3 results
On the instability of buoyancy-driven flows in porous media
- Shyam Sunder Gopalakrishnan
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- Journal:
- Journal of Fluid Mechanics / Volume 892 / 10 June 2020
- Published online by Cambridge University Press:
- 03 April 2020, A13
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The interface between two miscible solutions in porous media and Hele-Shaw cells (two glass plates separated by a thin gap) in a gravity field can destabilise due to buoyancy-driven and double-diffusive effects. In this paper the conditions for instability to arise are presented within an analytical framework by considering the eigenvalue problem based on the tools used extensively by Chandrasekhar. The model considered here is Darcy’s law coupled to evolution equations for the concentrations of different solutes. We have shown that, when there is an interval in the spatial domain where the first derivative of the base-state density profile is negative, the flows are unstable to stationary or oscillatory modes. Whereas for base-state density profiles that are strictly monotonically increasing downwards such that the first derivative of the base-state density profile is positive throughout the domain (for instance, when a lighter solution containing a species A overlies a denser solution containing another species B), a necessary and sufficient condition for instability is the presence of a point on either side of the initial interface where the second derivative of the base-state density profile is zero such that it changes sign. In such regimes the instability arises as non-oscillatory modes (real eigenvalues). The neutral stability curve, which delimits the stable from the unstable regime, that follows from the discussion presented here along with the other results are in agreement with earlier observations made using numerical computations. The analytical approach adopted in this work could be extended to other instabilities arising in porous media.
Dynamics of pulsatile flow through model abdominal aortic aneurysms
- Shyam Sunder Gopalakrishnan, Benoît Pier, Arie Biesheuvel
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- Journal:
- Journal of Fluid Mechanics / Volume 758 / 10 November 2014
- Published online by Cambridge University Press:
- 07 October 2014, pp. 150-179
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To contribute to the understanding of flow phenomena in abdominal aortic aneurysms, numerical computations of pulsatile flows through aneurysm models and a stability analysis of these flows were carried out. The volume flow rate waveforms into the aneurysms were based on measurements of these waveforms, under rest and exercise conditions, of patients suffering abdominal aortic aneurysms. The Reynolds number and Womersley number, the dimensionless quantities that characterize the flow, were varied within the physiologically relevant range, and the two geometric quantities that characterize the model aneurysm were varied to assess the influence of the length and maximal diameter of an aneurysm on the details of the flow. The computed flow phenomena and the induced wall shear stress distributions agree well with what was found in PIV measurements by Salsac et al. (J. Fluid Mech., vol. 560, 2006, pp. 19–51). The results suggest that long aneurysms are less pathological than short ones, and that patients with an abdominal aortic aneurysm are better to avoid physical exercise. The pulsatile flows were found to be unstable to three-dimensional disturbances if the aneurysm was sufficiently localized or had a sufficiently large maximal diameter, even for flow conditions during rest. The abdominal aortic aneurysm can be viewed as acting like a ‘wavemaker’ that induces disturbed flow conditions in healthy segments of the arterial system far downstream of the aneurysm; this may be related to the fact that one-fifth of the larger abdominal aortic aneurysms are found to extend into the common iliac arteries. Finally, we report a remarkable sensitivity of the wall shear stress distribution and the growth rate of three-dimensional disturbances to small details of the aneurysm geometry near the proximal end. These findings suggest that a sensitivity analysis is appropriate when a patient-specific computational study is carried out to obtain a quantitative description of the wall shear stress distribution.
Global stability analysis of flow through a fusiform aneurysm: steady flows
- Shyam Sunder Gopalakrishnan, Benoît Pier, Arie Biesheuvel
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- Journal:
- Journal of Fluid Mechanics / Volume 752 / 10 August 2014
- Published online by Cambridge University Press:
- 02 July 2014, pp. 90-106
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The global linear stability of steady axisymmetric flow through a model fusiform aneurysm is studied numerically. The aneurysm is modelled as a Gaussian-shaped inflation on a vessel of circular cross-section. The fluid is assumed to be Newtonian, and the flow far upstream and downstream of the inflation is a Hagen–Poiseuille flow. The model aneurysm is characterized by a maximum height $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ and width $W$, non-dimensionalized by the upstream vessel diameter, and the steady flow is characterized by the Reynolds number of the upstream flow. The base flow through the model aneurysms is determined for non-dimensional heights and widths in the physiologically relevant ranges $0.1 \leq H \leq 1.0$ and $0.25 \leq W \leq 2.0$, and for Reynolds numbers up to 7000, corresponding to peak values recorded during pulsatile flows under physiological conditions. It is found that the base flow consists of a core of relatively fast-moving fluid, surrounded by a slowly recirculating fluid that fills the inflation; for larger values of the ratio $H/W$, a secondary recirculation region is observed. The wall shear stress (WSS) in the inflation is vanishingly small compared to the WSS in the straight vessels. The global linear stability of the base flows is analysed by determining the eigenfrequencies of a modal representation of small-amplitude perturbations and by looking at the energy transfer between the base flow and the perturbations. Relatively shallow aneurysms (of relatively large width) become unstable by the lift-up mechanism and have a perturbation flow which is characterized by stationary, growing modes. More localized aneurysms (with relatively small width) become unstable at larger Reynolds numbers, presumably by an elliptic instability mechanism; in this case the perturbation flow is characterized by oscillatory modes.