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Stokes flow in a rectangular cavity by rotlet forcing
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10.1063/1.2742679
/content/aip/journal/pof2/19/8/10.1063/1.2742679
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/8/10.1063/1.2742679

Figures

Image of FIG. 1.
FIG. 1.

The flow field associated with the single rotlet is decomposed into two flow fields associated with two corotating and counter-rotating rotlets, respectively, described by the elemental stream functions and .

Image of FIG. 2.
FIG. 2.

Stream function contour plots of the Stokes flow due to a single rotlet, positioned in the origin of the domain, with (a) , (b) , (c) , (d) . Dashed contours represent negative values of the stream function, and solid contours represent positive values. The contour level increment is 0.15 for the dashed lines of the central cell; 0.005 for the solid lines in (b); 0.001 for the solid lines in frames (c) and (d); and for the dashed lines of the outer cells in frame (d). For the corner cells of frame (a), the contour increment is .

Image of FIG. 3.
FIG. 3.

Contour plot of the vorticity for aspect ratio (a) and (b). Dashed contours represents negative values of the vorticity, and solid contours represent positive values. The contour level increment is 0.1 for the dashed lines and 0.01 for the solid lines.

Image of FIG. 4.
FIG. 4.

The scatter-plot shows that there is no functional relation between and . (a) , (b) .

Image of FIG. 5.
FIG. 5.

Formation of the first secondary cell from the corner cells: (a) , ; (b) , ; (c) , ; (d) , ; (e) , ; (f) , ; (g) , . Here, denotes the contour increment of the corner cells. The dashed contours of the central cell have negative values; the increment is 0.01.

Image of FIG. 6.
FIG. 6.

Evolution of the position of the axial stagnation point during the formation process of the first secondary cell.

Image of FIG. 7.
FIG. 7.

From the left to the right: isolines of , , and for different rotlet positions and two values of the aspect ratio. Dashed contours represent negative values of the stream function, and solid contours represent positive values. For all plots, it holds that the contour level increment is 0.03 for the main cells and 0.001 for the secondary cells. The contour increment in the plots of the corner cells is (b) and (c). For the plots we used in addition for the corner cells (a), for the corner cells (b), and for the (bottom) corner cells and for the top cell (c).

Image of FIG. 8.
FIG. 8.

Time-dependent fluid forcing by two rotating rotlets (solid line) and (dashed line).

Image of FIG. 9.
FIG. 9.

Stirring of an initially circular blob (initial position indicated by the black circular area in the first plot) due to two blinking rotlets (indicated by dots) after several periods of time.

Image of FIG. 10.
FIG. 10.

Evolution of the distribution patterns of an initially circular blob in a rectangular domain with aspect ratio ; , , . The position of the rotlets is indicated by dots. For the blinking rotlet model, we used .

Image of FIG. 11.
FIG. 11.

Evolution of an initially circular blob containing passive tracer particles after several blinking periods (from left to right: , , , and ) for , , and different values of the phase difference . The position of the rotlets is indicated by dots.

Image of FIG. 12.
FIG. 12.

Evolution of an initially circular blob containing passive tracer particles after several periods (from left to right: , , ) for . Comparison between a square wave time-dependent and sine wave time-dependent forcing protocol. The parameter set , , and ensures “forcing area” preservation. The position of the rotlets is indicated by dots. The relative contour length reads , , and for the square wave; and , , for the sine wave.

Tables

Generic image for table
Table I.

Numerical values of the first eight free terms and expansion coefficients of problem 1 for the case in which , ,and . Any value of could be used for this purpose, provided . However, the particular choice of here is related with a few applications; see Sec. IV.

Generic image for table
Table II.

Numerical values of the first eight free terms and expansion coefficients of problem 1 for the case in which , , and . Once again, any value of could be used for this purpose, provided . However, the particular choice of here is related with a few applications; see Sec. IV.

Generic image for table
Table III.

Rotlet positions producing symmetrically located stagnation points for various values of the aspect ratio .

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/content/aip/journal/pof2/19/8/10.1063/1.2742679
2007-08-14
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Stokes flow in a rectangular cavity by rotlet forcing
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/8/10.1063/1.2742679
10.1063/1.2742679
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