Video Lectures

(A) Fluid Mechanics & Rate Processes (click here)

(B) Finite Element Methods for Fluid Dynamics (click here)

(A) Fluid Mechanics & Rate Processes

ESO204A is an undergraduate course, offered primarily to second year students, on Fluid Mechanics and Rate Processes. I taught this course in 2020-2021/I in the period of lock-down as a completely online course. Below are the lectures that are recorded for the students. Hope you find them interesting. Criticism is most welcome.

Lecture Number



Title of Lecture

url (Duration of Lecture, each part) [Total Time]

0 (1)

Introduction to the Course: course content, text, policies etc

Lecture 0: Introduction to the Course. Its conduct and policies [20:21]

1 (1)

Introduction to Fluid Mechanics and rate processes and its diverse applications

Introduction to Fluid Mechanics and Rate Processes (24:02), (23:22)


2 (1)

Distinction between fluids and solids, Continuum hypothesis, Concept of a property, Viscosity, Newtonian and Non-newtonian fluids

The continuum hypothesis and property (28:28), (26:22)


3 (1)

Vectors and tensors, Gradient of a vector, Symmetric and skew-symmetric components of a tensor, Body and Surface forces, Stress tensor, Shear and normal components, Symmetry of stress tensor

The stress tensor (21:04), (20:26)


4 (2)

Fluid statics: show pressure is isotropic, net pressure force on a fluid element, absolute vs gauge pressure, basic equation of fluids statics, hydrostatic pressure distribution, U-tube manometer, Hydrostatic force on a submerged surface, Buoyancy

Fluid statics (24:16) (27:26) (07:38)


5 (2)

Kinematics: Lagrangian and Eulerian descriptions, Substantial derivative: relation between Eulerian (local) and Lagrangian (material) rates of change, Steady vs unsteady flows, Graphical description of flows: path lines, streak lines and stream lines. An example to show the difference between the three.

Kinematics-1 (24:51) (33:13)


6 (3)

Kinematics: Rate of deformation of a fluid element, Vorticity and angular rotation, strain rate tensor, decomposition of velocity gradient into shear strain rate and rotation. Constitutive model for a Newtonian Fluid

Kinematics-2 (22:58) (21:15) (19:34) (13:43)


7 (3)

Control Mass vs Control Volume analysis, Reynolds Transport Theorem, Integral form of Conservation of Mass

Reynolds Transport Theorem: conservation of mass (10:53) (19:41)


8 (3)

Conservation of mass continued; Differential form of conservation of mass: continuity equation; incompressible flows.

Conservation of mass: integral and differential forms (24:36) (18:17) (23.17)


9 (4)

Momentum Theorem: force balance. Tips on choice of control volume.

Integral form of momentum theorem (51:01)


10 (4)

Derivation of the differential form of momentum equation. Application for a Newtonian fluid.

Derivation of differential form of momentum equation (47:54)


11 (4)

Continuty and Momentum Equations: details of the various terms; cylindrical coordinates; far field boundary conditions and velocity on solid surface.

Continuity and Momentum Equations in cartesian and cylindrical coordinate system (50:04)


12 (5)

Boundary conditions at the interface of two fluids; Various types of flows: Uniform, steady, full-developed, one-dimensional, two-dimensional, three- dimensional, spatially periodic, temporally periodic

Boundary conditions and various kind of flows. (57:04)


13 (5)

Exact solution to flow equations: flow between two parallel plates

Flow between two parallel plates (32:46)


14 (5)

Hagen Poiseuelle Flow; Skin Friction Coefficent for laminar and turbulent flows; Reynolds experiments to demonstrate transition to turbulence

Flow in a pipe of circular section (26:10) (32:25)


15 (6)

First Law of Thermodynamics, Application to Control Volume via Reynolds Transport Theorem, Discussion on work done by surface forces: shaft work, work done by normal stresses, shear stresses etc.

Integral Form of Total Energy Equation-I (25:25) (27:45)


16 (6)

Simplified form of integral energy balance with assumptions of steady flow, incompressible flow, 1D flow at inlet and exit ports; Heads; Application to Tank with orifice

Integral Form of Total Energy Equation-II (20:25) (21:28) (21:33)


17 (6)

Application of integral energy balance to (a) flow across a pump, (b) flow in a ducted fan and (c) flow in a wind tunnel.

Integral Form of Total Energy Equation-III (22:48) (22:06)


18 (7)

Bernoulli equation and its application to flow measurements: venturi meter, orifice meter, flow nozzle

Bernoulli Equation (24:02) (25:45) (14:21)


19 (7)

Similitude and Modeling: Motivation for dimensional analysis, Buckingham-Pi Theorem, its application to flow past an aircraft to show dependence of drag coefficient to reynolds number; Non-dimensionlization of flow equations, Reynolds number

Similitude and Modeling-I (17:11) (19:59)


20 (7)

Similitude and Modeling: Non-dimensionlization of flow equations and Boundary Conditions continued, Reynolds number, Froude Number, Euler Number, Strouhal Number, Geometrical and Dynamical Similarity

Similitude and Modeling-II (25:21) (18:46)


21 (8)

Similtude and Modeling: Example problems. Incomplete Similarity wherein it is not possible to match the non-dimensional numbers

Similitude and Modeling-III (20:39) (23:38) (23:16)


22 (8)

Flow through pipe: review of Hagen Poiseuelle Flow, Entrance Length, Laminar and turbulent flows, Surface roughness, non-dimensional formulation of pressure loss in a pipe, Darcy’s friction factor, Moody’s chart

Flow in a pipe-I (22:50) (27:12)


23 (8)

Flow through pipe: review of friction factor, correlations, losses in fittings (minor losses)-elbow bends, valves, sudden expansion and contraction, example problem illustrating the application of Moody’s chart and losses.

Flow in a pipe-II (25:07) (19:44) (18:32)


24 (9)

Flow regimes and approximations: combining pressure gradient term and force due to gravity in the force equation, non-dimenionsal equations, Low Re flow-creeping flow, Stokes approximation, Stokes flow past a sphere, force coefficient, pressure coefficient at stagnation points.

Low Reynolds Number Flows: Stokes equations (22:08) (23:58)


25 (9)

Flow regimes and approximations:Inviscid Flow: Euler Equations & Boundary Conditions; High Re flows: concept of Boundary Layer; Order of magnitude analysis to estimate the boundary layer thickness, example of boundary layer on the wing of Airbus A320.

Viscous flow: concept of a boundary layer (24:45) (32:10)


26 (9)

Inviscid Flows: Irrotational flows, Circulation and its relation to vorticity via Stokes’ Theorem, Statement of Kelvin’s Theorem on Circulation, Velocity Potential, Stream Function

Inviscid Flows: velocity potential (31:11) (26:49)


27 (10)

Derivation of Potential flow equations for both velocity potential and stream function, linearity of equations and superposition, boundary conditions, orthogonality of equipotential- and stream-lines, derivation of the Bernoulli equation for irrotational flows from Euler equations

Potential Flow Equations (24:49) (25:57)


28 (10)

Some simple 2-D Potential Flows: Review of potential flow quations in Cartesian and cylindrical/polar coordinates, Uniform flow, Source-Sink, Free Vortex. For each one, the velocity potential, stream function and pressure distribution. Explaination on why the source/sink and free vortex are singularities.

Some simple 2D-Potential Flows (22:36) (16:36) (20:09)


29 (10)

Velocity potential and Streamfunction for a source not located at origin; Superposition of simple 2D potential flows: Rankine half body (source & uniform flow), Doublet (source & sink)

Potential Flows: Rankine half body and Doublet (24:23) (16:25)


30 (11)

Flow past a cylinder: Potential flow via superposition of doublet and uniform flow, pressure distribution, D’Alembert’s paradox, Real flow, separation, base pressure and its role in drag, drag-crisis, bluff body flows and role of laminar/turbulent state of boundary layer

Flow past a cylinder: potential and real flow (24:29) (23:57) (26:07)


31 (11)

Flow past a rotating cylinder: Potential flow via superposition of uniform flow, doublet free vortex, modeling spin rate via vortex strength, stagnation points and streamlines for various spin ratios, Lift coefficient as a function of spin rate-Robins-Magnus effect; Viscous effects and three-dimensionality.

Flow past a rotating cylinder: Robins-Magnus effect (25:27) (26:41) (10:09)


32 (12)

Boundary Layer: Boundary Layer Equations, Application to flat plate, Blasius profile, Skin friction and drag coefficient, Turbulent boundary layer, Transition, Displacement thickness, Momentum thickness, Karman Momentum Integral

Boundary Layer (23:44) (20:49) (14:44) (19:48)


33 (12)

Sports Aerodynamics: Flow past a smooth sphere-drag crisis due to transition of boundary layer from laminar to turbulent state, oil flow visualization, laminar separation bubble; flow past a cricket ball, role of seam in swing and reverse swing; contrast swing; drag coefficient vs Re for other sports projectiles; fluid-structure interactions in a badminton shuttle cock

Sports Aerodynamics: Conventional- and Reverse-Swing of a cricket ball (19:59) (27:35) (30:25)



Heat Transfer: Introduction, Conduction-Fourier’s Law, Convection-Newton’s Law of cooling, Radiation-Stefan-Boltzmann Law; Free and Forced Convection; Equation for unsteady conduction; Initial and Boundary Conditions

Introduction to Heat Transfer (20:46) (19:56) (32:11)



Non-dimensionalization of conduction equations, Biot Number, Fourier Number, Heisler Charts, Fins; Convection from a hot body in a stream: flow and energy equations, Boussinesq Approximation, Non-dimensional Equations, Nusselt Number, Grashof number, Prandtl Number, Eckert Number

Non-dimensional equations for heat conduction and convection (20:16) (20:56) (21:48) (25:31)



Introduction to Mass Transfer, Examples, Mass and molar density, Mass and molar fraction, Mass- and Molar-averaged velocity, Diffusion Velocity, Fluxes, Dilute mixture, Fick’s Law, Species conservation equation, Steady-state diffusion in stationary medium with no source.

Introduction to Mass Transfer (20:33) (20:16) (19:49)


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(B) Finite Element Methods for Fluid Dynamics

AE618A is a graduate course. However, almost half the registered class is undergraduate students. I taught this course in 2020-2021/II in the period of lock-down as a completely online course. Below are the lectures that are recorded for the students. The course ran for one semester with two lectures a week. Hope you find them interesting. Criticism is most welcome.

LectureNumber (Week)


Name of Lecture

Url (Duration of Lecture, each part) [Total]

1 (1)

Introduction to Computational Fluid Dynamics (CFD) and High Performance Computing (HPC), Some applications of CFD, Evolution of Computing Power, Speed up, Fluid-Structure Interactions

Introduction to CFD & HPC [21:22] [18:45] [16:48]


2 (1)

Introduction to the Course, its objectives, contents, course policies

Overview of the course: Finite Element Methods for Fluid Dynamics [23:50] [22:23]


3 (2)

Simple Finite Difference Methods to solve first order ODE’s; Introduction to PDE’s

Simple Finite Difference Methods to solve ODE’s [24:45] [28:45]


4 (2)

Simple Finite Difference Methods to solve second order ODE’s and Elliptic PDE’s; Example problem; ADI method

Simple Finite Difference Methods to solve PDE’s [22:58] [24:56] [18:41]


5 (3)

Preliminary concepts: Vector spaces, linear independence, basis, Infinite vs finite dimensional spaces, inner product, norms, completeness

Vector Spaces, Basis, Completeness [19:27] [22:09] [26:45]


6 (3)

Well- vs Ill-conditioned equation systems. Concepts of Continuity, Differentiability and Smoothness. Sobolev Spaces

Continuity, Differentiability and Smoothness [23:48] [21:06] [14:30]


7 (4)

Method of Weighted Residual (MWR), Trial and Weight Functions, Various methods: Subdomain/Least-squares/Collocation/Bubnov-Galerkin/Method of moments

Method of Weighted Residual (MWR) [23:39] [25:31]


8 (4)

Stong and Weak Forms, Trial- and Weight-function spaces, Essential and Natural Boundary Conditions, Equivalence of strong and weak forms, Properties of weak form, Alternate weak forms and related function spaces

Weak Form of a steady, 1D problem [23:16] [24:56] [22:22]


9 (5)

Galerkin’s Approximation, Finite Dimensional Spaces, Basis Functions, Matrix Form, Symmetry of stiffness matrix.

Galerkin’s Approximation & Matrix Form [24:10] [17:41] [19:43]


10 (5)

Example to illustrate Galerkin’s Approximation with n=1 and 2 using peicewise linear functions.

Example to illustrate Galerkin’s Approximation [29:25] [27:26] [20:23]


11 (6)

Piecewise Linear Finite Element Space, Shape functions, Stiffness matrix and force vecor, Positive Definiteness of stiffness matrix, Element level matrices & vectors, Location Matrix (LM) array

Piecewise Linear Finite Element Space: Element Level Matrices & Vectors [21:05] [27:37] [24:42] [21:53]


12 (6)

Element level stiffness matrix and force vector, Assembly, Example to illustrate Assembly for a 1D, steady problem with uniform mesh and spatially constant force, Comparsion of the resulting equation with a simple finite difference method.

Assembly of Element Level Matrices & Vectors [25:06] [25:22] [22:51]


13 (7)

Two- and Three-Dimensional problems, Notation, Strong form of the Linear Heat Conduction Problem, Weak form, Equivalence of Strong and Weak forms, Galerkin’s Approximation

Strong, Weak and Galerkin Form of Two- and Three-Dimensional Problems [22:06] [23:48] [24:15]


14 (7)

Two- and Three-Dimensional problems: Galerkin’s Approximation, Matrix Form, Approximations in Boundary Conditions, Properties of Stiffness Matrix, Element level matrices and vectors

Matrix form of Two- and Three-Dimensional Problems [20:39] [25:47] [24:43]


15 (8)

Linear Heat Conduction in Two- and Three-Dimensions: Matrix Form, Data processing arrays: Connectivity (IEN) array, Destination (ID) array and Location Matrix (LM) array, Assembly of element level stiffness matrix and force vector. Strong form of Classical Linear Elastodynamics-vector unknown

Assembly of element level matrices and vectors for Two- and Three-Dimensional Problems [22:47] [24:29] [30:24]



Revisit Method of Weighted Residual for a vector unkown, Symmetric Matrices and their product with general matrices, Classical Linear Elastostatics: strong form, weak form, their equivalance, matrix form, Data processing arrays for a vector unknown, Assembly, Illustration with an example

Classical Linear Elastostatics: vector unknown [23:50] [24:18] [21:27] [28:33]



Linear shape functions in 1D and Bilinear shape functions in 2D, Isoparametric Representation, Conditions on shape functions for convergence, Trilinear shape functions in 3D, Lagrange polynomials, Higher order shape functions in 1D and 2D.

Finite Element Interpolation/Basis Functions [21:03] [24:46] [22:05] [14:13]



Numerical Integration: Motivation, Set-up. Trapezoidal and Simpson’s Rule; Gauss Quadrature Rule in 1D, Generalization to higher dimensions; Derivatives of Shape Function; Shape Function Subroutine

Numerical Integration; Derivatives of Shape Function [17:04] [15:51] [19:28] [18:02]



Time Dependent Problems: various kinds of unsteadiness, Unsteady Linear Heat Conduction, Strong, Weak and Galerkin Form, Function Spaces, Matrix Form: Semi-Discrete nature of equations. Mass Matrix: element level and global, Initial Condition

Introduction to Time Dependent Problems [19:59] [22:47] [20:58]



1D unsteady linear heat conduction, derivation of element level matrices and vector, assembly for a uniform mesh, comparison with a simple finite difference method; The Generalized Trapezoidal Method, special cases of Euler Foward, Euler Backward and Crank-Nicolson Methods, Velocity and Displacement Implementation, Implicit-Explicit Methods

1D Unsteady Linear Heat Conduction; Generalized Trapezoidal Method [20:37] [21:49] [34:32]



Convergence, Consistency and Stability; Stability analysis of time integration methods via a 1 degree of freedom modal problem, stability of exact solution, amplification factor, conditional and unconditional stability, Courant condition, physical interpretation

Stability analysis of time integration methods [23:58] [27:53]



Consistency Analysis using a 1dof modal equation,truncation error, order of accuracy, Proof of Convergence of the Generalized Trapezoidal Method using consistency and stability, Steady state and unsteady calculations, effect of reduction of time step and order of accuracy.

Convergence of Generalized Trapezoidal Method [16:11] [18:44] [17:26]



Nonlinear Analysis: introduction to the non-linear problem and the need to use an iterative method, Fixed-point interation, contraction operator, order of convergence; Newton-Raphson Method, order of convergence, generalization to a vector unknown

Methods for Non-linear problems [29:28] [27:41]



Advection-Diffusion Equation, Stencils for Finite Difference and Galerkin method, Peclet number, Numerical instability of the Galerkin method in the advection limit, demonstration of numerical instability via node-to-node oscillations for calculations with high Peclet number, Artifical Diffusion, Upwinding

Numerical instability in the Galerkin Formulation for Advection Dominated Flows [21:21] [20:49] [24:15]



Review of upwinding, Consistency of the Galerkin Method, SUPG (Streamline-Upwind/Petrov-Galerkin) Method for Advection dominated flows in 1D, Generalization to multi dimensions

SUPG (Streamline-Upwind/Petrov-Galerkin) Method for Advection dominated flows [23:41] [21:43] [15:35]



Incompressible & Compressible Flows: governing equations, stabilized finite element formulations; Comments on parallel computing; Case studies.

Stabilized Finite Element Formulations for Flows [21:00] [17:41] [19:07] [21:00]


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