PHYSICAL EFFECTS OF VARIABLE FLUID PROPERTIES ON GASEOUS SLIP-FLOW THROUGH A MICRO-CHANNEL HEAT SINK

Physical effects induced in micro-convective gaseous slip-flow due to variation in fluid properties are numerically examined in this paper. The problem is particularly simulated for slip-flow through a micro-channel heat sink (MCHS) having constant heat flux supplied from the wall under hydrodynamically and thermally fully developed flow (FDF) conditions. It is observed that the Nusselt number (Nu) for slip-flow is significantly higher than the no-slip-flow condition and Nu is significantly affected due to variable fluid properties (VFP). Four different cases of VFP are studied in order to investigate their effects individually. Pressure and temperature dependent density (ρ(p, T)) variation flattens the axial velocity profile in radial direction (u(r)) profile which promotes fastermoving particles close to the wall which considerably enhances Nu. The incorporation of temperature-dependent viscosity (μ(T)) variation marginally enhances Nu along the flow. Incorporation of temperature-dependent thermal conductivity (k(T)) variation highly augments Nu due to higher ρ and higher k fluid near to the wall and the incorporation of temperature-dependent specific heat at constant pressure (Cp(T)) variation reduces Nu due to lower k fluid near to the wall. The investigation also shows that the pressure drop significantly deviates from no-slip to slip condition. Furthermore, the effects of VFP on the gauge static pressure drop (Δpg) and slip velocity are also examined. The incorporation of μ(T) and k(T) variations trivially affects the Δpg and slip velocity. However, the incorporation of Cp(T) variation significantly affects the Δpg and slip velocity.


INTRODUCTION
Gas micro-convection is an important active research area in transport phenomena since it is the basis for a broad range of miniaturized high-performance applications like Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical Systems (NEMS). Many practical devices like ducts, valves, pumps, turbines, heat sinks, etc. have been shrunk towards the microscale. The characteristic size involved in such applications can vary from 1 mm to less than 1 micron. Microscale devices are finding an important place in our day-to-day lives; however, the fundamental science at the microscale is still not well known. The main problem to expect the flow through microand nano-scale channels can be attributed to the rarefaction effects which take place in the flow when the continuum approach breaks down as the characteristic length of the flow becomes comparable to the mean free path between molecules. The physical effects induced due to rarefaction influence the heat transfer (HT), velocity profile, and pressure drop (Δp) in the channels [1]. The Knudsen number (Kn) is a measure of the degree of rarefaction which is defined as the ratio of the molecular mean free path (λ) to the characteristic length scale of the system. The microscale gas flow regimes can be classified into different categories according to the value of Kn, which can also be expressed in terms of Mach number (Ma) and Reynold number (Re) as [2,3]: Kn =� /2 Ma/Re, where is the ratio of specific heats, Ma is Mach number; Ma = um/c, and Re is Reynolds number; Re = ρm·um·D/μm. The different categories are as follows: (1) Kn < 10 −3 , continuum flow significantly influence both the flow and HT in the MCHS. Nonino et al. [38,39] and Giudice et al. [40] performed a parametric analysis to find the effects of temperature-dependent viscosity (TDV), thermal conductivity (TDTC), and VD on FC in simultaneously developing laminar flow of a liquid in straight ducts. It was confirmed that the effects of TDV and VD cannot be neglected in a wide range of operative conditions of the laminar FC. Mahulikar and Herwig [41,42] reported the physical effects due to variations in viscosity and thermal conductivity of liquid on laminar micro-convection. It was concluded that the effects of FPV become highly significant from macro-to-micro-scale convection. Herwig and Mahulikar [43] investigated the variable property effects on the flows through micro-sized channels. Gulhane and Mahulikar [44] numerically investigated the effect of property variations of air in laminar forced micro-convection with the entrance effect. Kumar and Mahulikar [45] numerically investigated the effects of TDV on FDF through a micro-channel. The frictional flow characteristics of water flowing through a circular micro-channel with VFP were investigated by Kumar and Mahulikar [46]. Kumar and Mahulikar [47] investigated the physical effects of VFP on flow and thermal development in micro-channel. It was also observed that the effects of VFP on static gauge pressure drop are highly significant for micro-convective flow. Kumar and Mahulikar [48] investigated the physical effects of VFP on HT and frictional flow characteristics of laminar gas microconvective flow. It was concluded that the physical effects need to be well considered in the applications of laminar gas microconvection based on large temperature gradients, for example, the design of MCHS, and the flow cannot be generally considered as a constant property flow, as in conventional channels.

OBJECTIVE AND SCOPE OF THE INVESTIGATION
So far the investigators did not explore the effects of ρ(p, T), μ(T), k(T) and Cp(T) variations in gaseous slipflow through a MCHS. This aspect stimulates researchers to pinpoint the ultimate changes in fluid flow and HT through a MCHS due to supplementary physical mechanisms induced due to ρ(p, T), μ(T), k(T) and Cp(T) variations. Therefore, the present work aims to numerically investigate the effects of these mechanisms on laminar FDF through a MCHS in the slip-flow regime. Incorporation of these mechanisms leads to upgrading the knowledge of microconvection physics within the slip-flow regime. The presented results reveal the influence of these mechanisms in micro-convection characteristics. These results are expected to be useful in the analysis and design of the microscale HF devices. Figure 1 illustrates the schematic diagram of the physical model and the BCs used in this study. Radius (R) of the micro-tube is 25 μm and the length (L) of the micro-tube is 2.5 mm. Aspect ratio = L/D = 50. CWHF BC ( w " = 7.5 W/cm 2 ) is imposed on the outer surface of the MCHS. The values of aspect ratio (L/D = 50), heat flux ( w " = 7.5 W/cm 2 ), inlet fluid temperature at the axis of micro-tube (T0,in = 5°C) and inlet mean axial velocity (um,in = 20 m/s) are chosen on the basis of following reasons: (i) the extreme temperature of air (Tw,ex) in the computational field, that should not go above its dissociation temperature (Tw,ex < 2000 K) (ii) the range of Kn should be in between 0.001 and 1.

Governing Equations
The 2-dimensional, steady-state, governing equations in cylindrical coordinates (with axisymmetry) are numerically solved for the above field. These equations incorporating ρ, µ, k and Cp variations for convective-flow through a uniform cross-section, in the dimensional form are as follows: Continuity Energy equation where ρ is the fluid density, u and v are the axial and radial velocities respectively, z and r are the axial and radial directions respectively, p is the pressure, T is the temperature, µ, Cp and k are the dynamic viscosity, specific heat, and thermal conductivity of fluid respectively.

Boundary Conditions
At the inlet upstream, z = 0 -, u(r) and T(r) profiles are for laminar FDF with constant fluid properties. These profiles are respectively given as [49]; u(r) = 2um·[1− (r/R) 2 ], and T(r) = T0,in+( w " ·R/k).[(r/R) 2 −((r/R) 4 /4)]. The T0,in is the inlet gas total temperature at the axis of the tube and um is the mean inlet velocity. Inlet BC exposes the role of variation in properties without mixing the entrance effect. From z = 0+ to z = L (inlet-downstream to exit) property variations are modelled as according to the different cases for non-reacting air. The pressure at the exit of the tube (pex = patm = 1.01325 × 10 5 N/m 2 ) is equal to standard atmospheric pressure. The symmetric BC is imposed at the axis of the micro-tube; hence, (∂u/∂r) = (∂p/∂r) = (∂T/∂r) = (∂ρ/∂r) = 0. The CWHF BC [k(∂T/∂r)] w = w " ] is applied with proper VS and TJ at the wall of the MCHS. According to the slip-flow theory, the VS and TJ at the wall are proportional to normal velocity and temperature gradients, respectively. In the slip flow regime, the N-S and energy equations are solved by including VS and TJ BCs.

NUMERICAL SOLUTION AND VALIDATION
Equations (1) -(4) are solved numerically along with the ideal gas equation (p = ρ·Ra·T) by ANSYS FLUENT solver, using the SIMPLE scheme. Second-order upwind advection scheme is used to discretize the convective terms in the momentum and energy equations. Maxwell's models are adopted for VS and TJ phenomena in the FLUENT solver for their simplicity and effectiveness. The low-pressure slip boundary formulation is used for VS and TJ at the wall.

VS boundary condition
The 1 st -order slip BC is used which was presented by Maxwell [50] as: TJ boundary condition In the same approach, TJ BC is used in the 1 st -order form as: where, uw and Tw are the reference wall velocity and temperature, respectively. The ua and Ta are the air velocity and temperature at the wall, respectively. The λ is the mean free path between molecules and is the coordinate normal to the wall. Here v and T are the momentum and thermal accommodation coefficients, respectively. The coefficients v and T describe the interaction of the fluid molecules with the wall. Normally, the values of these coefficients depend on the surface finish, temperature, and velocity at the fluid-wall interface. The values of these coefficients are close to unity for most engineering applications [16]. The value of v varies from near zero to unity for specular and diffuse reflections, respectively. In the present investigation, the values of v and T are the same that is equal to 0.9137 [4].
To simulate the problem, a graded mesh with finer grid density in the vicinity of the inlet and the wall, in order to capture abrupt changes in flow and temperature fields, is used for discretization. The graded mesh comprises 10000 cells [= 200 (in axial direction) × 50 (in radial direction)]. This grid system is conservatively selected on the basis of the grid independence test of final results i.e. NuD value. The correctness of the numerical solution is checked by validating NuD results for laminar FDF with constant fluid property for no-slip and no-temperature jump BCs. The results are confirmed with a benchmarked solution (NuCP = 4.363) and show a relative error lower than 0.1%. The solution is deemed converged if the plots of residuals for continuity, z and r momentum, and energy equations are less than 10 -15 or independent of a number of iterations. Additional details relating to the convergence of the solution, the correctness of the numerical results, and validation with benchmark cases for constant fluid properties are reported in [42, 44, and 45].

Variable Physical Properties
Air is used as a working fluid in the present simulations. Density (ρ) variation is as per ideal gas equation of state: p = ρ·Ra·T. For non-reacting and perfect gas: air, µ(T) = 1.462×10 −6 1.5 kg/(m·s) and ( ) = The average value of Cp (= 1018.2 J/kg·K) is used for the functioning temperature range of 273-550 K, because the deviation from the average value is less than 3%. For the range of 550-2100 K, Cp(T) variation is achieved by least-square error 4 th order polynomial fit within correctness of 0.12% in the following form [52]:

RESULTS AND DISCUSSION
In this section, the effects of VFP on the slip-flow and HT characteristics are presented. Firstly, the governing equations are solved for the case of ρ(p, T) variation with no-slip and with slip boundary conditions. A meaningful comparison between no-slip BC simulations and slip BC simulations is performed for ρ(p, T) variation. The ρ(p, T) variation helps to pinpoint the effects of density variation on slip-flow and HT characteristics independently. Table 1 shows wall temperature (Tw), bulk mean fluid temperature (Tm) and various flow properties at different locations of geometry and Table 2 shows dimensionless numbers at inlet and outlet of micro-tube for the slip-flow case. The Re is constant along the flow for ρ(p, T) variation since the mass flux (ρm·um) is constant. The incorporation of µ(T) variation reduces Re along the flow since air viscosity increases with temperature and (ρm·um) is constant. The Re is insignificantly affected by the incorporation of k(T) variation. A slight increment is observed in Re when Cp(T) variation is incorporated. This is because the Cp(T) variation leads to diminishing the rate of augmentation in Tm causes the reduction in the rate of increase in µm and decrease in ρm. It is also noted that the um also reduces due to the incorporation of Cp(T) variation. In the present research, Ma is much less compared to 0.3; hence, compressibility effects can be neglected. The Ma increases along the flow for ρ(p, T) variation and this is due to flow acceleration. The Ma is insignificantly affected by incorporating µ(T) and k(T) variations. However, Ma reduces when Cp(T) variation is incorporated due to the lowering of Tm and um. The density and pressure declines and Kn augments along the air heated flow. The value of Kn is maximum at the exit of the tube where ρ is lowest. The incorporation of µ(T) variation increases Kn, since Re reduces due to an increase in µm. The Kn is insignificantly affected by the incorporation of k(T) variation. The Kn slightly reduces when Cp(T) variation is incorporated due to a small reduction in Ma and a small increment in Re. Figure 2a shows the variation in Nu along the flow for ρ(p, T) variation with no-slip and slip BCs. In the case of slip-flow, the presence of the VS and TJ significantly affects the local Nu number. The VS enhances the advection near the wall which augments HT however the TJ increases the conduction thermal resistance at the wallfluid interface which degrades HT. Therefore, the collective effect of VS and TJ could augment or degrade the HT depending on their relative magnitude [16]. The effect of VS remains leading throughout the micro-tube resulting in an augment in Nu throughout the micro-tube as shown in Figure 2a. In the vicinity of the inlet, a rapid increment in Nu is observed for slip-flow as illustrated in Figure 2. This is due to the presence of large TJ in the locality of the inlet as shown in Figure 3. Figure 3 shows the difference in wall temperature (Tw) along the flow with no-slip and with slip BC.  Table 2. Dimensionless numbers at inlet and outlet of micro-tube for slip-flow   The variation in Nu along the flow for the case of slip-flow is shown in Figure 2b, for four different cases:

ρ(p, T) & µ(T) (iii) ρ(p, T), µ(T) & k(T) (iv) ρ(p, T), µ(T), k(T), & Cp(T).
The effect of ρ(p, T) variation is to flatten u(r) profile which promotes faster-moving particles close to the wall which considerably enhances the convection. The ρ(p, T) variation develops radially outward flow which increases thermal resistance in the fluid, thereby degrading the convection [6]. Figure 4a shows the u(r, z) profile for ρ(p, T) variation with no-slip and slip BCs at an axial location z/D = 5. Incorporation of slip-flow with ρ(p, T) variation flattens u(r) profile, which reduces axial velocity at the centerline of the micro-tube. This leads to larger mass flux near to the wall which enhances Nu as shown in Figure 2b Table  2. The µ(T) variation slightly flattens u(r) profile (see Figure 4b) which promotes faster-moving particles close to the wall which slightly enhances the convection. Figure 5 shows the ρ gradients over the cross-section at z/D = 5. In the case of heated air, ρ(r) profile is an inverted 'U' shape and lower ρ closer to the wall is less effective in heat transport which degrades convection. The ρ(r) profile is a converse of T(r) profile as shown in Figure 6 and 'U' shape ρ(r) profile augments convection. For slip-flow, the higher ρ closer to the wall is more effective in heat transport which augments Nu. The higher ρ closer to the wall is due to the lower temperature near to the wall as shown in Figure 6a. Incorporation of µ(T) variation slightly increases ρ closer to the wall which slightly augments Nu. This slight increment in ρ near the wall is due to the slight reduction in the temperature of fluid near the wall as shown in Figure  6b  Incorporation of k(T) variation highly augments Nu as given in Table 2 and shown in Figure 2b. The k(r) profile is U-shaped due to a higher temperature near to the wall. This leads to higher k-fluid near to the wall which is more effective to transfer more heat compared to higher k-fluid near to the centerline. Therefore, k(r) variation leads to enhance NuD for air heated case. The k(z) variation is considerably greater than k(r) variation. The k(z) variation induces axial conduction for w " = Constant BC, which considerably affects gas micro convection [44]. Heat flow at cross-section is given by, w " = kw·(∂T/∂r)w. For constant w " , the augmenting kw along the heated flow declines the corresponding temperature gradient near to the wall as shown in Figure 6b. This leads to a higher ρ closer to the wall (see Figure 5b) which is more effective in heat transport, thereby promoting convection. Therefore, higher ρ and higher k-fluid near to the wall lead to augment in Nu.
Incorporation of Cp(T) variation lessens the Nu as shown in Figure 2b and given in Table 2. Incorporating Cp(T) variation increases Cp closer to the wall which causes a lessening in Tw for a given w " as illustrated in Figures  6b and 7a. Figure 7b shows that the rate of change of Tm is lower for the case of Cp(T) variation only than for other cases of property variation. This is due to the increase in Cp(T) that lowers um (as um ∝ Tm/pm) [6]. Incorporating Cp(T) variation leads to lessening the rate of augmentation in Tm and Tw as shown in Figure 7 and given in Table 1 It is noted that the Δpg is nonlinear along the flow which is attributed to the role of temperature-sensitivity of gas density SρT (= ∂ρ/∂T), rather than the widely reported role of compressibility associated with pressure-sensitivity of gas density Sρp (= ∂ρ/∂p) [53]. The Δpg due to ρ(p, T) variation is 24967.6 Pa with the no-slip BC, which nearly reduces to 778.247 Pa with slip BC. Figure 8b shows the Δpg along the flow due to a combination of variation in ρ(p, T), µ(T), k(T), and Cp(T) for the slip-flow case. It is observed that the incorporation of µ(T) and k(T) variations insignificantly affects the Δpg. However, the incorporation of Cp(T) variation significantly affects the Δpg. This is because the incorporation of Cp(T) variation lessens μm due to a reduction in Tm.   Figure 9 shows the variation of slip velocity (uslip) along the flow due to a combination of variation in ρ(p, T), µ(T), k(T), and Cp(T). The uslip increases along the flow. It is noted that the uslip is trivially affected by the incorporation of µ(T) and k(T) variation. This is because the incorporation of µ(T) and k(T) variation trivially affect the Tm. However, the uslip is significantly affected by incorporating Cp(T) variation. This is because the incorporation of Cp(T) variation reduces μm due to a reduction in Tm.

CONCLUSIONS
Physical effects induced in micro-convective gaseous slip-flow due to variations in gas properties with CWHF BC are studied numerically for hydrodynamically and thermally developed flow. The flow and energy Temperature profile in radial direction u(r) Axial velocity profile in radial direction Greek symbols Sρp Pressure-sensitivity of gas density (∂ρ/∂p) SρT Temperature-sensitivity of gas density (∂ρ/∂T) Δp Pressure drop, Pa μ(T) Temperature dependent viscosity, Ns/m 2 ρ(p, T) Pressure and temperature dependent density, kg/m 3 Specific heat ratio