The primary source is the course summary prepared by Ryan Newson (Winter 2005), which covers the material in a compact but mathematically precise form. The exposition of viscoelasticity and elasticity follows A.J.M. Spencer, Continuum Mechanics (Dover Publications, 2004), one of the clearest undergraduate treatments of the subject. The fluid mechanics chapters draw on D.J. Acheson, Elementary Fluid Dynamics (Oxford University Press, 1990), and G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 1967). Readers seeking a more geometrically modern treatment of kinematics should consult M.E. Gurtin, An Introduction to Continuum Mechanics (Academic Press, 1981).

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Chapter 1: One-Dimensional Viscoelasticity

1.1 The Principle of Fading Memory

The central question of mechanics is deceptively simple: if you apply a force to a material, how does it deform? For a perfectly elastic spring the answer is instantaneous and local — the deformation is proportional to the force at that moment, with no reference to the past. For a Newtonian viscous fluid the force is proportional to the rate of deformation. Real materials, however, belong to neither category. A polymer gel pulled quickly responds like a stiff elastic solid; pulled slowly, it flows. Biological tissue deforms more under a sustained load than under a brief one of the same magnitude. These materials are viscoelastic: their response depends on the entire history of forcing.

The simplest mathematical framework that captures this history-dependence is the convolution integral. If \(f(t)\) is the applied force (stress) and \(r(t)\) is the resulting deformation (strain), we write

\[ r(t) = \int_0^t \alpha(\tau)\, f(t - \tau)\, d\tau \]

The kernel \(\alpha(\tau)\) assigns a weight to the force applied \(\tau\) seconds ago. The physical principle that guides the choice of \(\alpha\) is fading memory: remote history contributes less than recent history.

This is not a mathematical convenience but a physical observation: a rubber band stretched five minutes ago contributes negligibly to its current state compared to what happened in the last second. The shape of the decay of \(\alpha\) — exponential, power-law, or otherwise — characterises the material.

The general linear viscoelastic constitutive relation between \(f\) and \(r\) is a linear ODE with constant coefficients:

\[ q_2 \frac{d^2 r}{dt^2} + q_1 \frac{dr}{dt} + q_0 r = p_2 \frac{d^2 f}{dt^2} + p_1 \frac{df}{dt} + p_0 f \]

The six constants \(p_0, p_1, p_2, q_0, q_1, q_2 \geq 0\) characterise the material. Different choices give rise to the classical mechanical models.

1.2 Mechanical Analogues: Springs and Dashpots

Before analysing the general ODE, it is instructive to study the two elementary building blocks.

\[ f_{sp}(t) = G\, r_{sp}(t) \]

where \(G\) [N/m] is the stiffness. The response is instantaneous and reversible; there is no memory whatsoever.

\[ f_{dp}(t) = \eta\, \frac{dr_{dp}}{dt}(t) \]

where \(\eta\) [N·s/m] is the viscosity. The dashpot resists change: the force is proportional to the velocity, not the position. It dissipates energy irreversibly.

These two elements are combined in two canonical configurations.

1.3 The Maxwell Fluid

In the Maxwell model, a spring and dashpot are connected in series. When two elements are in series, they share the same force (the force must be transmitted through both) and their individual deformations add:

\[ f(t) = f_{sp}(t) = f_{dp}(t), \qquad r(t) = r_{sp}(t) + r_{dp}(t) \]

Differentiating the deformation and substituting the element laws:

\[ \frac{dr}{dt} = \frac{1}{G}\frac{df}{dt} + \frac{f}{\eta} \]

This is a first-order ODE. To understand the Maxwell model physically, perform a stress relaxation test: impose a sudden constant deformation \(r(t) = r_0 H(t)\), where \(H\) is the Heaviside step function, and ask what force is required to maintain it. Solving gives:

\[ f(t) = G r_0 \exp\!\left(-\frac{G}{\eta}\, t\right) \]

The stress relaxes exponentially to zero with relaxation time \(\tau_M = \eta/G\). For \(t \gg \tau_M\), no force is needed to maintain the deformation — the dashpot has accommodated all the strain and the spring is at rest. This is fluid-like behaviour: the Maxwell model eventually flows without limit under any sustained load.

Kelvin-Voigt (parallel)

fGη

Zener / SLS

fG₁G₂η

1.4 The Kelvin–Voigt Solid

In the Kelvin–Voigt model, a spring and dashpot are in parallel. Elements in parallel share the same deformation (they are attached at both ends) and their forces add:

\[ r(t) = r_{sp}(t) = r_{dp}(t), \qquad f(t) = f_{sp}(t) + f_{dp}(t) \]

This gives:

\[ \eta\, \frac{dr}{dt} + G r = f \]

To understand this model, perform a creep test: suddenly apply a constant force \(f(t) = f_0 H(t)\) and observe the deformation. Solving:

\[ r(t) = \frac{f_0}{G}\!\left(1 - e^{-Gt/\eta}\right) \]

The deformation grows exponentially from zero toward the equilibrium value \(f_0/G\) — exactly the deformation a spring of stiffness \(G\) would have in equilibrium. The dashpot retards the approach but cannot prevent the system from reaching the same elastic equilibrium. This is solid-like behaviour: the Kelvin–Voigt model deforms but eventually stops.

The contrast between the two models is fundamental. The Maxwell model has zero long-time stiffness: under a sustained force, it flows indefinitely. The Kelvin–Voigt model has zero instantaneous stiffness: it cannot deform discontinuously under a sudden impulse. Neither pathology matches most real materials, which motivates the next model.

1.5 The Standard Linear Solid (Zener Model)

The Standard Linear Solid (SLS), also called the Zener model, consists of a Kelvin–Voigt element with an additional spring placed in series. This seemingly minor modification has important consequences:

• Finite instantaneous stiffness (unlike pure K-V, which cannot respond instantly)
• Finite long-time stiffness (unlike pure Maxwell, which flows to zero force)
• A single relaxation time governing the smooth transition between the two regimes

The SLS is the simplest model that captures both elastic and viscous behaviour without pathology at short or long times. It serves as the first approximation for cartilage, tendon, polymer networks, and many biological soft tissues.

1.6 Laplace Transforms and the Convolution Kernel

The Laplace transform is the natural tool for linear viscoelasticity because it converts differential equations into algebra and convolution integrals into products.

\[ \mathcal{L}\!\left\{\frac{df}{dt}\right\} = s F(s) - f(0), \qquad \mathcal{L}\!\left\{\frac{d^2f}{dt^2}\right\} = s^2 F(s) - s f(0) - f'(0) \]

and the convolution theorem: if \(h(t) = \int_0^t g(\tau) f(t-\tau)\, d\tau\), then \(H(s) = G(s) F(s)\).

Applying the Laplace transform to the general viscoelastic ODE, assuming zero initial conditions, yields the transfer function:

\[ R(s) = \frac{p_2 s^2 + p_1 s + p_0}{q_2 s^2 + q_1 s + q_0}\, F(s) \]

By the convolution theorem, the creep kernel is:

\[ \alpha(\tau) = \mathcal{L}^{-1}\!\left[\frac{p_2 s^2 + p_1 s + p_0}{q_2 s^2 + q_1 s + q_0}\right]_{t=\tau} \]

This beautifully closes the loop: the ODE parameters \((p_i, q_i)\) and the memory kernel \(\alpha(\tau)\) are two equivalent ways of specifying the same viscoelastic material. The kernel representation is physically intuitive; the ODE representation is computationally convenient; the Laplace transfer function connects them.

1.7 Standard Experiments

Three canonical laboratory tests probe the memory function from different angles.

The creep test applies a step in force: \(f(t) = f_0 H(t)\). In Laplace space, \(F(s) = f_0 / s\), so \(R(s) = (f_0/s)\hat{\alpha}(s)\). The response \(r(t)/f_0\) is the creep compliance \(J(t)\), which increases monotonically with time and saturates (for solid-like models) or grows without bound (for fluid-like models).

The stress relaxation test applies a step in deformation: \(r(t) = r_0 H(t)\). Now we invert the roles: the deformation is prescribed, and we solve for the force required. The response \(f(t)/r_0\) is the relaxation modulus \(G(t)\), which decays with time toward zero (Maxwell) or a finite equilibrium value (SLS, K-V).

The oscillatory test applies \(f(t) = f_0 \sin(\omega t)\) and observes the steady-state response \(r(t) = r_0 \sin(\omega t - \delta)\). The loss angle \(\delta\) quantifies the phase lag. For a perfectly elastic material, \(\delta = 0\); for a purely viscous fluid, \(\delta = \pi/2\). The storage modulus \(G' = (f_0/r_0)\cos\delta\) measures elastic energy storage; the loss modulus \(G'' = (f_0/r_0)\sin\delta\) measures viscous dissipation.

1.8 Plasticity and Nonlinear Response

The viscoelastic theory above is fundamentally linear: superposition holds. Many materials, however, exhibit plasticity — irreversible deformation that occurs once a stress threshold is exceeded. The simplest model replaces the smooth spring with a frictional element obeying Coulomb friction:

• Static friction: no motion until the applied force exceeds the static yield stress \(f_s\)
• Dynamic (kinetic) friction: once the element is sliding, the resisting force is a constant \(f_k < f_s\)

This threshold behaviour means the constitutive law is nonlinear: the response depends on whether the system is currently yielding. From the standpoint of the memory kernel, plasticity introduces a dependence on the magnitude of the forcing, not merely its history — the integral representation fails. These ideas foreshadow the theory of elastoplasticity in three dimensions, where yield surfaces and flow rules replace the linear constitutive tensor.

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Chapter 2: Kinematics of Deformation

The previous chapter treated deformation as a scalar quantity. To describe real three-dimensional materials, we must develop a geometric framework for how a body deforms. This is kinematics: the study of motion without reference to forces. The key object is the strain tensor, which measures local stretching and shearing at each point of the material.

2.1 Displacement and Finite Strain in One Dimension

Let \(x\) denote the initial position of a material point and \(f(x)\) its position after deformation. The displacement is:

\[ u(x) = f(x) - x \qquad [m] \]

The natural measure of local deformation is the ratio of deformed to reference length. For an infinitesimal element \(dx\), the deformed length is \(f'(x)\,dx\), giving a stretch ratio of \(f'(x)\). The Green–Lagrange strain:

\[ E(x) = \frac{1}{2}\!\left[(f'(x))^2 - 1\right] = \frac{1}{2}\!\left[(u_x)^2 + 2u_x\right] \]

is constructed so that \(E = 0\) for rigid motions (where \(f'(x) = 1\) everywhere). The geometric factor of \(\frac{1}{2}\) is conventional and ensures that \(E\) reduces to \(e\) in the linear limit.

For small deformations, \(|u_x| \ll 1\), and the quadratic term \((u_x)^2\) is negligible. The linearised (infinitesimal) strain is:

\[ e(x) = u_x = f'(x) - 1 \]

This linear approximation is valid for metals under service loads, geological rocks under tectonic stresses, and most structural engineering problems. It breaks down for rubber, biological tissue under large deformations, and any problem where geometric nonlinearity matters.

2.2 Cartesian Tensors and Index Notation

The extension to three dimensions requires the machinery of tensors. The essential tool is the Einstein summation convention: a repeated index in a term implies summation over that index, so \(a_{ij} b_j \equiv \sum_{j=1}^{3} a_{ij} b_j\).

\[ \delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}, \qquad \delta_{ij} a_j = a_i \]

Under a rotation of coordinate axes, with rotation matrix \(\beta_{ij} = \mathbf{e}'_j \cdot \mathbf{e}_i\) (the direction cosines between old and new basis vectors), the components of vectors and second-order tensors transform as:

\[ a'_i = \beta_{ij} a_j, \qquad a'_{ij} = \beta_{im} \beta_{jn} a_{mn} \]

The fundamental principle is that physical laws written as tensor equations are frame-invariant: they hold in every coordinate system without change. This is why tensor notation is the language of continuum mechanics.

2.3 Deformation and Strain in Three Dimensions

Let \(a_i\) denote the reference (initial) position of a material point and \(x_i(\mathbf{a})\) its deformed position. The displacement is:

\[ u_i = x_i - a_i \qquad [m] \]

The deformation gradient \(F_{ij} = \partial x_i / \partial a_j\) encodes the local mapping from reference to deformed configuration. The Green–Lagrange strain tensor in three dimensions is:

\[ E_{ij} = \frac{1}{2}\!\left[\delta_{pq}\, x_{p,i}\, x_{q,j} - \delta_{ij}\right] = \frac{1}{2}\!\left[u_{j,i} + u_{i,j} + u_{l,i}\, u_{l,j}\right] \]

where \(u_{i,j} = \partial u_i / \partial a_j\). The last term is quadratic in displacement gradients. Linearising for small deformations:

\[ e_{kk} = e_{11} + e_{22} + e_{33} = \nabla \cdot \mathbf{u} \]

This measures the fractional change in volume: a positive dilatation means the material expanded; negative means it compressed.

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Chapter 3: Stress and Linear Elasticity

Having described deformation geometrically, we now connect it to forces through a constitutive law — the mathematical encoding of the material’s mechanical character.

3.1 Traction and the Stress Tensor

Imagine slicing the deformed body along an imaginary internal surface with outward unit normal \(\hat{n}\). The material on the positive side exerts a force per unit area — a traction vector — on the material on the negative side.

This is a profound result: it asserts that the traction depends linearly on the surface normal, regardless of the surface’s shape or orientation. The proof proceeds by applying Newton’s law to a shrinking tetrahedron with three faces aligned with the coordinate planes and one inclined face — Cauchy’s classical argument.

The diagonal entries \(\tau_{11}, \tau_{22}, \tau_{33}\) are normal stresses: positive means tension, negative means compression. The off-diagonal entries \(\tau_{12} = \tau_{21}\), etc., are shear stresses. The symmetry \(\tau_{ij} = \tau_{ji}\) is a consequence of angular momentum balance applied to an infinitesimal volume.

3.2 The Isotropic Linear Elastic Constitutive Law

For small deformations of an isotropic, homogeneous elastic solid, the most general linear relation between stress and strain is:

The first term is isotropic — it acts like a pressure proportional to the volumetric strain. The second term is the deviatoric (shear) contribution. Together, two scalar constants \(\lambda\) and \(\mu\) describe all elastic behaviour of an isotropic material.

The Lamé constants are related to more familiar engineering parameters:

Engineering Parameter	Symbol	Formula
Young’s modulus	\(E_Y\)	\(\mu(3\lambda + 2\mu)/(\lambda + \mu)\)
Poisson’s ratio	\(\nu\)	\(\lambda / (2(\lambda + \mu))\)
Bulk modulus	\(K\)	\(\lambda + 2\mu/3\)
Shear modulus	\(G\)	\(\mu\)

Young’s modulus is what you measure in a uniaxial tension test: the ratio of axial stress to axial strain. Poisson’s ratio measures the transverse contraction when a bar is pulled: rubber has \(\nu \approx 0.5\) (nearly incompressible), most metals have \(\nu \approx 0.3\), and cork has \(\nu \approx 0\) (which is why it compresses into a bottle without bulging out the sides).

3.3 Equilibrium Conditions

In the absence of body forces, force balance on any material volume requires:

This is simply Newton’s first law applied to an infinitesimal element: the net force per unit volume must vanish in equilibrium.

Substituting the constitutive law into the equilibrium equations yields the Navier equations of linear elasticity:

\[ (\lambda + \mu)\, \nabla(\nabla \cdot \mathbf{u}) + \mu\, \nabla^2 \mathbf{u} = \mathbf{0} \]

This is a system of three coupled elliptic PDEs for the three displacement components. Every solution of the Navier equations corresponds to an equilibrium elastic deformation field.

3.4 Boundary Value Problems

Boundary conditions for an elastic solid take two forms:

• Displacement (Dirichlet) conditions: \(u_i\) prescribed on a surface — modelling a clamped boundary or a fixed support
• Traction (Neumann) conditions: \(t_i = \tau_{ij} n_j\) prescribed on a surface — modelling an applied pressure or a free surface (\(t_i = 0\))

Two problem classes appear in the course:

Multi-layer unidirectional problems consider a material composed of parallel layers, each with different Lamé constants. Since deformation is constrained to be in one direction, the 3D equations reduce to 1D. Stress must be continuous across interfaces (force balance), and displacement must be continuous (no gaps or overlaps). These conditions uniquely determine the response throughout the layered medium.

Tethered tube problems arise in biomechanics. A pressurised cylindrical tube (blood vessel model) with both ends fixed has a different stress distribution than a free tube. The constraint that total length is fixed creates an axial stress throughout the wall, coupling the circumferential and axial equations in a non-trivial way.

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Chapter 4: Three-Dimensional Viscoelasticity

The elastic constitutive law of Chapter 3 can be systematically extended to viscoelasticity in three dimensions using the same mathematical machinery developed in Chapter 1, together with a structural insight about the decomposition of the stress tensor.

4.1 Dilatational and Deviatoric Decomposition

Any symmetric tensor can be split uniquely into an isotropic part (proportional to the identity) and a trace-free (deviatoric) part:

\[ \tau_{ij} = \frac{1}{3}\tau_{kk}\, \delta_{ij} + s_{ij}, \qquad e_{ij} = \frac{1}{3}e_{kk}\, \delta_{ij} + \varepsilon_{ij} \]

where \(s_{ij} = \tau_{ij} - \frac{1}{3}\tau_{kk}\delta_{ij}\) and \(\varepsilon_{ij} = e_{ij} - \frac{1}{3}e_{kk}\delta_{ij}\) are deviatoric (trace-free). The trace \(e_{kk} = \nabla \cdot \mathbf{u}\) measures volumetric change; the deviatoric part measures shape change at constant volume.

\[ \tau_{kk} = 3K\, e_{kk} \]\[ s_{ij} = 2\mu\, \varepsilon_{ij} \]

Volumetric and shear responses are governed by completely independent elastic constants \(K\) and \(\mu\). This decoupling is special to isotropy and linearity.

4.2 The Correspondence Principle

The deviatoric constitutive relation \(s_{ij} = 2\mu\,\varepsilon_{ij}\) has exactly the same structure as the 1D elastic spring \(f = G\,r\). The extension to viscoelasticity follows by the same principle applied in 1D: replace the proportionality by the general linear ODE.

The procedure is: (1) solve the elastic boundary value problem, expressing the answer in terms of \(\mu\); (2) take the Laplace transform; (3) substitute the viscoelastic operator for \(2\mu\); (4) invert the Laplace transform.

This principle is not obvious — it rests on the fact that the boundary conditions do not mix dilatational and deviatoric quantities, and that the governing equations are linear. It is a true theorem, not an approximation. The models considered in Chapter 1 (Maxwell, Kelvin–Voigt, SLS) are all special cases of the general operator above.

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Chapter 5: From Solids to Fluids — Kinematics of Flow

The four preceding chapters treated materials with a preferred reference configuration: a body deforms from its natural state, and displacement measures departure from that state. A fluid has no such preferred configuration — it deforms continuously without limit. The appropriate kinematic variable shifts from displacement to velocity.

5.1 Eulerian and Lagrangian Descriptions

The velocity field \(\mathbf{u}(\mathbf{x}, t)\) is naturally Eulerian: it tells you the velocity of whatever fluid particle happens to be at position \(\mathbf{x}\) at time \(t\). Almost all of fluid mechanics is formulated in the Eulerian frame. The Lagrangian frame appears in numerical simulations (particle tracking), in oceanography (Lagrangian floats), and in the theoretical analysis of transport and mixing.

5.2 Streamlines and Pathlines

For steady flows (\(\partial \mathbf{u}/\partial t = 0\)), streamlines and pathlines coincide: since the velocity snapshot is frozen, a particle moves exactly along the streamline it is currently on. For unsteady flows they diverge, sometimes dramatically — a dye streak in an unsteady flow provides a streakline that is neither a streamline nor a pathline.

5.3 The Material Derivative

When a quantity \(Q(\mathbf{x}, t)\) is observed in the Eulerian frame, its rate of change at a fixed point is simply \(\partial Q / \partial t\). But the rate of change following a fluid particle involves an additional term from the particle moving through a spatially varying field:

The material derivative is the Eulerian representation of the Lagrangian time derivative. The acceleration of a fluid particle is \(D\mathbf{u}/Dt\), not \(\partial\mathbf{u}/\partial t\). The difference \((\mathbf{u}\cdot\nabla)\mathbf{u}\) is the nonlinear advective acceleration — the term responsible for much of the richness and difficulty of fluid dynamics.

5.4 Rate of Strain in a Fluid

Fluids resist not strain but rate of strain. The velocity gradient tensor \(\partial u_i / \partial x_j\) decomposes into its symmetric and antisymmetric parts:

\[ \frac{\partial u_i}{\partial x_j} = \dot{e}_{ij} + \Omega_{ij} \]\[ \dot{e}_{ij} = \frac{1}{2}\!\left(u_{i,j} + u_{j,i}\right) \quad [s^{-1}] \]

is the rate-of-strain tensor and \(\Omega_{ij} = \frac{1}{2}(u_{i,j} - u_{j,i})\) is the spin tensor (local rotation rate). The rate-of-strain tensor is directly analogous to the strain tensor in elasticity; it measures how fast a fluid element is being deformed.

The spin tensor is related to the vorticity vector \(\boldsymbol{\omega} = \nabla \times \mathbf{u}\) by \(\omega_k = -\varepsilon_{kij}\Omega_{ij}\), where \(\varepsilon_{kij}\) is the Levi–Civita permutation symbol.

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Chapter 6: Inviscid Fluid Mechanics

6.1 Conservation of Mass

The first fundamental law is that matter is neither created nor destroyed. For a fluid with density \(\rho(\mathbf{x},t)\):

Incompressibility is an excellent approximation for liquids and for gas flows at Mach number \(Ma = U/c_s < 0.3\) (where \(c_s\) is the speed of sound). The condition \(\nabla \cdot \mathbf{u} = 0\) is a constraint on the velocity field — it has no time derivative, so it is not a dynamical equation but a kinematic restriction that eliminates compressional degrees of freedom.

6.2 Conservation of Momentum and the Euler Equations

\[ \rho\, \frac{D\mathbf{u}}{Dt} = -\rho\nabla\Pi + \nabla \cdot \boldsymbol{\tau} \]

where \(\Pi\) is the gravitational potential (\(\nabla\Pi = g\hat{z}\) for a uniform field) and \(\boldsymbol{\tau}\) is the stress tensor.

For an inviscid fluid (\(\mu = 0\)), the stress is purely isotropic: \(\tau_{ij} = -p\,\delta_{ij}\). Writing the dynamic pressure \(\tilde{p} = \Pi + p/\rho_0\):

Despite neglecting viscosity entirely, the Euler equations describe a vast range of phenomena: surface ocean waves, tidal flows, acoustics, aerodynamics at high Reynolds number, and the large-scale dynamics of geophysical fluids. The reason is that viscosity matters primarily near solid boundaries (in thin boundary layers) and at small scales; away from boundaries and at large scales, the Euler equations are an excellent model.

6.3 Vorticity and Potential Flow

\[ \frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega} \cdot \nabla)\mathbf{u} \]

The right-hand side is the vortex stretching term: if vortex lines are stretched by the flow, their vorticity intensifies (just as a spinning skater pulls in their arms to spin faster). In two dimensions, vortex stretching vanishes and vorticity is simply advected: \(D\omega/Dt = 0\).

Kelvin’s circulation theorem states that for an inviscid barotropic fluid, the circulation around any closed material loop is conserved. This means that if a flow starts irrotational (\(\boldsymbol{\omega} = 0\) everywhere), it remains irrotational.

The theory of potential flow is therefore the theory of harmonic functions, enriched with physical boundary conditions. On a solid boundary: \(\partial\phi/\partial n = 0\) (no normal flow through the wall). Far from a finite body: \(\mathbf{u} \to \mathbf{U}_\infty\) (uniform stream).

\[ \frac{\partial\phi}{\partial t} + \frac{1}{2}|\nabla\phi|^2 + \tilde{p} = \text{const} \]

6.4 The Stream Function

In two-dimensional incompressible flow \(\mathbf{u} = (u(x,z), w(x,z))\), the incompressibility condition \(u_x + w_z = 0\) is automatically satisfied if we write:

\[ u = \psi_z, \qquad w = -\psi_x \]

where \(\psi(x,z)\) is the stream function. The level curves \(\psi = \text{const}\) are streamlines: since \(\nabla\psi \cdot \mathbf{u} = \psi_z u + \psi_x(-w) = 0\), the velocity is tangent to curves of constant \(\psi\).

For irrotational 2D flow, \(\omega = w_x - u_z = -\nabla^2\psi = 0\), so \(\psi\) also satisfies Laplace’s equation. Moreover, the combination \(\phi + i\psi\) is a complex analytic function of \(x + iz\), bringing the full power of complex analysis — conformal mappings, the Cauchy integral formula, residues — to bear on 2D potential flow. This is exploited extensively in AMATH 463 for aerofoil theory.

6.5 Surface Gravity Waves and Dispersion

The most important example of Euler-equation dynamics in this course is the propagation of small-amplitude waves on a free water surface. We linearise the Euler equations about a state of rest, seeking wave solutions of the form \(\phi \propto e^{kz}\cos(kx - \omega t)\) (in deep water, \(z < 0\) below the surface). Applying the kinematic and dynamic free-surface boundary conditions leads to:

Three facts are immediately striking. First, longer waves (\(k\) small) travel faster: \(c_p \propto k^{-1/2}\). Second, the group velocity is exactly half the phase speed: energy propagates at half the rate of the wave crests. Third, the dispersion relation is irrational in \(k\), meaning the dispersion is strong — very different from the linear dispersion \(\omega = ck\) of sound or light.

The physical consequence: a pebble dropped in a pond creates a circular wave packet. Watching the packet, one sees individual crests appear at the inner (trailing) edge, travel forward through the group, and disappear at the outer (leading) edge — because the crests travel at twice the speed of the energy they carry. This remarkable phenomenon is a direct consequence of the dispersion relation.

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Chapter 7: Viscous Fluid Mechanics

Real fluids are never inviscid. Viscosity dissipates kinetic energy, enforces the no-slip condition at walls, and generates drag. The Euler equations are excellent far from boundaries and at large scales; the Navier–Stokes equations describe the full picture, including the critical region near solid walls where velocity must transition from the no-slip value to the free-stream value.

7.1 The Navier–Stokes Equations

For a Newtonian fluid, the viscous stress tensor is \(2\mu\,\dot{e}_{ij}\). The incompressible Cauchy momentum equation becomes:

The viscous term \(\nu\nabla^2\mathbf{u}\) diffuses momentum: regions of high velocity lose momentum to neighbouring regions of low velocity. The no-slip boundary condition \(\mathbf{u} = \mathbf{0}\) at solid boundaries is an empirical law consistent with molecular sticking and confirmed by countless experiments down to nanometre scales.

The Navier–Stokes equations are among the most important PDEs in science. They are nonlinear (through the advective term \((\mathbf{u}\cdot\nabla)\mathbf{u}\)) and coupled, making exact solutions rare. Nevertheless, several exact solutions exist for flows where geometry forces a dramatic simplification.

7.2 The Reynolds Number and Scaling

To understand when viscosity matters and when it does not, we nondimensionalise the equations. Scaling lengths by \(L\), velocities by \(U\), time by \(L/U\), and pressure by \(\rho U^2\), the Navier–Stokes equations take the form:

\[ \frac{D\mathbf{u}^*}{Dt^*} = -\nabla^*\tilde{p}^* + \frac{1}{Re}\,{\nabla^*}^2\mathbf{u}^* \]

At low \(Re\) (slow flows, small scales, high viscosity), viscosity dominates and the flow is smooth and laminar — Stokes flow. At high \(Re\) (fast flows, large scales, low viscosity), inertia dominates and the flow tends to become turbulent.

Representative values: blood in a capillary, \(Re \sim 10^{-3}\); swimming bacteria, \(Re \sim 10^{-4}\); a person swimming, \(Re \sim 10^6\); flow over an aircraft wing, \(Re \sim 10^7\). The onset of turbulence in a straight pipe occurs around \(Re \approx 2300\), though laminar flow can persist to much higher \(Re\) in carefully controlled experiments.

7.3 Exact Solution: Poiseuille Flow

For steady, fully developed flow between two infinite parallel plates at \(x = \pm h\), driven by a pressure gradient \(dp/dz\) in the \(z\)-direction, the Navier–Stokes equations reduce to a single ODE. Symmetry forces \(\mathbf{u} = u(x)\hat{z}\), and incompressibility is automatically satisfied. The momentum equation becomes:

\[ \nu\, \frac{d^2 u}{dx^2} = \frac{1}{\rho}\frac{dp}{dz} = \text{const} \]

with no-slip \(u(\pm h) = 0\). Integrating twice and applying boundary conditions:

\[ u(x) = \frac{1}{2\nu}\frac{dp}{dz}\!\left(x^2 - h^2\right) \]

This parabolic profile is the hallmark of Poiseuille (pressure-driven) flow. The maximum velocity is at the centreline \(x = 0\); the flow rate per unit width is \(Q = -\frac{2h^3}{3\nu}\frac{dp}{dz}\).

\[ u(r) = \frac{1}{4\nu}\left(-\frac{dp}{dz}\right)(R^2 - r^2) \]

with flow rate \(Q = \pi R^4 (-dp/dz) / (8\mu)\). The \(R^4\) dependence is dramatic: halving a blood vessel’s radius (by arterial plaque) reduces flow by a factor of 16, which is why arterial stenosis causes such severe haemodynamic consequences.

7.4 Vorticity Dynamics

Taking the curl of the Navier–Stokes equations yields the vorticity equation:

\[ \frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega} \cdot \nabla)\mathbf{u} + \nu\,\nabla^2\boldsymbol{\omega} \]

The three contributions are:

• \(D\boldsymbol{\omega}/Dt\): rate of change of vorticity following a fluid particle
• \((\boldsymbol{\omega}\cdot\nabla)\mathbf{u}\): vortex stretching — vorticity amplified by differential advection
• \(\nu\nabla^2\boldsymbol{\omega}\): viscous diffusion of vorticity

Vortex stretching is absent in two-dimensional flows (since \(\boldsymbol{\omega}\) is perpendicular to the plane of motion and cannot be stretched by an in-plane velocity gradient). In 2D, vorticity simply advects and diffuses:

\[ \frac{D\omega}{Dt} = \nu\,\nabla^2\omega \]

This is structurally identical to the advection-diffusion equation for a passive scalar (such as temperature or dye concentration). The analogy is deep: just as dye spreads by diffusion and is carried by the flow, so does vorticity. Vorticity is created at solid boundaries (where the no-slip condition imposes a discontinuity in velocity), diffuses away from the wall, and is swept downstream. The entire structure of viscous boundary layers can be understood as the diffusion of wall-generated vorticity into the flow interior.