Classification of Fluid Flows using Vector Calculus

1. Classification of Fluid Flows using Vector Calculus P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Recognition of Speical Flow Fields……

2. Characterization of Flow Fields based Velocity Field • Diverging Flows : • Converging Flows : • Solenoidal Flows :

3. Flow field with Convergence / divergence • Atmospheric vertical motion is caused mainly by air mass convergence or divergence from horizontal motion. • Meteorological convergence indicates retardation in air flow with an increase in air mass in a given volume due to net three-dimensional inflow. • Meteorological divergence, or negative convergence, indicates acceleration with a decrease in air mass. Convergence is the contraction and divergence is the spreading of a field of flow.

4. The Influence of Divergence/Convergence

5. Characterization of Flow Fields based Velocity Field • Cyclonic Flows: • Anti Cyclonic Flows:

6. Irrotational Flow Field • Flows with vorticity are said to be rotational flows. • Flows without 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, .

7. Irrotational Solenoidal Flow Field • Irrotational flow • Solenoidal flow • Irrotational, Solenoidal flow • Laplacian flow

8. Electroosmotic Flows • When electric fields are applied across capillaries or microchannels, bulk fluid motion is observed. • The velocity of this motion is linearly proportional to the applied electric field, and dependent on both • (a) the material used to construct the microchannel and • (b) the solution in contact with the channel wall. • This motion is referred to aselectroosmosis.

9. Complex Lamellar Flow • A complex-lamellar velocity field is defined as a flow field whose streamlines are intersect orthogonally to vortex lines. Complex-lamellar flows are characterized by the fact that there exists a family of surfaces orthogonal to the stream lines. Irrotational flow field is a subset of complex-lamellar flows. This is a special class of flow fields, where streamlines are steady in a accelerating flow. The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The lamellae to which "lamellar flow" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field.

10. Creation of Uniform Flow Through A Diverging Duct

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12. BELTRAMI FLOW FIELDS • A three-dimensional vector field V is called as Beltrami if • A Beltrami flow field is characterized by the fact that the vorticity vector is collinear with velocity through out the flow field. • Beltrami fields are known as Magnus force-free fields. • The ”surface” of the sun (called the photosphere) shows Beltrami behaviour. • Reconstruction of the three-dimensional field above the photosphere is possible using Beltrami model.

13. Classification of Approximate Nature of Beltrami Field • From a mathematical viewpoint, above equation can be roughly rewritten in the form where λ(x) is a scalar function which varies in general with position λ ≡ 0 is an irrotational field, also known as potential field. Linear Beltrami fields: they are characterized by a constant λ. Non-linear Beltrami fields: they corresponds to a variable function λ(.) This can be considered to be a prototype for the study of turbulence

14. Mathematically Understood Flows