# Linear stability of the flow of a second order fluid past a wedge

The linear stability analysis of Rivlin-Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called Order fluids, which is one of the first classes proposed to model departures from Newtonian behaviour. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterises the nonNewtonian response of the fluid. Stabilisation is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0, in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilises streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the nonmodal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth which is enhanced when K > 0 and diminished when K < 0.

The following files generate Figures 2-15 and Tables I-II in the paper:

‘meanFlow’ generates the mean flow profiles for a given value of γ and a chosen range of K, then saves the results to .csv files. The .csv files are then loaded and used by the following files.

'velProfiles' generates figures 2(a)-(f): Velocity profile and relative variation with respect to the Newtonian

profile for increasing and decreasing values of the parameter K; (a), (b) γ = 0 (flat plate); (c), (d) γ = 0.5

(flow past a wedge); (e), (f) γ = 1 (stagnation flow).

‘eigenvalues’ generates figures 3(a)-(b): Comparison between Newtonian and non-Newtonian eigenspectrum for the temporal problem for flow over a flat plate (γ = 0) with α* = 0.179, Re = 580 and (a) K = 0.03, (b) K = −0.03. The least damped eigenvalues are those in the grey circle.

‘growthRatesSpatial’ generates figure 4(b): Spatial growth rates for flow over a flat plate (γ = 0) and Re =

580. Newtonian and non-Newtonian cases with K = 0.01, −0.05.

‘growthRatesTemporal’ generates figure 4(a) and 5(a)-(d). Figure 4(a): Temporal growth rates for flow over a flat plate (γ = 0) and Re = 580, Newtonian and non-Newtonian cases with K = ±0.01. Figure 5: Temporal

growth rates for a flow past a wedge, Newtonian and non-Newtonian cases; (a) γ = 0.5, Re = 10000, K =

3e−04; (b) γ = 1 (stagnation point), Re = 27000, K =1e−04 ; (c) γ = 1.2, Re = 27000, K = 1e−04 ; (d) γ =

−0.14 (inflection point), Re = 300, K = 0.05.

‘neutralCurves’ generates figures 6(a)-(e): Temporal neutral curves in the Newtonian and non-Newtonian

cases. (a) γ = 0 (flat plate), K̃ = ±1e+03 ; (b) γ = 0.5, K̃ = ±1e+04 , x0 = 1; (c) γ = 1 (stagnation point), K̃ =±2.5e+04 , x0 = 1; (d) γ = 1.2, K̃ = ±5e+04 , x0 = 1; (e) γ = −0.14 (inflection point), K̃ = ±100; x0 = 1.

‘neutralCurves’ can also be used, with appropriate modifications, to calculate the critical Reynolds numbers of Table I.

‘stab3DNewt’ generates figures 7(a)-(b): Contour plot for the temporal growth rate, ωi, in the Newtonian

case (K = 0) for the flat plate (γ = 0). The red asterisk (∗) represents max ) represents max α,β ωi. The black line represents the neutral curve. (a) Re0 = 500; (b) Re0 = 1000.

‘stab3D’ generates figures 8-10. Figures 8(a),(b): Contour plots for ωi in the non-Newtonian cases for

the flat plate (γ = 0) and Re0 = 500. The red asterisk (∗) represents max ) represents maxα,β ωi. The black line represents the neutral curve. (a) K = −0.001; (b) K = 0.001. Figures 8(c),(d): Comparison of Newtonian (-) and non-Newtonian (- -) temporal growth rates for (c) α = 0.02; (d) β = 0.2.

Figures 9(a),(b): Contour plots for ωi in the non-Newtonian cases for the flat plate (γ = 0) and Re0 = 1000. The red asterisk (∗) represents max ) represents maxα,β ωi . The black line represents the neutral curve. (a) K = −0.0001; (b) K = 0.0001. Figures 9(c),(d): Comparison of Newtonian (-) and non-Newtonian (- -) temporal growth rates for (c) α = 0.02; (d) β = 0.2.

Figures 10(a),(b): Contour plots for ωi in the non-Newtonian cases for the flow past a corner (γ = −0.14) and Re0 = 150. The red asterisk (∗) represents max ) represents maxα,β ωi. The black line represents the neutral curve. (a) K = −0.003; (b) K = 0.003. Figures 10(c),(d): Comparison of Newtonian (-) and non-Newtonian (- -) temporal growth rates for (c) α = 0.02; (d) β = 0.2.

‘transientGrowthAlphaBeta’ generates figures 11-12. Figure 11: Contour plot of G max for γ = 0 (flat plate)

and Re0 = 1000. The black line indicates where an exponentially unstable mode exists. (a) K = 0; (b) K = 10−4 ; (c) K = −1e−04. Figure 12: Contour plot of G max for γ = 0.5 and Re0 = 500. (a) K = 0; (b) K = 1e−04; (c) K = −1e−04.

‘transientGrowthAlphaBeta’ can also be used, with appropriate modifications and change of scalings, to

generate Table II: Largest global optima for Reθ = 166 and Reθ = 385. The asterisk (∗) represents max ) indicates where an exponentially unstable mode exists and GΓ is calculated excluding the TS wave. The missing values indicate where an exponential unstable mode exists also as β→0.

‘transientGrowthAlphaWi’ generates figure 13(a)-(b): Ratio of non-Newtonian to Newtonian maximum

possible amplification for the flat plate γ = 0 and Re0 = 500, β = 0.6. (a) Gmax/Gmax,Newt ; (b) tmax/tmax,Newt .

‘optimalDistCompare’ generates figure 14(a)-(d): Comparison between Newtonian and non-Newtonian

optimal disturbances for γ = 0.5, Re0 = 500, α = 0.6, β = 0. (a) wall-normal, |v0|, and spanwise, |w0|, initial velocities; (b) wall-normal, |vmax|, and spanwise, |wmax|, velocities at t = tmax ; (c) streamwise, |u0|, initial velocities; (d) streamwise, |umax |, streamwise velocity at t = tmax .

‘optimalDist’ generates figure 15(a)-(d): Optimal disturbance for the stagnation point flow with γ = 1, Re0 = 500, α = 0.6, β = 0 and a non-Newtonian parameter K = 0.0001. (a),(b) disturbance at t = 0; (c),(d)

disturbance at t = tmax.

The figures can be obtained by specifying parameters (gamma – angle parameter, Re – Reynolds numberetc.) given in the captions of the figures.

Research results based upon these data are published at https://doi.org/10.1063/5.0018300