Rigorous theory for transient capillary imbibition in channels of arbitrary cross section

Bhattacharya, S.; Azese, Martin Ndi; Singha, Sagnik

doi:10.1007/s00162-016-0409-6

Cited by 7 publications

(16 citation statements)

References 38 publications

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“…As shown, these sections are connected through the continuity and momentum balance equation ( CD, DF, with EC being the inlet region). This system happens to be similar to that presented in previous papers by one of the authors (Bhattacharya et al 2017;Sumanasekara et al 2017), and as a result, some details of its descriptions are left out. However, we present a summary in the subsequent paragraphs.…”

Section: Description Of the Flow System And Governing Equationsmentioning

confidence: 70%

“…This leads us to consider the normal flow gradient as non-existent. As a consequence, the inlet flow is uniform and unidirectional (1-D flow), limited by the surface ABC as shown in figure 2, a consideration also followed in previous papers (Bhattacharya et al 2017;Sumanasekara et al 2017). We ultimately have P A ≈ P B ≈ P C ∼ P o = atmospheric pressure.…”

Section: Entry Regionmentioning

confidence: 76%

“…However, for both conduits, we chose to present only cases having pipetted depth ranging from h * 0 = 0.2 to h * 0 = 5, considered here as a moderate-to-high range, allowing us to also examine the effects of initial over filled. Worthy of note is that our choices within this range are also motivated by those used in Bhattacharya et al (2017) and Sumanasekara et al (2017). Using typical experimental values described in § 3.2 this range corresponds to h * 0 ∼ 3 mm − h * 0 ∼ 70 mm.…”

Section: Data and Results: Rectangular Ductmentioning

confidence: 99%

“…where v and p are respectively the dimensionless velocity and pressure within this region, and ∇ is the gradient vector non-dimensionalised with l c . In previous research (Bhattacharya et al 2017;Sumanasekara et al 2017), however, we showed that δ is the effective capillary number defined as…”

Section: Front Region: 3-d Flow H F << H(t)mentioning

confidence: 96%

“…Lucas 1918;Washburn 1921;Fries & Dreyer 2008), which is oftentimes valid only for certain conditions (see, e.g. Das et al 2012;Bhattacharya et al 2017) and even extends to some investigations that involve viscoelastic liquids (Sumanasekara et al 2017). Ultimately, for this heuristic approach, we avoid lamenting the specific forms in (2.6) and (2.5a-d) and instead surmised a time-monotonic dependence of γ (t) consequently resulting in the following two forms:…”

Section: Surface Tension Transfigurationmentioning

confidence: 99%

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Examining capillary dynamics in rectangular and circular conduits subject to unsteady surface tension

et al. 2022

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The unsteady effects of transitioning surface tension, $\gamma (t)$ , on the dynamics of capillary imbibition in channels of arbitrary shape are analytically investigated with a focus on rectangular and circular channels. With proper scaling, two unsteady models for $\gamma (t)$ are defined and used to highlight this transient behaviour. The convoluted dynamics at the flow front are correctly captured in the governing equations, which are rigorously analysed using unsteady eigenfunction expansion. Then, the final solution and data are obtained by employing the Runge–Kutta fourth-order scheme elegantly applied simultaneously to two derived nonlinear ordinary differential equations. Ultimately, the results are more accurate compared with previous studies. Dynamics and kinematic similarity between rectangular and circular channels are investigated and discussed and the conditions for equivalence in both channels are highlighted. Using a small parameter ( $\epsilon$ ) that stretches the time scale, we successfully use a robust asymptotic analysis to develop and capture the long-time dynamics. Ultimately, we recover the Lucas–Washburn regime analysed in Washburn (Phys. Rev., vol. 17, 1921, pp. 273–283), Lucas (Kolloidn. Z., vol. 23, 1918, pp. 15–22) for steady surface tension where the variations of depth and rate with time result in $h\thicksim t^{1/2}$ and $v \thicksim t^{-1/2}$ . In the end, the three forces, namely the inertia, $F_{v}$ , the viscous, $F_{\mu }$ , and the surface tension, $F_{\gamma }$ , are briefly analysed and used to highlight three main distinct regimes. We show that at early times, $F_{v}/F_{\gamma } \thicksim 1$ , whereas at a long time, $F_{\mu }/F_{\gamma } \thicksim -1$ .

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Section: Description Of the Flow System And Governing Equationsmentioning

confidence: 70%

Section: Entry Regionmentioning

confidence: 76%

Section: Data and Results: Rectangular Ductmentioning

confidence: 99%

Section: Front Region: 3-d Flow H F << H(t)mentioning

confidence: 96%

Section: Surface Tension Transfigurationmentioning

confidence: 99%

See 3 more Smart Citations

Examining capillary dynamics in rectangular and circular conduits subject to unsteady surface tension

et al. 2022

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Transient penetration of a viscoelastic fluid in a narrow capillary channel

2017

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This article describes an unexplored transport phenomenon where a mildly viscoelastic medium encroaches a narrow capillary channel under the action of surface-tension force. The ultimate goal of the study is to provide the penetration length and the intrusion rate of the liquid as functions of time. The resulting analysis would be instrumental in building an inexpensive and convenient rheometric device which can measure the temporal scale for viscoelastic relaxation from the stored data of the aforementioned quantities. The key step in the formulation is a transient eigenfunction expansion of the instantaneous velocity profile. The time-dependent amplitude of the expansion as well as the intruded length are governed by a system of integro-differential relations which are derived by exploiting the mass and momentum conservation principles. The obtained integro-differential equations are simultaneously solved by using a fourth-order Runge–Kutta method assuming a start-up problem from rest. The resulting numerical solution properly represents the predominantly one-dimensional flow which gradually slows down after an initial acceleration and subsequent oscillation. The computational findings are independently verified by two separate perturbation theories. The first of these is based on a Weissenberg number expansion revealing the departure in the unsteady imbibition due to small but finite viscoelasticity. In contrast, the second one explains the long-time behaviour of the system by analytically predicting the decay features of the dynamics. These asymptotic results unequivocally corroborate the simulation inferring the accuracy of the numerics as well as the utility of the simplified mathematical models.

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Measurement and characterization of slippage and slip-law using a rigorous analysis in dynamics of oscillating rheometer: Newtonian fluid

Azese

2018

Physics of Fluids

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This article presents a rigorous calculation involving velocity slip of Newtonian fluid where we analyze and solve the unsteady Navier-Stokes equation with emphasis on its rheological implication. The goal of which is to model a simple yet effective non-invasive way of quantifying and characterizing slippage. Indeed this contrasts with previous techniques that exhibit inherent limitations whereby injecting foreign objects usually alter the flow. This problem is built on the Couette rheological flow system such that μ-Newton force and μ-stress are captured and processed to obtain wall slip. Our model leads to a linear partial differential equation and upon enforcing linear-Navier slip boundary conditions (BC) yields inhomogeneous and unsteady “Robin-type” BC. A dimensional analysis reveals salient dimensionless parameters: Roshko, Strouhal, and Reynolds while highlighting slip-numbers from BC. We also solve the slip-free case to corroborate and validate our results. Several graphs are generated showing slip effects, particularly, studying how slip-numbers, a key input, differentiate themselves to the outputs. We also confirm this in a graphical fashion by presenting the flow profile across channel width, velocity, and stress at both walls. A perturbation scheme is introduced to calculate long-time behavior when the system seats for long. More importantly, in the end, we justify the existence of a reverse mechanism, where an inverse transformation like Fourier transform uses the output data to retrieve slip-numbers and slip law, thus quantifying and characterizing slip. Therefore, we not only substantiate our analysis, but we also justify our claim, measurement and characterization, and theorize realizability of our proposition.

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Rigorous theory for transient capillary imbibition in channels of arbitrary cross section

Cited by 7 publications

References 38 publications

Examining capillary dynamics in rectangular and circular conduits subject to unsteady surface tension

Examining capillary dynamics in rectangular and circular conduits subject to unsteady surface tension

Transient penetration of a viscoelastic fluid in a narrow capillary channel

Measurement and characterization of slippage and slip-law using a rigorous analysis in dynamics of oscillating rheometer: Newtonian fluid

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