A critical re-examination of Reynolds analogy for micro-convective flow

The present research numerically investigates the validity of the Reynolds analogy for microconvective water flow between Stanton number ( St ) and Fanning friction factor ( f f ), taking into account combined fluid properties variations such as temperature-dependent density, viscosity, and thermal conductivity. The Reynolds analogy is suggested to be valid when St increases for thermophysical fluid properties (TFP) with a decrease in f f . This analogy, therefore, helps to find the flow regime that increases heat transfer while shear stress decreases for TFP. Hence, the Reynolds analogy for TFP helps to design and improve the performance of the different devices, including micro-scale heat exchangers for electronics cooling, internal cooling passages of turbine airfoils, and many biomedical devices. Three modified non-dimensional parameters ( Π SρT , Π SμT , and Π SkT ) appear from the non-dimensionalization of the governing conservation equations. Using dimensional analysis, the dependence of the friction factor on these parameters is examined. Cite this article as: K umar R . A critical re-examination of reynolds analogy for micro-convective flow. J Ther Eng 2022;8(4): 515-528.


INTRODUCTION AND REVIEW
The temperature gradients along and across the flow in micro-convective flows are much steeper due to high heat fluxes and low Reynolds number (Re). Consequently, the effects of TFP variations in micro-convective flows can not be neglected. In many industrial applications, heat transfer (HT) from a wall to a flowing fluid stream is an essential process and HT rates affect the performance of the overall system. Therefore, precious attempts were made to raise HT rates and reduce shear stress (τ) to improve system efficiency. Reynolds analogy helps to discover the flow regime in which HT improves while τ drops for TFP. Reynolds analogy for TFP helps to design and optimize the performance of different devices, including micro-scale heat exchangers, micro-scale passages for electronics cooling, internal cooling passages of turbine airfoils, external surfaces of gas turbine airfoils, and many biomedical devices.
Firstly, Sieder and Tate [1] studied the impact of variation in μ(T) on forced convection (FC). The method of property ratio was used to find μ(T) variation effect. An asymptotic theory was used to examine the influence of TFP on the fully developed flow (FDF) [2,3]. The effects of μ(T) variation on Nusselt number (Nu) and friction factor ( f ) for laminar flow through a circular tube were analyzed by Kakac et al. [4]. Harms et al. [5] studied laminar FDF in a semicircular duct with µ(T) variations. It was observed that the µ(T) variations create significant distortions in both the velocity and temperature distributions.
The influences of TFP on the local temperature, heat flux, and Nu were numerically analyzed [6]. The curvature effect and the effect of variation in μ(T) on laminar FC and FDF were analyzed [7]. The dependence of Nu and f on μ(T) variation was analyzed under both cooling and heating conditions. Pure continuum-based micro-convective gas flow with ρ(T) variation was numerically simulated [8]. The physical special effects induced due to μ(T) and k(T) variations were investigated for the case of laminar microconvective water flow [9,10]. The physical effects induced due to variable gas properties in micro-convective flow were reported [11]. The effects of μ(T) variation can not be overlooked for a wide range of operating conditions in the entry region of straight ducts [12]. Herwig and Mahulikar [13] examined the effects of TFP on single-phase incompressible micro-convective flow. Mahulikar et al. [14] suggested the need to examine the impacts of fluid variation on f. The fluid friction characteristics in laminar FDF were studied and the Reynolds analogy was reexamined for TFP [15]. The effects of TFP on thermally developing flow were numerically studied by Liu et al. [16], in the cooling passages of micro-channel. The effects of fluid variation on single-phase micro-convective compressible flow were investigated [17]. The physical mechanisms induced due to TFP in laminar micro-convective FDF were examined [18]. Because of TFP, a significant difference in pressure drop from macro to micro scale was measured. Gulhane and Mahulikar [19] studied the hydrodynamic and thermally developing flow problem and the Graetz problem due to fluid properties variations. Harley et al. [20] provided theoretical and experimental research on compressible gas flow in micro-channels with a large subsonic Mach number. Kumar and Mahulikar [21] explored the effects of μ(T) variation on laminar micro-convective FDF. The abnormal HT and fluid flow observations were recognized owing to variability in μ(T) and these observations were clarified using the concept of thermal and hydrodynamic undevelopment of flow. Frictional flow characteristics of microconvective flow for TFP were investigated [22]. Recently, Kumar and Mahulikar [23] numerically re-examined the validity of the Chilton-Colburn analogy between St·Pr 2/3 and f f for laminar micro-convective flow with μ(T) and k(T) variations. Kumar and Mahulikar [24] numerically investigated the heat transfer characteristics of convective water flow through a micro-tube. The effects of variation in inlet temperature and wall heat flux on heat transfer are studied for variable fluid properties. The results show that the Nu decreases with an increase in inlet temperature for variable fluid properties. The deviations produced by temperaturedependent properties on heat transfer and frictional flow characteristics of water flowing through a microchannel are numerically investigated by Kumar and Mahulikar [25]. The Nu displays a significant deviation from conventional theory due to flattening of the radial temperature profile due to thermal conductivity variation. The performance of the heat sink is optimized with the help of the entropy generation minimization (EGM) method [26][27][28].
Kumar and Mahulikar [29] and Kumar [30] analyzed the rarefaction and non-rarefaction effects on heat transfer characteristics of hydrodynamically and thermally developing airflow through microtubes. Keepaiboon et al. [31] experimental investigated boiling heat transfer characteristics of a refrigerant in a microfluidic channel at a high mass flux. They proposed the new boiling heat transfer correlation of a refrigerant for two-phase flow at the microfluidic scale Gaikwad et al. [32] discussed the EGM in a slip-modulated electrically actuated transport through an asymmetrically heated microchannel. Optimum values of geometric and thermo-physical parameters were introduced for which a change in the thermal transport of heat caused by viscous dissipation and Joule heating effect leads to EGM in the system. Sarma et al. [33] analyzed the entropy generation characteristics under the influence of interfacial slip for a non-Newtonian microflow. The optimum value of the geometric parameter such as the channel wall thickness and the thermophysical parameters such as the Peclet number (Pe) and Biot number (Bi), were determined, leading to a minimum rate of entropy generation in the system. Sarma et al. [34] examined the prominent role of the Debye-Hückel parameter, viscoelastic parameter, the thermal conductivity of the wall, channel wall thickness, Bi, Pe, and axial temperature gradient on the entropy generation rate. They established the optimum values of the above parameters, leading to the EGM method.

OBJECTIVE AND SCOPE OF INVESTIGATION
The important step for the analysis of water cooling with forced convection is the use of a similar argument, the Reynolds analogy. The Reynolds analogy is a powerful analytical tool since it was first proposed in the late 1800s. This states that the f f due to fluid flowing over the wall is proportional to the convective heat transfer coefficient (h). It's most simple form is, St = f f /2 for CFP. Earlier, the Reynolds analogy is valid for incompressible and laminar flows. It has been extended to turbulent flows in different computational as well as analytical forms by many researchers. Mahulikar and Herwig [9] reported that the effect of μ(T) and k(T) variations for water is highly significant in microconvection. Gulhane and Mahulikar [18] and Kumar and Mahulikar [35] reported that the effect of ρ(T) variation for water is significant in micro-convection due to a rapid increment in fluid bulk mean temperature. Therefore, the present research is an extension of the earlier work reported by Kumar and Mahulikar [23], to include the effect of ρ(T) variation in addition to µ(T) and k(T) variations. Hence, the first objective of the present research is to reexamine the Reynolds analogy for the case of combined ρ(T), μ(T), and k(T) variations without entrance effect. Due to TFP, three modified non-dimensional parameters "Π SρT , Π SμT , and Π SkT " are emerged from the dimensionless form of governing conservation equations. The role of Π SμT and Π SkT in flow friction was analyzed by Kumar and Mahulikar [23]. It is also thought that in micro-convection, Π SρT creates a powerful impact on flow friction. Therefore, the second objective of the investigation is to examine the role of Π SρT , Π SμT , and Π SkT in flow friction. The Poiseuille number (Po = 4 f f ·Re D ) correlates with Π SρT , Π SμT , and Π SkT to find the effects of TFP on laminar liquid micro-convection. This research would be very helpful in improving micro-convection knowledge that offers better efficiency of micro-devices.

DESCRIPTION OF THE PROBLEM
A circular cross-sectional micro-tube with an aspect ratio (L/D) = 50 is subjected to constant wall heat flux (CWHF) boundary condition (BC) as shown in figure 1. The following data is fixed for all investigated cases: Radius of micro-tube R = 50 × 10 -6 m, length of micro-tube L = 5 × 10 -3 m, and inlet temperature at axis T 0,in = 293 K. The smaller diameter and higher aspect ratio are selected to analyze the effect of steeper temperature gradients on laminar microconvection characteristics.

Problem Formulation
Attention is focused on the calculation of St, Nu, Re, Prandtl number (Pr), Peclet number (Pe), Po given as: St = h/(ρ m ·u m ·c p ), Nu = h·D/k m , Re = ρ m ·u m ·D/μ m , Pr = c p ·μ m /k m , Pe = Re·Pr, Po = 4f f ·Re D . The subscript 'm' shows the mean value of the properties evaluated at bulk mean temperature (T m ). The cross-sectionally weighted averaged axial velocity (mean velocity) u m is defined as [17]: u u r dA (1/ρ·A) and T m is the enthalpy-average temperature (bulk mean temperature) of the bulk fluid, which represents the total energy of the flow at a reference point as [17]: with CFP, f f ·Re D = 16 and Po = 64. Therefore, St can be written as: St Nu From this relation, it is concluded that St increases with an increase in f f . Therefore, the Reynolds analogy is popularly regarded as holding when St increases for CFP with an increase in f f . The Reynolds analogy valid region illustrates the region in which convective HT is more emphatic at the cost of augmented f f . As per Reynolds analogy, the flattening of velocity profile improves St and Nu, it also increases f f . However, the flattening or sharpening of the velocity profile is affected due to TFP, which affects Nu [10]. It was supposed that the Reynolds analogy was invalid for liquids (when Pr ≠ 1) and particularly when TFP were considered [36]. Therefore, the Reynolds analogy is revisited for TFP. From the study of Gulhane and Mahulikar [18], It is found that h increases as a result of flattening the velocity profile, which increases St along the flow. The flattening of the velocity profile results in radially outward flow, which can not be ignored, and radial convection has a significant effect on convective HT, because Nu α (∂u/∂r) w . Owing to flattening of u(r, z) profile, (∂u/∂r) w increases, therefore Nu also increases [10]. Due to TFP, τ w α μ m [15]. The reduction in μ m along the flow causes a reduction in f f , when Reynolds analogy is valid. Hence, Reynolds analogy is now valid when St increases with a reduction in f f for TFP. It is concluded that the Reynolds analogy helps to find the flow regime in which h increases while τ w decreases for TFP. By putting St = Nu/ (Re·Pr) in equation (1), can be written in this form: (1/Re) = (f f /16). Therefore, the Reynolds analogy is qualitatively valid for that region, where (1/Re) and f f are directly dependent.

Momentum equation [Axial direction]
Dimensional form

Momentum equation [Radial direction]
Dimensional form

Energy equation
Dimensional form Nondimensional form flow. The governing equations (2)-(9) with BCs are solved by ANSYS FLUENT software. FLUENT is based on a finite volume differencing scheme which is 2 nd -order accurate. The algorithm "Semi-Implicit Method For Pressure-Linked Equations" (SIMPLE) is used to achieve a good convergence behavior. When the residuals for continuity, r, and z momentum equations are less than 10 -12 and less than 10 -15 for the energy equation, the solution is deemed converged. Additional information related to the accuracy of the numerical results, the convergence of solution, and validation with benchmark cases for CFP are in [10, 15, and 23].

Inference from Theoretical Studies based on TFP
For water in the temperature range of 273-373 K, the variations in ρ(T), μ(T), and k(T) are 4%, 84%, and 21% respectively [38]. The ρ(T) variation for pure water in the temperature range of 274-372 K is given by the Thiesen-Scheel-Diesselhorst relation as [39]:  (10) where T is in °C. The pressure effect on density is normally ignored in the case of water. The μ(T) variation for singlephase water is given as [40]: The non-dimensional static pressure and non-dimensional temperature are given as p -= p·D/(µ m ·u m ) and θ = k m ·(T−T m )/(q" w ·D) respectively. The S ρT (= ∂ρ/∂T) is the density-temperature sensitivity, S μT (= ∂μ/∂T) is the viscosity-temperature sensitivity and S kT (= ∂k/∂T) is the thermal-conductivity-temperature sensitivity. The Re D and Pe D are Re and Pe based on diameter respectively. The laminar micro-convective flow with TFP depends upon the following non-dimensional parameters: The Π parameters are the magnitude of the product of the temperature perturbation parameter [= f(q" w ·D/k)] and non-dimensional property sensitivities, S ρT (T/ρ), S μT (T/μ) and S kT (T/k). The modified non-dimensional parameter "Π SρT " shows the comparative importance of momentum transport due to S ρT over energy transport due to fluid conduction. The "Π SμT " indicates the significance of cross-flow momentum transport over energy transport due to S μT . The Π SkT gives the comparative importance of momentum transport due to S kT and energy flow due to fluid conduction. Higher values of Π parameters show a stronger influence on micro-convection due to temperature-dependent fluid properties. The Brinkman number "Br qw " appears in the non-dimensional energy equation as the product of ] is the modified Br based on S kT [9]. The other non-dimensional parameters are involved in the governing equations as:

Boundary Conditions
The computational field is subjected to four flow and thermal boundary conditions (BCs) as follows: 1. Inlet (z = 0): The laminar, FDF profiles of u(r) and T(r) at the inlet-upstream for CFP are given as: u in (r) = 2u m,in (1-r -2 ) and T in (r) = T 0,in + (q" w ·R/k)·[r -2 -(r -4 /4)] respectively [37]. The u m,in is the inlet mean axial velocity and T 0,in is the inlet water temperature at the axis. The variations in ρ(T), μ(T) and k(T) are turned on from inlet downstream (z = 0 + ). The influence of thermophysical properties on micro-convective flow without considering entrance effects is expressed by this inlet BC [10]. The present study focuses primarily on the effects of fluid thermophysical properties on micro-convective flow only. Thus, the entrance effects are ignored. However, in general, the micro-convection characteristics are determined jointly by taking into account the effects of the entrance and fluid thermophysical properties. imposed at the nonporous rigid wall of the tube; therefore, u w = v w = 0. The constant is applied at the wall, q" w = k w ·(∂T/∂r) w .

Computational Domain and Numerical Methodology
The computational field is split into the graded mesh [10,000 cells = 200 (in the axial direction) × 50 (in the radial direction)] with a finer grid spacing near the inlet and the wall. The finer grid spacing near the inlet and the wall is used to capture a rapid change in fields of temperature and The k(T) variation for single-phase water is calculated by least-squares error third-order polynomial fitting of data in the operating temperature range of 274-372 K as [38]: . .
The c p (T) variation is less than 1% within a temperature range of 274-372 K, hence, c p is assumed to be constant. The micro-convective flow has the following characteristics: The S ρT > 0 for 0 to 4 °C, because water density increases with increasing temperature. However, after 4 °C, S ρT < 0 is negative because water density decreases with increasing temperature.
(2) Viscosity-temperature sensitivity . T kg m s K (14) In liquids, water has very high S μT . The S μT < 0 since water viscosity decreases with increasing temperature.   and after that, f f decreases along the flow as shown in figure  4. The following reasons are attributed to this, (1) The flow undevelopment happens in the locality of the inlet due to μ(T) variation as ∂/∂z(∂u/∂r) w > 0. (2) The water viscosity reduces with increasing temperature which decays f f along the flow as (∂μ/∂z) < 0 [18]. As q" w increases, the axial location zff,max moves towards the exit of micro-tube for same u m,in as given in Table 1 and also shown in Figure 4.
The variation of Po along the flow is illustrated in figure  5. It is observed that Po decreases with increasing q" w and u m,in . The main cause behind this is: the rate of increase in Re is less than the rate of reduction in f f . It is also noted that the rate of change of Re increases with an increase in q" w , which is confirmed from Table 1. The deviation in Po from 64 is smaller in the locality of the inlet for the case of lower q" w as illustrated in Figure 5(a). As q" w increases, the deviation in Po from 64 also increases which is clearly shown in figures 5(a, b, c).

Significance of the modified non-dimensional parameters, Π SρT , Π SμT , and Π SkT
Three modified non-dimensional parameters "Π SρT , Π SμT , and Π SkT " appear from the dimensionless form of there are 5 (n-m) independent non-dimensional groups. Selecting ρ, u m , D and k as the repeating variables (RV) and leaving the remaining parameters as; μ, S ρT , S μT , S kT and q" w , five non-dimensional groups are obtained as follows: Π 1 = Re D = ρ·u m ·D/μ, Π 2 = Br qw = μ·u 2 m /(q" w ·D), Π 3 = Br SρT = S ρT ·u 2 m ·μ/(ρ·k), Π 4 = Br SμT = S μT ·u 2 m /k, Π 5 = Br SkT = S kT ·μ·u 2 m /k 2 . The Br qw is Br based on q" w , Br SρT is modified Br based on S ρT ; and Π SρT = |Br SρT /Br qw | = |S ρT · q" w ·D/(ρ·k)|, Br SμT is modified Br based on S μT ; and Π SμT = |Br SμT /Br qw | = |S μT ·q" w ·D/(μ·k)| and Br SkT is modified Br based on S kT ; and Π SkT = |Br SkT /Br qw | = |S kT ·q" w ·D/k 2 | [9]. The dimensionless form of the governing equations and above dimensional analysis gives a short form of equation (16)   emerged in continuity equation yields a strong effect on Po. Figure 6 gives the variation of Po versus Π SρT , only Reynolds' analogy valid data has been taken for u m,in = 0.075, 1, and 3 The role of Π SρT , Π SμT , and Π SkT in flow friction is investigated considering combined ρ(T), μ(T), and k(T) variations. For large q" w which is mainly allowed at high u m,in , Π SρT  micro-convective flow. Therefore, in figure 6, the regression equations have been proposed to correlate Po with Π SρT at different u m,in , in the Reynolds analogy valid region as given in Table 2. m/s. Figure 6 (a, b, c) illustrates that a similar pattern for different permissible is followed in the Reynolds analogy valid region. As increases, the value of Π SρT also increases, which indicates that the influence of ρ(T) variation increases on  region. As q" w increases, the value of Π SμT also increases, which shows the effect of μ(T) variation increases on microconvective flow. Table 3 gives the regression equations which mathematically represent the correlation between Figure 7 gives the variation of Po versus Π SμT , only Reynolds analogy valid data has been taken for the same cases as in figure 6. Figure 7 (a, b, c) indicates a similar pattern for different allowable, in the Reynolds' analogy valid  For large q" w that is enabled at high u m,in , Π SkT also produces a strong effect on Po. Figure 8 illustrates the variation of Po versus Π SkT , only Reynolds analogy valid data has been taken for the same cases as in figures 6, 7. From figure 8, it is observed that Π SkT increases with an increase in q" w , which shows the effect of k(T) variation increases on micro-convective flow. Again a similar trend is observed for different allowable q" w in the Reynolds analogy valid region as shown in figure 8 (a, b, c). Table 4 gives the estimated regression equations which mathematically represent the correlation between Po and Π SkT for u m,in = 0.075, 1, and 3 m/s, in the Reynolds' analogy valid region as illustrated in figure 8.
The role of Π SρT , Π SμT , and Π SkT in flow friction is expressed with the help of estimated regression equations. From these equations, it is observed that as u m,in increases, the effect of ρ(T), μ(T), and k(T) variations increases on micro-convective water flow.

CONCLUDING REMARKS
1. The performance of the overall system is affected by HT rates. Therefore, significant efforts have been devoted to increase the HT rates and decrease the shear stress for improvement in the performance of the system. The Reynolds analogy is valid only for that portion of the flow regime, where St increases with decreasing f f for TFP. Therefore, the Reynolds   analogy helps to find the flow regime in which HT increases while τ decreases for TFP. 2. The Reynolds analogy is largely valid at low mean velocities; however, the Reynolds analogy is largely invalid at high mean velocities. 3. Direct proportionality of (1/Re) with f f for a significant section of the flow regime, also validates the Reynolds analogy. 4. The significance of mass transport due to ρ(T) variation is given in governing equation (3) and momentum transport in axial and radial directions due to μ(T) variation is given in governing equations (5,7). The significance of energy transport by the axial fluid conduction due to k(T) variation is given in the governing equation (9). 5. Higher values of the Π SρT , Π SμT , and Π SkT parameters show a stronger effect on micro-convection due to TFP. 6. The regression equations have been proposed to correlate Po with Π SρT , Π SμT , and Π SkT , in the Reynolds analogy valid region.

ACKNOWLEDGMENTS
The author is grateful to Prof. Shripad P. Mahulikar, Professor, Department of Aerospace Engineering, IIT Bombay, India, for giving valuable suggestions to improve the quality of the manuscript.

DATA AVAILABILITY STATEMENT
The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

CONFLICT OF INTEREST
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ETHICS
There are no ethical issues with the publication of this manuscript.