Fluid dynamics and thermal physics are foundational theoretical pillars of aerospace engineering, propulsion systems, energy conversion, chemical processing, and microelectronic thermal management. From the macroscopic continuity and momentum conservation of fluid streams to the microscopic statistical distributions and thermodynamic limits of molecular heat motion, physical equations construct a rigorous analytical framework.
This cheat sheet systematically organizes 21 core physical laws and foundational equations in fluid dynamics and thermal physics, partitioned into Fluid Dynamics, Laws of Thermodynamics, Kinetic Theory & Statistical Physics, and Thermal Radiation & Heat Transfer, presenting standard mathematical formulas, key variable definitions, physical principles, validity domains, and industrial applications.
🔗 مصفوفة قوانين ومعادلات الفيزياء: الميكانيكا الكلاسيكية · الديناميكا الحرارية والموائع · الكهرومغناطيسية · البصريات والصوتيات · ميكانيكا الكم · النسبية وعلم الكون · الفيزياء النووية والجسيمات · الفيزياء الفلكية والجيوفيزياء
1. Fluid Dynamics (7 Laws)
Fluid dynamics describes the conservation of mass, momentum transfer, energy conversion, and pressure response mechanisms of fluids in static and flowing states.
01 · Continuity Equation
Continuity Equation (Conservation of Mass)
$$A_1 v_1 = A_2 v_2 \iff \rho_1 A_1 v_1 = \rho_2 A_2 v_2$$
- ①الجوهر الفيزيائي: Law of conservation of mass in steady fluid flow; fluid velocity increases where a conduit constricts and decreases where it widens.
- ②تعريف المتغيرات: $A$ is conduit cross-sectional area ($\text{m}^2$), $v$ is area-averaged flow velocity ($\text{m/s}$), and $\rho$ is fluid density ($\text{kg/m}^3$). For incompressible flow ($\rho = \text{const}$), volumetric flow rate $Q = Av$ is conserved.
- ③الأهمية الجوهرية: The primary constraint equation for piping and streamtube analysis, representing the 1D integral form of $\nabla\cdot(\rho \mathbf{v}) + \partial\rho/\partial t = 0$.
- ④نطاق الصلاحية والتطبيقات: Incompressible assumption holds for liquids and low-speed gas flows (Mach number $\text{Ma} < 0.3$); high-speed compressible flows require density-varying mass flux. Applied in hydraulic piping, firefighting hose nozzles, and river canyon torrent dynamics.
02 · Bernoulli's Equation
Bernoulli's Principle
$$p + \rho g h + \frac{1}{2}\rho v^2 = \text{const}$$
- ①الجوهر الفيزيائي: Conservation of mechanical energy along a streamline in steady, inviscid, incompressible flow; regions of higher flow speed experience lower static pressure, and vice versa.
- ②تعريف المتغيرات: $p$ is static pressure ($\text{Pa}$), $\rho$ is density, $g$ is gravitational acceleration, $h$ is geometric elevation, $v$ is flow speed, and $\frac{1}{2}\rho v^2$ is dynamic pressure.
- ③الأهمية الجوهرية: The foundational baseline of low-speed aerodynamics and hydrodynamics, illustrating lossless exchange among pressure potential, gravitational potential, and kinetic energy.
- ④نطاق الصلاحية والتطبيقات: Strictly valid along a single streamline in inviscid, steady, incompressible flow; real viscous fluids suffer irreversible total head loss. Applied in airfoil aerodynamic lift generation, Venturi flowmeters, Pitot tube airspeed indicators, and atomizers.
03 · Pascal's Law
Pascal's Principle
$$\Delta p = \text{const} \iff \frac{F_1}{A_1} = \frac{F_2}{A_2}$$
- ①الجوهر الفيزيائي: Any pressure increment applied to an enclosed incompressible fluid is transmitted undiminished in all directions throughout the entire fluid and onto the container walls.
- ②تعريف المتغيرات: $\Delta p$ is pressure change, $F_1, F_2$ are applied forces on respective pistons, and $A_1, A_2$ are the corresponding piston surface areas.
- ③الأهمية الجوهرية: The axiomatic foundation of fluid power transmission and hydraulics, enabling substantial mechanical force multiplication proportional to area ratio.
- ④نطاق الصلاحية والتطبيقات: Static, enclosed, incompressible liquids. Applied in automotive hydraulic brake systems, industrial hydraulic presses, bottle jacks, heavy excavator arms, and aircraft flight control surface actuators.
04 · Archimedes' Principle
Buoyancy Principle
$$F_b = \rho g V_{\text{disp}}$$
- ①الجوهر الفيزيائي: Any object immersed in a static fluid experiences a vertically upward buoyant force equal in magnitude to the weight of the fluid displaced by the object. Microscopically arises from the hydrostatic pressure differential between bottom and top surfaces.
- ②تعريف المتغيرات: $F_b$ is buoyant force ($\text{N}$), $\rho$ is ambient fluid density ($\text{kg/m}^3$), $g$ is gravitational acceleration, and $V_{\text{disp}}$ is displaced fluid volume ($\text{m}^3$).
- ③الأهمية الجوهرية: The definitive principle governing static equilibrium of floating and submerged bodies, fluid density metrology, and naval hydrostatics.
- ④نطاق الصلاحية والتطبيقات: Static or uniformly moving immersed bodies. Floats when mean body density is less than fluid density, neutral at equal density, sinks when denser. Applied in ship displacement engineering, submarine ballast tank diving control, hot air balloon buoyancy, and hydrometer density readings.
05 · Poiseuille's Law
Hagen-Poiseuille Equation
$$Q = \frac{\pi r^4 \Delta p}{8 \eta L}$$
- ①الجوهر الفيزيائي: In steady laminar flow of a viscous fluid through a cylindrical pipe, volumetric flow rate is directly proportional to the driving pressure gradient and the fourth power of pipe radius ($r^4$), and inversely proportional to fluid viscosity and pipe length.
- ②تعريف المتغيرات: $Q$ is volumetric flow rate ($\text{m}^3/\text{s}$), $r$ is internal pipe radius, $\Delta p$ is driving pressure drop, $\eta$ is dynamic viscosity ($\text{Pa}\cdot\text{s}$), and $L$ is pipe length.
- ③الأهمية الجوهرية: The cornerstone equation of microfluidics and cardiovascular hemodynamics. Extreme sensitivity to radius: halving the lumen radius ($r \to 0.5r$) increases hydraulic resistance 16-fold and reduces flow to $1/16$.
- ④نطاق الصلاحية والتطبيقات: **Strictly valid only for incompressible Newtonian fluids in laminar regime**; turbulent flow shifts pressure loss closer to quadratic flow dependence. Explains hypertension induced by arterial stenosis, guides hypodermic needle sizing, and governs microchannel lab-on-a-chip architecture.
06 · Reynolds Number
Reynolds Number (Flow Regime Criterion)
$$\mathrm{Re} = \frac{\rho v L}{\eta} = \frac{v L}{\nu}$$
- ①الجوهر الفيزيائي: Dimensionless ratio of inertial forces to viscous shear forces in a fluid flow, serving as the fundamental criterion that distinguishes smooth, predictable laminar flow from chaotic, fluctuating turbulent flow.
- ②تعريف المتغيرات: $\mathrm{Re}$ is the dimensionless Reynolds number, $\rho$ is density, $v$ is characteristic velocity, $L$ is characteristic length scale (e.g., pipe diameter), $\eta$ is dynamic viscosity, and $\nu = \eta/\rho$ is kinematic viscosity.
- ③الأهمية الجوهرية: The core foundation of dynamic similarity in fluid mechanics, allowing scaled experimental wind tunnel models to accurately reproduce full-scale prototype aerodynamics.
- ④نطاق الصلاحية والتطبيقات: For circular pipe flow, critical Reynolds number $\mathrm{Re}_{\text{crit}} \approx 2300$ ($\mathrm{Re} < 2300$ laminar, $\mathrm{Re} > 4000$ fully turbulent). Applied in aircraft and bullet train wind tunnel scaling, towing tank ship model testing, and pipeline pumping power estimation.
07 · Navier-Stokes Equations
Navier-Stokes Equations (N-S Equations)
$$\rho \frac{\mathrm{D}\mathbf{v}}{\mathrm{D}t} = -\nabla p + \eta \nabla^2 \mathbf{v} + \mathbf{f}$$
- ①الجوهر الفيزيائي: The continuum formulation of Newton's second law (conservation of momentum) for viscous Newtonian fluids; the convective material acceleration on the left balances pressure gradients, viscous shear diffusion, and body forces on the right.
- ②تعريف المتغيرات: $\rho$ is density, $\mathbf{v}$ is velocity vector field, $\mathrm{D}/\mathrm{D}t = \partial/\partial t + (\mathbf{v}\cdot\nabla)$ is the material derivative, $p$ is static pressure, $\eta$ is dynamic viscosity, and $\mathbf{f}$ is body force per unit volume (e.g., $\rho \mathbf{g}$).
- ③الأهمية الجوهرية: The constitutional governing equations of modern fluid mechanics and computational fluid dynamics (CFD). The mathematical existence and smoothness of 3D Navier-Stokes solutions remains a Millennium Prize Problem.
- ④نطاق الصلاحية والتطبيقات: Continuous Newtonian fluids. Applied in numerical weather prediction models, full-vehicle aerodynamic CFD simulations, Formula 1 ground-effect downforce optimization, and 3D pulsatile arterial hemodynamics.
2. Laws of Thermodynamics (6 Laws)
Thermodynamics establishes energy conservation, the irreversible directional arrow of time, the absolute zero limit, and theoretical thermodynamic engine efficiency bounds.
08 · Zeroth Law of Thermodynamics
Thermal Equilibrium Transitivity
$$T_A = T_B, \ T_B = T_C \implies T_A = T_C$$
- ①الجوهر الفيزيائي: If two thermodynamic systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other; thermal equilibrium satisfies mathematical transitivity.
- ②تعريف المتغيرات: $T_A, T_B, T_C$ represent the state parameter "thermodynamic temperature". Systems in mutual thermal equilibrium share identical temperature values.
- ③الأهمية الجوهرية: Axiomatically validates temperature as a well-defined scalar state function, providing the sole logical justification for thermometry.
- ④نطاق الصلاحية والتطبيقات: Universally valid for macroscopic states of thermal equilibrium. Applied in calibrating contact temperature sensors (mercury thermometers, thermocouples, resistance temperature detectors RTD) and establishing international temperature scales (ITS-90).
09 · First Law of Thermodynamics
Conservation of Energy & Internal Energy
$$\Delta U = Q + W \iff \mathrm{d}U = \delta Q + \delta W$$
- ①الجوهر الفيزيائي: The law of conservation and transformation of energy applied to thermodynamic systems; the change in internal energy equals the heat added to the system plus the work done on the system.
- ②تعريف المتغيرات: $\Delta U$ is change in internal energy ($\text{J}$), $Q$ is net heat absorbed (exothermic is negative), and $W$ is work performed on the system (for expansion work done by system, $\delta W = -p\mathrm{d}V$).
- ③الأهمية الجوهرية: The fundamental balance sheet of thermodynamic machinery. Internal energy $U$ is a state function (an exact differential $\mathrm{d}U$), whereas heat $Q$ and work $W$ are path-dependent inexact differentials ($\delta$).
- ④نطاق الصلاحية والتطبيقات: Universally holds for all closed and isolated systems. **Strictly disproves perpetual motion machines of the first kind** (machines that produce net work without energy input). Applied in internal combustion engine power stroke modeling, power plant boiler thermal balance, and refrigeration compressor sizing.
10 · Second Law of Thermodynamics
Entropy Increase Principle
$$\Delta S_{\text{isolated}} \ge 0 \iff \mathrm{d}S \ge \frac{\delta Q}{T}$$
- ①الجوهر الفيزيائي: All spontaneous real macroscopic thermodynamic processes are irreversible; heat cannot spontaneously flow from a colder to a warmer body without external intervention. The total entropy of an isolated system never decreases.
- ②تعريف المتغيرات: $S$ is entropy ($\text{J/K}$), equality holds for idealized reversible processes, strict inequality applies to real irreversible processes, and $\delta Q/T$ is Clausius heat-temperature quotient.
- ③الأهمية الجوهرية: Defines the forward "arrow of time" in physical nature, sets theoretical limits on converting heat into mechanical work, and **rules out perpetual motion machines of the second kind**.
- ④نطاق الصلاحية والتطبيقات: Macroscopic multi-particle statistical ensembles. Applied in exergy availability loss analysis, Gibbs free energy spontaneity criteria ($\Delta G = \Delta H - T\Delta S < 0$), and industrial friction energy waste diagnostics.
11 · Third Law of Thermodynamics
Nernst Heat Theorem
$$S \to 0 \ (T \to 0\text{ K})$$
- ①الجوهر الفيزيائي: As the thermodynamic temperature of a system approaches absolute zero ($0\;\text{K} = -273.15\;^\circ\text{C}$), the entropy of any pure, perfectly crystalline substance in ground state approaches zero; absolute zero cannot be attained in a finite sequence of operations.
- ②تعريف المتغيرات: $S$ is thermodynamic entropy ($\text{J/K}$), and $T$ is absolute temperature ($\text{K}$). At absolute zero, the ground state becomes non-degenerate ($W = 1$, $\ln W = 0$).
- ③الأهمية الجوهرية: Fixes the absolute reference baseline of entropy, enabling calculation of standard absolute entropies for chemical reactions.
- ④نطاق الصلاحية والتطبيقات: Perfect crystalline equilibrium ground states. Near absolute zero, isobaric heat capacity and thermal expansion coefficients vanish. Applied in superconducting magnet cryogenic cooling, helium dilution refrigeration limits, and quantum computing cryostats.
12 · Statistical Interpretation of Entropy
Boltzmann's Entropy Formula
$$S = k_B \ln W$$
- ①الجوهر الفيزيائي: The entropy of a macroscopic state is proportional to the natural logarithm of the number of accessible microscopic states (thermodynamic probability $W$); entropy is a quantitative measure of micro-state multiplicity and statistical disorder.
- ②تعريف المتغيرات: $S$ is Boltzmann entropy, $k_B \approx 1.380649 \times 10^{-23}\;\text{J/K}$ is Boltzmann's constant, and $W$ (or $\Omega$) is the total count of microstates consistent with the macrostate.
- ③الأهمية الجوهرية: The great bridge linking macroscopic phenomenological thermodynamics to microscopic molecular statistical mechanics; reveals that irreversible entropy increase is simply a statistical progression from less probable to overwhelmingly probable configurations.
- ④نطاق الصلاحية والتطبيقات: Microcanonical ensembles and statistical equilibrium systems. Inscribed on Boltzmann's tombstone in Vienna. Formed the mathematical foundation for Shannon's information entropy, rubber elasticity conformational models, and nanoscale thermal physics.
13 · Carnot Efficiency
Carnot Cycle Maximum Efficiency
$$\eta = 1 - \frac{T_c}{T_h}$$
- ①الجوهر الفيزيائي: All reversible heat engines operating between a high-temperature heat reservoir $T_h$ and a low-temperature cold reservoir $T_c$ attain the theoretical physical maximum thermal efficiency; no real irreversible cycle can exceed this upper bound.
- ②تعريف المتغيرات: $\eta$ is thermal efficiency, $T_h$ is hot reservoir thermodynamic temperature ($\text{K}$), and $T_c$ is cold sink thermodynamic temperature ($\text{K}$, strictly in Kelvin).
- ③الأهمية الجوهرية: The "speed-of-light limit" of heat engine engineering, dictating that improving power plant cycle efficiency requires **maximizing peak heat addition temperature $T_h$ and minimizing heat rejection temperature $T_c$**.
- ④نطاق الصلاحية والتطبيقات: Idealized closed reversible cycle consisting of two isothermal and two adiabatic stages. Applied in ultra-supercritical boiler steam power cycle optimization, aerospace gas turbine single-crystal blade cooling limits, and nuclear power plant condenser design.
3. Kinetic Theory & Statistical Physics (4 Laws)
Kinetic theory models macroscopic gas properties—pressure, temperature, transport, and equipartition—from statistical molecular collision dynamics and velocity distributions.
14 · Ideal Gas Law
Ideal Gas Equation of State
$$pV = nRT = N k_B T$$
- ①الجوهر الفيزيائي: Unifies Boyle's, Charles's, Gay-Lussac's, and Avogadro's laws into a unified equation of state, defining the mathematical relation among pressure, volume, and temperature for dilute gases in equilibrium.
- ②تعريف المتغيرات: $p$ is pressure ($\text{Pa}$), $V$ is volume ($\text{m}^3$), $n$ is amount of substance ($\text{mol}$), $R \approx 8.314\;\text{J/(mol}\cdot\text{K)}$ is universal gas constant, $N$ is total molecular count, $k_B$ is Boltzmann's constant, and $T$ is absolute temperature ($\text{K}$).
- ③الأهمية الجوهرية: Baseline equation for gas behavior, derived from microscopic kinetic theory via $p = \frac{1}{3}nm\overline{v^2}$ and mean kinetic energy $\frac{1}{2}m\overline{v^2} = \frac{3}{2}k_B T$.
- ④نطاق الصلاحية والتطبيقات: Dilute gases at high temperature and low pressure where molecular volume and intermolecular potentials are negligible; dense or cold regimes require the van der Waals equation $[p + a(n/V)^2](V - nb) = nRT$. Applied in compressed gas cylinders, gas thermometers, and barometric altimetry.
15 · Boltzmann Distribution
Maxwell-Boltzmann State Statistics
$$N \propto e^{-E / k_B T} \iff P(E) = \frac{1}{Z}e^{-E / k_B T}$$
- ①الجوهر الفيزيائي: In a classical particle system in thermal equilibrium, the probability of a state with energy $E$ being occupied is proportional to the Boltzmann factor $e^{-E / k_B T}$; occupation probability decays exponentially with energy, while higher temperatures broaden high-energy occupancies.
- ②تعريف المتغيرات: $P(E)$ is probability of microstate occupancy, $E$ is microstate energy, $k_B$ is Boltzmann's constant, $T$ is temperature, and $Z = \sum e^{-E_i / k_B T}$ is canonical partition function.
- ③الأهمية الجوهرية: The core postulate of equilibrium statistical physics, generating all macroscopic thermodynamic potentials from microscopic energy states.
- ④نطاق الصلاحية والتطبيقات: Distinguishable particles and weakly degenerate classical systems. Applied in chemical kinetics Arrhenius activation energy ($k \propto e^{-E_a/RT}$), semiconductor thermal carrier generation, barometric density decay with altitude, and simulated annealing algorithms.
16 · Maxwell-Boltzmann Speed Distribution
Molecular Velocity Distribution
$$f(v) = 4\pi\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-mv^2 / 2k_B T}$$
- ①الجوهر الفيزيائي: Gas molecules in equilibrium possess isotropic thermal velocity vectors; the probability density of finding a molecule with speed in $[v, v+\mathrm{d}v]$ is given by an asymmetric skewed bell curve, where $v^2$ stems from the 3D spherical velocity volume element $4\pi v^2\mathrm{d}v$.
- ②تعريف المتغيرات: $f(v)$ is speed probability density function, $m$ is single molecular mass, $v$ is molecular speed, $k_B$ is Boltzmann constant, and $T$ is absolute temperature.
- ③الأهمية الجوهرية: The first statistical distribution in physics describing microscopic velocity kinematics. The characteristic speeds satisfy: **most probable speed $v_p = \sqrt{2k_B T/m}$ < mean speed $\overline{v} = \sqrt{8k_B T/(\pi m)}$ < root-mean-square speed $v_{\text{rms}} = \sqrt{3k_B T/m}$**.
- ④نطاق الصلاحية والتطبيقات: Classical equilibrium ideal gases. Explains why liquids evaporate below their boiling points (high-kinetic-energy tail molecules overcome surface potential barriers), applied in gas isotope centrifugal separation and molecular beam epitaxy.
17 · Mean Free Path
Mean Free Path
$$\lambda = \frac{1}{\sqrt{2}\pi d^2 n} = \frac{k_B T}{\sqrt{2}\pi d^2 p}$$
- ①الجوهر الفيزيائي: The average distance traversed by a gas molecule between successive elastic collisions with neighboring molecules; the factor $\sqrt{2}$ accounts for relative velocity between simultaneously moving colliding partners.
- ②تعريف المتغيرات: $\lambda$ is mean free path ($\text{m}$), $d$ is effective collision diameter ($\sim 10^{-10}\;\text{m}$), $n$ is molecular number density, $p$ is gas pressure, and $T$ is temperature.
- ③الأهمية الجوهرية: Microscopic physical yardstick for gas transport phenomena (viscosity, thermal conduction, mass diffusion); dictates whether flow operates in continuum ($\text{Kn} < 0.01$) or free molecular regime ($\text{Kn} > 10$).
- ④نطاق الصلاحية والتطبيقات: Hard-sphere elastic scattering. At standard ambient conditions, air molecules have $\lambda \approx 68\;\text{nm}$; under ultra-high vacuum ($p < 10^{-4}\;\text{Pa}$), $\lambda$ reaches kilometers, colliding primarily with chamber walls. Applied in semiconductor PVD/CVD deposition chamber vacuum calibration and vacuum insulation panel design.
4. Thermal Radiation & Heat Transfer (4 Laws)
Thermal energy transfers through conduction, convection, and electromagnetic radiation, governing industrial heat exchangers, cooling systems, and interstellar radiation.
18 · Stefan-Boltzmann Law
Total Blackbody Emissive Power
$$j = \sigma T^4 \quad (j = \varepsilon \sigma T^4)$$
- ①الجوهر الفيزيائي: Total electromagnetic radiation power emitted per unit surface area into hemispherical space by a blackbody over all wavelengths (radiant emittance) is strictly proportional to the fourth power of its absolute thermodynamic temperature ($T^4$).
- ②تعريف المتغيرات: $j$ is radiant emittance ($\text{W/m}^2$), $\sigma = 2\pi^5 k_B^4 / (15c^2 h^3) \approx 5.670374 \times 10^{-8}\;\text{W/(m}^2\cdot\text{K}^4)$ is the Stefan-Boltzmann constant, $T$ is temperature ($\text{K}$), and $\varepsilon$ is real surface emissivity ($0 < \varepsilon < 1$).
- ③الأهمية الجوهرية: Macroscopic radiation law derived by integrating Planck's spectral distribution, demonstrating the steep $T^4$ non-linear growth in radiative emission.
- ④نطاق الصلاحية والتطبيقات: Diffuse emitting surfaces in thermal equilibrium. Applied in determining stellar radii from luminosity and temperature ($L = 4\pi R^2 \sigma T^4$), optical infrared pyrometer calibration, and space radiator sizing in deep-space cryogenic environments.
19 · Wien's Displacement Law
Wien's Peak Emission Law
$$\lambda_{\max} T = b \quad (b \approx 2.898\times 10^{-3}\text{ m}\cdot\text{K})$$
- ①الجوهر الفيزيائي: The wavelength $\lambda_{\max}$ at which blackbody spectral radiance attains its maximum is inversely proportional to absolute temperature; peak emission shifts toward shorter wavelengths (higher frequencies) as temperature rises.
- ②تعريف المتغيرات: $\lambda_{\max}$ is peak emission wavelength ($\text{m}$), $T$ is absolute temperature ($\text{K}$), and $b \approx 2.89777 \times 10^{-3}\;\text{m}\cdot\text{K}$ is Wien's displacement constant.
- ③الأهمية الجوهرية: Established optical pyrometry and color temperature estimation from thermal spectra, providing a historical empirical catalyst for Planck's quantum hypothesis.
- ④نطاق الصلاحية والتطبيقات: Blackbody emission maxima. Explains color progression of heated metal from dull red to bright orange to white-hot; the Sun ($T \approx 5778\;\text{K}$) peaks in visible green light ($\sim 500\;\text{nm}$); human body ($310\;\text{K}$) peaks around $9.3\;\mu\text{m}$ mid-infrared, motivating pyroelectric infrared motion sensors.
20 · Fourier's Law of Heat Conduction
Fourier's Law
$$\mathbf{q} = -k \nabla T \iff q = -k \frac{\mathrm{d}T}{\mathrm{d}x}$$
- ①الجوهر الفيزيائي: In pure thermal conduction within an isotropic continuum, the local heat flux vector is proportional to the negative spatial temperature gradient; heat spontaneously flows down temperature gradients from hotter to colder regions.
- ②تعريف المتغيرات: $\mathbf{q}$ is heat flux density ($\text{W/m}^2$), $k$ (or $\lambda$) is thermal conductivity ($\text{W/(m}\cdot\text{K)}$), and $\nabla T$ is temperature gradient vector.
- ③الأهمية الجوهرية: The constitutive governing equation for conductive heat transfer; combining with internal energy conservation yields the transient heat diffusion equation $\partial T/\partial t = \alpha \nabla^2 T$ (where $\alpha = k/(\rho c_p)$ is thermal diffusivity).
- ④نطاق الصلاحية والتطبيقات: Dense solids and quiescent fluids. Applied in architectural thermal insulation design, CPU heat pipe and vapor chamber conduction modeling, and geothermal energy thermal depletion analysis.
21 · Newton's Law of Cooling
Convective Heat Transfer
$$\frac{\mathrm{d}T}{\mathrm{d}t} = -h(T - T_a) \iff T(t) = T_a + (T_0 - T_a)e^{-ht}$$
- ①الجوهر الفيزيائي: When temperature difference between a surface and ambient flowing fluid is moderate, surface heat loss rate (convective flux $q = h_c(T_s - T_a)$) is proportional to temperature differential, yielding negative exponential thermal decay.
- ②تعريف المتغيرات: $T$ is instantaneous body temperature, $T_a$ is ambient medium temperature, $T_0$ is initial temperature, $h$ is overall cooling rate parameter ($\text{s}^{-1}$), and $t$ is elapsed time.
- ③الأهمية الجوهرية: Fundamental phenomenological model for convective heat transfer, showing rapid early cooling that asymptotically approaches ambient temperature.
- ④نطاق الصلاحية والتطبيقات: Small-to-moderate temperature differences dominated by convection; high temperatures or vacuum require $T^4$ radiative adjustments. Applied in industrial metal casting quench control, blast chilling time estimation, forensic postmortem time of death estimation, and electronics heatsink air cooling analysis.
5. Physical Quantities, Symbols, and SI Units
| Physical Quantity | Symbol | Scalar / Vector | SI Base or Derived Unit | SI Symbol | Equivalence & Conversions |
|---|---|---|---|---|---|
| Pressure / Stress | $p, P$ | Scalar (Pressure) / Tensor (Stress) | Pascal | Pa | $1;\text{Pa} = 1;\text{N/m}^2 = 10^{-5};\text{bar} = 1/101325;\text{atm}$ |
| Thermodynamic Temperature | $T$ | Scalar | Kelvin (SI Base Unit) | K | $T(\text{K}) = t(^\circ\text{C}) + 273.15$ |
| Celsius Temperature | $t, \theta$ | Scalar | Degree Celsius | °C | $1;^\circ\text{C}$ temperature difference $\equiv 1;\text{K}$ difference |
| Heat / Internal Energy / Work | $Q, U, W$ | Scalar | Joule | J | $1;\text{J} = 1;\text{N}\cdot\text{m} = 1;\text{W}\cdot\text{s} \approx 0.239;\text{cal}$ |
| Thermodynamic Entropy | $S$ | Scalar | Joules per Kelvin | J/K | $S = \int(\delta Q/T)$ |
| Dynamic Viscosity | $\eta, \mu$ | Scalar | Pascal-second | $\text{Pa}\cdot\text{s}$ | $1;\text{Pa}\cdot\text{s} = 1;\text{N}\cdot\text{s/m}^2 = 10;\text{P} = 1000;\text{cP}$ |
| Kinematic Viscosity | $\nu$ | Scalar | Square meters per second | $\text{m}^2/\text{s}$ | $1;\text{m}^2/\text{s} = 10^4;\text{St} = 10^6;\text{cSt}$ |
| Volumetric Flow Rate | $Q, \dot{V}$ | Scalar | Cubic meters per second | $\text{m}^3/\text{s}$ | $1;\text{m}^3/\text{s} = 1000;\text{L/s} = 3600;\text{m}^3/\text{h}$ |
| Mass Flow Rate | $\dot{m}$ | Scalar | Kilograms per second | kg/s | $\dot{m} = \rho \cdot Q$ |
| Thermal Conductivity | $k, \lambda$ | Scalar | Watts per meter-Kelvin | $\text{W/(m}\cdot\text{K)}$ | Silver $\sim 429$, Copper $\sim 400$, Water $\sim 0.6$, Air $\sim 0.026$ |
| Heat Transfer Coefficient | $h, \alpha$ | Scalar | Watts per square meter-Kelvin | $\text{W/(m}^2\cdot\text{K)}$ | Air free convection $5\text{--}25$, Water forced $1000\text{--}15000$ |
| Radiant Emittance / Heat Flux | $j, q$ | Scalar (Emittance) / Vector (Flux) | Watts per square meter | $\text{W/m}^2$ | $1;\text{W/m}^2 = 1;\text{J/(s}\cdot\text{m}^2)$ |
| Boltzmann Constant | $k_B$ | Physical Constant | Joules per Kelvin | J/K | $k_B = 1.380649 \times 10^{-23};\text{J/K}$ (exact) |
| Universal Gas Constant | $R$ | Physical Constant | Joules per mole-Kelvin | $\text{J/(mol}\cdot\text{K)}$ | $R = N_A k_B \approx 8.314462;\text{J/(mol}\cdot\text{K)}$ |
| Stefan-Boltzmann Constant | $\sigma$ | Physical Constant | Watts per square meter-Kelvin$^4$ | $\text{W/(m}^2\cdot\text{K}^4)$ | $\sigma \approx 5.670374 \times 10^{-8};\text{W/(m}^2\cdot\text{K}^4)$ |
| Wien Displacement Constant | $b$ | Physical Constant | Meter-Kelvin | $\text{m}\cdot\text{K}$ | $b \approx 2.897772 \times 10^{-3};\text{m}\cdot\text{K}$ |
| Reynolds Number | $\mathrm{Re}$ | Dimensionless | Dimensionless (1) | - | $\mathrm{Re} = \rho v L / \eta$ (inertial force / viscous force) |