Interfacial Fluid Mechanics: A Mathematical Modeling by Vladimir S. Ajaev

By Vladimir S. Ajaev

Interfacial Fluid Mechanics: A Mathematical Modeling Approach presents an advent to mathematical versions of viscous stream utilized in speedily constructing fields of microfluidics and microscale warmth move. the fundamental actual results are first brought within the context of easy configurations and their relative value in general microscale purposes is mentioned. Then, numerous configurations of significance to microfluidics, such a lot particularly skinny films/droplets on substrates and constrained bubbles, are mentioned intimately. themes from present examine on electrokinetic phenomena, liquid circulate close to established stable surfaces,evaporation/condensation, and surfactant phenomena are mentioned within the later chapters.

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8), we find 1 u = − (hxxx + 1)(y2 − 2yh). 11) we obtain an evolution equation for the interface shape in the form ht + 1 3 h hxxx 3 x + h2hx = 0. 1 only when the contact angle θ is sufficiently small. In fact, since the contact angle is the inverse tangent of the absolute value of the interface slope at the contact line, the value of θ has to be of the same order of magnitude as Ca1/3 , as discussed in more detail in Goodwin and Homsy [58]. From now on, we use the re-scaled contact angle Θ defined by Θ = θ Ca−1/3 .

22 1 Basic Phenomena and Applications to Thin Films a y n t b air n B liquid h(x,t) A’ A x Fig. 8 (a) Sketch of a curved liquid–air interface showing the Cartesian coordinates, the unit normal and tangential vectors to the interface, and the slope angle. (b) Sketch for the derivation of the kinematic boundary condition Let us now discuss the boundary conditions for the Stokes flow equations at a curved liquid–air interface, shown in Fig. 8a, with h(x,t) denoting the distance from the interface to the x-axis.

The contributions to the force balance which remain finite in the limit of infinitely small δ h are the force Fs , due to the surface tension acting on the side boundary of the control volume, and the forces in the two fluids acting at the top and bottom boundaries, so (1) Fis = (2) Ti j − Ti j S n j dS. 101) 28 1 Basic Phenomena and Applications to Thin Films Fig. 11 Sketch of a two-dimensional interface Γ showing the local Cartesian coordinates used to express the normal stress condition in terms of mean curvature.

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