# Weakly nonlinear theory of shear-banding instability in a granular plane Couette flow: analytical solution, comparison with numerics and bifurcation

##### Citations

26 citations

### Cites background from "Weakly nonlinear theory of shear-ba..."

...…constitutive relations for the stress tensor (§ 4.2) and the heat flux (§ 6) along with extended hydrodynamic equations (2.11)–(2.13) can also be tested in dynamic simulations of granular flows, including the stability analyses of shear flows (Gayen & Alam 2006; Shukla & Alam 2009, 2011a,b)....

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19 citations

### Cites background from "Weakly nonlinear theory of shear-ba..."

...As has been discussed in Drazin & Reid (2004) and Shukla & Alam (2011), when αci and (a1)r are of the opposite sign, the existence of a finite amplitude solution is guaranteed....

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18 citations

### Cites background or methods or result from "Weakly nonlinear theory of shear-ba..."

...They employed both the direct method of centre manifold reduction (Shukla & Alam 2009) and the indirect method of amplitude expansion (Shukla & Alam 2011), and showed the equivalence between these two methods that result in the same expression for the first Landau coefficient, which is the…...

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...In addition to the well-known shear-banding instability (kx = 0) whose nonlinear saturation has been studied recently by us (Shukla & Alam 2009, 2011), there are long-wave (kx ∼ 0) stationary and travelling instabilities in a granular plane Couette flow....

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...Starting with granular hydrodynamic equations, Shukla & Alam (2009, 2011) derived the Landau–Stuart equation for the shear-banding instabilities (for the first time in granular flows) via weakly nonlinear analyses....

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...In our previous papers (Shukla & Alam 2009, 2011), we considered one-dimensional streamwise-independent equations which can be obtained by putting ∂/∂x(·) = 0 in (2.11)–(2.14) since our focus was on the shear-banding instability, for which the associated patterns have no variations along the…...

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...For the special case of shear-banding modes (kx = 0), it has been shown (Shukla & Alam 2011) that the second harmonic X[2;2] is real and X[2;2] = X̃[2;2] = X[0;2]....

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18 citations

12 citations

### Cites methods from "Weakly nonlinear theory of shear-ba..."

...a physical boundary condition starting from a set of granular hydrodynamic equations [54–56] by the method of Reynolds and Potter [57]....

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##### References

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17,845 citations

### "Weakly nonlinear theory of shear-ba..." refers methods in this paper

...We employed the method of singular value decomposition (Press et al. 1992) for solving AX = b system in each case of (4.25), (4.26) and (4.31)....

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...We employed the method of singular value decomposition (Press et al. 1992) for solving AX = b system in each case of (4....

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6,145 citations

### "Weakly nonlinear theory of shear-ba..." refers methods in this paper

...The Ginzburg–Landau-type order-parameter equations have been widely used to study the bifurcation scenario, nonlinear waves and patterns in fluid mechanics (Newell, Passot & Lega 1993; Manneville 1990; Cross & Hohenberg 1993)....

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...…complex Ginzburg–Landau equation has been used to obtain a qualitative (and sometimes quantitative) understanding of a host of phenomena, namely phase transitions, superconductivity, superfluidity, liquid crystals, vortex glass and defect turbulence (Cross & Hohenberg 1993; Aranson & Kramer 2002)....

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5,427 citations

### "Weakly nonlinear theory of shear-ba..." refers background or methods in this paper

...1993) by calculating the first 10 or more Landau coefficients, and then finding the nearest singularity from Domb–Sykes plots (Hinch 1991) to estimate the radius of convergence....

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...Such studies have been carried out for incompressible shear flows of Newtonian fluids (Herbert 1980; Newell et al. 1993) by calculating the first 10 or more Landau coefficients, and then finding the nearest singularity from Domb–Sykes plots (Hinch 1991) to estimate the radius of convergence....

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4,632 citations

3,753 citations