Fluid Mechanics

1. Material (Substantial) Derivative

In fluid mechanics we track physical quantities using two complementary viewpoints. The Lagrangian description follows individual fluid parcels as they move, while the Eulerian description measures quantities at fixed points in space as fluid flows past. The material derivative connects the two: it gives the rate of change experienced by a moving fluid element in terms of Eulerian field variables.

Derivation from the Chain Rule

Consider a scalar field \( \phi(\vec{x}, t) = \phi(x, y, z, t) \) -- this could be temperature, density, concentration, or any other intensive property. A particular fluid parcel follows a trajectory\( \vec{x}(t) = \big(x(t),\, y(t),\, z(t)\big) \) determined by the local velocity. The total rate of change of \( \phi \) as experienced by this parcel is, by the multivariable chain rule:

Since \( dx/dt = v_x \), \( dy/dt = v_y \), \( dz/dt = v_z \) are the velocity components, we can write this compactly using the gradient operator\( \nabla = (\partial/\partial x,\, \partial/\partial y,\, \partial/\partial z) \):

The capital-D notation \( D/Dt \) (or sometimes written \( d/dt \) with context) is reserved for this total derivative following the fluid, to distinguish it from the partial \( \partial/\partial t \) at a fixed point. The last form uses Einstein summation convention (repeated index j is summed over x, y, z).

Physical Meaning of Each Term

Example -- steady converging nozzle: Consider steady, incompressible flow through a nozzle. The velocity field is time-independent (\( \partial\vec{v}/\partial t = 0 \)), yet a fluid parcel accelerates as it enters the throat because it moves into a region of higher velocity:

Material Derivative of a Vector Field

For a vector quantity such as velocity \( \vec{v} \) itself, the material derivative applies component-by-component. In index notation:

or equivalently in vector notation:

This quantity is the acceleration of a fluid parcel and appears on the left-hand side of every momentum equation (Newton's second law for fluids). Note that\( (\vec{v}\cdot\nabla)\vec{v} \) is a nonlinear term -- the velocity advects itself -- and is the primary source of complexity in fluid dynamics.

Explicit Cartesian Components

Writing out the x-component of \( D\vec{v}/Dt \) in full (the y and z components follow by symmetry):

Lamb Form (Useful Vector Identity)

The convective acceleration can be rewritten using a key vector identity that separates irrotational and rotational contributions:

where \( \vec{\omega} = \nabla\times\vec{v} \) is the vorticity. This Lamb form is essential for deriving Bernoulli's equation (along streamlines the cross-product term vanishes) and the Crocco-Vazsonyi theorem relating entropy gradients to vorticity.

Cylindrical Coordinates \((r, \theta, z)\)

Many flows have axial or rotational symmetry. In cylindrical coordinates, the material derivative of a scalar and the radial component of \( D\vec{v}/Dt \) are:

The extra term \( -v_\theta^2/r \) is the centripetal acceleration -- it arises because the unit vectors \( \hat{e}_r, \hat{e}_\theta \) themselves rotate as a parcel moves in the \( \theta \)-direction.

Reynolds Transport Theorem (RTT)

The material derivative is the pointwise version of a more general result for material volumes. For any extensive property \( B \) with intensive counterpart \( \beta = dB/dm \), the Reynolds Transport Theorem states:

This theorem converts the Lagrangian (system) viewpoint into the Eulerian (control volume) viewpoint for integral quantities. When applied to mass (\( \beta = 1 \)), momentum (\( \beta = \vec{v} \)), and energy (\( \beta = e + v^2/2 \)), it yields the integral conservation laws that are the foundation of all fluid dynamics.

The material derivative operator appears on the left-hand side of every fluid dynamical equation of motion. It converts the Eulerian field description into the acceleration experienced by a fluid parcel (Newton's second law).

2. Continuity Equation (Mass Conservation)

The continuity equation is the mathematical statement that mass is neither created nor destroyed. It is the simplest of the conservation laws and must be satisfied by every physical flow, regardless of whether the fluid is viscous or inviscid, compressible or incompressible.

Derivation from a Fixed Control Volume

Consider a fixed (Eulerian) control volume V bounded by surface S in the flow. The total mass inside is \( M = \int_V \rho\,dV \). Since mass cannot be created or destroyed (no nuclear reactions), the only way M can change is through mass flux across S. The outward mass flux through an area element \( d\vec{A} = \hat{n}\,dA \) is \( \rho\vec{v}\cdot\hat{n}\,dA \). Therefore:

This is the integral form of mass conservation -- it is exact and applies toany control volume, even one containing shocks or discontinuities.

Differential (Eulerian) Form

Since V is fixed, we can bring the time derivative inside the integral. Then apply the divergence theorem to convert the surface integral: \( \oint_S \rho\vec{v}\cdot\hat{n}\,dA = \int_V \nabla\cdot(\rho\vec{v})\,dV \). This gives:

Since V is arbitrary, the integrand must vanish everywhere (by the localization lemma). This yields the conservative (Eulerian) form:

In index notation (summation over repeated indices):

Non-Conservative (Lagrangian) Form

Expanding the divergence product rule: \( \nabla\cdot(\rho\vec{v}) = \rho\,\nabla\cdot\vec{v} + \vec{v}\cdot\nabla\rho \), and recognizing the material derivative \( D\rho/Dt = \partial\rho/\partial t + \vec{v}\cdot\nabla\rho \):

Physical interpretation: Following a fluid parcel, its density changes at a rate proportional to \( \nabla\cdot\vec{v} \) -- the volumetric dilation rate. If \( \nabla\cdot\vec{v} > 0 \) (diverging flow), the parcel expands and its density decreases. If \( \nabla\cdot\vec{v} < 0 \) (converging flow), the parcel is compressed and density increases.

Incompressible Flow

A flow is incompressible when the density of each fluid parcel remains constant: \( D\rho/Dt = 0 \). (Note: this does not require uniform density -- stratified flows like the ocean can be incompressible with spatially varying \( \rho \).) The continuity equation then reduces to:

This states that the velocity field is solenoidal (divergence-free): fluid parcels preserve their volume as they move. Geometrically, the flow maps every subregion to another of equal volume. The condition \( \nabla\cdot\vec{v} = 0 \) acts as a constraint on the velocity field and, combined with the momentum equations, determines the pressure field.

Component Forms in Common Coordinate Systems

Stream Function (2D Incompressible)

For 2D incompressible flow, \( \nabla\cdot\vec{v} = 0 \) is automatically satisfied by introducing the stream function \( \psi(x,y) \) such that:

One can verify: \( \partial v_x/\partial x + \partial v_y/\partial y = \partial^2\psi/\partial x\partial y - \partial^2\psi/\partial y\partial x = 0 \)identically. The stream function reduces the number of unknowns from two velocity components to one scalar field. Curves of constant \( \psi \) are streamlines, and the volume flux between two streamlines equals \( \Delta\psi \).

The continuity equation is the most fundamental constraint in fluid mechanics. Every analytical solution, numerical simulation, and experimental measurement must satisfy it. When checking a proposed velocity field, verifying \( \nabla\cdot\vec{v} = 0 \) (incompressible) is always the first sanity check.

3. Cauchy Momentum Equation

The Cauchy momentum equation is Newton's second law applied to a continuous medium. It is the most general form of the momentum equation and applies to anymaterial -- Newtonian and non-Newtonian fluids, elastic and plastic solids, viscoelastic polymers. The specific material behavior enters only through the constitutive relation for the stress tensor.

Forces on a Fluid Element

Consider an infinitesimal fluid element of volume \( dV \) and mass \( dm = \rho\,dV \). Two categories of forces act on it:

The Stress Tensor and Cauchy's Stress Theorem

The traction vector \( \vec{t}^{(\hat{n})} \) is the force per unit area exerted on a surface element with outward unit normal \( \hat{n} \). Augustin-Louis Cauchy proved (1823) that \( \vec{t} \) depends linearly on \( \hat{n} \) through a second-order tensor:

Here \( \sigma_{ij} \) is the Cauchy stress tensor: a 3×3 matrix where \( \sigma_{ij} \) represents the force per unit area in the i-direction exerted on a surface whose outward normal points in the j-direction. The full tensor is:

The diagonal elements (\( \sigma_{xx}, \sigma_{yy}, \sigma_{zz} \)) are normal stresses (tension/compression). The off-diagonal elements (\( \sigma_{xy}, \sigma_{xz}, \ldots \)) are shear stresses.

Symmetry of the Stress Tensor

Conservation of angular momentum for the fluid element (in the absence of body torques) requires:

This means \( \sigma_{xy} = \sigma_{yx} \), etc. The stress tensor is symmetric, reducing the number of independent components from 9 to 6. (Exceptions: ferrofluids, micropolar fluids, and other media with intrinsic angular momentum can have asymmetric stress tensors.)

Derivation of the Momentum Equation

Integral form (control volume). Apply Newton's second law via the Reynolds Transport Theorem with \( \beta = \vec{v} \) (momentum per unit mass):

The left side is the rate of change of momentum plus the momentum flux out through S. The right side is the total surface force plus body force. Converting the surface integrals via the divergence theorem and localizing (V arbitrary), we obtain the differential form:

In index notation (with the convention that the divergence of a tensor acts on the second index):

Conservative (Flux) Form

Expanding \( \rho D\vec{v}/Dt \) and using the continuity equation, the momentum equation can be rewritten in conservation form (important for numerical methods and for flows with shocks):

or more compactly: \( \partial(\rho\vec{v})/\partial t + \nabla\cdot(\rho\vec{v}\otimes\vec{v}) = \nabla\cdot\boldsymbol{\sigma} + \rho\vec{g} \), where \( \rho\vec{v}\otimes\vec{v} \) is the momentum flux tensor (a dyadic product).

Decomposition: Pressure + Deviatoric Stress

It is universal practice to decompose the stress tensor into an isotropic part (pressure) and a deviatoric part (viscous or extra stress):

where \( p = -\tfrac{1}{3}\sigma_{kk} \) is the mechanical (mean) pressure (negative sign because compressive stress is negative in the physics convention) and \( \tau_{ij} \) is thedeviatoric (viscous) stress tensor, which is traceless:\( \tau_{kk} = 0 \). Substituting into the Cauchy equation:

This form separates the roles of pressure (acts to compress/expand) and deviatoric stress (acts to shear and deform). The system is still open -- we need a constitutive relation specifying how \( \tau_{ij} \) depends on the velocity field to close the equations. Different constitutive choices yield:

• \(\tau_{ij} = 0\): Perfect (inviscid) fluid → Euler equations
• Newtonian: \( \tau_{ij} \propto \) strain rate → Navier-Stokes equations (Section 4)
• Power-law: \( \tau \propto \dot{\gamma}^n \) → shear-thinning (\( n < 1 \)) or shear-thickening (\( n > 1 \))
• Bingham plastic: Yields only above a threshold stress → toothpaste, mud, lava
• Viscoelastic: Stress depends on deformation history → polymers, biological fluids

The Cauchy momentum equation is completely general and applies to any continuous medium. Augustin-Louis Cauchy (1789-1857) published the foundational theory of stress in 1822-1828, establishing continuum mechanics as a rigorous mathematical discipline.

4. Navier-Stokes Equations

The Navier-Stokes equations are obtained by inserting the Newtonian constitutive law into the Cauchy momentum equation. They are the governing equations for the vast majority of fluid flows encountered in nature and engineering -- from blood flow in capillaries to ocean currents to the aerodynamics of aircraft.

The Rate-of-Strain Tensor

Before writing the constitutive law, we need the kinematic quantity that measures how fast fluid elements deform. The velocity gradient tensor \( \partial v_i/\partial x_j \) decomposes into a symmetric part (strain rate) and an antisymmetric part (rotation):

where the rate-of-strain (deformation) tensor is:

and the rotation tensor is \( \Omega_{ij} = \frac{1}{2}(\partial v_i/\partial x_j - \partial v_j/\partial x_i) \), related to vorticity by \( \omega_k = -\epsilon_{ijk}\Omega_{ij} \). The key physical insight: viscous stress depends only on the strain rate, not on rigid-body rotation. A fluid undergoing solid-body rotation (\( e_{ij} = 0 \)) experiences no viscous stress.

Newtonian Constitutive Law (Stokes' Hypothesis)

For a Newtonian fluid, the deviatoric stress is assumed to be a linear, isotropic function of the strain rate. The most general such tensor that respects material symmetry is:

Equivalently, writing out the strain rate explicitly:

The three material parameters are:

• p -- thermodynamic pressure (from equation of state)
• \(\mu\) -- dynamic (shear) viscosity [Pa·s]. Resistance to shearing deformation. For water at 20°C: \( \mu \approx 10^{-3} \) Pa·s; for air: \( \mu \approx 1.8\times10^{-5} \) Pa·s.
• \(\lambda\) -- second (bulk) viscosity. Resistance to volumetric compression/expansion. Stokes' hypothesis sets\( \lambda = -\tfrac{2}{3}\mu \), making the bulk viscosity \( \kappa = \lambda + \tfrac{2}{3}\mu = 0 \). This is exact for monatomic gases and a good approximation in most cases.

Full Compressible Navier-Stokes Equations

Substituting the Newtonian constitutive law into the Cauchy momentum equation\( \rho Dv_i/Dt = \partial\sigma_{ij}/\partial x_j + \rho g_i \) yields:

This is the full compressible form, needed for high-Mach-number aerodynamics, combustion, astrophysics, and acoustics. For constant \( \mu \) and \( \lambda \), the viscous terms simplify:

Incompressible Navier-Stokes Equations

For an incompressible Newtonian fluid (\( \nabla\cdot\vec{v} = 0 \), constant \( \rho \)and \( \mu \)), the bulk-viscosity term vanishes identically. The equations simplify to their most celebrated form:

Term-by-Term Physical Identification

Component Form (Cartesian)

Writing out all three components explicitly (incompressible, constant \( \mu \), gravity in -z):

Cylindrical Coordinates \((r, \theta, z)\)

The r-component of the incompressible Navier-Stokes equations in cylindrical coordinates (essential for pipe flow, vortex dynamics, rotating machinery):

where \( \nabla^2 = \frac{1}{r}\frac{\partial}{\partial r}\!\left(r\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2}{\partial\theta^2} + \frac{\partial^2}{\partial z^2} \). The extra terms (\( -v_\theta^2/r \) centripetal acceleration, \( -v_r/r^2 \) metric correction) arise from the curvature of the coordinate system.

Pressure Poisson Equation

An important consequence of incompressibility: taking the divergence of the momentum equation and using \( \nabla\cdot\vec{v} = 0 \) yields an elliptic equation for pressure:

This shows that in incompressible flow, pressure adjusts instantaneously to enforce the divergence-free constraint -- it is determined globally by the velocity field, not by a local equation of state. Physically, pressure waves propagate at infinite speed (the incompressible limit of \( c \to \infty \)).

Important Exact Solutions

Despite the nonlinearity, several exact solutions are known for special geometries and boundary conditions:

Boundary Layer Theory & the Blasius Solution

At high Reynolds number, viscous effects are confined to thin boundary layers near solid surfaces. Prandtl (1904) showed that in these layers, the N-S equations simplify to the boundary layer equations by an order-of-magnitude analysis. For steady, 2D flow over a flat plate with zero pressure gradient:

Blasius (1908) found a similarity solution by introducing the variable\( \eta = y\sqrt{U_\infty/(\nu x)} \) and stream function \( \psi = \sqrt{\nu x U_\infty}\,f(\eta) \). The PDE reduces to the Blasius ODE:

This nonlinear ODE has no closed-form solution but is solved numerically by the shooting method: guess \( f''(0) \), integrate as an IVP, and iterate until \( f'(\infty) = 1 \). The exact value is \( f''(0) = 0.332057 \). Key results:

• Velocity profile: \( v_x/U_\infty = f'(\eta) \) -- a universal self-similar profile
• BL thickness: \( \delta_{99} \approx 5.0\sqrt{\nu x/U_\infty} \propto x^{1/2}\,Re_x^{-1/2} \)
• Wall shear stress: \( \tau_w = 0.332\,\mu U_\infty\sqrt{U_\infty/(\nu x)} \)
• Skin friction: \( C_f = 0.664/\sqrt{Re_x} \)
• Displacement thickness: \( \delta^* = 1.7208\sqrt{\nu x/U_\infty} \)

The Fortran program below solves the Blasius equation using a shooting method with 4th-order Runge-Kutta integration, compiled and executed at runtime:

5. Euler Equations (Inviscid Flow)

When viscous effects are negligible (\( \mu = 0 \)), the Navier-Stokes equations reduce to the Euler equations, derived by Leonhard Euler in 1755 -- predating Navier (1822) and Stokes (1845) by nearly a century. This is the appropriate limit at high Reynolds number away from solid boundaries (outside boundary layers).

Incompressible Euler Equations

Setting \( \mu = 0 \) in the incompressible Navier-Stokes equations:

Equivalently, expanding the material derivative:\( \rho\!\left(\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot\nabla)\vec{v}\right) = -\nabla p + \rho\vec{g} \). In index notation: \( \rho(\partial v_i/\partial t + v_j\,\partial v_i/\partial x_j) = -\partial p/\partial x_i + \rho g_i \).

Compressible Euler Equations (Conservation Form)

For compressible inviscid flow (aerodynamics, shock waves, gas dynamics), the Euler equations are written in conservation form, coupled with an energy equation and an equation of state:

where \( E = \rho(e + \tfrac{1}{2}|\vec{v}|^2) \) is the total energy per unit volume (internal + kinetic) and for an ideal gas \( p = (\gamma - 1)\rho e \). This is a system of hyperbolic conservation laws admitting wave-like solutions, including shock waves and expansion fans.

Key Properties

Lamb Form & Connection to Bernoulli

Using the Lamb vector identity on the convective term, the steady Euler equation for an incompressible fluid becomes:

Along a streamline (\( d\vec{s} \parallel \vec{v} \)), the right-hand side vanishes, immediately yielding Bernoulli's equation. In irrotational flow (\( \vec{\omega} = 0 \)), the Bernoulli constant is the same on every streamline.

The Euler equations remain central to modern computational fluid dynamics (CFD). High-resolution shock-capturing schemes (Godunov, WENO, discontinuous Galerkin) are designed specifically for the compressible Euler system and form the backbone of aerospace CFD codes.

6. Bernoulli's Equation

Bernoulli's equation is one of the most widely used results in fluid mechanics. It expresses conservation of mechanical energy along a streamline for ideal flow. Despite its simplicity, it describes venturi meters, pitot tubes, lift on airfoils (qualitatively), siphons, and countless engineering applications.

Derivation Along a Streamline

Start with the steady Euler equation divided by \( \rho \):\( (\vec{v}\cdot\nabla)\vec{v} = -\frac{1}{\rho}\nabla p - g\hat{z} \). Apply the Lamb identity \( (\vec{v}\cdot\nabla)\vec{v} = \nabla(\tfrac{1}{2}v^2) - \vec{v}\times\vec{\omega} \)and take the dot product with \( d\vec{s} \) along a streamline (parallel to \( \vec{v} \)). Since \( \vec{v}\times\vec{\omega} \) is perpendicular to\( \vec{v} \), that term vanishes:

For incompressible flow (\( \rho = \text{const} \)), integration yields:

Energy Interpretation

Each term represents an energy per unit volume [J/m³ = Pa]:

Alternative Forms

Assumptions & Limitations

Required assumptions:

• (1) Steady (\( \partial/\partial t = 0 \)) -- unless using the unsteady form
• (2) Inviscid (\( \mu = 0 \)) -- cannot apply inside boundary layers or wakes
• (3) Incompressible (\( \rho = \text{const} \)) -- valid for \( Ma \lesssim 0.3 \); compressible form exists for higher Mach
• (4) Along a streamline -- unless the flow is irrotational (\( \vec{\omega} = 0 \)), in which case the constant is the same everywhere

Applications

The Python visualization below demonstrates Bernoulli's equation in a converging-diverging nozzle, showing the trade-off between velocity and pressure as the cross-section varies:

7. Reynolds Number & Non-Dimensionalization

The Reynolds number is the single most important dimensionless parameter in fluid mechanics. It determines whether a flow is laminar or turbulent, whether viscous effects dominate or can be neglected, and how flows at different physical scales can be related through dynamic similarity.

Derivation from Non-Dimensionalization

Non-dimensionalize the incompressible N-S equations using characteristic scales: length \( L \), velocity \( U \), time \( L/U \), and pressure \( \rho U^2 \). Defining dimensionless variables \( \vec{x}^* = \vec{x}/L \), \( \vec{v}^* = \vec{v}/U \),\( t^* = tU/L \), \( p^* = p/(\rho U^2) \):

where the single parameter controlling the entire flow is:

This is remarkable: two geometrically similar flows with the same Re will have identical dimensionless velocity and pressure fields, regardless of the actual physical scales. This is the principle of dynamic similarity that underlies all wind tunnel and scale model testing.

Physical Meaning

The inertial term \( \rho(\vec{v}\cdot\nabla)\vec{v} \) scales as \( \rho U^2/L \)(convective acceleration). The viscous term \( \mu\nabla^2\vec{v} \) scales as\( \mu U/L^2 \) (momentum diffusion). Their ratio:

Flow Regimes

Laminar-Turbulent Transition

The critical Reynolds number for transition depends on the geometry and the level of disturbances:

Other Important Dimensionless Numbers

The non-dimensionalized N-S equations with additional physics (heat transfer, gravity, compressibility) introduce further dimensionless groups:

Osborne Reynolds (1842-1912) demonstrated the transition from laminar to turbulent flow in his famous 1883 pipe flow experiment at the University of Manchester, using dye injection to visualize the flow. The dimensionless group bearing his name was formally introduced by Arnold Sommerfeld in 1908.

8. Vorticity Equation

The vorticity formulation of the Navier-Stokes equations eliminates pressure entirely, revealing the fundamental dynamics of rotation in fluids. Vorticity is the key concept connecting fluid mechanics to aerodynamic lift, tornadoes, ocean gyres, and turbulence.

Definition and Physical Meaning

The vorticity is defined as the curl of the velocity field:

In 2D: \( \omega_z = \partial v_y/\partial x - \partial v_x/\partial y \) (a scalar). The vorticity equals twice the angular velocity of a local fluid element: \( \vec{\omega} = 2\vec{\Omega}_{\text{local}} \). A flow with \( \vec{\omega} = 0 \)everywhere is called irrotational (potential flow). Importantly, a fluid element can move in a curved path and still have zero vorticity (think of a free vortex:\( v_\theta = \Gamma/(2\pi r) \) is irrotational for \( r > 0 \)).

Derivation of the Vorticity Transport Equation

Take the curl of the incompressible Navier-Stokes equations. Using the vector identities\( \nabla\times\nabla p = 0 \) (curl of a gradient vanishes) and\( \nabla\times[(\vec{v}\cdot\nabla)\vec{v}] = (\vec{v}\cdot\nabla)\vec{\omega} - (\vec{\omega}\cdot\nabla)\vec{v} + \vec{\omega}(\nabla\cdot\vec{v}) - \vec{v}(\nabla\cdot\vec{\omega}) \), with \( \nabla\cdot\vec{v} = 0 \) and \( \nabla\cdot\vec{\omega} = 0 \) (div of curl vanishes):

Term-by-Term Interpretation

Special Case: 2D Vorticity Equation

In two dimensions, vorticity is a scalar \( \omega = \omega_z \) and the stretching term vanishes. The equation simplifies to a scalar advection-diffusion equation:

For inviscid 2D flow (\( \nu = 0 \)), this becomes\( D\omega/Dt = 0 \): vorticity is conserved following each fluid parcel. This is a powerful constraint -- once you know the initial vorticity distribution, it is simply carried along by the flow.

Kelvin's Circulation Theorem

For inviscid (\( \nu = 0 \)), barotropic (\( \rho = \rho(p) \)) flow with conservative body forces, the circulation around any material (fluid) contour is conserved:

Profound consequence: If a flow starts irrotational (e.g., fluid at rest), it remains irrotational for all time in the absence of viscosity. Vorticity cannot be created in the interior of an inviscid fluid -- it can only be generated at boundaries (no-slip condition requires viscosity). This explains why potential flow theory works so well in the outer flow away from walls.

Helmholtz Vortex Theorems

Hermann von Helmholtz (1858) proved three fundamental theorems for inviscid barotropic flow:

• I. The strength (circulation) of a vortex tube is constant along its length.
• II. A vortex tube always consists of the same fluid particles (vortex lines move with the fluid).
• III. The strength of a vortex tube is constant in time (equivalent to Kelvin's theorem).

Together: vortex lines are "frozen" into the fluid. They cannot start or end in the fluid interior; they must either form closed loops or extend to boundaries/infinity.

Vortex Models

The Python visualization below compares vorticity fields in canonical flows -- Rankine vortex, Lamb-Oseen viscous decay, potential flow around a cylinder, and radial velocity/vorticity profiles:

The vorticity formulation is foundational for vortex methods in CFD, the understanding of turbulence (Kolmogorov 1941 theory), and the design of vortex generators, winglets, and other aerodynamic devices. In 2D, the vorticity-stream function formulation (\( \nabla^2\psi = -\omega \)) eliminates pressure entirely and reduces the system to a single scalar equation.

9. Stokes (Creeping) Flow

Low Reynolds number limit. When Re ≪ 1, the inertial term \( \rho(\vec{v}\cdot\nabla)\vec{v} \) is negligible compared to the viscous term. Formally, non-dimensionalizing the Navier-Stokes equations with scales \( U, L \):

In the limit Re → 0, the left-hand side vanishes, giving the Stokes equations:

Stokes Flow Past a Sphere

The celebrated exact solution for uniform flow \( U \) past a sphere of radius \( a \). Using the stream function ansatz \( \psi(r,\theta) = f(r)\sin^2\theta \) and the biharmonic equation \( \nabla^4\psi = 0 \), the velocity components are:

The stream function is:

Stokes Drag Law

Integrating the surface stress (pressure + viscous) over the sphere yields the Stokes drag:

Remarkably, the drag decomposes as:

Two-thirds from viscous shear stress and one-third from pressure, regardless of \( \mu \) or \( U \). The drag coefficient is:

Extensions & Limitations

Applications: sedimentation of small particles, microorganism locomotion (bacteria, spermatozoa), flow in porous media (Darcy's law), lubrication theory, microfluidics, Brownian motion of colloids, paint and polymer rheology.

10. Potential Flow Theory

Irrotational flow. If the vorticity vanishes everywhere,\( \vec{\omega} = \nabla\times\vec{v} = 0 \), then by the Helmholtz decomposition the velocity can be written as the gradient of a scalar potential:

Substituting into the incompressibility condition \( \nabla\cdot\vec{v} = 0 \) yields Laplace's equation for the velocity potential:

Stream Function & Complex Potential

In 2D incompressible flow, a stream function \( \psi \) exists such that:

For irrotational flow, \( \psi \) also satisfies \( \nabla^2\psi = 0 \). The functions\( \phi \) and \( \psi \) are harmonic conjugates, satisfying the Cauchy-Riemann equations. This motivates the complex potential:

where \( z = x + iy \). Any analytic function \( w(z) \) represents a valid potential flow.

Elementary Solutions

Flow Past a Cylinder

Superposing uniform flow + doublet gives non-lifting flow past a cylinder of radius \( a \):

Adding a point vortex of circulation \( \Gamma \) introduces asymmetry and lift:

Kutta-Joukowski Theorem

For any 2D body in a potential flow with circulation \( \Gamma \), the lift per unit span is:

This result is exact and independent of body shape. For an airfoil, the Kutta condition (flow leaves the trailing edge smoothly) uniquely determines \( \Gamma \). The drag in potential flow is always zero (d'Alembert's paradox).

Conformal Mapping

Any conformal (angle-preserving) map \( \zeta = f(z) \) transforms one potential flow into another. The Joukowski transform:

maps a circle to an airfoil shape, allowing exact solutions for lift on Joukowski airfoils. The Schwarz-Christoffel transform handles polygonal boundaries.

11. Energy Equation

Total Energy Conservation

First law of thermodynamics for a fluid element. The total energy per unit mass is \( e_{\text{tot}} = e + \tfrac{1}{2}|\vec{v}|^2 \), where \( e \) is the specific internal energy. Applying Reynolds transport to the total energy:

where \( \vec{q} = -k\nabla T \) is the heat flux (Fourier's law) and \( \boldsymbol{\sigma} \) is the Cauchy stress tensor.

Mechanical Energy Equation

Taking the dot product of the momentum equation with \( \vec{v} \) gives the kinetic energy balance:

Expanding \( \nabla\cdot(\boldsymbol{\sigma}\cdot\vec{v}) = \vec{v}\cdot(\nabla\cdot\boldsymbol{\sigma}) + \boldsymbol{\sigma}:\nabla\vec{v} \) and subtracting the mechanical equation from the total energy equation:

Thermal Energy Equation

For a Newtonian fluid with \( \boldsymbol{\sigma} = -p\mathbf{I} + \boldsymbol{\tau} \), decomposing the stress power \( \boldsymbol{\sigma}:\nabla\vec{v} = -p(\nabla\cdot\vec{v}) + \boldsymbol{\tau}:\nabla\vec{v} \). Using\( De = c_v\,dT + [T(\partial p/\partial T)_\rho - p]\,d\rho/\rho \) and Fourier's law:

Incompressible Newtonian Form

For incompressible flow (\( \nabla\cdot\vec{v} = 0 \)), using \( de = c_p\,dT \) (since \( c_p \approx c_v \)):

Viscous Dissipation Function

The viscous dissipation \( \Phi = \boldsymbol{\tau}:\nabla\vec{v} \) represents the irreversible conversion of kinetic energy to heat through viscous friction. For a Newtonian fluid:

In Cartesian coordinates (incompressible):

The positive-definiteness (\( \Phi \geq 0 \)) is guaranteed by the second law of thermodynamics.

Compressible Form (Enthalpy)

For compressible flow, using enthalpy \( h = e + p/\rho \) and the ideal gas law:

For adiabatic (\( k = 0 \)), inviscid (\( \Phi = 0 \)) flow, this gives \( Dh/Dt = (1/\rho)Dp/Dt \). Along a streamline in steady flow, this integrates to the compressible Bernoulli equation: \( h + \tfrac{1}{2}v^2 = h_0 \) (total enthalpy is conserved).

Coupling & Dimensionless Numbers

12. Dimensional Analysis & Buckingham Pi Theorem

Principle of Dimensional Homogeneity

Every physically meaningful equation must be dimensionally homogeneous — each term must have the same dimensions. This principle constrains the functional form of physical laws and is the foundation of dimensional analysis. In mechanics, the three fundamental dimensions are mass (M), length (L), and time (T); thermal problems add temperature (Θ).

Buckingham Pi Theorem (1914)

Statement: If a physical problem involves \( n \) dimensional variables \( q_1, q_2, \ldots, q_n \) and \( k \) independent fundamental dimensions, then the problem can be described by \( n - k \) independent dimensionless groups:

Systematic Procedure

Example: Drag on a Sphere

The drag force \( F \) on a sphere depends on \( (\rho, U, D, \mu) \) — five variables (\( n = 5 \)) with three dimensions M, L, T (\( k = 3 \)), giving\( n - k = 2 \) dimensionless groups. The dimensional matrix:

Choosing \( \rho, U, D \) as repeating variables, forming the Pi groups:

Therefore the drag coefficient depends only on the Reynolds number:

This reduces a five-variable problem to a single curve \( C_D(Re) \).

Similitude & Model Testing

Dimensional analysis underpins model testing in wind tunnels and tow tanks. For dynamic similarity, all relevant Pi groups must match between model and prototype:

Key Dimensionless Numbers in Fluid Mechanics

Incomplete Similarity & Self-Similarity

Often it is impossible to match all dimensionless numbers simultaneously (e.g., matching both Re and Fr requires changing \( g \)). In practice, one matches the dominant Pi group and corrects for others. When a problem has no characteristic length or time, self-similar solutions exist — the solution depends only on a combined variable (e.g., Blasius: \( \eta = y\sqrt{U/\nu x} \)).