friction factor, keeping it distinct from the Fanning convention.

Conceptual model

Reynolds number compares inertial advection with viscous diffusion of momentum. For pipe flow, Re=ρVD/μ; it is not a universal threshold for every geometry. Friction factor definitions must be declared, as Darcy f is four times the Fanning factor.

For fully developed laminar circular-pipe flow, Darcy f=64/Re. In turbulent conditions friction depends on both Reynolds number and relative roughness ε/D. A useful explicit turbulent approximation is Haaland, but transition and non-Newtonian behavior need a different treatment.

Darcy–Weisbach loss must preserve velocity squared and L/D. If your chart predicts pressure loss decreases when length increases at fixed velocity and diameter, the implementation or units are wrong.

Governing relationship

Re = ρVD/μ; Δp = f(L/D)ρV²/2

Engineering interpretation

The relevant phenomena are inertia; viscosity; wall pressure loss. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.010 m. Increase the driving gradient by 20% and evaluate Re = ρVD/μ; Δp = f(L/D)ρV²/2. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.140 m. Increase the driving gradient by 30% and evaluate Re = ρVD/μ; Δp = f(L/D)ρV²/2. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.270 m. Increase the driving gradient by 40% and evaluate Re = ρVD/μ; Δp = f(L/D)ρV²/2. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 2.400 m. Increase the driving gradient by 50% and evaluate Re = ρVD/μ; Δp = f(L/D)ρV²/2. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. Waterlike fluid has ρ=1000 kg/m³, μ=0.001 Pa·s, V=0.020 m/s and D=0.025 m in a 10 m round pipe. Estimate laminar frictional loss.

Solution with units. Re=ρVD/μ=500. The Darcy friction factor is f=64/500=0.128. Then Δp=f(L/D)(ρV²/2)=0.128×400×0.20=10.24 Pa. If flow speed increases substantially, the laminar friction factor changes and the same formula for f may cease to apply.

Model audit. Always identify Darcy rather than Fanning friction factor; using the latter without a factor-of-four conversion produces a fourfold error.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in reynolds number and friction, identifying dimensions for every quantity in Re = ρVD/μ; Δp = f(L/D)ρV²/2.
2. What observable field or flux would you measure to check the claimed relationship for reynolds number and friction? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for reynolds number and friction simplifies, and another in which the equation becomes unreliable. Explain why.

Applied Project 2 — Momentum transport

Develop an engineering model. Supply geometry, conservation equations, units, two operating points, simulation plots, uncertainty, and a validation plan.

Detailed project brief

Study a proposed laboratory pipe carrying a 1000 kg/m³ liquid of viscosity 0.003 Pa·s through 6 m of 25 mm internal diameter. Compare mean speeds of 0.1, 0.4 and 1.0 m/s. Calculate Reynolds number at each operating point and decide which friction-factor assumptions can be defended. Estimate pressure loss using a declared Darcy convention, and compute hydraulic power from ΔpQ. Next, investigate a smooth-pipe diameter increase and record effects on velocity and pump demand when volumetric throughput—not speed—is held fixed. Explain why it is incorrect to reuse laminar Poiseuille flow above its Reynolds applicability range. Discuss how upstream elbows, temperature variation and wall roughness could affect the trial.

Unit 2 Assessment — Momentum transport

1. Explain the physical principle supporting Viscosity and Newtonian stress and state all assumptions in τ_xy = μ ∂u/∂y.
2. Derive a dimensionless or limiting-form check of τ_xy = μ ∂u/∂y; what happens when the driving gradient tends to zero?
3. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Viscosity and Newtonian stress.
4. Explain the physical principle supporting Plane Couette and Poiseuille flow and state all assumptions in u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y).
5. Derive a dimensionless or limiting-form check of u(y) = Uy/h − (1/2μ)(dp/dx)y(h−y); what happens when the driving gradient tends to zero?
6. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Plane Couette and Poiseuille flow.
7. Explain the physical principle supporting Laminar pipe flow and state all assumptions in Q = πR⁴Δp/(8μL).
8. Derive a dimensionless or limiting-form check of Q = πR⁴Δp/(8μL); what happens when the driving gradient tends to zero?
9. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Laminar pipe flow.
10. Explain the physical principle supporting Reynolds number and friction and state all assumptions in Re = ρVD/μ; Δp = f(L/D)ρV²/2.
11. Derive a dimensionless or limiting-form check of Re = ρVD/μ; Δp = f(L/D)ρV²/2; what happens when the driving gradient tends to zero?
12. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Reynolds number and friction.

Laboratory manual The laboratory environment is fully local: no tracking, paid API or network dependency. Each model lists assumptions and solves an analytic expression or an explicitly labeled numerical approximation. Graphs show a physical profile and parameter sensitivity. How to investigate Copy the equation and state variable units. Derive a limiting or conservation check. Change only one input, then record both plots and the key numerical output. Export the CSV and independently reproduce a calculation. Identify an assumption that could fail in practice and an experiment to test it. Safety and scope All property and equipment data in this book are illustrative and intended solely for learning. None of the models selects relief equipment, predicts hazardous releases, sizes industrial units or replaces measured properties and qualified process-safety review.

Engineering measurement, uncertainty and model validation

Why equations need measured inputs

An equation can be mathematically exact under its assumptions and still predict a real process poorly if its inputs were measured incorrectly. In transport calculations, viscosity, effective diffusivity, thermal conductivity and boundary conditions can all carry uncertainty. Before comparing model and experiment, list each measured quantity, its units, calibration source and expected accuracy. Keep instrument resolution separate from repeatability: repeated readings may agree closely while every reading shares the same calibration bias.

Case A: pipe radius controls flow

For a pressure-controlled laminar circular pipe, the Hagen–Poiseuille relation has Q proportional to R to the fourth power. Taking logarithmic differentials gives approximately δQ/Q = 4δR/R + δ(Δp)/(Δp) − δμ/μ − δL/L for small signed changes. If radius is uncertain by ±2% while all else is treated as exact, its first-order contribution alone is roughly ±8% in Q. This is why reporting a very precise flow from a loosely measured pipe diameter creates false confidence. In a real laboratory, measure diameter at multiple locations and inspect noncircularity; the physical pipe may not have a single well-defined radius.

Case B: heat through a layered wall

For known surface temperatures, total steady heat rate through resistances in series is q=ΔT/ΣR. If insulation conductivity is temperature dependent, evaluating k at one arbitrary temperature can systematically bias q. A physically appropriate estimate may integrate k(T) or iterate layer temperatures with measured property data. Contact resistances introduce additional drops that are absent from a perfect-contact wall model. Measure surface temperatures on both sides of each contact where possible; otherwise infer the combined thermal resistance and state that it may not be uniquely separable into component properties.

Case C: diffusivity from a release experiment

A one-dimensional Gaussian tracer pulse offers an experimental estimate of effective diffusivity because its variance grows linearly as 2Dt. Suppose measured variance increases from 0.002 m² at 10 s to 0.014 m² at 70 s. The estimated slope is (0.014−0.002)/(70−10)=0.0002 m²/s, hence D=0.0001 m²/s. This conclusion assumes a uniform infinite medium without advection, reaction or unmodeled mixing. If the mean position also shifts, convective transport must be considered. A single measurement of peak concentration cannot reliably identify both injected mass and diffusivity.

Case D: boundary-layer coefficient identification

A reported convective coefficient h is often computed as measured heat rate divided by exposed area and temperature difference. If the actual wall temperature varies across the surface, substituting a single thermocouple value may be inaccurate. In an external-flow heated-plate trial, place sensors at multiple downstream distances. Repeat at varied mean velocity and compare Nusselt trends with a correlation whose Reynolds and Prandtl limits match the equipment. Surface contamination, radiant exchange and conduction through the support can all masquerade as convection. Correcting for these parasitic paths is necessary before interpreting h as a clean fluid transfer coefficient.

Case E: numerical uncertainty versus model uncertainty

A computational solver can exhibit excellent grid convergence while its material-property model is wrong. Suppose a refined model predicts 12.0 W, a still finer mesh predicts 11.9 W, but the only experimentally measured heat flow is 9.5 W. The mesh difference is 0.1 W, yet the model–experiment discrepancy is 2.4 W. Refining the mesh further is unlikely to resolve a missing convection resistance or a conductivity error. Check residuals, boundary settings and energy conservation first, then challenge physical assumptions and measurements separately. State which uncertainty sources were quantified and which remain qualitative.

Practical uncertainty propagation

For a prediction y=f(x₁,…,xₙ) with small independent input uncertainties, one common first-order method uses u_y²≈Σ(∂f/∂x_i)²u_i². For correlated inputs there are additional covariance terms. The approximation may fail with strongly nonlinear responses, broad uncertainties, discontinuous regimes or parameters near physical limits. Monte Carlo sampling can help if plausible input distributions are available, but arbitrary normal distributions must not be treated as evidence. In all uncertainty analyses, specify whether quoted intervals are standard uncertainties, expanded uncertainties or observed ranges.

Engineering reporting checklist

A reproducible transport calculation should make it possible for a separate reader to reconstruct its main result. Record the domain geometry; draw the direction of positive flux; write all governing laws and their units; identify every boundary and initial condition; document material properties and their sources; and state any empirical correlation and validity limits. Include at least one independent limiting-case calculation, such as zero gradient causing zero pure conductive flux or a vanishing reaction rate giving nearly complete catalyst utilization. Report numerical solver settings if applicable, including grid resolution, timestep and residual definition. Finally, distinguish what was directly measured, what was fitted, what was computed and what was inferred. This separation prevents a simplified teaching simulation from being mistaken for a validated industrial design tool.