Thermal Creep Slip 28032011

1

BOUNDARY ELEMENT METHOD FOR EVALUATION OF THERMAL CREEP FLOW IN

MICROGEOMETRIES

C. Nieto1

, H. Power*2

and M. Giraldo1

1

Universidad Pontificia Bolivariana, Grupo de Energía y Termodinámica, Facultad de Ingeniería

Mecánica 2

The University of Nottingham, Faculty of Engineering, Department of Mechanical, Materials and

Manufacturing Engineering.

* Corresponding Author

ABSTRACT

Flow in rarefied gases can be caused by a temperature gradient in the tangential direction without the

presence of any pressure gradient. In that case, the fluid is subjected to a driven force that allows its flow

from colder to hotter regions. This phenomenon is known as thermal creep and has gained importance in

recent years in connection with gas flow in micro scale systems. Examples of Micro Electro Mechanical

Systems (MEMS) based on the thermal creep phenomenon, like Knudsen type compressor and gas

vacuum pumps, can operate subjected to temperature gradients without moving parts. Prediction of flow

behaviour in micro systems had been done by using continuum based models (i.e. Stokes system of

equations) under appropriate boundary conditions. Maxwell boundary condition accounts the slip velocity

due to tangential shear rate and tangential heat flux effects. These boundary conditions has been

misapplied to predict flow behaviours reported in some numerical works dealing with flow fields

confinement by curved or moving surfaces, due to wrong evaluation of the local tangential shear rate at

the wall surfaces. In this work a boundary integral equation formulation for Stokes slip flow, based on the

normal and tangential projection of the Green’s integral representational formulae for the Stokes velocity

field, which directly incorporates into the integral equations the local tangential shear rate and heat flux at

the wall surfaces, is presented. The tangential heat flux is evaluated in terms of the tangential gradient of

the temperature integral representational formulae presenting singularities of the Cauchy type, which are

removed by the use of an auxiliary field. These formulations are used to evaluate microchannel flow

performance with complex geometries often encountered in MEMS subjected to jump temperatures.

KEYWORDS

Linear slip boundary conditions, boundary element method, rotating mixers, thermal creep

1. Introduction

The role of Nano-scale fluids has become more prevalent after the introduction of chemical and biological

analysis and micro electrometrical system fabrication technologies, as they are the key to the mechanism

that takes place within these new fields of technologies. When geometry devices are scaled down, the

surface-to-volume ratio increases dramatically so that the surface related phenomena become increasingly

dominant, e.g. micro heat exchangers and micro mixers present higher heat and mass transfer rates than

macro systems of equal capacity [1]. Some new features emerge when mechanical structures are

sufficiently small, and it becomes important to understand the various types of interactions that arise

between the fluid flow constituents and the solid surfaces that contain it. For instance, a phenomenon

known as the slip flow regime could emerges as consequence of an insufficient number of molecules in

the sampling region [2] or hydrophilic and hydrophobic recovering quality of surfaces in contact with

fluids [3]. Neto et al [4], on a review article, highlights the need of properly describing the flow near the

fluid-solid interfaces, because of its relevance to a wide range of applications, from lubrication to micro

fluidics.

In the case of micro fluid flow of rarefied gases, the thermal creep or transpiration also appears as a

consequence of inequalities in temperatures that force the fluid to slide over a surface from colder to

hotter regions [5]. Rarefied gas flows are general associated with low-density conditions, such as high-

2

altitude and vacuum. However, the small length scales commonly encountered in micro fluid flows imply

that rarefaction effects will be important at much higher pressures, for more details see [6].

Taking into account the advantages of the Boundary Integral Methods for the numerical simulation of

micro fluid flow under slip condition, this work presents a numerical approach based on the use of the

normal and tangential projections of the velocity integral representation formula for Stokes flows,

resulting in weakly singular mixed system of integral equations of the first and second kinds for the

normal and tangential components of the surface traction. The proposed approach is used to study flows

in a different geometries regular present micro fluid flow systems (i.e. eccentric Couette rotating cylinder,

cavity flow, stepped channels) under linear slip conditions with the thermal jump effect at surfaces. The

development of these formulations permits the inclusion of linear slip conditions into the boundary

integral expressions directly, allowing the evaluation of micro flow in plane and curved geometries not

subjected to symmetry conditions. Validation of numerical results is done by direct comparison with

exact solution for the tangential heat flux and interpolation of the temperature derivatives in tangential

direction from the temperature filed.

2. Governing equations

The fluid flow of an incompressible viscous Newtonian fluid with viscosity , can be modelled by the

Navier-Stokes equations of motion, written in dimensionless form as follows:

(1)

being the Reynolds number, defined by where and are some characteristic velocity and

scale length respectively. Further, the fluid flow also satisfy the continuity equation, written in

dimensionless form

(2)

being the velocity vector, the pressure and the Kronecker delta. At micro scale conditions, the

fluid flow occurs at small Reynolds numbers , since the characteristic length scale of the

systems is below 100 µm and the velocity flow is of the order of 1 mm/s [7]. Then, the above system of

equations reduces to its linear quasi-static form defined by the Stokes system of equations, written as

(3)

The use of governing equations based on the continuum for the study of micro and nano flow, requires the

definition of appropriate velocity slip conditions at the wall surfaces to still satisfy the continuum

hypothesis. In order to simulate micro fluid flow of rarefied gases, besides the slip condition due to the

tangential projection of the wall shear rate, the thermal creep effect must be considered. Thermal creep is

a consequence of inequalities in temperatures that forces the fluid to slide over a surface from colder to

hotter regions, [5]. Rarefied gas flows are normally correlated with low-density conditions, for instance

high altitude and vacuum. Nevertheless, the small length scales are commonly encountered in micro fluid

flows, which imply that rarefaction effects will be significant at much higher pressures, for more details

see [6]. In this case, Maxwell slip condition relates the tangential gas velocity at the boundary contours to

the tangential shear stress and heat flux, as:

(4)

with

(5)

where and are respectively the i components of the normal and tangential vectors to a boundary

surface and is the dimensionless tangential heat flux at the boundary contours. In the

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above expression, is a coefficient constant proportional to the product of the Reynolds number times

the square of the Knudsen number and inversely proportional to the Eckert number.

For the implementation of the slip condition in the velocity integral representational formula for Stokes

flows, the tangential shear rate at solid-fluid interfaces can be evaluated in terms of the surface traction

force which its tangential projection can be expressed as:

(6)

where for consistence with our formulation, the above expression has been written in dimensionless form,

with a characteristic traction force . To complete the boundary conditions at the solid surfaces, the

following non-flux condition across any points x on the boundary surfaces needs to be considered:

(7)

where the following integral relation needs to be satisfied, according to the conservation of mass for an

incompressible fluid:

(8)

with S as the union of the external and internal surfaces and , respectively. For the case of

pressure driven flows, the inlet boundary is defined by a pressure inlet in the normal direction as follows:

(9)

The solution of equation (3) under boundary condition (5) requires the evaluation of the temperature field

with the aim of obtaining the temperature gradient in the tangential direction at the boundary contours.

The temperature field is found by solving the energy equation, written here in dimensionless form as:

(10)

where is the Peclet number and the thermal conductivity. The Peclet number can

be expressed in terms of the Reynolds number through the Prandtl number as . For

cases of , our previous assumption of small Reynolds number implies that , reducing

the above energy equation (at the first order of approximation in ) to the Laplace equation, i.e. quasi-

static approximation;

(11)

In the slip flow region, the temperature field is also subject to a jump condition at the boundary contours,

[8], given by the following equation:

(12)

with as the normal heat flux to the surface and is a coefficient constant function of the

Knudsen and Prandtl numbers and the energy accommodation coefficient. The above jump condition

represents a Robin type boundary condition for the temperature field.

3. Integral formulation for the normal and tangential heat flux with thermal creep

The integral representation formula for the energy equation defined in (11) (i.e. Laplace equation) is

defined as:

(13)

The kernels of the above integral representational formula are for the surface single

layer potential and for the surface double layer

potential, which are weak singular and regular, respectively, where is the Euclidean distance between

point and , i.e. . The temperature jump condition (12) is replaced into the integral

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representational formula for the temperature (13), leading to the following boundary integral equation for

the unknown normal heat flux:

(14)

The above integral equation can be solved by a classical BEM procedure, quadratic scheme in the present

work, been possible to be written in matrix form as:

(15)

where is the number of elements with surface and the matrices of influence coefficients

and

are defined as:

(16)

(17)

where the integral densities along the element are approximated by quadratic elements,

(18)

being , , the values of on the three nodes of the element with as the local coordinate

of an isoparametric element and interpolation functions given by

(19)

The surface heat flux in the normal direction obtained through the solution of the equation (15) can be

substituted into the temperature jump condition (12) to find the corresponding fluid temperature at contact

with the solid surfaces.

The tangential heat flux at the boundary contours is obtained by taking the limiting value of gradient of

the temperature field, at an internal point, approaching a boundary point, and then multiplying the

resulting expression by the surface tangential vector. In this way the surface gradient of the temperature

field is given by:

(20)

where the first integral is a singular integral of the Cauchy type and the second a hyper singular integral.

However by multiplying the above relation by the surface tangential vector, the resulting integral relation

for the tangential derivative only possesses singularities of the Cauchy type:

(21)

where the limiting value of the above kernels as the distance r tends to zero is given by:

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The evaluation of the above integrals needs to be considered in the sense of Cauchy principal value.

Various regularization methods to reduce the order of singularity of this type of integrals are available in

the literature; most of them are based on expanding the kernel around the singular point and subtraction of

the most singular part as original suggested by Miklin [9]. A simple alternative is to find the integral

representational formulae of a known potential field, with the same singularity than the field in

consideration. Subtraction of both integral equations removes the most singular part of the integral

operators (for more details see [10]). In general this is not a simple task, but in the present case it is

possible to define the potential

(22)

solution of the Laplace equation , with values of the constants and equal to the

values of the temperature field (13) and its gradient at a given evaluation point . Therefore at the

evaluation point and . By subtracting the integral

representation formula of the tangential derivative of both fields, we obtain:

(23)

At an evaluation point , the above integral relation reduces to the following regular integral equation:

(24)

Given the values of the temperature and its normal derivative , obtained from (12) and (14),

equation (24) provides a linear relationship between the directional derivatives of the temperature at the

evaluation point , i.e. between the values of and . A second relation between

these two values is given by the known normal derivative at , i.e. . From these two expressions, it is possible to obtain the directional derivatives of the

temperature field at each boundary point and find the corresponding tangential derivative to be used in the integral equation (35) and (36) for the normal and

tangential components of the surface velocity in order to determine the corresponding effect of the

thermal creep on the flow field.

4. Integral formulation for slip Stokes flows with thermal creep

The Stokes velocity field has the following direct integral representation formulae for an arbitrary point

in a closed domain filled with a Newtonian fluid [11]:

(25)

where is a constant dependent on the position of the source point. For internal points , for

point at a smooth boundaries and for external points .

The kernels of the above integral representational formula are the Stokeslet and its corresponding surface

tractions or Stresslet, which two dimension expressions are given by:

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(26)

(27)

Normal and tangential projections of the direct boundary integral representational formula for the Stokes

velocity field can be obtained after multiplying equation (25) by the local normal and tangential vectors

and , at a surface point, and expressing the velocity and surface traction vectors in terms of their

normal and tangential components , and , , i.e. and with and , ), the following surface

integral equations are obtained, for the normal and tangential projections, respectively [12]:

(28)

(29)

These two equations represent a system of surface integral equations for the normal and tangential

velocities and surface tractions , , that can be written in matrix form as:

(30)

where and correspond to the normal and tangential projections vectors respectively and the

matrices of influence coefficients

and

are defined as:

(31)

(32)

where the integral densities along the element are approximated by quadratic elements like in previous

section. The equation (30) is finally written as:

(33)

taking into account that

is defined by:

(34)

The limiting value of the integral kernels in (28) and (29) as the radius r tends to zero is given by:

7

with only logarithmic singularities on and

. In the above limiting

values, the term tends to as the radius r tends to zero, where is the curvature at a

point x on the boundaries (see Courant and Hilbert [13], Vol. 2, page 299).

The equation (33) can be solved when the normal velocity (i.e. boundary condition defined in (7)) and

pressure (i.e. boundary condition defined in (9)) are known, been possible to solve for the unknown

values of , , . The corresponding values of and are directly obtained from the

relations and .

In order to consider the slip regime with thermal creep the slip condition with thermal creep effect,

equation (4), is substituted into the boundary integral equation (28) and (29) leading to the following

system of integral equations for the normal and tangential components of the unknown surface traction:

(35)

and

(36)

that can be written in matrix form as:

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(37)

and the matrices of influence coefficients

and

are those defined in (31) and (32), and the

coefficient is defined as:

(38)

The additional matrix of influence coefficients

for the thermal creep term is defined as follows:

(39)

Finally, the velocity field is obtained by solving equations (35) and (36) at the boundaries subjected to

slip condition with thermal creep (i.e. solid wall boundaries). The tangential heat transfer to be replaced at

the last term on the right of equations (36) and (37) is evaluated after solving equation (24) for the

temperature gradient components, and , where the values of the normal heat flux

and the temperature jump are obtained through the solution of (14) and the evaluation of (12),

respectively. At the inlet and outlet boundaries, like in the case of pressure driven flows, the velocity and

pressure values are obtained by solving equations (28) and (29), while the temperature is obtained

thorugh the solution of the energy equation (13).

5. Numerical results

In this section is presented the performance of the proposed numerical scheme in terms of the normal and

tangential projections of the direct boundary integral formulation for Stokes velocity fields to predict the

behaviour of slip flow with thermal creep in micro geometries. Initially, a temperature field that satisfy

Laplace equation is used to evaluate the behaviour of the boundary integral formulation proposed for the

solution of the tangential heat flux evaluation in both curved (circle) and corner geometries (square).

Having validated the proposed numerical approach for the tangential heat flux, numerical results for the

eccentric Couette rotating mixer, the cavity flow, and the backward and forward facing steps are

presented with the aim of show the performance of the formulation under mixed boundary conditions.

Additionally, the tangential heat flux at the boundary contours obtained with the integral formulation is

compared with that obtained by direct differentiation of the temperature field in the tangential direction of

the surface interpolation functions, at each element. Since the temperature at the elements in terms of

the interpolation functions is given by:

(40)

then the directional derivative of the temperature field along its surface can be written as:

(41)

where the super index in the temperature denotes the node position along the quadratic elements, and the

interpolation functions derivatives are given by:

(42)

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From the expression (40), the tangential heat flux can be evaluated from the temperature field at each

boundary point. These results can be used to evaluate velocity field defined by (3) under the thermal

creep effect when is replaced in boundary condition (4) as shown before.

5.1 Assessment of tangential heat flux boundary integral formulation

The assessment of tangential heat flux boundary can be done analytically by using a proposed expression

for the temperature field that satisfy the Laplace equation and then take the tangential derivatives to

compare with results obtained through the numerical approach. In this order of ideas, the following

temperature field is a solution of the Laplace equation (11):

(43)

where the direction derivatives in and directions are, respectively:

(44)

Two cases are proposed with the aim to probe the behaviour of the numerical approach for the evaluation

of the tangential heat flux in curved and corner geometries. The first one is a circle of radios and

centre at the origin, defined by: . For this geometry, the normal and tangential vector are

and , leading to the following expressions for the normal an tangential

heat flux, respectively:

(45)

(46)

The second case is a square centred at the origin with sides’ length , where the normal and

tangential vector and its respectively normal and tangential derivatives at each face are presented in Table

1.

Table 1. . Normal and tangential vectors and temperature derivatives for a square centred at the

origin

Face

Results for temperature, normal and tangential heat flux at the circle’s boundary are plotted in Figure 1,

where open circles represent the numerical solution by the boundary integral approach and the solid line

correspond the expression (45) and (46). In Figure 2 is plotted the same results for the square centred at

the origin. In both cases, the boundary integral approach satisfactory matches the exact solution of the

tangential heat flux. In that way, we can verify that the boundary integral formulation defined in equation

(24) for the tangential heat flux can be used for both types of curved and corner flows.

5.2 Eccentric Couette flow between rotating cylinders

Since the numerical approach for the evaluation of the tangential heat flux has already validate with the

previous examples and the normal and tangential projections of the boundary integral approach has been

used before for the study of linear slip flow (in [12] we present a comparison between numerical results

and exact solution for both concentric and eccentric Couette mixers under the linear slip condition), in

this case is considered the flow between eccentric cylinders as shown in Figure 3. The eccentric Couette

mixer has been previously studied by several authors (see [14], [15] and [16]) due its ability to develop

complex laminar flow: e.g. pressure gradients, Taylor vortices, and eddies. In the present work, an

extension to the numerical approach of the micro fluid flow with BEM is done with the aim to study the

effect of the thermal creep with linear slip condition over the flow behaviour.

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(A) (A)

(B) (B)

(C) (C)

Figure 1. Assessment of boundary integral

formulation for the evaluation of the boundary

temperature (A), normal heat flux (B) and tangential

heat flux (C) at a circle of radii . (solid line:

exact solution; open circle markers: boundary integral

approach)

Figure 2. Assessment of boundary integral

formulation for the evaluation of the boundary

temperature (A), normal heat flux (B) and tangential

heat flux (C) at a square of side 2 by 2 (solid line:

exact solution; open circle markers: boundary integral

approach)

Figure 3. Eccentric Couette rotating cylinders for evaluation of slip boundary and thermal creep effect

Figures 4 to 6 show the obtained fluid temperature, normal and tangential heat flux distribution at the

rotor and housing boundaries of the eccentric Couette mixer with a given constant temperature at the solid

walls. Two different cases are reported in the figures, corresponding to a hotter and colder rotor, i.e.

and respectively. Values for the same variable under no slip-no thermal creep

conditions are also plotted, showing how the consideration of the thermal creep effect, drastically modify

the patterns of the heat transfer variables. The temperature profiles in Figure 4 (A) and (B) show the

development of temperature of profile under the thermal creep effect. It can be noted that as the no

thermal creep effect is considered, the nondimensional temperature shift down. This can be atribuitted to

the increase in the jump temperature at the wall, and then less heat is transferred from the wall to the

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adjacent fluid as can be observed in Figures 5 (A) and (B). In Figure 6 can be observed how the results

for the tangential heat flux through the boundary integral approach (filled circle markers) match those

obtained by interpolation evaluation (solid line), allowing us to conclude that the first one satisfactory

reproduce thermal creep effect, since the last one results from the direct differentiation of the

interpolation functions as presented before.

(A)

(B)

Figure 4. Temperature profiles for the eccentric Couette mixer ( with thermal creep and

(solid line with open markers) and non thermal creep effect and (solid line with

open markers): (A) and , (B). Circle markers: inner rotor; triangle markers: static

housing

(A) (B)

Figure 5. Normal heat flux for the eccentric Couette mixer ( with thermal creep and

(right side - solid line with open markers) and non thermal creep effect and

(left side - solid line with filled markers): (A) and , (B). Circle markers: inner rotor;

triangle markers: static housing.

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(A)

(B)

Figure 6. Tangential heat flux for the eccentric Couette mixer ( with thermal creep and

(solid line with open markers): (A) and , (B). Circle markers: inner rotor;

triangle markers: static housing.

In Figure 7, we report the corresponding streamlines inside the mixer in the case of a rotating rotor and

the interaction between both the slip regime and the thermal creep induced fluid motion. Figure 7 (A)

presents the reference case of the no slip-no thermal creep condition with the characteristic vortex

formation at the mixers bottom. The consideration of the slip condition tends to reduce the vortex size due

to the reduction in momentum difussion at both rotating and stationary boundary walls (see Figures 7 (B))

and tends to dissapear as the slip condition is increased (e.g when is aumented from to

as shown in Figure 7 (C)). In the other hand, the inclusion of the thermal creep with no slip condition

conduce to the original recirculation zone observed under only no slip condition, plus an asymmetric

recirculation region around the rotor side, where the position of the recirculation changes according to the

direction of the heat flux, i.e. or (see Figures 7 (G) and (M)). Similar recirculation

patterns are reported by Lockerby et.al. [6] in the case of eccentric circular cylinders, however in their

analysis they do not use a jump temperature condition, as in (12), but instead they use a high order

thermal stress slip condition, where the tangential velocity is proportional to the tangential derivative of

the normal heat flux. As the slip over boundaries is considered, the original recirculation zone disappear

at values of , while the second recirculation induced by the thermal creep increase its size

detaching from the rotor’s head and moves to the mixer´s bottom (see Figures 7 (H) and (N)). Finally,

when the slip length is increased from to , a recirculation zone corresponding in position

to that one observed in the reference case (i.e. the no slip one condition) reapers while the thermal creep

one remains as beforehand (see Figures 7 (I) and (O)). The direction and position of the recirculation

zones are associated with temperature gradient direction. In Figures 7 (D) to (F) and (J) to (L) the

streamlines for the both conditions, i.e. or , are poltted. The flow pattern in both cases

shown iddentical behviour but the flow direction (see vector field) is inverted since the temperature

gradient is inverted too.

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(A) (B) (C)

(D) (E) (F)

(G) (H) (I)

(J) (K) (L)

(M) (N) (O)

Figure 7. Streamlines for the eccentric Couette mixer. (A) to (C) are for , , (

(A), (B), (C)); (D) to (F) are for , , : ( (D),

(E), (F)); (G) to (I) are for , , , ( (G), (H),

(I)); (J) to (L) are for , , , : ( (J), (K), (K)); (M) to

(O) are for , , , ( (M), (N), (O)).

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5.3 Cavity flow

To test the developed numerical scheme for slip flow with thermal creep in corner geometries, we also

considered the problem of a shear driven cavity flow (see Figure 8). In this type of flow, the motion is

caused by horizontal velocity imposed at the upper horizontal wall. In our case, the slip flow condition is

considered and the thermal creep effect is prompted by a difference in temperatures between the

horizontal and the vertical boundaries. In this work two cases are considered: and , where

refers to the horizontal flat walls and is defined at the vertical flat walls. The effect of the variation

in the heat flux will be analysed over the velocity field.

Figure 8. Cavity flow for evaluation of slip boundary and thermal creep effect. Th temperature at

horizontal boundary condition and Tv temperature at vertical boundary condition

In Figures 9 and 10 results for the temperature, normal and tangential heat flux are plotted for both

temperature cases, respectively. The temperature profiles plotted in Figure 9 (A) and Figure 19 (A)

present a similar behaviour than for the eccentric Couette flow, since the nondimensional temperature

shift down as the thermal creeping appears due to reduction in the normal heat transfers between the walls

and the adjacent fluid (see Figure 9 (B) and Figure 10 (B)). It possible to make a similar conclusion

regarding the tangential heat flux since it was evaluated by both boundary integral and interpolation

approach of the tangential heat flux, showing a perfect match of the first one in relation with last one, as

can be observed in Figure 9 (C) and Figure 10 (C).

The streamlines for the shear driven cavity flow are plotted in Figure 11. Figures 11 (A) to (C) presents

the results for the flow without the thermal effect, where Figure 11 (A) correspond the well-known no slip

condition that had been studied for other authors before [17]-[18]. As the slip is considered the

recirculation pattern is slightly modified respect to the no slip condition due to the reduction in the

momentum diffusion, tending to produce an almost symmetric recirculation vortex (see Figures 11 (B)

and (C)). Once the thermal creep is considered it leads to the formation of four recirculation zones into

the cavity for both the no slip and slip conditions. The dominated recirculation zone is up to the direction

of the heat flux, i.e. or , as can be observed in Figures 11 (G) to (I) for the case when

and in Figures 11 (M) to (O) for the case when . For both the no slip and slip condition

with slip length of the recirculation cells are asymmetrical, while when the slip length is

increased to , the four recirculation zones becomes almost symmetrical showing that the thermal

creep induced movement can overcome the shear driven flow. The degree of asymmetry in the vortex

formation (e.g in the cases when and ) can be explained by the flow formation associated

to the only thermal creep driven flow ( ) as shown in Figures 11 (D) to (F) for and in

Figures 11 (J) to (L) for . Identically to the eccentric Couette flow, in the case the flow pattern for

both temperature cases are equivalents but the flow direction is inversed as the temperature gradient is

opposite leading to a flow behaviour principally dominated by thermal creep effect in both stationary and

moving wall cases.

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(A) (A)

(B) (B)

(C) (C)

Figure 9. Evaluation of dimensionless temperature (A) and normal (B) and tangential

(C) heat flux with boundary integral formulation (open circles) and interpolation

differentiation (solid line) approaches for the cavity flow with: = 1. Temperature of

horizontal boundaries is greater than the vertical boundaries

Figure 10. Evaluation of dimensionless temperature (A) and normal (B) and tangential

(C) heat flux with boundary integral formulation (open circles) and interpolation

differentiation (solid line) approaches for the cavity flow with: = 1. Temperature of

vertical boundaries is greater than the horizontal boundaries

16

(A) (B) (C)

4

(D) (E) (F)

(G) (H) (I)

(J) (K) (L)

(M) (N) (O)

Figure 11. Streamlines for the cavity flow. (A) to (C) are for , , ( (A),

(B), (C)); (D) to (F) are for , , , : ( (D), (E),

(F)); (G) to (I) are for , , , : ( (G), (H), (I));

(J) to (L) are for , , : ( (J), (K), (L)); (M) to (O) are

for , , : ( (M), (N), (O))

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The horizontal velocity distribution at vertical central line and vertical velocity distribution at horizontal

central line are plotted in Figure 12 (A), that shown a comparative behaviour with results presented in

previous works [17]-[18] for the no slip condition . As the slip condition is considered, the

velocity profiles at both vertical and central lines are modified due to the expected reduction in

momentum transfer at the fluid-walls interface. The result plotted in Figure 12 (B) correspond to the case

when thermal creep is analyzed at the central lines. The velocity profiles for this cases are the same for

the both temperature gradient cases due the tangential heat flux presents a zero value at this points (see

Figure 9 (C) and Figure 10 (C)) corresponding to a maximum and minimum values in the temperature

distribution as shown in Figure 9 (A) and Figure 10 (A), not allowing any thermal creep driven force at

this points.

(A)

(B)

Figure 12. Horizontal velocity distribution at vertical central line and vertical velocity distribution at

horizontal central line with slip condition and (A), with slip condition with thermal creep

(B).

5.4 Backward and forward facing steps

Micro channel has been studied previously by a few authors [19]-[20] due to their significant role in

micro fluidic industry. In this case, the simulation of micro fluid flowing through backward and forward

facing stepped channels with the geometry as shown in Figure 13 is studied due to its strong presence in

various applications such as Y and T type micro mixers [22],[23], heat exchangers [8], Knudsen type

compressor and gas vacuum pumps [21]. The analysis of both stepped channels is considered under linear

slip condition coupled with the thermal creep effect. In this type of problem geometry, we will

concentrate on the flow behaviour under three slip length conditions ( ) and will see the effect

of the thermal creep movement induced over the fluid flow behaviour. In a similar way like in previous

problems, two different temperature conditions will be imposed (temperature of the lower boundary, ,

greater than the upper boundary , , and temperature of the lower boundary greater than the upper

boundary) to allows the study of the heat flux direction effect over velocity patterns. At the inlet and

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outlet boundaries a Neumann type boundary condition is applied , meaning that we will

consider that temperature field will be fully developed inside the stepped channels.

(A) (B)

Figure 13. Backward facing step (A) and forward facing step (B) channels for evaluation of slip flow

with thermal creep effect

The temperature profiles in Figure 14 (A) and (B) show the development of temperature of profile under

the thermal creep effect. Similar than to the previous results for the eccentric Couette and the cavity flow,

it can be noted that as the no thermal creep effect is considered, the nondimensional temperature shift

down. The reduction in the temperature can be associated to two major aspects: the slip condition implies

that more fluid passes through the channel (i.e., the heat source at the wall must heat a greater amount of

fluid); as the thermal creep effect is considered (i.e. is changed from 0 to 1), the jump in temperature

at the wall will increase, which means that less heat is transferred from the wall to the adjacent fluid).

(A) (B)

Figure 14. Temperature profiles at the facing steps (backward and forward) under different temperature

gradients: (A) and (B). Solid line open circles are for no thermal creep results; dashed

line filled circles are for thermal creep results ( ).

The flow pattern for the no slip-no thermal creep flow is shown in Figures 15 (A) and 16 (A) for the

backward and forward facing step channels, respectively. The results follow a very reasonable trend since

we are dealing with the incompressible viscous Stokes flow and no recirculation zones are expected after

(before) the backward (forward) step. As the slip length increases, the velocity of the flow increases

proportionally with the slip length, tending to completely attach to the step’s walls despite the flow

direction (backward and forward) as can be observed in Figure 15 (B) and 16 (B).

Figures 15 (E) and (F), illustrates the effect of thermal creep as recirculation appears for the backward

facing step when . In Figure 15 (E) can be observed two thermal creep induced recirculation zones

downstream the step (one at the bottom and another at the top) for the settings and and

, which can be explained with the flow pattern present in the no pressure gradient-thermal

creeping flow plotted in Figure 15 (C). This behaviour tends to reduce its effect when the slip condition is

included leading to the elimination of the upper vortex (see Figure 15(F) for and and

), despite the induced vortex observed when the temperature gradient is the only driven force

19

(Figure 15 (D)). This can be understood when the flow direction is observed in both cases when

and or , since the former is against the flow while the latter contributes to it.

When the heat flux direction is inverted (i.e. ) and the no slip condition and the thermal creep are

considered, one recirculation zone appears downstream the step´s bottom rear as can be observed in

Figure 15 (I). As the thermal creep effect is included alongside the slip regime (Figure 15 (J)), the

previous behaviour is inverted and the thermal creep recirculation moves upstream the step. This explain

by the inversion in flow velocities respect to the previous case when thermal creep is the only driven

force, as can be observed in Figures 15 (G) and (H). Similar behaviours are observed in the case of the

forward facing step. However, in these cases the formation of recirculation is given in a manner contrary

to what happens in the backward facing step as can be observed in Figure 16.

(A) (B)

(C) (D)

(E) (F)

(G) (H)

(I) (J)

Figure 15. Streamlines for the backward facing step. Results in (A) and (B) are under linear slip

condition, pressure gradient ( ) and no thermal creep condition , ( (A),

(B)); (C) and (D) are under linear slip condition, no pressure gradient ( ) and thermal creep

condition , with : ( (C), (D)); (E) and (F) are under linear slip

condition, pressure gradient ( ) and thermal creep condition , with : (

(E), (F)); (G) to (H) are under linear slip condition, no pressure gradient ( ) and thermal

creep condition , with : ( (G), (H)); (I) and (J) are under linear slip

condition, pressure gradient ( ) and thermal creep condition , with : (

(I), (J))

20

(A) (B)

(C) (D)

(E) (F)

(G) H)

(I) (J)

Figure 16. Streamlines for the forward facing step. Results in (A) and (B) are under linear slip condition,

pressure gradient ( ) and no thermal creep condition , ( (A), (B)); (C)

and (D) are under linear slip condition, no pressure gradient ( ) and thermal creep condition

, with : ( (C), (D)); (E) and (F) are under linear slip condition,

pressure gradient ( ) and thermal creep condition , with : ( (E),

(F)); (G) to (H) are under linear slip condition, no pressure gradient ( ) and thermal creep

condition , with : ( (G), (H)); (I) and (J) are under linear slip

condition, pressure gradient ( ) and thermal creep condition , with :

( (I), (J))

6. Conclusions

In this work, has been evaluated the thermal creep effect at micro scale flows. This phenomenon usually

takes place due to micro fluid flow of rarefied gases and could affect the velocity field due to a heat flux

and a temperature difference into the domain. An integral equation approach based on the normal and

tangential projections of the direct boundary integral representational formula for the Stokes velocity field

is developed for the numerical simulation of creeping flow under linear slip boundary conditions and

applied to analysis the performance of a three different cases under variable boundary conditions. The slip

condition was included in the boundary integral formulation by expressing the tangential shear rate in

terms of tangential component of the surface traction vector. The projection of the velocity integral

representational formula on the normal and tangential directions smoothest the singularity of the integral

kernels resulting only in a weak singular kernel of the logarithm type, which can be numerical integrated

by using Telles' transformation and standard Gaussian Quadrature formulae.

21

The integral formulation has been modified to account the thermal creep effect which is function of the

tangential heat flux. For the latter, an integral representational formulae was obtained by taking the

limiting value of gradient of the temperature field and multiplying the resulting relation by the surface

tangential vector, presenting only singularities of the Cauchy type. The order of singularity of this type of

integrals was reduced by finding the integral representational formulae of a known potential field, with

the same singularity than the field in consideration, reducing the expression to a regular integral equation

at the evaluation point. The tangential heat transfer attained through this formulation where satisfactorily

validated in the first instance with results obtained for a circular and square region shape where exact

solution was developed for a temperature field proposed as solution of the Laplace equation.

Additionally, a second comparison was done by evaluation of the temperature derivatives in tangential

direction by using the surface interpolation functions, at each element and the temperature field

obtained through the boundary integral representation of Laplace equation with temperature jump. In the

case, the boundary integral formulation proposed to evaluate the tangential heat transfer also matched the

results obtained by the interpolation approach values allowing us its use for the study of flows confined

by even both curved and corner geometries. The formulation gave appropriate results for geometries with

moving boundary conditions as well as with input and output boundary conditions.

ACKNOWLEDGEMENTS

The first author acknowledges the support of the PhD program at the Universidad Pontificia Bolivariana

and COLCIENCIAS, Colombia, under the support provided with the project "Desarrollo de geometrías

para aplicación industrial en microintercambiadores de calor" code 1210-479-21999, contract 436-2008.

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