Finite difference analysis of viscous laminar converging flow in conical tubes

Sutterby, J. L.

doi:10.1007/bf00411560

Cited by 12 publications

(9 citation statements)

References 3 publications

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“…This was done in two ways-the first is using the ANSYS-FLUENT software, and PHYSICAL REVIEW E 91. 043001 (2015) the second is using the numerical method of Sutterby [25]; the results Irom the two procedures are in agreement. The inertial terms are included in the flow computation due to the variation in the wall diameter.…”

Section: Introductionmentioning

confidence: 76%

“…The first was by using a n s y sf l u e n t 13.0.0 in order to determine the modification of the flow and pressure profiles due to tube deformation. The second was by using the finite-difference formulation of Sutterby [25] for axisymmetric flows. It was verified that there is good agreement between the predictions of the two simulations for the developing flow in a tube starting with a plug flow at the inlet, and the maximum difference in the velocity is less than 1 % for the grid resolutions used here.…”

Section: A Base-state Fluid Flowmentioning

confidence: 99%

“…The laminar velocity profiles are reconstructed using two techniques, the ANSYS-FLUENT 13.0.0 software and the finite-difference formulation ofSutterby [25] for axisymmetric flows. The results of the two were found to be in agreement to within 1 %, thus verifying the consistency of the flow profiles obtained by the two methods.…”

Section: Introductionmentioning

confidence: 99%

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Stability of the flow in a soft tube deformed due to an applied pressure gradient

Verma

Kumaran

2015

Phys. Rev. E

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A linear stability analysis is carried out for the flow through a tube with a soft wall in order to resolve the discrepancy of a factor of 10 for the transition Reynolds number between theoretical predictions in a cylindrical tube and the experiments of Verma and Kumaran [J. Fluid Mech. 705, 322 (2012)]. Here the effect of tube deformation (due to the applied pressure difference) on the mean velocity profile and pressure gradient is incorporated in the stability analysis. The tube geometry and dimensions are reconstructed from experimental images, where it is found that there is an expansion and then a contraction of the tube in the streamwise direction. The mean velocity profiles at different downstream locations and the pressure gradient, determined using computational fluid dynamics, are found to be substantially modified by the tube deformation. The velocity profiles are then used in a linear stability analysis, where the growth rates of perturbations are calculated for the flow through a tube with the wall modeled as a neo-Hookean elastic solid. The linear stability analysis is carried out for the mean velocity profiles at different downstream locations using the parallel flow approximation. The analysis indicates that the flow first becomes unstable in the downstream converging section of the tube where the flow profile is more pluglike when compared to the parabolic flow in a cylindrical tube. The flow is stable in the upstream diverging section where the deformation is maximum. The prediction for the transition Reynolds number is in good agreement with experiments, indicating that the downstream tube convergence and the consequent modification in the mean velocity profile and pressure gradient could reduce the transition Reynolds number by an order of magnitude.

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Section: Introductionmentioning

confidence: 76%

Section: A Base-state Fluid Flowmentioning

confidence: 99%

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Stability of the flow in a soft tube deformed due to an applied pressure gradient

Verma

Kumaran

2015

Phys. Rev. E

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“…The transition Reynolds number in Shankar & Kumaran (2000, 2001a was calculated for a parabolic velocity profile. However, the flow in a converging tube is much flatter than the parabolic profile (Sutterby 1965;Shankar & Kumaran 1999), and the velocity gradient at the wall is higher than the value of (2U max /R) for a parabolic flow, where U max is the maximum velocity and R is the tube radius. If α is the slope of the tube wall, there is a significant departure from a parabolic profile when Re α > 1, even when the slope α is small.…”

Section: Discussionmentioning

confidence: 94%

A dynamical instability due to fluid–wall coupling lowers the transition Reynolds number in the flow through a flexible tube

Verma

Kumaran

2011

J. Fluid Mech.

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A flow-induced instability in a tube with flexible walls is studied experimentally. Tubes of diameter 0.8 and 1.2 mm are cast in polydimethylsiloxane (PDMS) polymer gels, and the catalyst concentration in these gels is varied to obtain shear modulus in the range 17-550 kPa. A pressure drop between the inlet and outlet of the tube is used to drive fluid flow, and the friction factor f is measured as a function of the Reynolds number Re. From these measurements, it is found that the laminar flow becomes unstable, and there is a transition to a more complicated flow profile, for Reynolds numbers as low as 500 for the softest gels used here. The nature of the f -Re curves is also qualitatively different from that in the flow past rigid tubes; in contrast to the discontinuous increase in the friction factor at transition in a rigid tube, it is found that there is a continuous increase in the friction factor from the laminar value of 16/Re in a flexible tube. The onset of transition is also detected by a dye-stream method, where a stream of dye is injected into the centre of the tube. It is found that there is a continuous increase of the amplitude of perturbations at the onset of transition in a flexible tube, in contrast to the abrupt disruption of the dye stream at transition in a rigid tube. There are oscillations in the wall of the tube at the onset of transition, which is detected from the laser scattering off the walls of the tube. This indicates that the coupling between the fluid stresses and the elastic stresses in the wall results in an instability of the laminar flow.

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“…It was reported that the drag force in a flexible-walled tube is higher than that for a laminar flow for Reynolds numbers as low as 700. However, there has been a subsequent study by Yang et al (2000), using a simulation method developed by Sutterby (1965), which suggested that the higher drag force in Krindel & Silberberg (1979) could be due to a slow convergence of streamlines, rather than a dynamical instability. The low-Reynolds-number instability in the flow past a soft polyacrylamide gel was demonstrated by Kumaran & Muralikrishnan (2000) and Muralikrishnan & Kumaran (2002), and hysteresis and oscillations were also reported by Eggert & Kumar (2004).…”

Section: Introductionmentioning

confidence: 96%

A multifold reduction in the transition Reynolds number, and ultra-fast mixing, in a micro-channel due to a dynamical instability induced by a soft wall

2013

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A dynamical instability is observed in experimental studies on micro-channels of rectangular cross-section with smallest dimension 100 and 160 µm in which one of the walls is made of soft gel. There is a spontaneous transition from an ordered, laminar flow to a chaotic and highly mixed flow state when the Reynolds number increases beyond a critical value. The critical Reynolds number, which decreases as the elasticity modulus of the soft wall is reduced, is as low as 200 for the softest wall used here (in contrast to 1200 for a rigid-walled channel). The instability onset is observed by the breakup of a dye-stream introduced in the centre of the micro-channel, as well as the onset of wall oscillations due to laser scattering from fluorescent beads embedded in the wall of the channel. The mixing time across a channel of width 1.5 mm, measured by dye-stream and outlet conductance experiments, is smaller by a factor of 10 5 than that for a laminar flow. The increased mixing rate comes at very little cost, because the pressure drop (energy requirement to drive the flow) increases continuously and modestly at transition. The deformed shape is reconstructed numerically, and computational fluid dynamics (CFD) simulations are carried out to obtain the pressure gradient and the velocity fields for different flow rates. The pressure difference across the channel predicted by simulations is in agreement with the experiments (within experimental errors) for flow rates where the dye stream is laminar, but the experimental pressure difference is higher than the simulation prediction after dye-stream breakup. A linear stability analysis is carried out using the parallel-flow approximation, in which the wall is modelled as a neo-Hookean elastic solid, and the simulation results for the mean velocity and pressure gradient from the CFD simulations are used as inputs. The stability analysis accurately predicts the Reynolds number (based on flow rate) at which an instability is observed in the dye stream, and it also predicts that the instability first takes place at the downstream converging section of the channel, and not at the upstream diverging section. The stability analysis also indicates that the destabilization is due to the modification of the flow and the local pressure gradient due to the wall deformation; if we assume a parabolic velocity profile with the pressure gradient given by the plane Poiseuille law, the flow is always found to be stable.

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Finite difference analysis of viscous laminar converging flow in conical tubes

Cited by 12 publications

References 3 publications

Stability of the flow in a soft tube deformed due to an applied pressure gradient

Stability of the flow in a soft tube deformed due to an applied pressure gradient

A dynamical instability due to fluid–wall coupling lowers the transition Reynolds number in the flow through a flexible tube

A multifold reduction in the transition Reynolds number, and ultra-fast mixing, in a micro-channel due to a dynamical instability induced by a soft wall

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