^{1}, H. J. H. Clercx

^{1}, G. J. F. van Heijst

^{1}and V. V. Meleshko

^{2}

### Abstract

The Stokes flow inside a two-dimensional rectangular cavity , is analyzed for a highly viscous, incompressible fluid flow, driven by a single rotlet placed at position . Specifically, a rigorous solution of the governing two-dimensional biharmonic equation for the stream function is constructed analytically by means of the superposition principle. With this solution, multicellular flow patterns can be described for narrow cavities, in which the number of flow cells is directly related to the value of the aspect ratio . The solution also shows that for a certain rotlet position , which depends on and , the flow has a stagnation point symmetrically placed inside the rectangle. As the flow would not be affected by placing a second (inactive) rotlet in this stagnation point, this allows us to construct a blinking rotlet model for the rectangular cavity, with the inactive rotlet in the stagnation point of the flow induced by the active rotlet. For rectangular cavities, it holds that more than one of these special rotlet positions can be found for cavities that are elongated to sufficiently large aspect ratios. The blinking rotlet model is applied to illustrate several aspects of stirring in a Stokes flow in a rectangular domain.

We would like to thank Dr. Alexei Galaktionov for providing software to perform contour kinematics simulations.

I. INTRODUCTION

II. STOKES FLOW BY A SINGLE ROTLET

A. Problem 1

B. Problem 2

III. MULTICELLULAR FLOW PATTERNS IN A SLENDER CAVITY

IV. CHAOTIC TRANSPORT IN A BOUNDED STOKES FLOW

V. SUMMARY AND DISCUSSION

### Key Topics

- Stokes flows
- 60.0
- Viscosity
- 14.0
- Boundary value problems
- 12.0
- Rotating flows
- 8.0
- Time dependent material structure
- 7.0

## Figures

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 .

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 .

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 .

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 .

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.

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.

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

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

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.

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.

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

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

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).

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).

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

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

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.

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.

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 .

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 .

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.

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.

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.

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

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.

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.

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.

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.

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

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

Article metrics loading...

Full text loading...

Commenting has been disabled for this content