Scalar-Vector Interaction in Animating but non-alive Fields …… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Special Vector Calculus.

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Presentation transcript:

Scalar-Vector Interaction in Animating but non-alive Fields …… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Special Vector Calculus for Fluid Flow Field

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:

Understanding of Material Derivative of A Scalar Field The operator D represents the substantial or material change of the quantity T(t:x,y,z). The first term on the right hand side of above equation represents the local or temporal change of the quantity T(t:x,y,z) with respect to a fixed position vector x. The operator d symbolizes the spatial or convective change of the same quantity with respect to a fixed instant of time. The convective change of T(t:x,y,z) may be expressed as:

Understanding of Material Derivative of A Vector Field  V as the gradient of the vector field which is a second order tensor.

Rate of Change of Material Derivative of A Vector Field Dividing above equation by dt yields the acceleration vector. The differential dt may symbolically be replaced by Dt indicating the material character of the derivatives. Material or substantial acceleration

Component of Material Acceleration

Visualization of Material Acceleration

Variety of Pumps due to Various Components of Material Acceleration

Turbo-Machines & GEOMETRIES

Variety of Compressors due to Various Components of Material Acceleration

Local Accounting of Accelerating vector field An essential responsibility of an engineer is to develop a measure to account for an accelerating flow field. A simplest but comprehensive measure of accounting is essential. What is it? Can we Define some measure? If A is thought to be a flux vector, a net flux out of the volume may be expressed as The is named as Divergence of a vector field: Divergence is an operator that measures the magnitude of a vector field's source or sink at a given point.

Divergence of Velocity in Various Coordinate Systems In different coordinate systems: Cartesian : Cylindrical: Spherical:

Divergence Rules Some “divergence rules”:

What is the Divergence Effect in Fluid Dynamics? Think of a vector field as a velocity field for a moving fluid. The divergence measures the expansion or contraction of the fluid. A vector field with constant positive or negative value of divergence.