Fluid Dynamics Fundamentals — Continuum Mechanics and Conservation Laws

Scope: rigorous development of fluid dynamics from first principles, connecting macroscopic conservation laws with microscopic kinetic interpretations. Includes the continuum hypothesis, stress tensors, constitutive laws, and derivation of the Navier–Stokes equations.

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1. The Continuum Hypothesis

In fluid mechanics, matter is treated as a continuous medium—its properties (density, velocity, temperature, etc.) are defined as continuously differentiable fields. This is valid when the characteristic length scale $L$ of the flow is much larger than the molecular mean free path $\lambda$: $\text{Knudsen number: } \mathrm{Kn} = \frac{\lambda}{L} \ll 1.$

If $\mathrm{Kn} < 0.01$, continuum mechanics applies accurately; higher values require kinetic or rarefied gas corrections.

Field variables: $\rho(\mathbf{x},t),\; \mathbf{v}(\mathbf{x},t),\; T(\mathbf{x},t),\; p(\mathbf{x},t).$

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2. Material (Substantial) Derivative

The time rate of change of a quantity following a fluid particle is: $\frac{D\phi}{Dt} = \frac{\partial\phi}{\partial t} + \mathbf{v} \cdot \nabla\phi.$

Thus, the acceleration field:

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This operator converts local field equations to the Lagrangian viewpoint.

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3. Conservation of Mass

Integral form over control volume $V$: $\frac{d}{dt}\int_V \rho\,dV + \oint_S \rho\mathbf{v}\cdot\mathbf{n}\,dA = 0.$

Applying divergence theorem:

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This is the continuity equation.

For incompressible flow (constant density):

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4. Conservation of Momentum (Cauchy Momentum Equation)

4.1 Integral Form

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4.2 Differential Form

Using divergence theorem and applying to an infinitesimal element:

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This is the Cauchy momentum equation, valid for all continua (solids or fluids).

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5. Stress Tensor Decomposition

The Cauchy stress tensor $\boldsymbol{\sigma}$ is decomposed as:

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where:

• $p$ is the thermodynamic pressure, isotropic part of stress.
• $\boldsymbol{\tau}$ is the deviatoric (viscous) stress tensor, associated with rate of deformation.

Symmetry of $\boldsymbol{\sigma}$ ($\sigma_{ij}=\sigma_{ji}$) follows from angular momentum conservation.

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6. Constitutive Relation for a Newtonian Fluid

For a simple, isotropic fluid, $\boldsymbol{\tau}$ depends linearly on the rate-of-strain tensor:

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Then:

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where:

• $\mu$ = dynamic (shear) viscosity,
• $\lambda$ = second (bulk) viscosity.

For incompressible flow ($\\nabla\cdot\mathbf{v}=0$):

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Stokes’ hypothesis: $\lambda = -\tfrac{2}{3}\mu$ often used for gases.

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7. Navier–Stokes Equation (Momentum Conservation for a Newtonian Fluid)

Substitute stress decomposition into Cauchy equation:

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Using the Newtonian constitutive relation:

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For incompressible flow ($\\nabla\cdot\mathbf{v}=0$):

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This is the Navier–Stokes equation.

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8. Conservation of Energy

Energy per unit mass: $e = u + \frac{v^2}{2} + gz$. Conservation law:

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Internal energy equation (subtract kinetic part):

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For Fourier conduction $\mathbf{q} = -k\nabla T$:

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where $\Phi = \boldsymbol{\tau}:\nabla\mathbf{v}$ is viscous dissipation.

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9. Boundary Conditions

Boundary Type	Condition	Physical Meaning
Solid wall	$\mathbf{v} = 0$	No-slip condition
Free surface	$\mathbf{n}\cdot\boldsymbol{\tau} = 0$	No tangential stress
Interface	Continuity of $v_n, \tau_{nn}, \tau_{nt}$	Stress balance
Far field	$\mathbf{v} \to \mathbf{v}_\infty, p \to p_\infty$	Ambient matching

The no-slip condition originates microscopically from momentum exchange between molecules and the wall.

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10. Vorticity and Streamfunction Formulations

10.1 Vorticity

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For incompressible flow:

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10.2 Streamfunction (2D Incompressible Flow)

Define $\psi(x,y)$ such that:

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Then $\\nabla\cdot\mathbf{v}=0$ automatically.

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11. Non-Newtonian and Complex Fluids

For viscoelastic or shear-dependent fluids, $\boldsymbol{\tau}$ depends nonlinearly on $\\nabla\mathbf{v}$:

Model	Constitutive Relation	Application
Power-law	$\tau_{ij} = K	e_{ij}
Bingham plastic	$\tau = \tau_y + \mu_p \dot\gamma$	Slurries, toothpaste
Maxwell	$\tau + \lambda d\tau/dt = 2\mu e$	Viscoelastic solutions

These extend Navier–Stokes to rheological fluids.

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12. Dimensionless Form and Similarity Parameters

Nondimensionalize using reference quantities $L, U, \rho, \mu, k$:

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Key Dimensionless Numbers

Symbol	Definition	Significance
Reynolds (Re)	$\rho UL/\mu$	Inertial vs viscous forces
Prandtl (Pr)	$\mu c_p/k$	Momentum vs thermal diffusion
Mach (Ma)	$U/a$	Compressibility effects
Peclet (Pe)	$Re \cdot Pr$	Advective vs conductive heat transfer
Froude (Fr)	$U/\sqrt{gL}$	Inertia vs gravity

Dynamic similarity requires matching these nondimensional groups.

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13. Microscopic Interpretation — Kinetic Theory Connection

From the Boltzmann equation, the viscous stress and heat flux emerge as first-order moments of velocity fluctuations:

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Molecular expressions:

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These link continuum parameters to molecular transport phenomena.

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14. Energy Coupling and Irreversibility

Combining thermodynamics and fluid mechanics, local entropy production in a viscous, heat-conducting fluid:

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This expresses mechanical-to-thermal energy conversion (viscous heating) and enforces the second law in continuum mechanics.

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15. Summary of Governing Equations

Conservation Law	Differential Form
Mass	$\partial\rho/\partial t + \nabla\cdot(\rho\mathbf{v}) = 0$
Momentum	$\rho D\mathbf{v}/Dt = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \rho\mathbf{g}$
Energy	$\rho Du/Dt = \boldsymbol{\tau}:\nabla\mathbf{v} - \nabla\cdot\mathbf{q}$
Entropy	$\rho Ds/Dt = \sigma_s \ge 0$

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16. Cross-Links

• 10_NonEquilibrium_Thermodynamics.md — entropy generation and coupled transport.
• Fluid_Dynamics/01_Potential_and_Viscous_Flows.md — applications of Navier–Stokes to laminar and inviscid flows.
• Fluid_Dynamics/transport-equations.md — PDE formulations and numerical solution methods for flow fields.