### Credits:

3L-0T-1P-0A (10 Credits)

### Course Content:

Introduction; Reynolds Transport Theorem; Integral form of continuity, momentum and energy equations; Eulerian and Lagrangian view-points; Constitutive relations; Navier Stokes equations; Exact solutions; Potential flow; Boundary layer theory; Separation and drag; Turbulent flow: Reynolds averaged equations; Turbulent flows in pipes and channels; compressible flows.

### Lecturewise Breakup:

I. Introduction: (2 Lectures)

• Fluid statics; Definition of continuum, statement of physical laws (mass, momentum, energy and second law of thermodynamics), Reynolds transport theorem. Distinction between a systems approach and a control volume approach.

II. Simplified Global Analysis: (6 Lectures)

• Integral form of the mass balance equation; steady and unsteady flows; examples
Integral form of the momentum equation; steady and unsteady flows; examples in fixed and moving coordinate systems; Examples involving static pressure variation in the vertical direction
Examples of force calculation for flow over vanes; flow in a variable-area bend
Derivation of energy equation from first law of thermodynamics; evaluation of losses using correlations and charts; examples
Application of energy equation for flow through pipes and pipe networks [steady flows]
Application of the energy equation for flow between reservoirs;

III. Dimensional Analysis: (1 Lecture)

• Buckingham-p theorem, nondimensional parameters, problem-solving using non-dimensionalization; Reynolds number and Mach number similarity.

IV. Detailed Analysis: (4 Lectures)

• Discussion on local versus global approach to solving flow problems; Derivation of conservation of mass equation using RTT, coordinate-free form, curvilinear coordinates, incompressibility
Newton's second law of motion via RTT, body and surface forces, Eulerian and Lagrangian form of acceleration; Meaning of the material derivative; flow kinematics: streamlines. Calculation of material derivatives through examples
Expression of surface forces in terms of the stress tensor; Properties of the stress tensor; Discussion on the most general form of the constitutive relation for a linear homogeneous isotropic material
Construction of the strain-rate tensor, stress-strain rate relationship, thermodynamic and mechanical pressures, Stokes hypothesis.

V. Navier-Stokes Equations: (3 Lectures)

• Special form of NS equations for constant property fluids, two dimensional Cartesian coordinates, steady flow
Mathematical properties of NS equations, Discussion on non-uniqueness of the solution of NS equations; Calculation of volume flow rate, forces and moments from the local solution
Boundary conditions for velocity and pressure, stream function and vorticity; vorticity transport equation; Surface tension and continuity condition for the traction vector at material interfaces.

VI. Exact Solutions: (6 Lectures)

• Creeping flow and fully developed flow approximations. Flow between parallel plates, friction factor relationship
Flow between concentric rotating cylinders, Taylor-Couette flow, application to viscometry Stability considerations; Flow in a tube of square cross-section
General discussion on separation-of-variables for solving PDEs; Application to square tubes and partially-filled tubes; Friction factor relations; Hydraulic diameter approach
Unsteady flow in a circular tube: effect of transients; (Introduction to Bessel functions.) Discussion on the realizability of the predicted flow field
Stokes 1st and 2nd problems; extension to a general transient problem via Duhamel's theorem
Stokes flow past a sphere; Analytical solution, streamline patterns, nature of pressure gradient, expression for drag
Data for drag and lift coefficients for spheres and cylinders as a function of Reynolds number. Determination of the trajectory of particles moving in a fluid medium
Theory of Lubrication: Reynolds equation, journal-bearing problem, load bearing capacity.

VII. Potential Theory: (7 Lectures)

• Inviscid, incompressible irrotational flow, utility and applications, Kelvin's theorem, governing equations; Bernoulli's equation (steady and unsteady)
Method of potentials: stream function and velocity potential, flow kinematics in terms of streamlines and isopotential contours; Cauchy-Riemann conditions, complex potential, complex velocity, solution by complex potentials using the method of superposition; boundary conditions
Elementary complex potentials for uniform flow, sources, sinks vortices; flow in a sector. Superposition of source and uniform flow, doublets, superposition of doublet and uniform flow
Flow past a circular cylinder. Flow past a cylinder with circulation, calculation of forces, Blasius integral laws, Kutta-Joukowski theorem
Development of complex potentials by conformal transformation, flow past an ellipse-shaped object; flow past a vertically oriented flat plate. Flow at sharp corners
Thin aerofoil theory: complex potential, flow patterns, Kutta condition, development of lift, angle of stall. Experimental results for aerofoils and comparison with theory in terms of pressure distribution and lift coefficients for circular cylinders and aerofoils.

VIII. Boundary-Layer Theory: (6 Lectures)

• Prandtl's wind tunnel experiments, BL approximation, notion of an impressed pressure field; Separation explained in terms of boundary-layers. CD versus Re curve for a circular cylinder; Vortex shedding, Wake structure; Qualitative description of Kelvin-Helmholtz instability
Boundary-layer growth in favorable and adverse pressure gradients; Dissipation capacity of a boundary-layer; Application to the design of wind tunnels; BL control by suction and blowing
Derivation of BL equations and boundary conditions; Momentum-integral (MI) approach; displacement and momentum thicknesses
MI approach for a flat plate, non-zero pressure gradient flows, Importance of a fourth order polynomial for inflexion point profiles.

IX. Turbulent Boundary-Layers : (3 Lectures)

• Notion of instability and transition, fully developed boundary-layers, effect on viscous drag, heat transfer and point of separation; BL control; Reynolds decomposition, time-averaged NS equations, closure, Reynolds stresses, cross-correlation and its physical significance, Boussinesq approximation, BL equations for a turbulent flat plate BL
Prandtl's mixing length theory, two-layer model, derivation of the log-law; 1/7th power law approximation. Calculation of wall shear stress from BL measurements
Numerical examples, Explanation of Moody's chart in terms of the log-law, effect of wall roughness, effect of pressure gradient.

X. Compressible Flow : (4 Lectures)

• High speed gas flow, special features, speed of sound, one-dimensional form of the governing equations; Isentropic gas relations; velocity measurement using a pitot tube at all Mach numbers Flow through nozzles, area-velocity relations, numerical examples; Non-ideal flow in nozzles, formation of shocks; Application to supersonic velocity measurement by a pitot tube; Shock-boundary layer interaction.

### Laboratory Sessions:

I. Flow visualization around streamlined and bluff objects (including multimedia resources).

II. Measurement of viscosity of liquids and gases.

III. Force acting on a circular cylinder placed in cross-flow.

IV. Probes and transducers – pitot static tube, 5-hole probe, manometers, hotwire anemometer, Wind tunnels (This material is to be covered in a one hour lecture.).

V. Measurement of velocity and velocity fluctuations in a turbulent mixing layer.

VI. Boundary-layer flow over a flat plate (laminar and turbulent).

### References:

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12. Houghton, E.L. and Carpenter, P.W., Aerodynamics for Engineering Students, Fourth edition, Edward Arnold, UK, 1993.

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16. Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, Pergamon, UK, 1984.

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19. Panton, R. L., Incompressible Flow, John Wiley, New York, 1984.

20. Papanastasiou, T.C. and Georgiou, G.C.  Viscous Fluid Flow, CRC Press, Washington D.C., 2000.

21. Platten, J. K. and Legros, J. C., Convection in Liquids, Springer-Verlag, New York, 1984.

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23. Shames, I.H., Mechanics of Fluids, McGraw-Hill, New York, 1989.

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29. Yuan, S.W., Foundations of Fluid Mechanics, Prentice Hall, New Jersey, 1976.