MATH313 – Preliminaries: Units, Dimensions & Conservation Laws

Vocabulary flashcards summarising key terms and definitions from the lecture notes on units, dimensions, conservation laws, diffusion, advection, heat conduction, boundary conditions, and nondimensionalisation.

Primary Quantity

A fundamental physical quantity that cannot be defined in terms of other quantities (e.g. length, mass, time, temperature).

Secondary Quantity

A physical quantity derived from combinations of primary quantities (e.g. velocity = length ⋅ time⁻¹).

Dimension [q]

Symbolic representation of the physical nature of a variable q, independent of the units used to measure it (e.g. [v] = length ⋅ time⁻¹).

SI Unit System

International system of units based on metre, kilogram, second, kelvin, etc., used for most scientific measurements today.

CGS Unit System

Centimetre–gram–second system; an older metric system where, for example, temperature is expressed in degrees Celsius.

Dimensional Homogeneity

Property that both sides of a physical equation have identical dimensions, providing a validity check.

Flux J(x,t)

Rate at which a quantity passes through a unit area per unit time; positive in the chosen forward direction.

Source Term f(x,t)

Rate of production (+) or removal (−) of a quantity per unit volume per unit time within a region.

Fundamental Conservation Law (1-D)

PDE ∂u/∂t + ∂J/∂x = f that locally balances rate of change, flux divergence, and sources/sinks.

Constitutive Relation

Additional empirical or physical law linking variables (e.g. u and J) needed to close a conservation equation.

Fick’s Law

Mass flux in diffusion is proportional to the negative concentration gradient: J = −D ∂C/∂x.

Diffusivity D

Proportionality constant in Fick’s Law with dimensions length² ⋅ time⁻¹, indicating how fast substances spread.

Diffusion Equation

PDE ∂C/∂t = D ∂²C/∂x² describing time evolution of concentration via diffusion.

Advection

Transport of a substance by bulk fluid motion; flux often proportional to concentration (J = sC).

Advection Equation

PDE ∂C/∂t + s ∂C/∂x = 0 modelling pure transport with wave speed s.

Travelling Wave Solution

Form C(x,t)=C₀(x−st) that moves unchanged at speed s, solving the advection equation.

Inviscid Burgers’ Equation

Non-linear PDE ∂C/∂t + C ∂C/∂x = 0 arising from flux J = ½ C² (advection with velocity C).

Advection–Diffusion Equation

Combined model ∂C/∂t + s ∂C/∂x = D ∂²C/∂x² incorporating both transport and diffusion.

Reaction Term

Expression such as −rC added to a conservation law to model decay, growth, or chemical reactions.

Specific Heat c

Energy required to raise one unit mass by one unit temperature; units J ⋅ kg⁻¹ ⋅ K⁻¹.

Thermal Conductivity k

Material property measuring ability to conduct heat; units W ⋅ m⁻¹ ⋅ K⁻¹.

Heat Diffusivity α

k / (ρc); indicates rate at which temperature disturbances spread through a material.

Fourier’s Law

Heat flux is proportional to negative temperature gradient: J = −k ∂T/∂x.

Heat Equation

PDE ∂T/∂t = α ∂²T/∂x² governing conduction in solids.

Dirichlet Boundary Condition

Boundary condition specifying the value of a field variable (e.g. T = constant at x = L).

Neumann Boundary Condition

Boundary condition specifying the value of the derivative/flux (e.g. ∂T/∂x = 0 for insulation).

Robin (Newton / Mixed) Boundary Condition

Boundary condition combining value and flux, e.g. −k ∂T/∂x = h(T − T_s).

Biot Number (Bi)

Dimensionless ratio hL/k comparing convective to conductive heat transfer resistances.

Newton’s Law of Cooling

Empirical law stating heat flux density is proportional to temperature difference between surface and surroundings.

Interface (Continuity) Conditions

At material junctions, temperature and heat flux remain continuous across the interface.

Moving Boundary

Boundary whose position changes in time (e.g. melting front), making the problem a free-boundary one.

Higher-Dimensional Conservation Law

Local form ∂u/∂t + div J = f derived using the Divergence Theorem.

Gradient ∇u

Vector of spatial partial derivatives pointing in direction of greatest increase of scalar field u.

Divergence div J

Scalar measure of a vector field’s net outflow per unit volume: ∂J₁/∂x₁ + ∂J₂/∂x₂ + ∂J₃/∂x₃.

Laplacian ∆u

Sum of second spatial derivatives; in 3-D, ∆u = ∂²u/∂x₁² + ∂²u/∂x₂² + ∂²u/∂x₃².

Nondimensionalisation (Scaling)

Process of converting variables to dimensionless form via characteristic scales to reveal key parameters.

Characteristic Length x_c

Chosen spatial scale (e.g. domain length L) used in nondimensionalisation.

Characteristic Time t_c

Time scale (e.g. L²/α) making leading coefficient in scaled PDE equal to 1.

Similarity Solution

Solution depending on variables through a combined similarity variable, often arising after scaling.

Embedding Method

Analytical technique (developed by the author) for solving varied heat/diffusion IBVPs on different domains.

Reynolds Number Re

Dimensionless number ρvL/µ comparing inertial to viscous forces in fluid flow; derived to be unitless via dimensional analysis.

Dimensional Analysis

Systematic use of dimensions to deduce relationships, check equations, and form dimensionless groups.