Mathematical modeling of capillary flow of viscous fluid

We consider flow of viscous fluid in the capillaries under assumption that the contact angle of the curve varies over the length of the capillary. For the two laws of change we obtain estimates of the distance of meniscus from its initial position and reduction of its movement speed. It was found that the results of the evaluations are consistent with the existing physical concepts.


Introduction
Fluid flow in capillaries (capillary flow) is often found in nature and technology; it plays an important role in the plant and animal life, when filtering the fluid media in the grounds, in the drying processes of dispersed bodies, media purification from mechanical impurities, impregnation of porous materials and in many other cases. Fundamental principles of capillary fluid flow theory are presented in [1-6 and etc.]. Let's name some of them.
In general case, the fluid transfer in the capillaries occurs by both molecular (diffusive) and convective mechanisms. First is caused by the difference in concentration of a substance along the length of the capillary. It is important to note that because of braking action of the capillary solid walls the diffusion coefficient is much smaller than that in a free fluid and this fluid transfer mechanism will be the most significant in very thin capillaries.
When capillary lateral dimensions are relatively large the convective transfer is dominating its intensity is determined, firstly, by the pressure drop along the length of the capillary, secondly, by the temperature gradient along it, thirdly, by the action of the surface tension forces at the interface between fluid and gaseous medium (meniscus), mass forces, in particular, gravitational forces.
The surface tension force is the force acting on a contour unit length of the media interface and is characterized by the surface tension coefficient σ also called surface tension. The magnitude of the surface tension coefficient depends on the material of media in contact, temperature, presence of adsorbed (surface active) agent on the fluid surface and other factors.

Part of mathematical modeling
Taking into account the surface tension, in the absence of mass forces at the contact boundary of immiscible viscous fluid and gaseous medium the following relations are written [3]: Here, the subscripts , mean the parameters of the fluid and gaseous medium, respectively; , normal and tangential components of stress ⃗ ; , ; , components of motion velocity of medium, interface surface; , = ( )capillary pressure, tangential force per unit area. In the simplest case, when ≈ − , ≈ − , the surface tension along the contact boundary is constant (( ) = 0), the capillarytube of radius , meniscus has a shape of a spherical segment, capillary pressure [5] = ±2 /̃. Here, ̃mean radius of the interface curvature, ̃= / cos ,limiting wetting angle; plus sign is selected for a convex fluid surface, minus signfor a concave ( < 0).
In accordance with the Young's equation [4][5][6] where , surface tension at the interfaces solid bodygas, solid bodyfluid. In work [6] it is shown that the contact angle determined according to (2) gives the angle value at the three phase boundary of the media in contact on a microscopic scale. The macroscopic contact angle depends on the system geometry which is, in turn, under a great influence of the mass forces, structure and material of the solid body surface, medium temperature.
When studying the capillary flow of fluid in the above-mentioned cases it is usually considered that capillary is a tube, probably, with varying along the length cross-section [5], the media in a capillary are under action of gravitational force, along the movement of meniscus the limiting wetting angle remains constant.
Analyzing the available result you can conclude that during the spraying of polymeric powder materials [7][8][9][10] at the stage of coalescence of fluid particles situated on the treated body surface, their spreading [8], the medium movement is similar to the capillary one. For better understanding of peculiarities of the given processes taking into account the specificity of particles coalescence we consider the capillary flow of fluid between two plates situated at a small distance from each other = 2 . Let's assume that the meniscus cross-section is close to a circular arc of radius its surface is under the influence of pressure from the gas side, pressure fluid. Let's also assume that during the meniscus movement the limiting wetting angle varies according to a particular law.
(3) Here ℎ ̅ = ℎ/ℎ ,ℎ = ℎ( )meniscus distance from the original position occupied by it in the capillary at the time = 0; ℎ = /∆ * , = cos , characteristic limiting wetting angle for the considered system solid body-fluid-gas, parameter ̅ ≥ , ∆ * = − *pressure drop of media in contact on the meniscus surface. The average velocity of the meniscus movement will be: ̃= ℎ/ = ℎ · ℎ ̅ / . Considering as it is customary in the capillary hydrodynamics that the given movement of fluid is laminar, close to the stationary (quasi-stationary) to describe the average velocity of its movement in the absence of the mass forces we use the Poiseuille solution: Hence, the average velocity of the meniscus movement in the capillary ̃=̃( ) = ( ̅ 2 /3 ℎ)(1 − ℎ ̅ / ̅ ). (6) If the ratio ℎ ̅ / ̅ ≪ 1, then from (5) and (6)  It is seen that immediately after the beginning of fluid movement in the capillary the remoteness of the meniscus increases proportionally to √ at that velocity of its displacement decreases with time, the movement is slowed down. According to (6) the meniscus stops when ℎ ̅ = ̅ i.e. at the distance from the movement start where ∆ ̅ = ∆ /dimensionless pressure drop and 0 < ∆ ̅ < 1 0 < ∆ ̅ < 1.
(8) From formula (8) it follows that the time to the meniscus stop is smaller the smaller are the fluid viscosity, the capillary width, the greater is the surface tension. The given regularities are quite consistent with the existing physical notions.
In the considered case the contact angle increases (see (3)) along the meniscus movement. However, the situation where on the contrary the angle decreases starting from is of great interest. In this case, we assume that dependence = (ℎ ̅ ) is described by the relationship: = (ℎ ̅ ) = + ( ̅ − )ℎ ̅ . By analogy with the previous case we introduce the auxiliary function ᴂ * = ᴂ * (ℎ ̅ ) = * + * ℎ ̅ , где * = / − ∆ , * = ( ̅ − )/ , parameter * = ( ( ̅ − )/ℎ ) 2 /(3 ), using which we obtain the equation for finding ᴂ * : ᴂ * − * ᴂ * /ᴂ * = * . (9) Taking into account that at the initial time = 0 (ℎ ̅ = 0) function ᴂ * = * , integrating (9), we determine: ℎ In this case, in contrast to the previous one when changing the variable from to ̅ (0 < ℎ ̅ < 1) the velocity of the meniscus movement is not equal to zero, the meniscus does not stop. After the lapse of control time from the beginning of the meniscus movement it will shift approximately at a distance ℎ determined by the formula (10) and velocity of its movement is ̃≈ ℎ /√2 . For fixed , , ∆ ̅ , time distance of the meniscus from the initial position increases with increasing of the surface tension and decreasing of the fluid viscosity proportionally to √ / .