One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary

Cavaccini, Giovanni; Pianese, Vittoria; Jannelli, Alessandra; Iacono, Salvatore; Fazio, Riccardo

doi:10.3233/jcm-2009-0252

Cited by 4 publications

(14 citation statements)

References 31 publications

(36 reference statements)

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“…The case λ = 1 was already treated in . We remark that our obtained result, that is, ℓ max ≈ 1.4 % , is thinly different from the one of Deutsch in Equation .…”

Section: Numerical Resultsmentioning

confidence: 54%

“…A formal derivation, for an open‐end capillary and two different liquids, from first principles, of a mathematical model describing the capillary dynamics can be found by the interested reader in . In particular, for a horizontal closed‐end capillary, the Newtonian equation of motion plus natural initial conditions is given by

({, \begin{matrix} falsenonefalsearray \\ arrayaxis ρ MathClass-open(ℓ + cR MathClass-close) \frac{d^{2} ℓ}{d t^{2}} + ρ {(\frac{dℓ}{dt})}^{2} = 2 \frac{γ \cos ϑ}{R} - 8 \frac{μℓ}{R^{2}} \frac{dℓ}{dt} + Ω MathClass-open(ℓ, L MathClass-close), \\ arrayaxis ℓ MathClass-open(0 MathClass-close) = \frac{dℓ}{dt} MathClass-open(0 MathClass-close) = 0, \end{matrix})

where ℓ and L are, respectively, the length of the liquid inside the capillary and the capillary length, whereas the remaining parameters assumed constant are ρ the density of the liquid, γ the surface tension, ϑ the contact angle, μ the viscosity, R the capillary radius of the capillary and c = O (1) the coefficient of apparent mass, introduced by Szekely et al in order to obtain a well‐posed problem .…”

Section: Mathematical Modellingmentioning

confidence: 99%

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An analytical and numerical study of liquid dynamics in a one‐dimensional capillary under entrapped gas action

Fazio

Iacono

2013

Math Methods in App Sciences

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Capillary dynamics has been and is yet an important field of research, because of its very relevant role played as the core mechanism at the base of many applications. In this context, we are particularly interested in the liquid penetration inspection technique. Due to the obviously needed level of reliability involved with such a non-destructive test, this paper is devoted to study how the presence of an entrapped gas in a close-end capillary may affect the inspection outcome. This study is carried out through a 1D ordinary differential model that despite its simplicity is able to point out quite well the capillary dynamics under the effect of an entrapped gas. The paper is divided into two main parts; the first starts from an introductory historical review of capillary flows modeling, goes on presenting the 1D second order * Corresponding author e-mail: rfazio@dipmat.unime.it 1 ordinary differential model, taking into account the presence of an entrapped gas and therefore ends by showing some numerical simulation results. The second part is devoted to the analytical study of the model by separating the initial transitory behavior from the stationary one. Besides, these solutions are compared with the numerical ones and finally an expression is deduced for the threshold radius switching from a fully damped transitory to an oscillatory one.

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“…The case λ = 1 was already treated in . We remark that our obtained result, that is, ℓ max ≈ 1.4 % , is thinly different from the one of Deutsch in Equation .…”

Section: Numerical Resultsmentioning

confidence: 54%

({, \begin{matrix} falsenonefalsearray \\ arrayaxis ρ MathClass-open(ℓ + cR MathClass-close) \frac{d^{2} ℓ}{d t^{2}} + ρ {(\frac{dℓ}{dt})}^{2} = 2 \frac{γ \cos ϑ}{R} - 8 \frac{μℓ}{R^{2}} \frac{dℓ}{dt} + Ω MathClass-open(ℓ, L MathClass-close), \\ arrayaxis ℓ MathClass-open(0 MathClass-close) = \frac{dℓ}{dt} MathClass-open(0 MathClass-close) = 0, \end{matrix})

Section: Mathematical Modellingmentioning

confidence: 99%

An analytical and numerical study of liquid dynamics in a one‐dimensional capillary under entrapped gas action

Fazio

Iacono

2013

Math Methods in App Sciences

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“…The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: the smaller is the capillary radius, the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence, the numerical solution to a prescribed accuracy becomes harder to achieve, see Cavaccini et al . . The model governing the dynamics of a liquid inside an open ended capillary is given by

\begin{matrix} eqnarrayright center left \\ eqnarray-1 \{\begin{matrix} ρ \frac{d}{dt} ([], (ℓ MathClass-bin+ cR) \frac{dℓ}{dt}) MathClass-rel= 2 \frac{γ normalcos ϑ}{R} MathClass-bin- 8 \frac{μℓ}{R^{2}} \frac{dℓ}{dt} \\ ℓ (0) MathClass-rel= 0 MathClass-punc, \frac{dℓ}{dt} (0) MathClass-rel= 0 MathClass-punc, \end{matrix} & eqnarray-2 & eqnarray-3 & eqnarray-4 (1) \end{matrix}

…”

Section: Mathematical Modelingmentioning

confidence: 99%

“…We have shown in Cavaccini et al . that, for a capillary problem, it would be advisable to apply an adaptive numerical method. In that paper, we used for the adaptive procedure the following monitor function

\begin{matrix} eqnarrayright center left \\ eqnarray-1 η MathClass-open(t_{k} MathClass-close) = \frac{|\frac{dℓ}{dt} MathClass-open(t_{k} + Δ t_{k} MathClass-close) - \frac{dℓ}{dt} MathClass-open(t_{k} MathClass-close)|}{Γ_{k}}, & eqnarray-2 & eqnarray-3 & eqnarray-4 (5) \end{matrix}

…”

Section: Extended Scaling Invariancementioning

confidence: 99%

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Extended scaling invariance of one‐dimensional models of liquid dynamics in a horizontal capillary

Fazio

Iacono

Jannelli

et al. 2012

Math Methods in App Sciences

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In this paper, we consider the adaptive numerical solution of one‐dimensional models of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: the smaller is the capillary radius, the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence, the numerical solution to a prescribed accuracy becomes harder to achieve. Therefore, in order to solve a capillary problem effectively, it would be advisable to apply an adaptive numerical method. Here, we show how an extended scaling invariance that can be used to define a family of solutions from a computed one. In the viscous case, the similarity transformation involves solutions of liquids with different density, surface tension, viscosity, and capillary radii, whereas in the inviscid case, we can generate a family of solutions for the same liquid and capillaries with different radii. With our study, we are able to prove that the monitor function, used in the adaptive algorithm, is invariant with respect to the considered scaling group. It follows, from this particular results, that all the solutions within the generated family verify the adaptive criteria used for the computed one. Moreover, all the solutions have the same order of accuracy even if the maximum value of the step size varies under the action of the scaling group. Copyright © 2012 John Wiley & Sons, Ltd.

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Surface tension driven flow of blood in a rectangular microfluidic channel: Effect of erythrocyte aggregation

et al. 2020

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Microfluidic platforms have increasingly been explored for in vitro blood diagnostics and for studying complex microvascular processes. The perfusion of blood in such devices is typically achieved through pressure driven setups. Surface tension driven blood flow provides an alternative flow delivery option, and various studies in the literature have examined the behaviour of blood flow in such fluidic devices. In such flows, the influence of red blood cell (RBC) aggregation, the phenomenon majorly responsible for the non-Newtonian nature of blood, requires particular attention. In the present work, we examine differences in the surface tension driven flow of aggregating, non-aggregating RBC, and Newtonian suspensions, in a rectangular micro channel. The velocity fields were obtained using micro-PIV techniques. The analytical solution for blood velocity in the channel is developed utilising the power law model for blood viscosity. The results showed that RBC aggregation has an impact at the late stages of the flow, observed mainly in the bluntness of the velocity profiles. At the initial stages of the flow the shearing conditions are found moderately elevated, preventing intense RBC aggregate formation. As the flow decelerates in the channel RBC aggregation increases, affecting the flow characteristics.

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One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary

Cited by 4 publications

References 31 publications

An analytical and numerical study of liquid dynamics in a one‐dimensional capillary under entrapped gas action

An analytical and numerical study of liquid dynamics in a one‐dimensional capillary under entrapped gas action

Extended scaling invariance of one‐dimensional models of liquid dynamics in a horizontal capillary

Surface tension driven flow of blood in a rectangular microfluidic channel: Effect of erythrocyte aggregation

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