# Chapter 18 notebook: Applied Mathematics: Scaling, Regimes, and Model Sanity

**Goal:** Reject inconsistent models and use dimensionless ratios to choose a plausible first approximation.

**Start with:** Rules 18.1.1, 18.2.4, 18.3.5. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data.

**Source:** [Finished Chapter 18](../skills/math-thumb-applied-mathematics/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions.

## 1. Frame your decision

Write a question in which an answer would change something you do. Gather: Output and accuracy target; variables with dimensions; geometry; characteristic length, speed and time; property values; relevant regimes.

- My question and intended decision:
- Known inputs and units:
- Required accuracy or threshold:
- What I expect before calculating:
- What I still need to find out:

Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step.

## 2. Choose a route

1. Define the output, domain, and decision accuracy. List variables with units.
2. Check dimensional homogeneity, signs, conservation, symmetries, and easy limits before solving.
3. Choose characteristic scales that make typical variables order one; allow different scales in different regions.
4. Form independent dimensionless groups and identify plausible dominant balances.
5. Compute relevant regime ratios with explicitly defined \(U,L,T\), properties, and geometry.
6. Select the simplest model consistent with those ratios, then substitute the approximation back into every discarded term.
7. Compare with data or a higher-fidelity model near thresholds; escalate when multiple ratios are order one or uncertainty crosses regimes.

**Chapter-specific stop check:** Dimensionally consistent does not mean physically correct. State characteristic scales explicitly and re-check discarded terms after choosing a regime.

## 3. Work the lab

A solid sphere with radius 0.01 m has conductivity 200 W/(m K), convection coefficient 20 W/(m^2 K), density 2,700 kg/m^3 and specific heat 900 J/(kg K). For lumped thermal capacitance use L_c=V/A=r/3. Bi=h × L_c/k≈0.000333, well below the usual 0.1 screen. The model time constant rho × c × V/(h × A)=rho × c × r/(3h)=405 s. These calculations support a nearly uniform internal-temperature model under the stated heat-transfer assumptions.

Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime.

```python
radius, conductivity, convection = 0.01, 200.0, 20.0
density, specific_heat = 2700.0, 900.0
assert min(radius, conductivity, convection, density, specific_heat) > 0
characteristic_length = radius/3
biot = convection*characteristic_length/conductivity
time_constant = density*specific_heat*characteristic_length/convection
print(f"Lc=V/A: {characteristic_length:.6f} m; Bi: {biot:.6f}")
print(f"Lumped time constant: {time_constant:.3f} s; passes Bi<0.1 screen: {biot < 0.1}")
```

**My prediction, observed result, and explanation:**

_Record your work here._

## 4. Practise without the answers

### Exercise 1

Check the dimensions of distance = speed × time + acceleration × time.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 2

If L=0.1 m and diffusivity alpha=1e-5 m^2/s, estimate the diffusion timescale.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 3

A computed Reynolds number is near a quoted transition threshold. Can you declare the regime without geometry or uncertainty?

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

**Coaching prompt:** “Use the Chapter 18 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.”

## 5. Answer key and reasoning

Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear.

### Answer 1

The first term is length; the second is length/time. They cannot be added as distances. A candidate repair needs a time-squared acceleration term, but dimensional consistency alone cannot determine its coefficient or prove the model.

### Answer 2

L^2/alpha=0.01/1e-5=1,000 s. Geometry and boundary conditions affect precise relaxation times.

### Answer 3

No. Thresholds depend on geometry, disturbances, and the definition of characteristic scales. Propagate uncertainty and compare with an appropriate model or data near transition.

## 6. Build the complete chapter toolkit

The new lab samples the chapter; the following checklist covers all 14 rules. Study one thematic group at a time. A large group can take several sessions.

- **Test the Model Before Solving It:** work with rules 18.1.1, 18.1.2, 18.1.3.
- **Strip a Model Down to Its Governing Scales:** work with rules 18.2.1, 18.2.2, 18.2.3, 18.2.4.
- **Use Ratios to Select the Right Physical Regime:** work with rules 18.3.1, 18.3.2, 18.3.3, 18.3.4, 18.3.5, 18.3.6, 18.3.7.

For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source.

Read each complete numbered profile in the [chapter reference](../skills/math-thumb-applied-mathematics/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below.

| Rule | Book role | First assumptions to inspect | Practice status |
|---|---|---|---|
| 18.1.1: Reject equations that are not dimensionally homogeneous | Workflow | quantities have defined dimensions;  consistent unit semantics | new |
| 18.1.2: Test every model in easy limiting cases | Workflow | known limit behavior;  specified limit path | new |
| 18.1.3: Report parameter sensitivity as a dimensionless elasticity | Independent | valid characteristic scales;  physical model validity | new |
| 18.2.1: Scale variables so typical dimensionless values are order one | Workflow | valid characteristic scales;  physical model validity | new |
| 18.2.2: Reduce dimensional variables with Buckingham Pi | Independent | valid characteristic scales;  physical model validity | new |
| 18.2.3: Find leading behavior by balancing the largest competing terms | Independent | valid characteristic scales;  physical model validity | new |
| 18.2.4: Compare process timescales before coupling models | Workflow | scale separation;  stable fast subsystem | new |
| 18.3.1: Use Reynolds number to compare inertia with viscosity | Workflow | valid characteristic scales;  physical model validity | new |
| 18.3.2: Use Peclet number to compare advection with diffusion | Workflow | valid characteristic scales;  physical model validity | new |
| 18.3.3: Use Damkohler number to compare reaction with transport | Workflow | valid characteristic scales;  physical model validity | new |
| 18.3.4: Use Fourier number to estimate diffusion penetration time | Independent | valid characteristic scales;  physical model validity | new |
| 18.3.5: Use lumped thermal capacitance only for small Biot number | Workflow | valid characteristic scales;  physical model validity | new |
| 18.3.6: Check Mach number before neglecting compressibility | Workflow | valid characteristic scales;  physical model validity | new |
| 18.3.7: Check Knudsen number before using continuum equations | Workflow | valid characteristic scales;  physical model validity | new |

For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key.

## 7. Apply it to your own problem

Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic).

- Selected rule number(s) and reason:
- Assumptions that hold, fail, or remain uncertain:
- Substitution with units:
- Result and error, uncertainty, or bound:
- Independent check or limiting case:
- Decision this supports:
- Stop here, do a named next calculation, or gather missing information:

## 8. Transfer and continue

Useful nearby chapters: Chapter 16: measurement; Chapter 22: ode; Chapter 23: pde. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need.

**Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.