Advanced Fluid Dynamics
This graduate-level course covers the physical foundations, governing equations and analytical solution methods of viscous flows. Flow kinematics, stress and deformation, nondimensionalisation, vorticity dynamics, low-Reynolds-number flows and boundary-layer theory are studied together. Emphasis is placed on justifying model assumptions, deriving equations and interpreting results physically.
Prerequisites
- Undergraduate knowledge of fluid mechanics.
- Knowledge of multivariable differential and integral calculus.
- Familiarity with vector analysis and differential equations.
Course Content
ChaptersThe continuum approximation, Eulerian and Lagrangian descriptions, material derivatives, streamlines and particle paths. Velocity gradients, deformation rates, rigid-body rotation and vorticity.
Surface forces, normal and shear stresses. Pressure and viscous components of the stress tensor, the Newtonian fluid model and incompressibility.
Differential mass, momentum and energy balances. Navier–Stokes equations, initial and boundary conditions, and problem formulation in different coordinate systems.
Characteristic scales, dimensionless parameters and dominant-term balances. Vorticity transport, stretching and diffusion; the vorticity–streamfunction formulation for two-dimensional incompressible flows.
The creeping-flow approximation and comparison of inertial and viscous effects. Stokes flow around a sphere, drag calculations and limits of validity.
Plane and circular channel flows, Couette and Poiseuille solutions, and combined pressure and wall-motion effects. Flow between coaxial rotating cylinders, shear stress and torque calculations.
Stokes’ first and second problems. Momentum diffusion, similarity variables, oscillating wall motion and viscous penetration depth.
Physical and mathematical foundations of similarity methods. Plane stagnation-point flow, suction and blowing at permeable surfaces, and flow over a rotating disk.
Derivation of boundary-layer equations using scaling analysis. Blasius and Falkner–Skan solutions; boundary-layer, displacement and momentum thicknesses. Wall shear stress and pressure-gradient effects.
The momentum integral equation and solutions using approximate velocity profiles. Skin friction, favourable and adverse pressure gradients, the separation condition and limitations of the boundary-layer approximation.
Jets, wakes and mixing layers. Similarity solutions, momentum transport, flow spreading and entrainment of the surrounding fluid.
Base flows and small disturbances, linear stability analysis and laminar–turbulent transition. Reynolds decomposition, mean-flow equations, Reynolds stresses and the closure problem. Basic characteristics of turbulent boundary layers.
Targeted Learning Outcomes
- Interpret flow kinematics through velocity gradients, deformation rates and vorticity.
- Formulate governing equations for viscous flows with appropriate constitutive relations and boundary conditions.
- Identify dominant physical processes through scaling analysis and justify model simplifications.
- Obtain analytical and similarity solutions for classical viscous-flow problems.
- Analyse boundary-layer problems using differential and integral methods.
- Assess the physical consistency of solutions and the limits of the approximations used.