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Linear instability of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid
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10.1063/1.2814385
/content/aip/journal/pof2/19/12/10.1063/1.2814385
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/12/10.1063/1.2814385

Figures

Image of FIG. 1.
FIG. 1.

Schematic of a two-layer flow in a channel of height , where represents the thickness of the lower, non-Newtonian fluid. Also shown here are profiles of the steady, streamwise velocity component generated with and .

Image of FIG. 2.
FIG. 2.

The effect of increasing the order of Chebyshev polynomials, , on the variation of with with , , , , , , , and .

Image of FIG. 3.
FIG. 3.

Eigenvalue distribution generated for the Newtonian case of South and Hooper (Ref. 23) with , , , , , , and .

Image of FIG. 4.
FIG. 4.

Comparison of the numerical solution (solid lines) with the predictions of the long-wave analysis (dotted lines) in terms of the variation of the complex part of the phase speed, , with the wavenumber, . (a) , , , ; (b) , , , . The rest of the parameters are , , and .

Image of FIG. 5.
FIG. 5.

The effect of varying the Bingham number, , on the dispersion curves; (a) and (b) . The rest of the parameters are , , , , , and . The labels A, B, C, D, and E are used to designate maxima in the dispersion curves; the energy “budgets” associated with these points are provided in Table I. The dotted line in (a) shows the dispersion curve obtained by linearizing about the base state for and setting in Eqs. (11) and (12).

Image of FIG. 6.
FIG. 6.

The effect of varying the flow index, , on the dispersion curves; (a) and (b) . Here, and the rest of the parameters are the same as in Fig. 5. The dotted line in (a) shows the dispersion curve obtained by linearizing about the base state for and setting and in Eqs. (11) and (12).

Image of FIG. 7.
FIG. 7.

Effect of varying the density ratio on the dispersion curves for small, (a), and large, (b), . The rest of the parameters are , , , , , , and . The labels A, B, and C are used to designate global maxima in the dispersion curves associated with , respectively; the energy “budgets” associated with these points are provided in Table II.

Image of FIG. 8.
FIG. 8.

Effect of varying the thickness ratio , (a), and viscosity ratio , (b), on the growth rate associated with the most dangerous mode, in the presence and absence of a yield stress. The rest of the parameters are , , , , , , and .

Image of FIG. 9.
FIG. 9.

Effect of varying the Reynolds number, , on the dispersion curves generated by plotting the imaginary parts of the two leading eigenvalues against with (dashed lines) and (solid lines). The rest of the parameters are , , , , , and .

Image of FIG. 10.
FIG. 10.

The effect of varying on the neutral stability curves of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and .

Image of FIG. 11.
FIG. 11.

Effect of varying on the dispersion curves with ; (a) and (b) . The rest of the parameters are the same as in Fig. 10. The curves associated with the “shear” modes are shown in the insets of panels (a) and (b).

Image of FIG. 12.
FIG. 12.

The cross-stream structure of the real and imaginary parts of associated with the most dangerous “interfacial” and “shear” modes in Fig. 11(a) shown in (a) and (b), respectively.

Image of FIG. 13.
FIG. 13.

The effect of varying the flow index, , on the neutral stability curve of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and . The dotted line in (a) was obtained by linearizing about a base state with and and by setting and in Eqs. (11) and (12).

Image of FIG. 14.
FIG. 14.

The effect of varying on the dispersion curves with ; (a) and (b) . The rest of the parameters are the same as Fig. 13.

Image of FIG. 15.
FIG. 15.

The effect of varying on the neutral stability curves of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and .

Image of FIG. 16.
FIG. 16.

The effect of varying on the neutral stability curves of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and .

Tables

Generic image for table
Table I.

Energy “budgets” for the points labelled A, B, C, D, and E in Fig. 5.

Generic image for table
Table II.

Energy “budgets” for the points labelled A, B, and C in Fig. 7.

Generic image for table
Table III.

Energy “budgets” for , and the parameter values used to generate Fig. 9.

Generic image for table
Table IV.

Energy budgets for the points labelled A–D in Fig. 11.

Generic image for table
Table V.

Energy budgets for the points labelled A–D in Fig. 14.

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/content/aip/journal/pof2/19/12/10.1063/1.2814385
2007-12-06
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Linear instability of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/12/10.1063/1.2814385
10.1063/1.2814385
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