3-0-0-9

### Course Content:

Vorticity, vorticity equation, Kelvin’s theorem, Helmholtz laws, kinematic decomposition of velocity fields, uniqueness of solenoidal and irrotational velocity fields, superposition of irrotational flows, complex analysis, conformal transformations, finite wings, three-dimensional potential flows, unsteady potential flows, virtual mass, cavitation, bubble dynamics, interfacial and free-surface waves, bores and hydraulic jumps.

### No. of Lectures (40 Lectures)

I. Introduction(1 lecture):

• Continuity equation, Navier-Stokes equations, Bernoulli equation

II. Vorticity dynamics (12 Lectures):

• Vorticity, vortex lines and tubes, vorticity equation, circulation, Kelvin’s circulation theorem, Helmholtz laws, creation of vorticity, Helmholtz’s kinematic decomposition, Biot-Savart law, inviscid motion of point vortices, line and sheet vortices, image vorticity, vortex momentum, uniqueness of solenoidal and irrotational velocity fields in singly- and multiply-connected domains, Kelvin’s minimum energy theorem, superposition of irrotational flows

III. Potential flows (17 Lectures):

• Stream function and velocity potential, complex analysis, standard flow patterns, circle theorem, Blasius integral laws, conformal transformations, Jowkowski transformation, introduction to thin-airfoil theory, finite wings, introduction to lifting-line theory, Schwarz-Christoffel theorem, axisymmetric and three-dimensional potential flows, sphere theorem, nature of 3D far-fields, unsteady potential flows, virtual mass

IV. Elements of bubble dynamics (3 Lectures):

• Rayleigh equation, Rayleigh-Plesset equation, cavitation and bubble collapse

V. Interfacial waves and free-surface flows (7 Lectures):

• Small amplitude waves at free surfaces and interfaces, non-linear theory of shallow-water waves, method of characteristics, bores and hydraulic jumps

### Target audience:

1. PG and 3rd, 4th year UG students

1. Milne-Thomson, L. M. “Theoretical Hydrodynamics”, 5th Ed., Dover

2. Milne-Thomson, L. M. “Theoretical Aerodynamics”, 4th Ed., Dover

3. Vallentine, H. R. “Applied Hydrodynamics”, 2nd Ed., Butterworths

4. Batchelor, G. K. “Introduction to Fluid Dynamics”, 1st Indian Ed., Foundation Books

5. Panton, R. L. “Incompressible Flow”, 3rd Ed., Wiley

6. Lighthill, J. “An Informal Introduction to Theoretical Fluid Mechanics”, Oxford Clarendon

7. Currie, I. G. “Fundamental Mechanics of Fluids”, 4th Ed., CRC Press