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Electromagnetic Fields in Transformers: Theory and Computations
Published in S.V. Kulkarni, S.A. Khaparde, Transformer Engineering, 2017
The curl is a measure of circulation of a vector field. The curl of a vector field is a vector; it may be interpreted as the circulation per unit area. Circulation is a concept from fluid dynamics, and it measures the ability of the field to rotate. Mathematically, it is determined as () ∇×A=|axayaz∂∂x∂∂y∂∂zAxAyAz|.
Coordinate Systems and Important Theorems
Published in Paolo Di Sia, Mathematics and Physics for Nanotechnology, 2019
A vector field a which has the above property (is path-independent) is called conservative. Therefore, the line integral above is zero along any close path and the curl of a conservative vector field is zero: () dφ=∇φ⋅dr→∇×(∇φ)=0.
Scalars and Vectors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
In the same fashion as the divergence describes the net flux through the surface of a small volume, the curl describes the circulation of a vector field around a small loop. The circulation integral for the small loop is proportional to the area of the surface spanned by the loop and to the scalar product between the surface normal and the curl, i.e., () dC=(∇×v→)⋅dS→.
Numerical simulation of geometry effect on mixing performance in L-shaped micromixers
Published in Chemical Engineering Communications, 2020
The curl of the velocity vector characterizes vorticity (that is double the rotation vector) and it represents the rotation of fluid elements (Munson et al., 1998). Vorticity can be used to show secondary flows at cross-sections affecting the mixing process. Figure 15 represents X-vorticity contours at A-A, B-B, C-C, and D-D planes in the serpentine microchannels at Re = 50. Two regions with peak vorticities in opposite directions are at yz planes in the L-shaped micromixer and near the upper wall of the A-A plane in the 90° V-shaped micromixer. By proceeding with the flow in the 90° V-shaped micromixer, an area with a peak value of X-vorticity is on the top middle of the cross-sections, and two regions with vorticities in the contrary direction are near the top of the side walls. X-vorticity distribution in the 60° V-shaped micromixer is approximately the same as that in the 90° V-shaped micromixer. The difference is the higher peak X-vorticity values in the 60° V-shaped micromixer that cover fewer proportions of the cross-sections.