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Journal ArticleDOI

Moving contact lines and rivulet instabilities. Part 1. The static rivulet

29 May 1980-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 98, Iss: 2, pp 225-242
TL;DR: In this paper, a linearized stability theory for small, static rivulets whose contact (common or three-phase) lines (i) are fixed, (ii) move but have fixed contact angles or (iii) have contact angles smooth functions of contact-line speeds is presented.
Abstract: A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Capillary instabilities are investigated by a linearized stability theory. The formulation is for small, static rivulets whose contact (common or three-phase) lines (i) are fixed, (ii) move but have fixed contact angles or (iii) move but have contact angles smooth functions of contact-line speeds. The linearized stability equations are converted to a disturbance kinetic-energy balance showing that the disturbance response exactly satisfies a damped linear harmonic-oscillator equation. The ‘damping coefficient’ contains the bulk viscous dissipation, the effect of slip along the solid and all dynamic effects that arise in contact-line condition (iii). The ‘spring constant’, whose sign determines stability or instability in the system, incorporates the interfacial area changes and is identical in cases (ii) and (iii). Thus, for small disturbances changes in contact angle with contact-line speed constitute a purely dissipative process. All the above results are independent of slip model at the liquid–solid interface as long as a certain integral inequality holds. Finally, sufficient conditions for stability are obtained in all cases (i), (ii) and (iii).
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors review the current state of understanding of the mechanisms of drop formation and how this defines the fluid properties that are required for a given liquid to be printable.
Abstract: Inkjet printing is viewed as a versatile manufacturing tool for applications in materials fabrication in addition to its traditional role in graphics output and marking. The unifying feature in all these applications is the dispensing and precise positioning of very small volumes of fluid (1–100 picoliters) on a substrate before transformation to a solid. The application of inkjet printing to the fabrication of structures for structural or functional materials applications requires an understanding as to how the physical processes that operate during inkjet printing interact with the properties of the fluid precursors used. Here we review the current state of understanding of the mechanisms of drop formation and how this defines the fluid properties that are required for a given liquid to be printable. The interactions between individual drops and the substrate as well as between adjacent drops are important in defining the resolution and accuracy of printed objects. Pattern resolution is limited by the extent to which a liquid drop spreads on a substrate and how spreading changes with the overlap of adjacent drops to form continuous features. There are clearly defined upper and lower bounds to the width of a printed continuous line, which can be defined in terms of materials and process variables. Finer-resolution features can be achieved through appropriate patterning and structuring of the substrate prior to printing, which is essential if polymeric semiconducting devices are to be fabricated. Low advancing and receding contact angles promote printed line stability but are also more prone to solute segregation or “coffee staining” on drying.

1,525 citations

Journal ArticleDOI
TL;DR: In this article, the authors describe which types of laser-induced consolidation can be applied to what type of material, and demonstrate that although SLS/SLM can process polymers, metals, ceramics and composites, quite some limitations and problems cause the palette of applicable materials still to be limited.

1,241 citations

Journal ArticleDOI
TL;DR: In this paper, the principles underlying common techniques for actuation of droplets and films on homogeneous, chemically patterned, and topologically textured surfaces by modulation of normal or shear stresses are reviewed.
Abstract: Development and optimization of multifunctional devices for fluidic manipulation of films, drops, and bubbles require detailed understanding of interfacial phenomena and microhydrodynamic flows Systems are distinguished by a large surface to volume ratio and flow at small Reynolds, capillary, and Bond numbers are strongly influenced by boundary effects and therefore amenable to control by a variety of surface treatments and surface forces We review the principles underlying common techniques for actuation of droplets and films on homogeneous, chemically patterned, and topologically textured surfaces by modulation of normal or shear stresses

474 citations

Journal ArticleDOI
TL;DR: In this paper, the Young equation is used to describe the motion of an interface between immiscible viscous fluids along a smooth homogeneous solid surface in the case of small capillary and Reynolds numbers, and an analytical expression for the dependence of the dynamic contact angle on the contact-line speed and parameters characterizing properties of contacting media is derived.
Abstract: A general mathematical model which describes the motion of an interface between immiscible viscous fluids along a smooth homogeneous solid surface is examined in the case of small capillary and Reynolds numbers. The model stems from a conclusion that the Young equation, σ1 cos θ = σ2 − σ3, which expresses the balance of tangential projection of the forces acting on the three-phase contact line in terms of the surface tensions σi and the contact angle θ, together with the well-established experimental fact that the dynamic contact angle deviates from the static one, imply that the surface tensions of contacting interfaces in the immediate vicinity of the contact line deviate from their equilibrium values when the contact line is moving. The same conclusion also follows from the experimentally observed kinematics of the flow, which indicates that liquid particles belonging to interfaces traverse the three-phase interaction zone (i.e. the ‘contact line’) in a finite time and become elements of another interface – hence their surface properties have to relax to new equilibrium values giving rise to the surface tension gradients in the neighbourhood of the moving contact line. The kinematic picture of the flow also suggests that the contact-line motion is only a particular case of a more general phenomenon – the process of interface formation or disappearance – and the corresponding mathematical model should be derived from first principles for this general process and then applied to wetting as well as to other relevant flows. In the present paper, the simplest theory which uses this approach is formulated and applied to the moving contact-line problem. The model describes the true kinematics of the flow so that it allows for the ‘splitting’ of the free surface at the contact line, the appearance of the surface tension gradients near the contact line and their influence upon the contact angle and the flow field. An analytical expression for the dependence of the dynamic contact angle on the contact-line speed and parameters characterizing properties of contacting media is derived and examined. The role of a ‘thin’ microscopic residual film formed by adsorbed molecules of the receding fluid is considered. The flow field in the vicinity of the contact line is analysed. The results are compared with experimental data obtained for different fluid/liquid/solid systems.

402 citations


Cites background or methods from "Moving contact lines and rivulet in..."

  • ...S models simultaneously trying to point out their common or specific features (Dussan V. 1976; Huh & Mason 1977b; Davis 1980; Zhou & Sheng 1990; Haley & Miksis 1991) or use purely empirical approaches (Kafka & Dussan V. 1979; Ngan & Dussan V. 1989; Dussan V. et al. 1991)....

    [...]

  • ...…boundary condition (Lamb 1932, p. 586), where for the coefficient of sliding friction one may use (S1a) β = const (Hocking 1977, 1981, 1990, 1992; Huh & Mason 1977b; Davis 1980; Lowndes 1980; Levine et al. 1980; Hocking & Rivers 1982; Zhou & Sheng 1990; Ehrhard & Davis 1991; Haley & Miksis 1991)....

    [...]

  • ...1976; Hocking 1977, 1981, 1992; Huh & Mason 1977b; Davis 1980; Hocking & Rivers 1982; Cox 1986; Durbin 1988; Zhou & Sheng 1990) (A2) U = k(θd−θs), where k is an empirical constant (Greenspan 1978; Greenspan & McCay 1981; Davis 1980; Hocking 1990; Haley & Miksis 1991)....

    [...]

  • ...586), where for the coefficient of sliding friction one may use (S1a) β = const (Hocking 1977, 1981, 1990, 1992; Huh & Mason 1977b; Davis 1980; Lowndes 1980; Levine et al. 1980; Hocking & Rivers 1982; Zhou & Sheng 1990; Ehrhard & Davis 1991; Haley & Miksis 1991)....

    [...]

  • ...…devoted to the subject, one can find the following approaches to Problem A: (A1) θd ≡ θs (Dussan V. 1976; Hocking 1977, 1981, 1992; Huh & Mason 1977b; Davis 1980; Hocking & Rivers 1982; Cox 1986; Durbin 1988; Zhou & Sheng 1990) (A2) U = k(θd−θs), where k is an empirical constant (Greenspan 1978;…...

    [...]

References
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Book
01 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.

7,426 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.

4,525 citations

Journal ArticleDOI
TL;DR: In this article, it is shown that the mutual interaction between the three materials in the immediate vicinity of a contact line can significantly affect the statics as well as the dynamics of an entire flow field.
Abstract: A contact line is formed at the intersection of two immiscible fluids and a solid. That the mutual interaction between the three materials in the immediate vicinity of a contact line can significantly affect the statics as well as the dynamics of an entire flow field is demonstrated by the behavior of two immiscible fluids in a capillary. It is well known that the height to which a column of liquid will rise in a vertical circular capillary with small radius, a, whose lower end is placed into a bath, is given by (2(j/apg) cos (), where (j is the surface tension of the air/liquid interface, f) is the static contact angle as measured from the liquid side of the contact line, p is the density, and g is the magnitude of the accelera­ tion due to gravity.! Thus, depending on the value of the contact angle, e, which is a direct consequence of the molecular interactions among the three materials at the contact line, the height can take on any value within the interval [ 2(J/apg, 2(J/apg]. In a sense, the influence of the contact angle is indirect: the contact angle, in capillaries with small radii, controls the radius of curvature of the meniscus which, in turn, regulates the pressure in the liquid under the meniscus. It is this pressure that determines the height of the column. In a similar manner, the dynamic contact angle can influence the rate of displacement of tbe meniscus through the capillary. The pressure drop

1,337 citations

01 Jan 1979
TL;DR: In this paper, it is shown that the mutual interaction between the three materials in the immediate vicinity of a contact line can significantly affect the statics as well as the dynamics of an entire flow field.
Abstract: A contact line is formed at the intersection of two immiscible fluids and a solid. That the mutual interaction between the three materials in the immediate vicinity of a contact line can significantly affect the statics as well as the dynamics of an entire flow field is demonstrated by the behavior of two immiscible fluids in a capillary. It is well known that the height to which a column of liquid will rise in a vertical circular capillary with small radius, a, whose lower end is placed into a bath, is given by (2(j/apg) cos (), where (j is the surface tension of the air/liquid interface, f) is the static contact angle as measured from the liquid side of the contact line, p is the density, and g is the magnitude of the accelera­ tion due to gravity.! Thus, depending on the value of the contact angle, e, which is a direct consequence of the molecular interactions among the three materials at the contact line, the height can take on any value within the interval [ 2(J/apg, 2(J/apg]. In a sense, the influence of the contact angle is indirect: the contact angle, in capillaries with small radii, controls the radius of curvature of the meniscus which, in turn, regulates the pressure in the liquid under the meniscus. It is this pressure that determines the height of the column. In a similar manner, the dynamic contact angle can influence the rate of displacement of tbe meniscus through the capillary. The pressure drop

1,169 citations