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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2011, 2 (3), P. 49-52
UDC 539.3
PLANAR FLOWS IN NANOSCALE REGIONS
S.A. Chivilikhin1, I.Yu. Popov1, V.V. Gusarov2
1 Saint Petersburg National Research University of Information Technologies, Mechanics, and Optics, Saint Petersburg, Russia 2 Saint Petersburg State Institute of Technology (Technical University),
Saint Petersburg, Russia sergey.chivilikhin@gmail.com
PACS 46.25.Cc, 68.35.Rh
Last years, fluid flows in nano-sized domains are intensively studied [1-4] due to nontriviality of observed effects and practical importance of this part of hydrodynamics. At present, there are no general equations of nano-hydrodynamics. Usually, the molecular dynamics is used for computations. As for analytical approaches, the simplest one involves introducing the slip condition at the boundary [5] together with classical hydrodynamics equations. The small scale of nanochannels gives us the possibility to use, in some cases, the Stokes approximation for motion equations [6].
In this work we apply the planar Stokes model [7] with slip boundary conditions for describing nano-flows. We have developed a method of flow calculation, which is based on the expansion of pressure in a complete system of harmonic functions. Using the pressure distribution, we calculate the velocity field and stress on the boundary. This method can be used for description of various problems of nanofluidics: hydrodynamics of nanochannels, flows along superhydrophobic surfaces, etc. Keywords: nanofluidics, Stokes flow, nanostructure.
1. Introduction
The equations of motion in the quasi-stationary Stokes approximation and the continuity equation in the region g have the form:
dßPaß = 0, x e g, (1)
dßVß = 0, x e g, (2)
where Paß = —P5aß + ß(daVß + dßVa) is the Newtonian stress tensor; Va are the components of the velocity; P is the pressure; ß is the coefficient of the dynamic viscosity, which is assumed to be constant, 5aß is the delta symbol of Kronecker. Summation over repeated indices is assumed.
We take in account four types of boundary parts: inlet, outlet, solid wall and free boundary. The total boundary 7 is the union of these parts. On the inlet we specify the velocity field
Va = V?, x e 7in. (3)
On the outlet we use soft boundary conditions
dnVa = 0, x e jout, (4)
where dn is the derivative along the normal to the boundary. On the solid wall we specify slip boundary conditions
LdnVT = VT,Vn = 0, x e 7wa11, (5)
where VT and Vn are the tangent and normal components of liquid velocity on the wall, L is the slip length.
On the free boundary we assume the action of the capillary force
Paßnß = -anadßnß, x e 7iree, (6)
where a is the coefficient of surface tension.
The law of free boundary evolution is determined from the condition of equality of normal velocity U of boundary and the normal component of liquid velocity at the boundary:
U = Vß nß, x e f*, (7)
The absence of the time derivative in the quasi-stationary motion equation (1) lets us specify the initial conditions only for the shape of the free boundary:
„free I „free io\
1 |t=o = 7o • (8)
In this work we take in account two-dimensional problems only (g c R2).
2. Calculation of the pressure and velocity with given force on the boundary
Let fa be the force applied to the total boundary 7. Then we can write one boundary condition
Paßnß = fa, X E J, (9)
instead of boundary conditions (3)-(6). We need to remark that we really know the force fa on the free boundary only. On other parts of the boundary we will calculate the force during the solution. Let x« and ^ be smooth fields in the region g related by
da Xß + dßXa = 2^Saß, (10)
Multiplying the motion equation (1) by x« , integrating over g , and using (2), (9), (10), we obtain
J P^dg = -0.5 J faXadr (11)
According to (10), ^ and x« are harmonic functions and
d(xi + x) = + iu )dz• (12)
where u is a harmonic function conjugate to -0. Let be a complete set of harmonic functions in the region g. Using (12) we obtain the correspondent functions Xak.
The complete set of analytical functions in the finite region g with simple connected boundary consists of functions zk ,k = 0,1,____ We obtain the complete set of harmonic functions ^k in the form re(u>fc), Im(wk). According to (1),(2) the pressure P is a harmonic function. We present it in the form
p = Y. p* ^ • (13)
k
Using the expressions (11), (13) we obtain the algebraic system for coefficients pk:
^ ^y ^k^ndg^Pk = -0.5 J faXandj, n = 0,1,... (14)
The stress tensor, expressed in terms of the Airy function tp,
Paß = dlß<P - (15)
satisfies the equation of motion (1) identically. The Airy function satisfies the biharmonical equation
av = 0, x E g. (16)
Planar flows in nanoscale regions 51
The boundary conditions (9) take the form
dTdap = eap fp, x e (17)
where dT is the derivative along the tangent to the boundary. Integrating (17) from a fixed point of the boundary to current one, we obtain
x
da p = eap J fpdj, x e j. (18)
xo
Using (15) and the expression for the Newtonian stress tensor, we obtain
d(daV) = 2^dVa + d$a, d($i + ) = P + iti, (19)
where
da ^ + dp = 2P5ap, (20)
ti = n(d2V1 — d1V2) is the harmonic function conjugate to P. Comparing (20) with (10) and using (13), we get the expression for $a :
= Pk • (21)
k
Therefore, we obtain the expression for velocity:
= X6?. (22)
On the boundary da>p was calculated above (18). To find >p (and, respectively, the velocity Va) in the region g, we solve the equation (16) with boundary conditions (18).
3. Calculation of the pressure and velocity with various boundary conditions
If the free boundary conditions are specified on the total boundary, then (22) gives us the solution of our problem. In a general case (22) is true too, but on the inlet, outlet and the wall the force fa is unknown.
On the inlet we can calculate the force with the help of (3) and differentiating (22) along of boundary:
fa = —eal3(2^dTV™ + dT), x e 7in- (23)
On the outlet we obtain the force using (4) by differentiating (22) along of boundary:
fa = —naP — TaUpdT, X E 7°"'• (24)
The slip boundary conditions (5) on the wall and (22) give us the force distribution
fa = —Ha fn + Tafr, X E 7to11, (25)
where fT = 1 ' npdT$p, fn = rdT ( + 7^777 ], K is the scalar curvature of the 1 + 2K L \ 1 + KL J
boundary.
Expressions (23)-(25) contain P = Pk^k and $a = PkXak. By substituting these
k k
expression into (14) we obtain the system of algebraic equations concerning coefficients pk. After solving this system we obtain the pressure (13) and velocity distributions (22).
Acknowledgements
The work was supported by Programs "Development of Scientific Potential of Russian High School" (project 2.1.1/4215), "Scientific Staff of Innovative Russia" (contracts P689 NK-526P, 14.740.11.0879, and 16.740.11.0030), and Russian Foundation for Basic Research (project No. 11-08-00267).
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