Vector Calculus for Measurements in Thermofluids P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Mathematical Description of Simple Flows….

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

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 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:

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

Engineering Use of Generalized Gradient Arbitrary Coordinate System Gradient of a scalar function: 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.

Fluid Mechanical Significance of Gradient of A Scalar

Gradient of a Vector

Divergence of a vector field Divergence is the outflow of flux from a small closed surface area (per unit volume) as volume shrinks to zero. Divergence of a vector field: In an orbitrary coordinate systems: In Cartesian coordinate system:

Definition of Divergence in fundamental CoS In Cartesian coordinate system: In Cylindrical coordinate system: In Spherical coordinate system:

Vector Calculus for Further Study of a River

What is divergence : Application to Rivers / Natural Reservoirs Think of a vector field as a velocity field for a moving fluid. The divergence measures sources and drains of flow: The divergence measures the expansion or contraction of the fluid. A vector field with constant positive or negative value of divergence. Presence of A source Presence of A Drain

What is divergence : Application to Turbo-machines Think of a vector field as a velocity field for a moving compressible fluid. The divergence measures the expansion or contraction of the fluid.

Common Knowledge about Flow Rate of a River The flow is higher close to the center and slower at the edges. V Let us insert small paddle wheels in a flowing river. A wheel close to the center (of a river) will not rotate since velocity of water is the same on both sides of the wheel. Wheels close to the edges will rotate due to difference in velocities. The curl operation determines the direction and the magnitude of rotation.

Flow Measurement in a River

Curl of a vector field The curl of vector field at a point in a medium is a measure of the net rotation of the vector as the area shrinks to zero. Curl of a vector field: If (in a Cartesian system) the vector is defined as

Thermofluid Significance of Curl Circulation is the amount of force that pushes along a closed boundary or path. It's the total "push" you get when going along a path, such as a circle. Curl is simply circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point) Curl is a vector field with magnitude equal to the maximum "circulation" at each point and oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of curl is the limiting value of circulation per unit area.

The Curl of Velocity Field Define the vorticity vector as being the curl of the velocity vorticity vector in cylindrical co-ordinates: vorticity vector in spherical co-ordinates:

Irrotational Flow Field Flows with non zero vorticity are said to be rotational flows. Flows with zero vorticity are said to be irrotational flows. If the velocity is exactly equal to gradient of a scalar, the flow filed is obviously irrotational. If an application calls for an irrotational flow, the problem is completely solved by finding a scalar field, .

Irrotational and Solenoid Fields Pure Irrotational Flow with non zero Divergence Pure Solenoidal Flow with non zero Rotation.