2. Faster Ways to Develop Balancing Skills for Omni Present & Non Countable Systems …… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Methodology For Analysis of Viscous Fluid Flows

3. Now I think hydrodynamics is to be the root of all physical science, and is at present second to none in the beauty of its mathematics. (William Thomson (Lord Kelvin) 1824±1907) You are not educated until you know the Second Law of Thermodynamics. Current Thinking of Intellectuals.

4. The Greatest Dispute !!!! The incomparable Newton's Principia Mathematica was published in 1687. Little more than a century and half after this, first principles of viscous fluid flows were affirmed in the form of the Navier-Stokes equations. Major contributions: Navier in 1823, Cauchy in 1828, Poisson in 1829, Saint Venant in 1843, and Stokes in 1845. There is always an easy solution to every human problem neat, plausible and wrong. (Henry Louis Mencken, 1880±1956)

5. The Greatest Agreement With very few exceptions, the Navier-Stokes equations provide an excellent model for both laminar and turbulent flows. The anticipated paradigm shift in fluid mechanics centers around the ability today as well as tomorrow of computers to numerically integrate those equations. We therefore need to recall (Realize) the equations of fluid motion in their entirety.

6. The Role of Mathematics in Teaching/Using Fluid Flows For Mechanical Engineers Fluid Flows - over and above its "physics" side offers an excellent opportunity to use mathematics. Fluid Flow is a best means even to clear the students' only formal understanding of the higher mathematical apparatus. The mathematical side of the subject has a big pedagogical value due to; the frequent use of mathematics, the construction of mathematical models, is also an engineer's task. The fast increase of the subject matter in Fluid Flows on one hand, and the limited, in some cases even decreasing time available for teaching it, calls for the more mathematics.

7. Viscous Fluid Flows for Post Graduate Students Correctly balancing the physics and mathematics is the important educational aim. By pushing it to extremes one may end up in a course of descriptive presentation, rules of thumb and table or graph readings fit for conventional routine jobs only. It is then far from what one might expect from a graduate engineering course. On the other hand, little application and a very big mathematical apparatus may feature a kind of theoretical physics which should not be the goal when training Thermal Scientists/Engineers.

8. Educational Aim of Teaching Viscous Fluid Flows To demonstrate when and how deep an engineer is bound to dive into the problem, where he should use exact mathematical methods and where approximations. To use the proper numerical apparatus, a pocket or desk calculator if a calculator is justified, or a thoroughly checked computer program if the problem requires it. To offer ample opportunities and utilise them consciously.

9. Books White, F.M. 1991 Viscous Fluid Flow (second edition), McGraw Hill. Boundary Layer Theory, H. Schlichting. Meinhard T. Schobeiri, 2010 Fluid Mechanics for Engineers : A Graduate Textbook Sherman, F.S. 1990 Viscous Flow. McGraw Hill. McCormack, P.S. & Crane, L.J. 1973 Physical Fluid Dynamics, Academic Press. Panton, R.L. 1996 Incompressible Flow (second edition), Wiley. Acheson, D.J. 1990 Elementary Fluid Dynamics. Clarendon Press, Oxford, 1990. Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge.

10. Syllabus Preliminary Concepts Fundamental Equations of Viscous Flow Solutions of the Newtonian Viscous-Flow Equations Laminar Boundary Layers The Stability of Laminar Flows Incompressible Turbulent Mean Flow Compressible-Boundary-layer Flow

11. Development of Fluid Flow Systems using a selected combination of Forces Systems only due to Body Forces. Systems due to only normal surface Forces. Systems due to both normal and tangential surface Forces. –Only mechanical forces. –Only electrical forces. –Electro-kinetic forces. –Thermo-dynamic Effects (Buoyancy forces/surface )….. –Physico-Chemical/concentration based forces (Environmental /Bio Fluid Mechanics

12. Major Flow Systems due to Mechanical Forces : Level 1 Incompressible – A vector dominated….. Compressible – Both vector and scalar ….

13. 1930’s Flying Story Cruising at High Altitudes ?!?!?! Aircraft were trying to approach high altitudes for a better fuel economy. This led to numerous crashes for unknown reasons. These included: The rapidly increasing forces on the various surfaces, which led to the aircraft becoming difficult to control to the point where many suffered from powered flight into terrain when the pilot was unable to overcome the force on the control stick. The Mitsubishi Zero was infamous for this problem, and several attempts to fix it only made the problem worse. In the case of the Super-marine Spitfire, the wings suffered from low torsional stiffness.

14. The P-38 Lightning suffered from a particularly dangerous interaction of the airflow between the wings and tail surfaces in the dive that made it difficult to "pull out“. Flutter due to the formation of thin high pressure line on curved surfaces was another major problem, which led most famously to the breakup of de Havilland Swallow and death of its pilot, Geoffrey de Havilland, Jr.

15. The Concept of Field The question we need to answer is how can a force occur without any countable finite bodies & apparent contact between them? Something must happen in the fluid to generate/carry the force, and we'll call it the field. Few basic properties along with surroundings must be responsible for the occurrence of this field. Let this field be . "Now that we have found this field, what force would this field place upon my system.“ What properties must the fields have, and how do we describe these field?

16. Fields & Properties The fields are sometimes scalar and sometimes vector in nature. There are special vector fields that can be related to a scalar field. There is a very real advantage in doing so because scalar fields are far less complicated to work with than vector fields. We need to use the calculus as well as vector calculus. Study of the physical properties of vector fields is the first step to ability to use Viscous Fluid Flow Analysis.

17. Preliminary Concepts Vector and Tensor Analysis, Applications to Fluid Mechanics Tensors in Three-Dimensional Euclidean Space Index Notation Vector Operations: Scalar, Vector and Tensor Products Contraction of Tensors Differential Operators in Fluid Mechanics Substantial Derivatives Differential Operator Operator Applied to Different Functions