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Stability and bifurcation analysis of a pendent drop using a novel dynamical model

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Abstract

Drop dynamics is often used to study liquid-fluid interfacial problems. Also, oscillatory pendent drops are a suitable and alternative tool to study non-linear dynamical systems. According to the paper by Ghorbanifar et al. (J Appl Fluid Mech 1:234, 10.47176/jafm.14.01.31313, 2021) it was predicted numerically that elastic force excreted by a pendent drop is a complete cubic polynomial function. The present work deals with the analytical analysis and proof of the force–displacement function of a pendent drop (which was realized by Ghorbanifar et al.) and using this function the bifurcation of drop was investigated. The oscillatory pendent drop equation of motion (OPDEM) presented here, can completely describe the force–displacement and damping functions of the pendent drop. In this article, the equation of motion of a pendent drop is presented, which paves the way for analytical investigations of drop dynamics. The presented novel dynamical model allows following the oscillation, growth, and detachment of a pendent drop and studying its elastic and non-linear behavior. Here, a drop was modeled as a combination of an elastic rod and a dashpot simulating the drop related surface tension and viscous damping forces, respectively. The displacement of the drop mass center was then related to the rod elongation. Then, the damped OPDEM was derived. It was found that the forcing and damping terms of OPDEM are complete cubic and quadratic polynomials, respectively. Using Lyapunov's method for stability analysis, non-linear dynamics of pendent drop was studied. It was shown the drop growth condition is that the displacement should lie between the local extrema of the forcing term of OPDEM. Exploiting the results presented by Ghorbanifar et al. effects of changing Bond and Capillary numbers on the bifurcation of the governing ODE of drop oscillations were investigated. It was realized that for the Bond numbers greater than about 0.11 saddle-point bifurcation occurs. This causes equilibrium points of OPDEM vanish and system becomes unstable. Also, increasing Capillary number to the values higher than 1.8E−5 causes transcritical bifurcation which leads to the removal of one of the balance points. For this case, by increasing Capillary number the system tends towards one equilibrium point. These results open a way to relate some dimensionless numbers in fluid flow to the dynamics of non-linear systems described by OPDEM and help in studying their stability.

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Abbreviations

A :

Constant

\(\overline{A}\) :

Second derivative of V with respect to y

a :

A stationary point of V

B :

Constant

\(\overline{B}\) :

Second derivative of V with respect to y and z

b :

Height of the static drop mass center from the pipe tip

C :

Constant

\(\overline{C}\) :

Second derivative of V with respect to z

D :

Constant

E(y):

Potential energy integral

f(y):

Restoring elastic force in OPDEM

f( t,y) :

A continuous vector function

H(a):

The Hessian matrix

i :

Unit vector in y direction

j :

Unit vector in z direction

k :

Positive constant

L :

Diameter of the pipe tip

L 0 :

Initial length of the rod

m :

Mass of the drop

q(y):

Damping force coefficient in OPDEM

T :

The resultant of elastic and damping forces

U(y):

Potential energy function of pendent drop

V(y, z):

The Lyapunov function

y :

Independent variable in vertical direction

y root1 :

Abscissae of the smaller root of q(y)

y root2 :

Abscissae of the bigger root of q(y)

y inflection_1 :

Abscissae of the first inflection point of U(y)

y inflection_2 :

Abscissae of the second inflection point of U(y)

y local_min :

Abscissae of the local minimum of f(y)

y rlocal_max :

Abscissae of the local maximum of f(y)

z :

Ordinate of the phase plane

α :

Coefficient of f(y)

β :

Coefficient of f(y)

γ :

Coefficient of f(y)

ε :

Coefficient of f(y)

α :

Coefficient of q(y)

β :

Coefficient of q(y)

γ :

Coefficient of q(y)

φ :

Elastic rod angle with horizontal direction

θ :

Stretched rod angle with its static state

Δ:

The strain of the elongated rod

µ :

Positive constant

ω :

External force frequency

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Engineering and Technology, Astara Branch, Islamic Azad University, Astara, Iran

    Shahram Ghorbanifar

  2. Aerospace Department, Sharif University of Technology, Tehran, Iran

    Mohammad Taeibi Rahni

  3. Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Masoud Zareh & Mohammad Hasan Nobakhti

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  1. Shahram Ghorbanifar
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  2. Mohammad Taeibi Rahni
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  4. Mohammad Hasan Nobakhti
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Correspondence to Masoud Zareh.

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Ghorbanifar, S., Rahni, M.T., Zareh, M. et al. Stability and bifurcation analysis of a pendent drop using a novel dynamical model. Arch Appl Mech 93, 487–501 (2023). https://doi.org/10.1007/s00419-022-02278-z

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