Thermal entry flow of power-law fluid through ducts with homogeneous slippery wall(s) in the presence of viscous dissipation
Introduction
In order to enhance the transport mechanism in nano-fluidic, bio-fluidic and micro-fluidic instruments, surfaces can be made slippery. Surface slip plays a significant role, not only in reducing the friction at the wall, that leads to enhancement in the flow rates for given driving forces, but also increases convection near the wall and thus enhances the heat/mass transfer. Navier's slip [1] condition is commonly used to quantify wall slip magnitude. Neto et al. [2] elaborated the intrinsic wall slip and proposed that typical slip lengths are of the order of ten nanometres. Maxwell [3] was the first one who quantify the slip length for a gas flowing past a solid boundary. When studying heat transfer analysis over homogeneous/heterogeneous surfaces, researchers have often derived and modelled effective boundary conditions. Larrode et al. [4] elaborated slip regime with heat transfer in cylindrical geometry for isothermal boundary conditions. Bayazitoglu and Kakac [5] studied the heat transfer analysis in circular tube with slippery wall for both isoflux and isothermal boundary conditions by using integral transform approach. Barron et al [6] presented a comprehensive study on heat transfer with slip at the wall for constant surface temperature boundary condition.
Recently, substantial interest in nano-fluidic, bio-fluidic and micro-fluidic equipments has been developed in the area of thermal transport. These types of devices are encountered in many applications in real life such as MEMS and flow sensors, mixing, environmental monitoring, drug delivery, biochemical investigation, synthetic chemistry, separation of chemical species and electronic cooling system [7]. Thus, the studies on heat transfer analysis have been reinforced with regard to the improvement in shape or design of several biomedical diagnostic devices which are used to handle many diseases. The complex non-Newtonian rheological fluids have enormous potentials in this field. It is also noted that the viscous dissipation term has strong impact on heat transfer rate in complex fluids which is observed in thermal processing of polymer melts and plastics. Furthermore, viscous dissipation is also significant in flows occurring at high temperature, for instance, flows in food processing applications where the end product is very important and accurate control is required. In such processes, we need to accurately estimate the heat transfer rate. In views of above discussion, here an analysis is carried out for thermal entry flow of a complex fluid through a confinement with homogeneous slippery wall(s) in the presence of viscous dissipation. It is important to mention that the theoretical studies pertaining to macroscale thermal transport processes over stretching/shrinking sheets, rotating porous disks and porous fins also have useful applications in extrusion processes, refrigeration and cooling of computer disk drives. The work carried out by Turkyilmazoglu [[8], [9], [10], [11]] is quite useful for illuminating discussion on these topics.
The thermal entry flow also known as classical Graetz-problem investigates the heat transfer between a flowing medium and a duct, based on the fully developed velocity profile. The fluid with specified inlet temperature enters the insulated region of the duct with a fully developed velocity profile. Then it suddenly comes in contact with the region of the duct which is heated or cooled or at a specified heat flux. Two regions are identified for the temperature field, namely, the entrance region and the fully developed region. The determination of fluid temperature in both regions is required as solution of the Graetz problem. The auxiliary quantities such as local and mean Nusselt number as also desired to be evaluated as part of the problem. Graetz [12] and Nusselt [13] investigated this problem for a Newtonian fluid with parabolic velocity and constant physical properties. Seller et al. [14] presented an excellent review of Graetz problem in term of Bessel function and formulated the correlation formula for the eigenvalues. Lyche and Bird [15] extended the Graetz problem for a fluid obeying power–law model. Siegel et al. [16] explored this problem for constant heat flux boundary condition. An extension of Graetz problem was made by Cess and Schaffer [17] for Bingham plastic fluid model for constant heat flux boundary condition. This problem is two-dimensional with more involved velocity profile as compared to the Newtonian problem. Hsu [18] investigated this problem for Newtonian fluid along with axial diffusion for both cases namely, uniform wall temperature (isothermal) or constant heat flux (isoflux) and explained that the influence of axial conduction on heat transfer could not be ignored for narrow slits.
Viscous dissipation effect on Graetz problem was studied by Ou and Cheng [19] for isoflux boundary condition. Due to its classical nature and fundamental importance, a comprehensive study on Graetz problem was carried out by Kays and Crawford [20], Shah and London [21] and White [22]. Johnson [23] studied the Graetz-Nusselt problem for isothermal and isoflux boundary conditions using power-law fluid. A semi-analytical analysis of Graetz-Nusselt problem for Bingham fluid was also made by Johnson [24]. Graetz-Nusselt problem for pipe under the influence of viscous dissipation and longitudinal conduction was elaborated by Jeong and Jeong [25]. A detailed analysis of the Graetz-Nusselt problem for Bingham liquid in the presence of axial conduction and viscous dissipation was reported by Min et al. [26]. Further extensions in classical Graetz problem incorporating the effect of axial conduction, existence of non-Newtonian liquids, rarefaction and surface slip were presented by several researchers [[27], [28], [29], [30], [31], [32]]. Norouzi et al. [33] examined the non-Newtonian effect in Graetz problem using FENE-P fluid and obtained the analytical solution in the form of Heun Tri-confluent function.
Recently Ali and Khan [34] presented the theoretical analysis of Graetz problem inside tube and channel for non-Newtonian Ellis fluid incorporating isothermal and isoflux conditions. However, still there is a corner of improvement in Graetz-Nusselt problem in term of complex rheological fluid models with viscous dissipation effects. Specifically, we propose to perform a comparative study for Graetz problem using power-law fluid under isoflux and isothermal conditions over homogeneous slippery surface(s). The integration of various thermal boundary conditions, complex nature of the fluid and slippery surface make this study versatile in the realm of thermal entry flows. This study further encompasses both cylindrical and plane geometries and thus covers comprehensively the essence of the Graetz problem.
Section snippets
Schematic sketch and constitutive equations
Let an incompressible power-law fluid enters the duct along with a fully developed velocity field and itemized inlet temperature (Ti) as portrayed in Fig. 1. The duct is either a channel or pipe with radius r0. A constant temperature (Ts) or uniform heat flux (qs)is specified at the wall (s). Our task is to obtain the temperature profile in the thermally developing and fully developed region of duct with homogeneous slippery surface(s) and in the presence of dissipation function.
To this end, we
Velocity profile
- (i)
Circular duct
The fully developed velocity profile for power-law fluid through a tube with slippery surface having slip length is given by [36]:
The mean flow rate and maximum velocity (velocity at the centreline) are given bywhere and .
- (ii)
Parallel plate duct
For this geometry, the following expressions of velocity, mean flow rate and maximum velocity are valid [36]:
Heat transfer analysis
The energy equation for incompressible fluid is
The initial / boundary conditions appropriate for the considered problem are:
The heat equation for cylindrical geometry with dissipation term can be written as [28].
The following dimensionless quantities are introduced.
Inserting Eq. (16) into Eq. (15) then
Constant wall temperature case
For solution of the Eq. (17) subject to the boundary conditions (19), one can use the following relation [28]:where θ∞is the fully developed temperature and θe is the thermally developing temperature.
For θ∞, the heat equation attains the following form
The straightforward integration of above equation yields the fully developed temperature field given by
For n = 1, Eq. (22) gives θ∞ for a Newtonian case [37]. For θe
Constant heat flux case
For this case the dimensionless temperature θ and Brinkman number can be defined as
Besides that the second boundary conditions for this case are given by (19b). The governing energy equation remains same as defined in Eq. (17).
- (i)
Solution for tube geometry
To obtain the solution of heat equation for constant heat flux boundary conditions, we again decompose the temperature as [38]:
For computation of θ∞, the following decomposition is made [38]:
After
Test case
Before, advancing to the discussions of the achieved solutions, it is mandatory to compare and validate our solutions against the already published work. A semi-analytical technique is employed to perform calculation of temperature profiles for the power-law fluid flow over
homogeneous slippery surface(s) under isoflux and isothermal boundary conditions. For establishing that our results can be used with great confidence, we compare these in the limiting case with the already reported results
Result and discussions
In order to obtain physical insight into the problem and to interpret the results, numerical calculations are carried out for average temperature and local Nusselt number by taking into account the variations in power - law index, slip length, Brinkman number for both isoflux and isothermal boundary conditions. Since the ensuing problem in (25) is a regular SLBVP, the eigenfunctions are mutually orthogonal with respect to corresponding weight function. We used standard orthogonality relation
Conclusions
The Graetz Nusselt- problem for an inelastic fluid obeying the power-law constitutive equation is investigated for plane and cylindrical geometries under specified wall temperature and prescribed heat flux conditions. The computations of bulk average temperature, fully developed temperature field and Nusselt number Nu(x∗) are based on infinite series of orthogonal eigenfunctions of the Sturm-Liouville boundary value problem (SLBVP). The coefficient of solution series are handled numerically
Nomenclature
- ρ
Fluid density
- r
Radial or transverse coordinate
- d/dt
Material derivative
- n
Power law index
- r0
Half width
- m
Consistency parameter
- wmax
Maximum velocity
- θ
Dimensionless temperature
- τ
xtra stress tensor
- Nu
Local Nusselt number
- w
Axial velocity
- cp
Specific heat
- Br
Brinkman number
- k
Thermal conductivity
- θm
Mean temperature
- Num
Average Nusselt number
- ∇
Gradient operator
- τR
Shear stress
- pr
Prandtl number
- Φ = trace(τ. ∇ V)
Viscous dissipation
- x
Axial coordinate
- ζ = 4
Circular tube
- ζ = 9/4
Flat channel
- λ
Eigenvalue
- p = 0
Flat channel
- ReD
Reynolds
CRediT authorship contribution statement
Muhammad Waris Saeed Khan Conceptualization, Methodology, Software, Writing - review & editing; Nasir Ali Conceptualization, Supervision, Writing - review & editing.
Declaration of Competing Interest
Authors have no conflict of interest regarding to this manuscript.
Acknowledgement
We are highly grateful for valuable suggestions of the anonymous reviewer.
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