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

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

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?

The Clue from Nature for Gradient

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.

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 .

Vector Calculus to Describe Characteristics of Fluid Mechanics

Human Capability to Imagine Geometry

Coordinate systems: Cylindrical (polar) An intersection of a cylinder and 2 planes An arbitrary vector: Diff. area: Diff. volume: Diff. length: r r

Coordinate systems: Spherical An intersection of a sphere of radius r A plane that makes an angle  to the x axis, A cone that makes an angle  to the z axis.

Properties of Coordinate systems: Spherical Properties: Diff. length: Diff. area: Diff. volume: An arbitrary vector:

System conversions 1. Cartesian to Cylindrical: 2. Cartesian to Spherical: 3. Cylindrical to Cartesian: 4. Spherical to Cartesian:

Gradient in Arbitrary Coordinate System Gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. Gradient of a scalar function:

Fluid Mechanical Significance of Gradient of A Scalar

Gradient of a Vector

System conversions for Gradient Cartesian : Cylindrical: Spherical: Gradient in different coordinate systems:

Properties of Gradient Operations Collect more of such relations, relevant to Thermo-Fluid Sciences.

Differential Operators in Fluid Mechanics In fluid mechanics, the particles of the working medium undergo a time dependent or unsteady motion. The flow quantities such as the velocity V and the thermodynamic properties of the working substance such as pressure p, temperature T, density  or any arbitrary flow quantity Q are generally functions of space and time. During the flow process, these quantities generally change with respect to time and space.

Material Derivatives A fluid element, often called a material element. Fluid elements are small blobs of fluid that always contain the same material. They are deformed as they move but they are not broken up. The temporal and spatial change of the flow/fluid quantities is described most appropriately by the substantial or material derivative. Generally, the substantial derivative of a flow quantity, which may be a scalar, a vector or a tensor valued function, is given by: