2. Scalar-Vector Interaction for better Life …… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Vector Calculus for Viscous Fluid Flows

3. Mathematical Model for Field For a function g whose derivative is G: the fundamental lemma of calculus states that where g(x) represents a well-defined function whose derivative exists. This is the integral developed in introductory calculus, but it is not the only definition of the integral.

4. The mother of Vector Field There are integrals called path integrals which have quite different properties. In general, a path integral does not define a function because the integral will depend on the path. For different paths the integral will return different results. In order for a path integral to deliver a mother (of a vector field), it must depend only on the end points. Then, a scalar field  will be related to the vector field F by

5. The Engine : An agent to generate Force

6. The Birth of A Special Operator

8. The First & Foremost Fluid Field Varible Compared to solids fluids seem almost alive, magical. They flow, change form to accommodate the surroundings, produce gurgling sounds, and refract light to produce shimmer. There are few things that can match the majesty of a waterfall or the serenity of a deserted beach. What causes (Primarily) fluids to flow? As with solids, motions can only be produced by unbalanced forces so what is the nature of the forces in a fluid?

9. The Clue from Nature for Gradient

10. Real Fluids : A Resource of Gradients At the end of the 1640s, Pascal temporarily focused his experiments on the physical sciences. Following in Evangelista Torricelli’s footsteps, Pascal experimented with how atmospheric pressure could be estimated in terms of weight.

11. Hydrostatics A Filed variable Recognized by the Pascal. Even based on pedagogical principle, to start with simple matters and turn later to the complicated ones, Fluid Mechanics traditionally starts with hydrostatics. These are the usually desired results picturing the connection between pressure p, conservative external force field potential  and density .

12. The Engineering Inversion of Integrals

13. The Need of A Textile Industry : More Power from same Large Hydro-Turbine with same amount of Water Flow Rate

14. Sir James B. France When the city of Lowell became the first successful planned industrial city in America, it not only revolutionized manufacturing but also created a new way of life: A life ruled by time, a life powered by new opportunities. Perhaps one of the most enduring legacies left to the city of Lowell was that of James Bicheno Francis. Throughout Francis’s life his bold and innovative ideas made him one of the driving minds behind America’s Industrial Revolution. The Chief of Police of water ….

15. THE CHIEF OF POLICE OF WATER In 1841 the Proprietors of Locks and Canals appointed Francis the task of determining the quantities of water drawn by the mills along the canal. Much impressed with Francis’s work, the proprietors, in addition to Chief Engineer, bestowed upon him the title of Manager of Locks and Canals in 1845. The people of Lowell also gave Francis another title. They nicknamed him “The Chief of Police of Water” due to his strict and scientific management of Lowell’s waterways.

16. Same Resource but More Power By 1846 Francis developed a new power canal to provide water power to the mills. Eighteen months later in 1848 the Northern Canal completed the 5.6 miles of canals in Lowell. It was also the largest canal in the system measuring 100 feet in width, 1.5 miles long and ranging from 17 to 20 feet in depth. With the completion of the Northern Canal, water flow increased by 50 percent.