Mathematical Prerequisites for Fluid Mechanics
This reference collects the essential mathematical tools you will need across fluid mechanics — from undergraduate viscous-flow analysis through graduate-level turbulence, stability theory, and compressible gas dynamics. Each formula is presented with precise conditions and a note on where it appears in the physics.
Why These Prerequisites?
Fluid mechanics is one of the most mathematically demanding branches of classical physics. The Navier-Stokes equations — a system of nonlinear partial differential equations — have resisted general analytical solution for nearly two centuries. Working effectively with these equations requires facility with vector calculus, tensor analysis, dimensional reasoning, complex variables, asymptotic methods, and thermodynamics.
This page is organized to build from foundational tools (vector calculus, integral theorems) through the construction of the governing equations (tensors, Navier-Stokes derivation) to the specialized techniques used to solve them (complex analysis, perturbation methods, similarity solutions). Each section is self-contained but cross-references the others where connections arise.
Quick Jump
§1 — Vector Calculus Identities
Vector calculus is the language of fluid mechanics. Every conservation law, every transport equation, and every flow visualization technique rests on the gradient, divergence, and curl operators. Mastery of these identities is non-negotiable.
$$\nabla f = \frac{\partial f}{\partial x_i}\hat{e}_i$$
The gradient of a scalar field points in the direction of steepest increase. In fluid mechanics it appears in the pressure gradient force term and in the definition of velocity potential for irrotational flows.
$$\nabla \cdot \mathbf{v} = \frac{\partial v_i}{\partial x_i}$$
The divergence measures the local rate of volume expansion of a fluid element. For incompressible flows this vanishes identically, giving the kinematic constraint that dominates low-speed fluid dynamics.
$$(\nabla \times \mathbf{v})_i = \epsilon_{ijk}\frac{\partial v_k}{\partial x_j}$$
The curl of the velocity field is the vorticity, the single most important kinematic quantity in fluid mechanics. Vorticity dynamics governs turbulence, boundary-layer separation, and aerodynamic lift generation.
$$\nabla^2 f = \frac{\partial^2 f}{\partial x_i \partial x_i}$$
The Laplacian appears in the viscous diffusion term of the Navier-Stokes equations and in potential flow theory where the velocity potential satisfies Laplace's equation.
$$\frac{Df}{Dt} = \frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla f$$
The material derivative is the rate of change following a fluid particle. The first term is the local (Eulerian) rate of change; the second is the convective rate of change due to transport by the flow. This operator bridges the Lagrangian and Eulerian descriptions.
$$\nabla \times (\nabla f) = \mathbf{0}$$
$$\nabla \cdot (\nabla \times \mathbf{v}) = 0$$
The first identity guarantees that irrotational flows can be described by a velocity potential. The second ensures that vorticity is always solenoidal — vortex lines cannot begin or end in the interior of the fluid.
$$\nabla \cdot (f\mathbf{v}) = f(\nabla\cdot\mathbf{v}) + \mathbf{v}\cdot(\nabla f)$$
$$\nabla \times (f\mathbf{v}) = f(\nabla\times\mathbf{v}) + (\nabla f)\times\mathbf{v}$$
These product rules are essential when deriving the continuity equation from the divergence of mass flux and when manipulating vorticity transport equations.
$$\epsilon_{ijk} = \begin{cases} +1 & \text{even permutation of } (1,2,3) \\ -1 & \text{odd permutation} \\ 0 & \text{repeated index} \end{cases}$$
The Levi-Civita (alternating) symbol converts cross products and curls into index notation. Combined with the Einstein summation convention, it allows compact manipulation of all vector identities and tensor expressions in fluid mechanics.
$$\mathbf{v}\cdot\nabla\mathbf{v} = \nabla\!\left(\tfrac{1}{2}|\mathbf{v}|^2\right) - \mathbf{v}\times(\nabla\times\mathbf{v})$$
This identity rewrites the nonlinear convective acceleration in terms of kinetic energy and vorticity. It is the key step in deriving Bernoulli's equation from the Euler equations and reveals why irrotational flows obey particularly simple energy relations.
§2 — Integral Theorems
Integral theorems convert between volume, surface, and line integrals. They are the mathematical backbone of every conservation law in fluid mechanics and the starting point for control-volume analysis.
$$\int_V \nabla\cdot\mathbf{F}\,dV = \oint_S \mathbf{F}\cdot\hat{n}\,dA$$
Converts a volume integral of divergence into a surface flux integral. This theorem underpins the integral forms of conservation of mass, momentum, and energy and is the basis of all control-volume analysis in engineering fluid mechanics.
$$\oint_C \mathbf{F}\cdot d\mathbf{l} = \int_S (\nabla\times\mathbf{F})\cdot\hat{n}\,dA$$
Relates the circulation around a closed curve to the flux of vorticity through the enclosed surface. This is the mathematical statement behind Kelvin's circulation theorem, the Kutta-Joukowski theorem for lift, and vortex dynamics.
$$\int_V \left(\phi\nabla^2\psi + \nabla\phi\cdot\nabla\psi\right)dV = \oint_S \phi\frac{\partial\psi}{\partial n}\,dA$$
Used in potential flow theory to establish uniqueness of solutions to Laplace's equation given appropriate boundary conditions.
$$\int_V \left(\phi\nabla^2\psi - \psi\nabla^2\phi\right)dV = \oint_S \left(\phi\frac{\partial\psi}{\partial n} - \psi\frac{\partial\phi}{\partial n}\right)dA$$
The symmetric form of Green's identity. Central to boundary-integral (panel) methods for solving potential flows around arbitrary bodies such as airfoils and ship hulls.
$$\frac{d}{dt}\int_{V(t)} f\,dV = \int_{V(t)}\frac{\partial f}{\partial t}\,dV + \oint_{S(t)} f\,\mathbf{v}\cdot\hat{n}\,dA$$
THE bridge from the Lagrangian (material) to the Eulerian (field) description. Every integral conservation law in fluid mechanics — mass, momentum, energy — is derived by applying the Reynolds transport theorem to a control volume moving with the flow.
$$\frac{d}{dt}\int_{a(t)}^{b(t)} f(x,t)\,dx = \int_{a(t)}^{b(t)}\frac{\partial f}{\partial t}\,dx + f(b,t)\frac{db}{dt} - f(a,t)\frac{da}{dt}$$
The one-dimensional precursor to the Reynolds transport theorem. Frequently used in deriving shallow-water equations and thin-film flow models where integration limits depend on time.
§3 — Tensor Analysis & Index Notation
Fluid stresses, strain rates, and the full Navier-Stokes equations are naturally expressed in tensor form. Index notation with the Einstein summation convention is the most compact and manipulation-friendly representation.
$$\sigma_{ij} = -p\delta_{ij} + \tau_{ij}$$
The total stress in a fluid decomposes into an isotropic pressure contribution and the deviatoric (viscous) stress tensor. This decomposition is fundamental to constitutive modeling of Newtonian and non-Newtonian fluids.
$$e_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)$$
The symmetric part of the velocity gradient tensor. It measures the rate of deformation of fluid elements and is the kinematic quantity to which viscous stress is proportional in Newtonian fluids.
$$\omega_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}-\frac{\partial v_j}{\partial x_i}\right)$$
The antisymmetric part of the velocity gradient tensor. Its dual vector is half the vorticity vector. This tensor represents the rigid-body rotation component of the local fluid motion.
$$\frac{\partial v_i}{\partial x_j} = e_{ij} + \omega_{ij}$$
Any velocity gradient decomposes uniquely into symmetric (strain) and antisymmetric (rotation) parts. This decomposition is a fundamental theorem of continuum mechanics and reveals the distinct physical roles of deformation and vorticity.
$$\epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}$$
This contraction identity is the workhorse for converting double cross-product expressions into index form. It is used repeatedly in deriving vorticity equations and in proving vector calculus identities such as the Lamb vector identity.
$$\sigma'_{ij} = Q_{ip}Q_{jq}\sigma_{pq}, \qquad I_1 = \sigma_{ii}, \quad I_2 = \tfrac{1}{2}(\sigma_{ii}\sigma_{jj}-\sigma_{ij}\sigma_{ji}), \quad I_3 = \det(\sigma_{ij})$$
Tensors transform via rotation matrices. Their invariants (trace, cofactor sum, determinant) are frame-independent and appear in constitutive models and turbulence closures.
$$t_i = \sigma_{ij}n_j$$
The traction (force per unit area) on a surface with outward normal n is given by the stress tensor contracted with the normal vector. This is the foundational relationship connecting internal stresses to surface forces in any continuum, and is the starting point for deriving the momentum equation.
§5 — Dimensional Analysis
Dimensional analysis reduces the number of independent parameters in any fluid problem. It determines which forces dominate, when terms can be neglected, and how laboratory models scale to full-size systems.
$$\text{If } f(q_1, q_2, \ldots, q_n) = 0 \text{ involves } k \text{ fundamental dimensions, then } f(\Pi_1, \Pi_2, \ldots, \Pi_{n-k}) = 0$$
Any physically meaningful relation among $n$ dimensional quantities can be rewritten in terms of $n - k$ dimensionless groups, where $k$ is the number of independent fundamental dimensions (typically mass, length, time, temperature).
$$Re = \frac{\rho U L}{\mu} = \frac{UL}{\nu}$$
The ratio of inertial to viscous forces — the single most important dimensionless number in fluid mechanics. It determines the flow regime: low Re gives laminar flow; high Re gives turbulent flow. The critical value depends on geometry.
$$Fr = \frac{U}{\sqrt{gL}}$$
Ratio of inertial to gravitational forces. Governs free-surface flows, ship waves, hydraulic jumps, and open-channel flow classification (subcritical vs. supercritical).
$$Ma = \frac{U}{c}$$
Ratio of flow speed to the speed of sound. Compressibility effects become significant for Ma > 0.3. Subsonic (Ma < 1), transonic, supersonic (Ma > 1), and hypersonic (Ma > 5) regimes each have distinct physics.
$$St = \frac{fL}{U}, \quad We = \frac{\rho U^2 L}{\sigma}, \quad Pr = \frac{\nu}{\alpha}, \quad Gr = \frac{g\beta\Delta T\,L^3}{\nu^2}$$
Strouhal: unsteady/oscillatory flows (vortex shedding). Weber: surface-tension-dominated flows (droplets, jets). Prandtl: thermal vs. momentum diffusion. Grashof: buoyancy-driven (natural convection) flows. Each controls a different physical mechanism.
$$Re \gg 1 \;\Rightarrow\; \text{neglect viscous terms (Euler equations)}, \qquad Re \ll 1 \;\Rightarrow\; \text{neglect inertia (Stokes flow)}$$
The art of fluid mechanics lies in knowing which terms to keep. At high Reynolds number, viscous effects are confined to thin boundary layers. At low Reynolds number (microfluidics, sedimentation), inertia is negligible and the equations become linear. The Mach number controls whether compressibility matters; the Froude number controls whether gravity matters relative to inertia.
$$\frac{\partial \mathbf{v}^*}{\partial t^*} + \mathbf{v}^*\cdot\nabla^*\mathbf{v}^* = -\nabla^* p^* + \frac{1}{Re}\nabla^{*2}\mathbf{v}^* + \frac{1}{Fr^2}\hat{\mathbf{g}}$$
Scaling all variables by characteristic quantities reveals that the Reynolds and Froude numbers are the only parameters controlling incompressible flow with gravity. At high Re the viscous term is negligible in the bulk, motivating boundary-layer theory.
§6 — Curvilinear Coordinates
Many canonical fluid flows — pipe flow, jets, vortices, flow over spheres — possess cylindrical or spherical symmetry. Expressing the governing equations in the natural coordinate system dramatically simplifies the problem.
$$\nabla f = \frac{\partial f}{\partial r}\hat{e}_r + \frac{1}{r}\frac{\partial f}{\partial\theta}\hat{e}_\theta + \frac{\partial f}{\partial z}\hat{e}_z$$
The gradient in cylindrical coordinates. The $1/r$ factor in the azimuthal component reflects the non-unit metric in the angular direction.
$$\nabla\cdot\mathbf{v} = \frac{1}{r}\frac{\partial(rv_r)}{\partial r} + \frac{1}{r}\frac{\partial v_\theta}{\partial\theta} + \frac{\partial v_z}{\partial z}$$
Setting this to zero gives the incompressibility condition for axisymmetric and fully three-dimensional pipe, jet, and vortex flows.
$$\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}\!\left(r\frac{\partial f}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2} + \frac{\partial^2 f}{\partial z^2}$$
Appears in the viscous term for pipe flow (Hagen-Poiseuille), the diffusion of vorticity in vortex cores, and potential flow problems with circular geometry.
$$\nabla\times\mathbf{v} = \left(\frac{1}{r}\frac{\partial v_z}{\partial\theta}-\frac{\partial v_\theta}{\partial z}\right)\hat{e}_r + \left(\frac{\partial v_r}{\partial z}-\frac{\partial v_z}{\partial r}\right)\hat{e}_\theta + \frac{1}{r}\left(\frac{\partial(rv_\theta)}{\partial r}-\frac{\partial v_r}{\partial\theta}\right)\hat{e}_z$$
The vorticity in cylindrical coordinates. The axial component is particularly important for swirling flows and the stability analysis of vortices.
$$\nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\!\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial f}{\partial\theta}\right) + \frac{1}{r^2\sin^2\!\theta}\frac{\partial^2 f}{\partial\phi^2}$$
Essential for Stokes flow past a sphere, potential flow over spherical bodies, and any problem with spherical symmetry. Separation of variables in this form leads to Legendre polynomials and spherical harmonics.
$$ds^2 = h_1^2\,dq_1^2 + h_2^2\,dq_2^2 + h_3^2\,dq_3^2, \qquad \nabla\cdot\mathbf{v} = \frac{1}{h_1 h_2 h_3}\sum_i \frac{\partial}{\partial q_i}(h_j h_k\,v_i)$$
The general orthogonal curvilinear framework. Scale factors $h_i$ encode the geometry; all differential operators follow from them. Cylindrical: $(h_r,h_\theta,h_z)=(1,r,1)$. Spherical: $(h_r,h_\theta,h_\phi)=(1,r,r\sin\theta)$.
$$\rho\!\left(\frac{\partial v_r}{\partial t} + v_r\frac{\partial v_r}{\partial r} + \frac{v_\theta}{r}\frac{\partial v_r}{\partial\theta} - \frac{v_\theta^2}{r} + v_z\frac{\partial v_r}{\partial z}\right) = -\frac{\partial p}{\partial r} + \mu\!\left(\nabla^2 v_r - \frac{v_r}{r^2} - \frac{2}{r^2}\frac{\partial v_\theta}{\partial\theta}\right) + \rho g_r$$
The radial momentum equation in cylindrical coordinates. Note the centripetal acceleration term $-v_\theta^2/r$ and the extra viscous terms that arise from the non-Cartesian geometry. Similar expressions hold for the azimuthal and axial components.
§7 — Complex Analysis for 2D Flows
Two-dimensional, incompressible, irrotational (potential) flow is elegantly described by complex-variable methods. The velocity field is encoded in a single analytic function, and conformal mappings transform simple geometries into airfoil shapes.
$$\frac{\partial\phi}{\partial x} = \frac{\partial\psi}{\partial y}, \qquad \frac{\partial\phi}{\partial y} = -\frac{\partial\psi}{\partial x}$$
For an analytic function $W(z) = \phi + i\psi$, the Cauchy-Riemann equations are exactly the conditions that the velocity potential $\phi$ and the stream function $\psi$ satisfy Laplace's equation and that the flow is incompressible and irrotational.
$$W(z) = \phi + i\psi$$
The complex potential encodes both the velocity potential (real part) and the stream function (imaginary part). Lines of constant $\phi$ are equipotential lines; lines of constant $\psi$ are streamlines.
$$\frac{dW}{dz} = u - iv$$
The derivative of the complex potential gives the conjugate velocity. The magnitude gives the speed, and the argument gives the flow direction. Stagnation points occur where this derivative vanishes.
$$W_{\text{uniform}} = U_\infty z, \quad W_{\text{source}} = \frac{m}{2\pi}\ln z, \quad W_{\text{vortex}} = \frac{i\Gamma}{2\pi}\ln z, \quad W_{\text{doublet}} = \frac{\mu}{2\pi z}$$
Every potential flow is built by superposing these elementary solutions: uniform stream, point source/sink, point vortex, and doublet. Their combinations produce flows past cylinders, Rankine bodies, and more complex shapes.
$$W(z) = U_\infty\!\left(z + \frac{a^2}{z}\right) + \frac{i\Gamma}{2\pi}\ln z$$
Superposition of a uniform stream, a doublet (which creates the cylinder of radius $a$), and a point vortex (which adds circulation $\Gamma$). The circulation breaks the symmetry and produces lift via the Kutta-Joukowski theorem.
$$\zeta = z + \frac{c^2}{z}$$
The Joukowski transform maps a circle in the $z$-plane to an airfoil shape in the $\zeta$-plane. By choosing the circle's center and radius appropriately, one generates airfoils with specified thickness and camber — the foundation of classical thin-airfoil theory.
$$F_x - iF_y = \frac{i\rho}{2}\oint_C \left(\frac{dW}{dz}\right)^{\!2} dz$$
The Blasius theorem computes the net force on a body directly from the complex velocity via a contour integral. Combined with the residue theorem, it yields the Kutta-Joukowski lift formula $L = \rho U_\infty \Gamma$ per unit span.
§8 — Ordinary & Partial Differential Equations
Fluid mechanics generates all three types of second-order PDEs — elliptic, parabolic, and hyperbolic — depending on the flow regime. The solution techniques range from separation of variables to similarity transforms to Green's functions.
$$A\frac{\partial^2 u}{\partial x^2} + 2B\frac{\partial^2 u}{\partial x\partial y} + C\frac{\partial^2 u}{\partial y^2} + \cdots = 0$$
$$B^2 - AC \begin{cases} < 0 & \text{Elliptic (Laplace, potential flow)} \\ = 0 & \text{Parabolic (diffusion, boundary layer)} \\ > 0 & \text{Hyperbolic (wave, compressible)} \end{cases}$$
The discriminant determines the equation type, which in turn dictates the appropriate boundary/initial conditions and solution methods. Potential flow is elliptic; the boundary layer equations are parabolic; supersonic flow equations are hyperbolic.
$$\nabla^2\phi = 0, \quad \phi(r,\theta) = \sum_{n=0}^{\infty}\left(A_n r^n + B_n r^{-n}\right)\!\left(C_n\cos n\theta + D_n\sin n\theta\right)$$
The classical approach for potential flow in cylindrical geometry. Each term in the series corresponds to a multipole: $n=0$ is the source, $n=1$ is the doublet/uniform stream, and higher terms give increasingly complex angular dependence.
$$f''' + \frac{1}{2}ff'' = 0$$
The Blasius equation is obtained by introducing the similarity variable $\eta = y\sqrt{U_\infty/(\nu x)}$ into the boundary-layer equations for flow over a flat plate. This third-order nonlinear ODE reduces the PDE to a single independent variable and is solved numerically with a shooting method.
$$\hat{f}(k) = \int_{-\infty}^{\infty} f(x)\,e^{-ikx}\,dx, \qquad f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(k)\,e^{ikx}\,dk$$
Fourier transforms convert differential equations into algebraic equations in wavenumber space. They are used extensively in stability analysis, turbulence theory (energy spectra), and acoustic wave propagation.
$$G_{ij}(\mathbf{x}) = \frac{1}{8\pi\mu}\left(\frac{\delta_{ij}}{r} + \frac{x_i x_j}{r^3}\right) \quad \text{(Stokeslet / Oseen-Burgers tensor)}$$
The Stokeslet is the fundamental solution for Stokes (creeping) flow driven by a point force. It is the building block for boundary-integral methods in microfluidics, biological locomotion, and colloidal suspensions.
$$\frac{dx}{dt} = u \pm c \quad \Longrightarrow \quad \text{Mach lines: } \mu = \arcsin(1/Ma)$$
For hyperbolic systems (supersonic flow, water waves), information propagates along characteristic curves. The method of characteristics converts PDEs into ODEs along these curves, enabling analytical solutions for nozzle design, shock interactions, and Riemann problems.
§9 — Stability & Perturbation Methods
Understanding when and why laminar flows become unstable is one of the central problems in fluid mechanics. Linear stability analysis, combined with asymptotic and perturbation methods, provides the theoretical framework for predicting transition to turbulence.
$$\mathbf{v} = \mathbf{V}(y) + \epsilon\,\mathbf{v}'(x,y,z,t), \qquad \epsilon \ll 1$$
Decompose the flow into a known base state and a small perturbation. Substitute into the Navier-Stokes equations and linearize by dropping terms of order $\epsilon^2$ and higher. The resulting linear system determines whether perturbations grow or decay.
$$\mathbf{v}'(x,y,t) = \hat{\mathbf{v}}(y)\,e^{i(\alpha x - \omega t)}$$
For parallel or nearly parallel base flows, perturbations are decomposed into Fourier modes in the streamwise direction and time. Temporal stability: $\alpha$ real,$\omega$ complex. Spatial stability: $\omega$ real, $\alpha$ complex. Growth occurs when $\text{Im}(\omega) > 0$ (temporal) or $\text{Im}(\alpha) < 0$ (spatial).
$$(U-c)(\phi''-\alpha^2\phi) - U''\phi = \frac{1}{i\alpha Re}\left(\phi''''-2\alpha^2\phi''+\alpha^4\phi\right)$$
The central eigenvalue problem of viscous stability theory. Given a base-flow profile $U(y)$ and Reynolds number, one solves for the complex phase speed $c = \omega/\alpha$. The locus of neutral curves $(\text{Im}(c)=0)$ in the $(\alpha, Re)$ plane defines the critical Reynolds number for instability.
$$(U-c)(\phi''-\alpha^2\phi) - U''\phi = 0 \quad \Longrightarrow \quad U''(y_c) = 0 \text{ is necessary for inviscid instability}$$
In the inviscid limit (Rayleigh equation), a necessary condition for instability is that the base-flow velocity profile has an inflection point. This explains why jets and wakes (which have inflection points) are far more unstable than boundary layers and pipe flows.
$$Re_{crit,\text{3D}} \geq Re_{crit,\text{2D}}$$
For every unstable three-dimensional perturbation there exists a more unstable two-dimensional perturbation at a lower Reynolds number. This theorem justifies restricting linear stability analysis to two-dimensional disturbances when seeking the critical Reynolds number.
$$f(\epsilon) = f_0 + \epsilon f_1 + \epsilon^2 f_2 + \cdots$$
When a small parameter $\epsilon$ appears and the solution depends smoothly on it, a straightforward power-series expansion yields successive corrections. Used for weakly nonlinear analysis, slow viscous flow corrections (Whitehead's paradox notwithstanding), and low-amplitude wave interactions.
$$f_{\text{outer}}(x;\epsilon) \quad \text{and} \quad f_{\text{inner}}(X;\epsilon), \quad X = x/\epsilon$$
$$\lim_{x\to 0} f_{\text{outer}} = \lim_{X\to\infty} f_{\text{inner}} \quad \text{(matching condition)}$$
Singular perturbation problems — where the small parameter multiplies the highest derivative — are the mathematical DNA of boundary-layer theory. The outer (inviscid) and inner (boundary-layer) solutions are constructed separately and matched in an overlap region. This is exactly how Prandtl resolved d'Alembert's paradox.
§10 — Thermodynamics & Equations of State
Compressible flow, heat transfer, and combustion all require coupling the Navier-Stokes equations to thermodynamic relations. The energy equation closes the system, and equations of state connect pressure, density, and temperature.
$$de = T\,dS - p\,dv$$
The fundamental thermodynamic identity relating changes in specific internal energy $e$ to entropy $S$ and specific volume $v = 1/\rho$. This is the starting point for deriving all thermodynamic relations used in compressible fluid mechanics.
$$h = e + \frac{p}{\rho}$$
The specific enthalpy combines internal energy and flow work. The stagnation (total) enthalpy $h_0 = h + \frac{1}{2}|\mathbf{v}|^2$ is conserved along streamlines in steady, adiabatic, inviscid flow — the compressible generalization of Bernoulli's equation.
$$p = \rho R T$$
The simplest equation of state. Valid for gases at moderate temperatures and pressures. Together with the caloric equation of state $e = c_v T$, it closes the compressible Navier-Stokes system for an ideal gas.
$$c = \sqrt{\left(\frac{\partial p}{\partial\rho}\right)_{\!s}} = \sqrt{\gamma R T} \quad \text{(ideal gas)}$$
The speed at which small pressure disturbances propagate through a fluid. It is defined as the isentropic derivative of pressure with respect to density. The Mach number $Ma = U/c$ measures the flow speed relative to this wave speed.
$$\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}Ma^2, \qquad \frac{p_0}{p} = \left(1+\frac{\gamma-1}{2}Ma^2\right)^{\!\gamma/(\gamma-1)}, \qquad \frac{\rho_0}{\rho} = \left(1+\frac{\gamma-1}{2}Ma^2\right)^{\!1/(\gamma-1)}$$
These relations connect static and stagnation quantities for isentropic (reversible, adiabatic) flow of an ideal gas. They are the foundation of compressible-flow analysis: nozzle design, shock relations, and wind-tunnel calibration all rely on them.
$$\rho\frac{De}{Dt} = -p\,\nabla\cdot\mathbf{v} + \Phi + \nabla\cdot(k\nabla T)$$
The internal energy equation for a Newtonian fluid. The three terms on the right are: reversible compression work, irreversible viscous dissipation, and heat conduction (Fourier's law). This equation, together with the continuity and momentum equations, completes the system for compressible, heat-conducting flow.
$$\Phi = \tau_{ij}\frac{\partial v_i}{\partial x_j} = \mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)\frac{\partial v_i}{\partial x_j} + \lambda(\nabla\cdot\mathbf{v})^2$$
The rate at which kinetic energy is irreversibly converted to internal energy (heat) by viscous stresses. Always non-negative (by the second law of thermodynamics). In incompressible flow it simplifies to $\Phi = 2\mu\,e_{ij}e_{ij}$. Important for high-speed flows and viscous heating in lubrication problems.
A Note on Mathematical Maturity
Fluid mechanics draws on virtually every branch of applied mathematics. You are not expected to have memorized every formula on this page before beginning the course. Rather, this page serves as a reference that you will return to repeatedly as new topics are encountered.
The most important prerequisites for getting started are: comfort with vector calculus (Section 1), the divergence and Stokes theorems (Section 2), and basic tensor/index notation (Section 3). The remaining sections become relevant as you progress through the course — complex analysis when you study potential flow, perturbation methods when you study boundary layers and stability, and thermodynamics when you study compressible flow.
If you find gaps in your background, the cross-course links at the top of this page point to dedicated courses on mathematics, linear algebra, classical mechanics, and thermodynamics.