Interfacial Thermocapillary Pressure of an Accelerated Droplet in Microchannels
Abstract
Fluid flow and thermocapillary heat transfer near a droplet/air interface in microchannels is studied analytically and numerically in this set of two companion papers. Thermocapillary forces generate a pressure difference across the droplet, which drives two symmetric re-circulation cells in the upper and lower half-portions of the droplet. The numerical formulation uses a sliding grid for the accelerating droplet, as well as an expanding/contracting grid in the gas region and an adaptive grid in the substrate below the closed-end microchannel. In contrast to past studies with a uniform interfacial pressure, this paper accommodates a varying interfacial pressure along the receding edge of the droplet.
Nomenclature
A interfacial area (m2), surface tension constant
B surface tension constant
cp specific heat (J/kgK)
d droplet height (m)
F force (N)
G microchannel constant
h convection coefficient (W/m2K)
H microchannel height (m)
k thermal conductivity (W/mK)
m mass of droplet (kg)
M mass of fluid within control volume (kg)
P pressure (Pa)
Pe Peclet number (vH/a)
q′′ heat flux (W/m2)
R gas constant (J/kgK)
t time (s)
T temperature (K)
u, v x and y-velocity components (m/s)
V volume (m3)
x, y Cartesian coordinates (m)
Greek
a Peclet weighted convection coefficient
b diffusion coefficient
f arbitrary scalar value
l relaxation coefficient
m dynamic viscosity (kg/ms)
q contact angle (rads)
r density (kg/m3)
s surface tension (N/m)
Subscripts
a air
b bulk
c capillary
d droplet
E, W, S, N east, west, south and north nodal points
f friction
i initial
L left edge
o surroundings
R right edge
s substrate
v vertical
w wall
Superscripts
k previous iteration
k+1/2 current (intermediate) iteration number
n+1 current time step
o previous iteration or time step
u velocity
1. Introduction
Effective methods of microfluidic transport have importance in various emerging technologies of micro and nano-systems. Effective flow control improves the thermal performance of micro heat exchangers [1] and microchannel heat sinks for electronics cooling [2]. Past methods of MEMS flow control have used pressure driven [3], electrically driven [4] or thermocapillary driven methods [5]. A comprehensive review of micro-pumping technologies was documented by Singhal et al. [6]. Benefits and limitations of about twenty different types of micro-pumps were described. The maximum flow rates of different micro-pumps were compared. Other comparisons involving actuation voltage, frequency of operation, cost of fabrication and electronics cooling capabilities were reported by the authors [6]. This paper examines a promising alternative to these past methods of flow control, by using thermocapillary forces to generate pressure differences that drive the micrfluidic motion.
Thermocapillary forces have been used successfully for flow control along micro-patterned surfaces [8]. Surface micro-grooves have additional benefits of reducing entropy production of convective heat transfer [9]. In a closed-end microchannel, thermocapillary pumping (TCP) involves oscillatory droplet motion during cyclic periods of heating and cooling. Fluid velocities of oscillatory thermocapillary convection have been reported by Agata et al. [10]. Thermocapillary forces have been used to generate pressure gradients across a vapor bubble in a capillary tube [11]. The difference of surface tension across a bubble leads to convective motion towards the region of higher temperature. In nucleate boiling problems, surface tension affects detachment size and departure frequency of bubbles along a surface [12]. Accurate modeling of force and heat balances across bubbles or droplets has importance in various technological problems related to multiphase flows [13 – 15].
Unlike other conventional methods of flow control in microdevices, thermocapillary control operates without any mechanical moving parts. A miniaturized optical switch involving thermocapillary transport was presented by Togo et al. [16]. Temperature variations within the liquid produce a spatial change of surface tension, thereby driving microfluidic motion for purposes of pumping or switching. The electrical analogy of this mechanism is electrocapillary transport, whereby an electrical potential gradient between two immiscible conducting fluids generates a spatial change of surface tension. This resulting fluid motion is called continuous electrowetting (CEW). Lee and Kim [17, 18] have reported CEW control of a liquid metal droplet in circular microchannels. Unlike CEW, this paper examines how thermal gradients across the fluid can generate fluid motion.
In this article, a 2-D computational fluid dynamics (CFD) model is developed to predict thermocapillary pumping (TCP) in closed-end microchannels. Predictions of TCP within a closed microchannel differ from open microchannels, as pressures upstream and downstream of the droplet are unequal. Large droplet velocities reported in past TCP studies with open-end channels cannot be obtained in a closed-end microchannel, since small droplet displacements create an opposing pressure gradient in the gas region, with a similar magnitude as the thermocapillary pressure. Also, open-end channel assumptions of steady-state Poiseuille flow are not fully applicable in closed microchannels, when cyclic heat input generates periodic acceleration of the droplet. This paper develops a coupled velocity/pressure formulation with a non-uniform interfacial pressure. This interfacial pressure generates thermocapillary forces that drive re-circulating motion near the corners of the droplet. Also, a theoretical model will be developed to predict overall trends arising from different geometrical configurations and thermophysical properties. Comparisons between theoretical and numerical results will be discussed.
2. Thermocapillary Pumping in a Closed Microchannel
Consider a micro-droplet enclosed by air regions on both sides of the droplet within a closed microchannel (see Fig. 1). A stationary heat source is applied at the back end of the droplet, which produces a temperature gradient and thermocapillary force within the droplet. The resulting pressure difference across the droplet generates fluid motion from left to right in Fig. 1. This pressure difference is similar to an externally applied fluid pressure, so certain portions of the flow region exhibit the characteristics of Poiseuille flow. The fluid velocity deviates from Poiseuille flow behavior near the ends of the droplet, due to re-circulation effects. If the interface u-velocity along the meniscus is nearly equal to the bulk velocity of the droplet, the shape of the droplet is approximately undistorted.
During thermocapillary pumping, the advancing contact angle is generally different than the receding contact angle during droplet translation. In this paper, the receding and advancing contact angles were both assumed as zero. This approximation is a reasonable assumption when the droplet length is much larger than the channel height, since the volume of the droplet ends becomes small, relative to the total volume of the droplet. It can be shown that the error of the predicted thermocapillary pressure for a water droplet, due to the assumption of a zero contact angle, is about 1.5% percent for a contact angle of 10 degrees. This deviation could be readily incorporated into the thermocapillary model of the CFD code, provided the contact angles were known. Unfortunately, such data in closed-end microchannels is not available in the archival literature (to our knowledge), so the current approximation of a zero contact angle was utilized.
The droplet velocity and displacement can be estimated with a slug-flow approximation (SA Model), which treats the droplet as a non-deforming slug. The net force on the droplet consists of a sum of three components, namely: (i) a thermocapillary force (Fc), (ii) external air force (Fa) and (iii) a frictional drag force (Ff). The thermocapillary force is the net force acting on the slug due to the thermocapillary pressure difference (see Fig. 1), i.e.,
The surface tension at the ends of the droplet depends on temperature, T, as follows, where A = 75.83 dyn/cm and B = 0.1477 dyn/cmK for water. Also, q is the contact angle between the liquid and solid and G = 2(1 + height/width) for rectangular microchannels.
An external air force arises from compression or expansion of the gas downstream and upstream of the droplet, respectively. The air surrounding the droplet was treated as a perfect gas according to the ideal gas law, so the external air force on the micro-droplet becomes where the subscript a refers to air and R = 0.287 kJ/kgK for air. It is assumed that liquid in the microchannel stays above the vapour pressure, so that no evaporation occurs and the gas phase remains dry, without moisture added to the air from the liquid.
Also, fluid friction within the droplet is approximated from the following velocity profile corresponding to Poiseuille flow [30], where y is the distance from the channel wall. Based on this velocity profile, the friction force at the wall becomes where b and ub refer to the channel depth and bulk droplet velocity, respectively.
Combining the previous friction, air and thermocapillary forces yields the following net force (F) on the micro-droplet,
[image] (6)
where R, b and ΔP refer to the gas constant (0.287 kJ/kgK for air), microchannel depth and total pressure difference across the droplet, respectively. Based on this net force, the velocity and displacement of the micro-droplet can be determined from temporal integration over a discrete time step, Dt, so that
[image] (7)
[image] (8)
These results will be called the analytical SA Model (slug-flow approximation). This theoretical formulation will be used for purposes of comparison and validation against the detailed numerical formulation in the next section.
3. Finite Volume Formulation of Fluid Flow
In this section, a numerical method (FVM; finite volume method) is developed to solve the detailed Navier-Stokes equations within the droplet in the closed-end microchannel. The microchannel region, adjoining air regions and surrounding substrate material were sub-divided into discrete control volumes (see Fig. 2). Within each region, the dependent variables are stored at nodal points that are centrally located within the control-volumes (see Fig. 3a). The following two-dimensional mass and momentum conservation equations are used to predict the fluid velocities within the micro-droplet.
[image] (9)
[image] (10)
[image] (11)
The algebraic governing equations will be derived after the above Navier-Stokes equations are integrated spatially over the control-volumes. Integrating Eq. (9) over a control volume, V, and discrete time step, Dt, it can be shown that the standard finite volume procedure leads to the following result,
[image] (12)
This equation was solved for each control volume within the moving droplet.
A similar discretization is used for the momentum equations. Integrating Eq. (10) over the discrete control volume and time step,
[image] (13)
The x-convection term is approximated by
[image] (14)
The integration point velocities can be determined by [31]
[image] (15)
[image] (16)
In these equations, a is a dimensionless coefficient weighted by the local Peclet number (Pe) [31]. This approach represents Peclet weighted upwinding in the convection terms. When calculating advection terms in Eq. (14), the velocity field must be evaluated in the moving droplet’s reference frame. The droplet’s bulk velocity will be subtracted from the local velocity field, before calculating the advection terms. This procedure involves a moving grid in the droplet, which must stay aligned with the grid points in the substrate. The next section will discuss further details regarding this moving grid formulation.
The diffusion term is approximated as
[image] (17)
The gradient of u at an interface leading to diffusion is written as
[image] (18)
[image] (19)
where b is a diffusion weighting coefficient that depends on Pe. Combining Eqs. (17) - (19),
[image] (20)
A similar formulation is used for the y-direction component of diffusion. In these approximations, the dependent variables are assumed to vary linearly between nodal points, so that piecewise linear profiles can be used and variables at a particular node depend only on nearby nodal values. In order to ensure accuracy of linear interpolations used for the above diffusion approximations, various grid refinements were performed until predicted results of droplet displacement and velocity became independent of further grid refinement. After each time step, the velocity mesh is re-generated so that the nodal points within the substrate remain aligned with nodes in the moving droplet. The moving grid procedure will be described in the next section.
The source term arising from thermocapillary forces is sub-divided into a constant portion, Sc, and a property dependent portion, SP, i.e.,
[image] (21)
where QP and RP represent the products of the volume and the variables Sc and SP, respectively. The source term is linearized with respect to an arbitrary scalar at node p, denoted by fp, which can denote a velocity component or temperature for the momentum or energy equations, respectively.
Assembling all terms and dividing by Dt yields the following x-momentum equation,
[image] (22)
where
[image] (23)
[image] (24)
[image] (25)
[image] (26)
[image] (27)
[image] (28)
[image] (29)
Equation (22) is the x-momentum equation and a similar result is obtained for y-momentum.
The coupled mass and momentum equations are solved iteratively by the SIMPLEC procedure (Patankar, [31]), with an iterative line Gauss-Seidel solver on a staggered grid. The staggered grid uses a u-velocity mesh with one fewer column than the pressure mesh (see Fig. 3b). A requirement of the u-velocity mesh is that the centers of the left and right columns of fictitious u-velocity control volumes are located along the left and right edges of the left-most and right-most control volumes, respectively. Grid refinement is performed near the droplet/air interface to improve accuracy of interfacial flux terms.
4. Results and Discussion
In this section, three example problems will be presented, namely (i) transient Poiseuille flow in a microchannel (validation problem), (ii) droplet flow in an open-end microchannel (comparison against experimental data) and (iii) droplet flow in a closed-end microchannel (application problem). Problem parameters and thermophysical properties are summarized in Tables 1, 2 (case 2) and 3 (case 3).
(i) Case 1: Transient Flow in a Microchannel with Open Ends and a Closed Top Surface
The numerical formulation was initially validated through simulation of transient Poiseuille flow in a microchannel. The flow is induced by a suddenly applied pressure gradient at time t = 0. Water is enclosed between two parallel plates separated by a distance of H = 30 microns. Initially (t = 0), the liquid is motionless and a constant pressure gradient is then applied across the channel, thereby setting the fluid into motion. The pressure drop across the channel was specified as 1 Pa and the time step size is 0.0001 ms.
The analytical solution for transient developing channel flow is given by
[image] (30)
This analytical result will be compared against predicted velocities from the finite volume method (FVM). No-slip conditions along the top and bottom walls and a zero flux along the inlet and outlet faces of the channel were applied for both velocity components. At the channel inlet, the pressure was specified in the wall control volumes as DPspec. The length used in the pressure gradient term is the distance between two pressure specification points in the staggered grid.
Figure 4 illustrates close agreement between analytical and predicted velocities at different times in the microchannel. The spatial velocity profile is symmetric about the centerline of the microchannel, so only results in a half-portion of the microchannel are depicted. For each time increment of plotted data (0.0002 ms), the fluid velocity in the core of the microchannel increases by about 6.7 microns/s. During this period, the thickness of the boundary layer increases due to transient diffusion of momentum along the wall. The flow becomes fully-developed when boundary layers along each wall merge in the centerline of the microchannel. Such fully-developed conditions are shown in Fig. 5. For both developing flow (Fig. 4) and fully-developed flow in the microchannel (Fig. 5), close agreement between analytical and predicted FVM results is obtained.
The solution error of velocity was calculated at each location and time step as follows,
[image] (31)
where up and ua refer to the predicted (FVM) and analytical velocities, respectively. A maximum error of about 8% occurred near the wall in the first time step. This error decreased with each time step and the steady-state results were nearly identical to the analytical solution. The average u-velocity error within the domain was below 0.05% at the steady-state. The steady-state centerline velocity may be calculated in the following manner,
[image] (32)
which yields a centerline velocity of 3.75 mm/s. Additional sensitivity results and grid refinement studies were conducted with 20 × 20 and 20 × 80 mesh simulations. For both grids, the percentage error was less than about 1% at different locations within the microchannel, so grid-independent results were confidently obtained with the 20 × 20 mesh discretization.
5. Conclusions
This article has reported new simulation data involving thermocapillary droplet motion in a closed microchannel. A finite volume method with a moving grid was developed to predict the coupled pressure/velocity fields at the droplet/air interface. The implicit pressure/velocity coupling yields new data of fluid re-circulation in the corners of the micro-droplet. This interfacial pressure becomes coupled to the re-circulating velocity field within the micro-droplet. Adaptive mesh refinement is applied at the moving interface with control points to adjust the grid spacing. The predicted results indicate that the external pressure difference across the micro-droplet decreases nearly linearly during the heating period. Predicted data in this paper has provided useful new insight regarding the fluid dynamics of thermocapillary droplet transport in microchannels.
Acknowledgements
Support from the Natural Sciences and Engineering Research Council of Canada, as well as a University of Manitoba Graduate Fellowship (P. Glockner), is gratefully acknowledged.
References
[1] S. Wu, J. Mai, Y. C. Tai, C. M. Ho, Micro Heat Exchanger Using MEMS Impinging Jets, Proceedings of the 12th Annual International Workshop on Micro Electro Mechanical Systems, pp. 171 – 176, Orlando, FL, January 17 – 21, 1999
[2] K. Vafai, L. Zhu, L., Analysis of a Two-Layered Micro Channel Heat Sink Concept in Electronic Cooling, International Journal of Heat and Mass Transfer, 42 (1999) 2287 – 2297
[3] S. Lim, H. Choi, Optimal Shape Design of a Pressure Driven Curved Micro Channel, AIAA Paper 2004-624, AIAA 42nd Aerospace Sciences Meeting and Exhibit, Reno, NV, 2004
[4] P. Dutta, A. Beskok, T. C. Warburton, Electroosmotic Flow Control in Complex Microgeometries, Journal of Microelectromechanical Systems, 11 (1) (2002), pp. 36 – 44
[5] C. J. Kim, The Use of Surface Tension for the Design of MEMS Actuators, Nanotechnology: Critical Assessment and Research Needs, pp. 239 – 246, edited by S. M. Hsu and Z. C. Ying, Kluwer Academic Publishers, Boston, MA, 2003
[6] V. Singhal, S. V. Garimella, A. Raman, Microscale Pumping Techniques for Microchannel Cooling Systems, Applied Mechanics Reviews, 57 (2004) 191 – 221
[8] S. Troian, Thermocapillary Flow on Patterned Surfaces: A Design Concept for Microfluidic Transport, ASME IMECE Microfluidics Symposium, New York, November, 2001
[9] G. F. Naterer, Adaptive Surface Micro-Profiling for Microfluidic Energy Conversion, AIAA Journal of Thermophysics and Heat Transfer, 18 (4) (2004) 494 – 501
[10] Y. Agata, K. Nishino, K. Torii, Characteristics of Surface Velocity in Oscillatory Thermocapillary Convection, Thermal Science and Engineering, Heat Transfer Society of Japan, 11 (4) (2003) 11 – 13
[11] N. T. Nguyen, S. T. Wereley, Fundamentals and Applications of Microfluidics, Artech House Publishers, Norwood, MA, 2002
[12] G. F. Naterer, W. Hendradjit, K. J. Ahn, J. E. S. Venart, Near-Wall Microlayer Evaporation Analysis and Experimental Study of Nucleate Pool Boiling on Inclined Surfaces, ASME Journal of Heat Transfer, 120 (3) (1998) 641 - 653
[13] G. F. Naterer, Reduced Flow of a Metastable Layer at a Two-Phase Limit, AIAA Journal, 42 (5) (2004) 980 - 987
[14] G. F. Naterer, Temperature Gradient in the Unfrozen Liquid Layer for Multiphase Energy Balance with Incoming Droplets, ASME Journal of Heat Transfer, 125 (1) (2003) 186 - 189
[15] G. F. Naterer, Heat Transfer in Single and Multiphase Systems, CRC Press, Boca Raton, FL, 2002
[16] H. Togo, M. Sato, F. Shimokawa, Multi-Element Thermo-Capillary Optical Switch and Sub-Nanoliter Oil Injection for Its Fabrication, Proceedings of the IEEE MEMS Conference, pp. 418 – 423, Orlando, FL, 1999
[17] J. Lee, C. J. Kim, Liquid Micromotor Driven by Continuous Electrowetting, Proceedings of IEEE MEMS Conference, Heidelberg, Germany, pp. 538-543, 1998
[18] J. Lee, C. J. Kim, Microactuation by Continuous Electrowetting Phenomenon and Deep RIE Process, Proceedings of MEMS Conference, DSC vol. 66, ASME IMECE Conference, pp. 475-480, Anaheim, CA, , 1998
Property
Value
Channel Height (d)
32 mm
Channel Depth (b)
500 mm
Droplet Length (L)
5276 mm
Receding Contact Angle (qR)
47.6o
Advancing Contact Angle (qA)
49.8o
Minimum Temperature Difference (DTmin)
19 oC
Geometry Constant (S)
37
Dynamic Viscosity (m)
26 × 109 Ns/m2
Table 1: Thermophysical properties and problem parameters (case 2: S = 37)
[image]
(a)
[image]
(b)
Figure 1: Schematic of (a) thermocapillary pumping and (b) recirculation cells
[image]
Figure 2: Schematic of adaptive grid refinement in the closed microchannel
[image]
(a)
[image]
(b)
Figure 3: Schematic of (a) overall grid and (b) pressure/velocity grids
[image]
Figure 4: Predicted velocity profile in a microchannel with open ends (case 1)
[image]
Figure 5: Steady-state velocity profile in a microchannel
[image]
Figure 6: Droplet velocity in a microchannel with open ends and top surface (case 2)
[image]
Figure 7: Droplet velocities at varying pressure differences
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