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research30 min studyLesson 992 in Physics

Identifying controlled parameters and scale hierarchies

Determine separately when convective inertia and unsteady acceleration can be neglected in a viscous flow.

Before you begin
On this page

The question for today

What you’ll understand

  • Compare advective, viscous and externally imposed time scales.
  • Distinguish small Reynolds number from a quasisteady approximation.
  • Rescale a proposed experiment while preserving two independent dimensionless parameters.

Calling viscosity large is incomplete because viscosity carries units. A controlled approximation compares the term containing viscosity with the terms one proposes to neglect, using the actual scales of the experiment. A fluid driven rapidly in a narrow channel can have negligible convective inertia while retaining substantial unsteady acceleration. These are different physical comparisons. Writing their dimensionless ratios exposes which approximation is justified and which measurements or boundary conditions determine the relevant scales. It also prevents a familiar steady-flow formula from being applied merely because the fluid seems thick.

Scale the equation before deleting terms

Take an incompressible Newtonian fluid with constant density and kinematic viscosity. Let U be a characteristic velocity variation, L the length over which it varies, and omega an imposed angular drive frequency. Estimate convective acceleration by U squared over L, viscous acceleration by viscosity times U over L squared, and unsteady acceleration by omega times U. These estimates presume that L and the imposed time actually describe the gradients and variation of interest.

Dividing by the viscous scale gives two independent ratios. Reynolds number compares convection with viscosity. The square of the Womersley parameter compares unsteady acceleration with viscosity. Small Reynolds number permits linear unsteady Stokes equations, but a quasisteady Stokes approximation also requires the second ratio to be small. Order-one numerical factors and the geometry still matter. A dimensionless estimate identifies a regime, not the full velocity profile or a universal error bound.

Re=ULν,α2=ωL2ν,∣(u⋅∇)u∣∣ν∇2u∣∼Re,∣∂tu∣∣ν∇2u∣∼α2\mathrm{Re}=\frac{UL}{\nu},\qquad \alpha^2=\frac{\omega L^2}{\nu},\qquad \frac{|(\mathbf u\cdot\nabla)\mathbf u|}{|\nu\nabla^2\mathbf u|}\sim\mathrm{Re},\qquad \frac{|\partial_t\mathbf u|}{|\nu\nabla^2\mathbf u|}\sim\alpha^2
ν is kinematic viscosity in m²/s, U is in m/s, L in m, and ω in radians per second with radians dimensionless. The comparisons concern a characteristic region; another length may govern a thin boundary layer.↔ Scroll wide equations horizontally.

Time ordering explains the physical mechanism

Momentum diffusion across the region takes a time of order L squared over viscosity. Advective transport takes L over U. The drive varies on the time scale one over omega, rather than its full period. Their ratios reproduce the two dimensionless parameters. When momentum diffuses much faster than the drive changes, a sequence of steady solutions can be plausible. When those two times are comparable, the velocity retains a history and phase lag even though nonlinear convection remains weak.

The assumptions surrounding the equation are separate again. Incompressibility, continuum transport, constant viscosity and no-slip boundaries require their own physical justification. Rapid driving can reveal wall compliance or frequency-dependent material response. A small Reynolds number by itself establishes none of these conditions. For a research design, name the observable and ask which omitted term could change it at the required accuracy; a small correction to a bulk velocity may still matter for a delicate phase measurement.

tν=L2ν,tadv=LU,tω=ω−1,Re=tνtadv,α2=tνtωt_\nu=\frac{L^2}{\nu},\qquad t_{\mathrm{adv}}=\frac{L}{U},\qquad t_\omega=\omega^{-1},\qquad \mathrm{Re}=\frac{t_\nu}{t_{\mathrm{adv}}},\qquad \alpha^2=\frac{t_\nu}{t_\omega}
These are scaling times, not exact relaxation eigenvalues or the oscillation period 2π/ω. Geometry supplies additional dimensionless factors. The ordering, rather than a dimensionful parameter alone, motivates approximation.↔ Scroll wide equations horizontally.

Similarity requires preserving the relevant ratios

Suppose a smaller apparatus is intended to reproduce a larger apparatus. Holding Reynolds number fixed constrains the product of velocity and length for the same fluid. Preserving the unsteady-to-viscous ratio independently constrains frequency times length squared. Matching only the first parameter can therefore turn an unsteady experiment into a quasisteady one. Dynamic similarity further requires matching geometry and other relevant dimensionless groups, including boundary or compressibility parameters when they enter the observable.

A controlled limiting argument specifies what is fixed while a small parameter decreases. Sending velocity toward zero at fixed driving frequency removes convection without removing unsteady acceleration. Sending frequency toward zero at fixed velocity does the opposite comparison. Such limits need not commute with long observation times or spatial boundary layers. Reporting both parameter values gives another researcher enough information to determine whether a disagreement is evidence against the underlying equation or simply a comparison between different regimes.

Work it through

Low Reynolds number with unsteady motion

Use L=1.0 mm, U=1.0 mm/s, ν=1.0×10⁻⁴ m²/s and ω=100 s⁻¹. Calculate Re, α² and the three scaling times. Which inertial term is small compared with viscosity, and is quasisteady Stokes flow justified by these values?

  1. Convert lengths and speeds to SI: L=10⁻³ m and U=10⁻³ m/s. The Reynolds number is 10⁻⁶/10⁻⁴=0.01, so characteristic convective inertia is small relative to viscous acceleration.

  2. The unsteady ratio is α²=100×10⁻⁶/10⁻⁴=1. Thus the time derivative is comparable to the viscous term and cannot be discarded on this scaling argument.

  3. The momentum-diffusion time is 0.01 s, the advection time is 1 s, and the drive time 1/ω is 0.01 s. These give the same ratios and explain why the fluid does not relax instantly relative to the imposed variation.

The result

Linear unsteady Stokes flow is a candidate approximation; quasisteady Stokes flow is not supported because α² is order one. This is a regime diagnosis, subject to the stated fluid and boundary assumptions.

The living notebook

Explore the model

Interactive lab

Interactive lab · heat

Compare diffusion with one fixed observation time

Use the exact heat mode as an analogy for viscous diffusion. Reset and set both controls to Spatial mode n=1 and Diffusivity κ=0.1, then Step once to t=0.5 s; predict amplitude exp(−0.05)≈0.951 and decay rate 0.1/s. For a fresh comparison, Reset and set both controls to n=1 and κ=1, then Step once; predict exp(−0.5)≈0.607 and rate 1/s.

Compare diffusion with one fixed observation timeThe exact periodic heat-equation mode is u(x,t) = exp(−κn²t) sin(nx), on [0,2π]. Playback advances physical time. The initial sine is shown separately, and the unnormalized L² norm is √π exp(−κn²t). This analytic instrument does not run a finite-difference solver or test discretization stability. Current readings: Current amplitude: 1; Spatial L² norm: 1.772; Decay rate · κn²: 3.6 /s. All parameters are adjustable with the labeled controls below.Periodic heat equation mode 3 at time 0: the amplitude is 1.0π/2π3π/22π−1−0.500.51x · periodic intervalTemperature perturbation uExact mode · n = 3
Current temperature modeInitial sineCurrent amplitude
Current amplitude
1
Spatial L² norm
1.772
Decay rate · κn²
3.6 /s
t = 0 s

The exact periodic heat-equation mode is u(x,t) = exp(−κn²t) sin(nx), on [0,2π]. Playback advances physical time. The initial sine is shown separately, and the unnormalized L² norm is √π exp(−κn²t). This analytic instrument does not run a finite-difference solver or test discretization stability. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThis normalized periodic mode shows a relaxation time changing relative to a fixed clock interval. It contains no fluid convection, oscillatory forcing, channel walls or Reynolds-number calculation.

Pause & reason

Check your understanding

A flow has Re=0.01 and α²=1 in the region of interest. Which reduction is supported by those ratios?

Your turn

Try it yourself

Build a geometrically similar device with L reduced by a factor of ten using the same fluid. Starting from the worked example, choose U and ω to preserve both Re and α². How do the three scaling times change?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Preserve UL first, then preserve ωL². Do not hold frequency fixed merely because the fluid is unchanged.

Compare with the solution

The new length is 0.10 mm=10⁻⁴ m. Keeping UL fixed requires U=10⁻² m/s=10 mm/s. Keeping ωL² fixed requires ω=10⁴ s⁻¹. The new diffusion time is 10⁻⁴ s, advection time 0.01 s and drive time 10⁻⁴ s; every scaling time is one hundredth of its original value. Their ratios remain Re=0.01 and α²=1. The numerical similarity is necessary for this reduction, but the faster drive may introduce material or wall effects that were negligible in the larger device.

Keep the idea

Take it with you

  • Dimensionless term ratios define the proposed approximation.
  • Weak convection and slow temporal variation are independent conditions.
  • A scaled experiment must preserve every dimensionless group relevant to its observable.

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