Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating

Abstract A mathematical model is developed to study laminar, nonlinear, non-isothermal, steady-state free convection boundary layer flow and heat transfer of a micropolar viscoelastic fluid from a vertical isothermal cone. The Eringen model and Jeffery’s viscoelastic model are combined to simulate the non-Newtonian characteristics of polymers, which constitutes a novelty of the present work. The transformed conservation equations for linear momentum, angular momentum and energy are solved numerically under physically viable boundary conditions using a finite difference scheme (Keller Box method). The effects of Deborah number (De), Eringen vortex viscosity parameter (R), ratio of relaxation to retardation times (λ), micro-inertia density parameter (B), Prandtl number (Pr) and dimensionless stream wise coordinate (ξ) on velocity, surface temperature and angular velocity in the boundary layer regime are evaluated. The computations show that with greater ratio of retardation to relaxation times, the linear and angular velocity are enhanced whereas temperature (and also thermal boundary layer thickness) is reduced. Greater values of the Eringen parameter decelerate both the linear velocity and micro-rotation values and enhance temperatures. Increasing Deborah number decelerates the linear flow and Nusselt number whereas it increases temperatures and boosts micro-rotation magnitudes. The study is relevant to non-Newtonian polymeric thermal coating processes.


INTRODUCTION
Coating hydrodynamics has been an area of considerable interest since the monumental paper by Landau and Levich (1942) in which an elegant formulation was developed for the thickness of the film of a fluid which is deposited on a plate withdrawn vertically from a bath at constant velocity. This work was however confined to Newtonian viscous fluids. In many modern industries non-Newtonian fluids are encountered e.g. in polymer coating processes (Lawrence and Zhou, 1991). Numerous researchers have therefore investigated coating dynamics of different stationary or rotating geometrical bodies (plates, cones, spheres, cylinders) with non-Newtonian liquids and have employed a range of mathematical constitutive equations. Jenekhe and Schuldt (1984) studied coating flows of power-law and Carreau fluids on spinning disks. Campanella et al. (1986) investigated dip coating of a circular cylinder in non-Newtonian power-law fluids. Zevallos et al. (2005) presented a finite element simulation of forward roll coating flows of viscoelastic liquids using both Oldroyd-B and FENE-P models. These studies however ignored heat transfer which may be critical in certain coating systems (Mitsoulis, 1986). The diffusion of heat can modify polymer properties significantly (Mark, 1996). Several authors have therefore studied thermofluid transport in non-Newtonian external coating flows. Cao and Cui (2015) examined Ostwald-DeWaele pseudo-plastic fluids in free convection flow in porous media. Choudhury and Das (2016) employed the Reiner-Rivlin second order differential viscoelastic model for reactive magnetized thermosolutal convection in porous media. . Thirumurugan and Vasanthakumari (2016) utilized the Walters B viscoelastic model to investigate the thermosolutal convection in twophase porous media transport. Prasad et al. (2013) and Subba  investigated computationally the momentum and heat transfer characteristics in external boundary layer slip flow of a viscoplastic fluid from a cylinder.
The classical Navier-Stokes theory does not describe sufficiently the flow properties of polymeric fluids, colloidal suspensions, and fluids having certain additives. Eringen (1966) proposed therefore the theory of micropolar fluids which exhibit micro-rotation effects as well as micro-inertia. Micropolar fluids are fluids with microstructure. They belong to a class of fluids with non-symmetric stress tensor also known as polar fluids. Micropolar fluids include as a special case the conventional Navier-Stokes model of classical viscous fluids that we shall call ordinary fluids. Physically micropolar fluids consist of arbitrarily oriented particles suspended in a viscous medium. The formulation for micropolar fluid theory allows their implementation in boundary layer flows, of relevance to materials processing, and has resulted in considerable activity among researchers. Yacob et al., (2011) proposed a mathematical model to characterise the melting heat transfer in both stretching and shrinking sheet flows of a micropolar fluid. Prakash and Sinha (1976) applied micropolar fluid theory to simulate lubrication squeeze films for circular disks, deriving analytical expressions for film characteristics. Papautsky et al. (1999) employed micropolar fluid theory to study low speed flows in micro-machined rectangular metallic pipette arrays. Bhargava et al. (2016) analyzed the thermo-solutal convection in a cavity containing micro polar fluid with a variation finite element method. Latiff et al. (2015) used Maple software to analyse the multiple slip effects on transient forced bio convection flow of a micro polar nanofluid from a stretching/shrinking sheet with applications in materials fabrication. Alam et al. (2015) used Eringen's micropolar model to study transient Falkner-Skan magnetic convection flows. Transport from external surfaces of curved bodies e.g. vertical cones has also stimulated some interest in recent years. Such flows are of relevance to chemical engineering systems and also materials synthesis.  studied the free convection heat transfer from a nonisothermal permeable cone with suction and temperaturedependent viscosity. Solchana et al. (2016) analysed the momentum, heat and mass transfer behaviour of magneto hydrodynamic flow towards a vertical rotating cone in porous medium with thermophoresis and Brownian motion effects.  examined the natural convection heat transfer about a vertical cone embedded in a porous medium with isothermal wall conditions. Yih (1999) studied the effect of thermal radiation flux on free convection about a truncated cone. Gorla et al. (1986) investigated the micro polar convection boundary layer flow from a cone.
An interesting model in non-Newtonian fluid mechanics is the viscoelastic Jeffrey fluid model. This model degenerates to a Newtonian fluid at a very high wall shear stress. This fluid model also approximates reasonably well the rheological behaviour of a wide range of industrial liquids including biotechnological detergents, physiological suspensions, dense foams, geological sediments, cosmetic creams, syrups, etc. Relatively few studies have considered polymeric external boundary layer flows from curved bodies where the fluid possesses both microstructural and viscoelastic characteristics. Prasad et al. (2015) have studied nanoscale micropolar convection from a cylinder. However they did not consider viscoelastic effects. Thus, the objective of the current study is to examine the viscoelastic micropolar heat transfer in external boundary layer flow from a cone using the Jeffery non-Newtonian (2005) and Eringen micropolar models (2001). This is the novelty of this article.

MATHEMATICAL MODEL
The regime under investigation is natural convection boundary layer flows of incompressible viscoelastic micropolar fluid (polymeric liquid) from a vertical isothermal solid cone, as shown in Fig. 1. Both cone and liquid are maintained initially at the same temperature. Fourier's law is assumed for heat conduction i.e. thermal relaxation effects are negated. Thermal stratification and dispersion are also neglected as is viscous heating. The flow is laminar, steadystate and the cone surface isothermal. These conditions correspond to steady polymeric convection in external boundary layer flow development on an axisymmetric cone geometry. Instantaneously the cone surface is raised to a temperature Tw> T∞, where the latter (ambient) temperature of the polymeric fluid is maintained as constant. The x-coordinate is measured along the slant surface of the cone from the origin (at the vertex) and the y -coordinate is directed normal to the cone surface. The cone has a semi-vertex angle of A.
With the Boussinesq buoyancy and boundary layer approximations, the governing equations for the conservation of mass, momentum, micro-rotation and energy can be written in two-dimensional Cartesian coordinates (x, y) as follows: Here r is the local radius of the truncated cone, Here T  is the free steam temperature. We introduce a stream function  defined by the Cauchy- Here all parameters are defined in the notation at the end of the article. The following non-dimensional partial differential equations for momentum and energy conservation therefore emerge: 4 74 Pr The eight corresponding dimensionless boundary conditions emerge as: Here primes denote the differentiation with respect to nondimensional transverse co-ordinate Here we also define the micropolar material parameters: The local Nusselt number of the inverted cone can be written as 1 4 ( , 0) In equation (12), Here h is the local heat transfer coefficient and k is the thermal conductivity of the micropolar viscoelastic fluid.

NUMERICAL SOLUTION WITH KELLER BOX IMPLICT METHOD
In this study the Keller-box implicit difference method is implemented to solve the non-linear eight order system of coupled boundary layer eqns. Step 1: Reduction of the n th order partial differential equation system to n first order equations New variables are introduced to Eqns. (7) -(9) subject to the boundary conditions (10) are first written as a system of first order equations. For this purpose, we reset Eqns. (7) -(9) as a set of simultaneous equations by introducing the new variables: ' uv  ' vq  ' gp  ' st  R v fv The associated boundary conditions are 0, 0, 0, 1, 0 , 0, 0, 0, 0

Phase b: Finite Difference Discretization
A two dimensional computational grid is imposed on the -η plane as sketched in Fig. 2. The stepping process is defined by: where kn and hj denote the step distances in the ξ and η directions respectively.
We now state the finite-difference approximation of equations.
(16) -(23) for the mid-point , below The boundary conditions are

Phase c: Quasilinearization of Non-Linear Keller Algebraic Equations
If we assume

Phase d: Block-tridiagonal Elimination of Linear Keller Algebraic Equations
The linear system is solved using the blockelimination method, since it possess a block-tridiagonal structure consists of variables or constants. The system consists of block matrices the complete linearized system is formulated as a block matrix system, where each element in the co-efficient matrix is a matrix itself. Then, this system is solved using the efficient Keller-box method. The numerical results are strongly influenced by the number of mesh points in both directions. After some trials in the  -directions (transverse i.e. radial coordinate) a larger number of mesh points are selected whereas in the  -directions (tangential coordinate) significantly less mesh points are necessary. The numerical algorithm is executed in MATLAB on a PC. The method demonstrates excellent stability, convergence and consistency (Keller, 1978).

RESULTS AND DISCUSSION
Effectively with greater relaxation time of the polymer the thermal boundary layer thickness is reduced.  fig. 9 that an increase in R strongly decelerates the flow i.e. damps linear velocity. Fig. 10 demonstrates that with stronger micro-polarity i.e. greater R value, the temperatures are elevated in the boundary layer. The regime is significantly heated and thermal boundary layer thickness is increased. The increased vortex viscosity encourages thermal diffusion i.e. energizes the flow. This increases the efficiency of thermal convection within the body of the fluid from the microscopic to the macroscopic scale and effectively transports heat with greater intensity from the cone surface into the fluid regime. Fig. 11 depicts that strong reversal of micro-element rotation is induced very close to the wall (cone surface) with increasing R values.  Furthermore, increasing Pr generates a substantial reduction in the fluid temperature and the thermal boundary layer thickness. At large Pr, the thermal boundary layer is thinner than at a smaller Pr. This is associated with the fact that for small Prandtl numbers the fluid has very high thermal conductivity. An increase in Pr from 0.5 through 0.71, 2.0, 10, 12 as shown in fig. 15, substantially decelerates the flow i.e. reduces hydrodynamic boundary layer thickness. Similarly higher Prandtl number, which corresponds to lower thermal conductivity of the polymer results in a marked depression in temperatures ( fig. 16). The lower thermal conductivity of denser polymers inhibits thermal diffusion via conduction heat transfer. Less heat is therefore conveyed from the cone surface (wall) into the boundary layer and thermal boundary layer thickness is decreased.

Figures 18 -20 presents the Nusselt Number comparisons for different values of
De, R and Pr respectively: Increasing Deborah number, decrease the Nusselt number i.e reduces heat transfer from the viscoelastic micropolar fluid to the cone wall. Increasing vortex viscosity parameter, R is also seen to decrease the Nusselt number. Increasing Prandtl number, Pr is observed to enhance the Nusselt number since temperatures are reduced in the fluid and greater heat is convected to the cone surface. This inevitably has implications for cooling of thin film coatings.

CONCLUSIONS
A mathematical model has been developed for the buoyancy-driven, natural convection boundary layer flow of a viscoelastic micropolar fluid from a vertical isothermal cone. The Jeffery non-Newtonian and Eringen micro-structural models have been used to simulate different rheological characteristics. The transformed boundary layer conservation equations have been solved with prescribed boundary conditions using the finite difference implicit Keller-box method which has second order accuracy. A comprehensive assessment of the effects of Deborah number De, ratio of retardation to relaxation time  , vortex viscosity parameter R , Prandtl number Pr, micro-inertia density parameter B. Excellent convergence and stability characteristics are demonstrated by the Keller box scheme which is capable of solving very strongly nonlinear rheological problems.
The present simulations have shown that: (i) Increasing Deborah number, De reduces velocity and Nusselt number whereas it increases temperature and angular velocity (micro-rotation of micro-elements).
(ii) Increasing ratio of retardation to relaxation time,  increases the velocity and angular velocity whereas it reduces the temperature (and thermal boundary layer thickness). (iii) Increasing Eringen vortex viscosity parameter, R decreases the velocity and angular velocity whereas it elevates temperature.

 
f  ()  7 (iv) Increasing micro-inertia density parameter, B, reduces velocity and temperature whereas it enhances angular velocity.
(v) Increasing Prandtl number, Pr decreases velocity, temperature and angular velocity. (vi) Increasing Deborah number and vortex viscosity parameter both depress Nusselt number whereas an increase in Prandtl number enhances the Nusselt number (heat transfer rate at the cone surface).
The current study has neglected rotation of the cone and therefore Coriolis body force effects, which are also relevant in polymer chemical processing operations, for example in spin coating processes. These will be considered in the near future.