1. KELLER BOX METHOD By V.Ramachandra Prasad Department of Mathematics, School of Advanced Sciences Vellore Institute of Technology 31 May 20211

2.  PRELIMINARY CONCEPTS  APPLICATIONS  OUT LINE OF THE TALK MATHEMATICAL MODEL NUMERICAL SOLUTION RESULTS AND DISCUSSION  CONCLUSIONS  REFERENCES OVERVIEW 31 May 20212

3. PRELIMINARIES  FLUID MECHANICS is the branch of science, which deals with the behavior of the fluids (liquids or gases) at rest as well as in motion.  The study of fluids in motion where the pressure forces are considered is called FLUID DYNAMICS. Fig. 1.1 Flow processes occur in many ways in our natural environment 31 May 20213 Fluid Flows and their Significance Flows are everywhere and there are flow- dependent transport processes that supply our body with the oxygen that is essential to life. In the blood vessels of the human body, essential nutrients are transported by mass flows and are thus carried to the cells, where they contribute, by complex chemical reactions, to the build-up of our body and to its energy supply. Without these flows, there would be no growth in nature and human beings would be deprived of their “natural food”.

4. PRELIMINARIES Contd…  As further vital processes in our natural environment, flows in rivers, lakes and seas have to be mentioned, and also atmospheric flow processes, whose influences on the weather and thus on the climate of entire geographical regions is well known.  Wind fields are often responsible for the transport of clouds and, taking topographic conditions into account, are often the cause of rainfall.  Other negative effects on our natural environment are the devastations that hurricanes and cyclones can cause.  In the technical environment, one finds here also a multitude of flow processes, that occur in aggregates, instruments, machines and plants in order to transfer energy, generate lift forces, run combustion processes or take on control functions.  Fluid flows coupled with chemical reactions that enable the combustion in piston engines to proceed in the desired way and thus supply the power that is used in cars, trucks, ships and aeroplanes. 31 May 20214 Fig. 1.2 Effects of flows on the climate of entire geographical regions Fig. 1.3 Fluid flows are applied in many ways in our technical environment

5. PRELIMINARIES Contd…  The continuous scientific development of fluid mechanics started with Leonardo da Vinci (1452–1519). Through his ingenious work, methods were devised that were suitable for fluid mechanics investigations of all kinds. Earlier efforts of Archimedes (287–212 B.C.) to understand fluid motions led to the understanding of the hydromechanical buoyancy and the stability of floating bodies.  Fluid mechanics developed only after the work of Leonardo da Vinci. His insight laid the basis for the continuum principle for fluid mechanics considerations and he contributed through many sketches of flow processes to the development of the methodology to gain fluid mechanics insights into flows by means of visualization. 31 May 20215 ArchimedesLeonardo da VinciLudwig Prandtl Isaac Newton

6. PRELIMINARIES Contd…  In thermodynamics, heat is defined as the energy that crosses the boundary of a system, when there is temperature difference between the system and its surroundings.  Heat always flows over the boundary of the system in the direction of falling temperature.  Heat transfer occurs by three mechanisms. 31 May 20216  Conduction Temperature gradients exists in a stationary medium, which may be a solid or a fluid.  Convection Heat flow occur between a surface and a moving fluid when they are at different temperatures.  Radiation The phenomenon or the mode of heat transfer without a media in the form of electromagnetic waves. INTRODUCTION TO HEAT TRANSFER

7.  Convection heat transfer may be classified according to the nature of the flow. When the flow is caused by an external means, such as by a fan or atmospheric winds, it is called forced convection. In contrast, for free (or natural) convection, the flow is induced by the buoyancy forces, which arise from density differences caused by temperature variations in the fluid.  The driving force for natural convection is buoyancy, a result of differences in fluid density.  In engineering applications, convection is commonly visualized in the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heat-dissipation fins, and solar ponds.  A very common industrial application of natural convection is free air cooling without the aid of fans: this can happen on small scales (computer chips) to large scale process equipment. 31 May 20217 PRELIMINARIES Contd…

8. Mass transfer occurs by two mechanisms. (i) Diffusion mass transfer (ii) Convective mass transfer  Diffusion mass transfer is the transfer of matter occurs by the movement of molecules or species or particles of one component into another. Diffusion mass transfer may occur either due to concentration gradient (Molecular Diffusion) or temperature gradient (Thermal Diffusion) or pressure gradient (Pressure Diffusion).  Convective mass transfer is a mechanism in which mass is transferred between the fluid and the solid surface as a result of movement of matter from the fluid to the solid surface or fluid.  Convective mass transfer is again classified as “Natural or Free convection mass transfer” and “Forced Convection mass transfer”.  In a natural convection mass transfer, the transfer of mass occurs by the motion of species due to the density differences resulting from temperature or concentration differences of the mixture of varying composition.  The study of convection with both heat and mass transfer is very useful to industry, agriculture, oceanography, etc. 31 May 20218 PRELIMINARIES Contd…

9. 31 May 20219 PRELIMINARIES Contd… POROUS MEDIUM Transport processes in porous media can involve fluid, heat and mass transfer in single or multi-phase scenarios. Such flows with and without buoyancy effects arise frequently in many branches of chemical engineering and owing to their viscous-dominated nature are generally simulated using the Darcy model. Most studies dealing with porous media have employed the Darcy law which is a linear empirical relation between the flow velocity and the pressure drop across the porous medium and is limited to slow, viscous-dominated flows. However, for high velocity flow situations, the Darcy law is inapplicable, since it does not account for inertial effects in the porous medium. In this situation, the relation between velocity and pressure drop is quadratic and the Darcy- Forchheimer drag force model is required. The appropriate non-Darcian model, following Bég et al is therefore:

10. 31 May 202110 PRELIMINARIES Contd… Magnetohydrodynamic transport phenomena arise in numerous branches of modern chemical engineering. The Lorentz electromagnetic force arises as a result of the interaction between the magnetic field and the electrical current. This latter is generated hydrodynamically by the bulk flow of the conducting liquid phase. In porous media applications such as packed beds, to sustain a given flow rate of the electrically conducting liquid in the bed, the pressure drop and the liquid holdup will be increased under magnetohydrodynamic conditions compared with the case of non-conducting fluids. In this study the magnetic field therefore acts radially to the external boundary layer flow. This has been studied by Nath (1970) and more recently by Makinde et al (2009).

11. 31 May 202111 PRELIMINARIES Contd… SORET/DUFOUR EFFECTS In nature and in many industrial and chemical engineering situations there are many transport processes which are governed by the joint action of the buoyancy forces from both thermal and mass diffusion that develop due to the coexistence of temperature gradients and concentration differences of dissimilar chemical species. When heat and mass transfer occur simultaneously in a moving fluid, the relation between the fluxes and the driving potentials may be of a more intricate nature. An energy flux can be generated not only by temperature gradients but also by a composition gradient. The energy flux caused by a composition gradient is called the Dufour or diffusion-thermal effect. On the other hand, mass fluxes can also be created by temperature gradients and this is the Soret or thermal- diffusion effect. For example, when species are introduced at a surface in fluid domain, with different (lower) density than the surrounding fluid, both Soret and Dufour effects can be influential.

12. 31 May 202112 APPLICATIONS hydromagnetic flow control of conducting transport in packed beds, magnetic materials processing, geophysical energy systems etc.,. Applications of the study arise in hydromagnetic flow control of conducting transport in packed beds, magnetic materials processing, geophysical energy systems etc.,. Industrial Applications Chromatography Technology, Polymeric Suspension, Polymeric Suspension, Lubricant Manufacture etc., Lubricant Manufacture etc.,

13. 31 May 2021 13 OUT LINE OF THE TALK This talk is divided into six chapters: NUMERICAL MODELLING OF MAGNETO-CONVECTIVE TRANSPORT FROM A HORIZONTAL CYLINDER IN A NON-DARCIAN REGIME WITH SORET/DUFOUR EFFECTS MHD FREE CONVECTION-RADIATION INTERACTION FROM A HORIZONTAL CYLINDER IN A NON-DARCY POROUS MEDIUM IN THE PRESENCE OF SORET/DUFOUR EFFECTS THERMO-DIFFUSION AND DIFFUSION-THERMO EFFECTS ON MHD FREE CONVECTIVE HEAT AND MASS TRANSFER FROM A SPHERE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM THERMO-DIFFUSION AND DIFFUSION-THERMO EFFECTS ON MHD FREE CONVECTION FLOW PAST A VERTICAL PLATE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM COMPUTATIONAL ANALYSIS OF FREE CONVECTION HEAT AND MASS TRANSFER FROM AN ISOTHERMAL HORIZONTAL CIRCULAR CYLINDER TO A MICROPOLAR REGIME WITH SORET/DUFOUR EFFECTS FREE CONVECTION HEAT AND MASS TRANSFER FROM AN ISOTHERMAL SPHERE IN A MICROPOLAR REGIME WITH SORET/DUFOUR EFFECTS

14. NUMERICAL MODELLING OF MAGNETO-CONVECTIVE TRANSPORT MODELLING FROM A CYLINDER IN A NON- DARCIAN REGIME WITH SORET/DUFOUR EFFECTS 1 31 May 202114

15. 1 Schematic diagram of the physical system 31 May 202115 CHAPTER 1: contd..

16. Model Assumptions 31 May 202116 fluid is viscous, incompressible and electrically-conducted Fluid flow is laminar, steady and two-dimensional All the thermophysical properties (such as viscosity, thermal conductivity, specific heat and permeability) are considered to be constant except the density variation in the buoyancy force term Ohmic dissipation and induced magnetic field effects are neglected i.e. magnetic Reynolds number is assumed to be very small. Electrical field is also absent and magnetic field is of such magnitude that Hall and ionslip effects are not invoked. Diffusion-thermo and thermo-diffusion effects are considered Viscous dissipation effects are neglected due to slow motion CHAPTER 1: contd..

17. vv Governing Equations The boundary conditions are follows: CHAPTER 1: contd.. 31 May 202117 Under the usual Bousinessq and boundary layer approximations, the equations for mass continuity, momentum, energy and concentration, can be written as follows:

18. vv The stream function is defined by Dimensionless variables are introduced : (6) The transformed coupled, nonlinear, dimensionless partial differential equations for momentum, energy and species conservation for the regime: The corresponding boundary conditions are: (7) (8) CHAPTER 1: contd.. 31 May 202118

19. vvNomenclature  the dimensionless radial coordinate,   the dimensionless tangential coordinate;   the azimuthal coordinate,  the local inertia coefficient ( Forchheimer parameter ),  is a Darcy parameter,  concentration to thermal buoyancy ratio parameter,  the Prandtl number,  the Schmidt number,  the Dufour number  the magnetic parameter,  the Soret number,  the blowing/suction parameter and Gr  the Grashof (free convection) parameter. CHAPTER 1: contd.. 31 May 202119

20. vv f w 0 ( the case of blowing), and f w > 0 for V w < 0 ( the case of suction). Of course the special case of a solid cylinder surface corresponds to f W = 0. The engineering design quantities of physical interest include the skin- friction coefficient, Nusselt number and Sherwood number, which are given by: (8) CHAPTER 1: contd.. 31 May 202120

21. NUMERICAL SOLUTION WITH IMPLICIT DIFFERENCE CODE  An efficient and accurate implicit numerical scheme was devised for parabolic partial differential equations by Keller in 1970 namely the Keller box method. He has been employed in a diverse range of nonlinear and coupled heat transfer problems.  This method is chosen since it seems to be the most flexible of the common methods, being easily adaptable for solving equations of any order by Cebeci and Bradshaw (1984). We therefore present a slightly more detailed exposition here.  Essentially 4 phases are central to the Keller Box Scheme. These are: a)Reduction of the Nth order partial differential equation system to N 1 st order equations b) Finite Difference Discretization c) Quasilinearization of Non-Linear Keller Algebraic Equations d) Block-tridiagonal Elimination of Linear Keller Algebraic Equations CHAPTER 1: contd.. 31 May 202121

22. A 2-dimensional computational grid is imposed on the  plane as sketched below. The stepping process is defined by:  o = 0 ;  n =  n-1 + k n, n = 1, 2, …N(9a)  0 = 0;  j =  j-1 + h j, j = 1,2…J(9b) where k n and h j denote the step distances in the  and  directions respectively CHAPTER 1: contd.. 31 May 202122

23. Denoting  as the value of any variable at station  n,  j and the following central difference approximations are substituted for each reduced variable and their first order derivatives, viz: (  ) n-1/2 j – ½ = [  n j +  n j-1 +  n-1 j +  n-1 j-1 ]/4(10a) (  /  ) n-1/2 j – ½ = [  n j +  n j-1 -  n-1 j -  n-1 j-1 ]/4k n (10b) (  /  ) n-1/2 j – ½ = [  n j +  n j-1 -  n-1 j -  n-1 j-1 ]/4h j (10c)  j-1/2 = [  j +   j-1 ]/2(10d)  n-1/2 = [  n +  n-1 ]/2.(10e) Equations (6) subject to the boundary conditions (7) are first written as a system of first-order equations. For this purpose, we introduce new dependent variables u( ,  )= f ’, v( ,  ) = f ’’, s( ,  ) = , t( ,  ) =  ‘ and g( ,  ) = , p( ,  ) =  ’ for velocity, temperature and concentration respectively. CHAPTER 1: contd.. 31 May 202123

24. vv Phase A: Reduction of the Nth order partial differential equation system to N 1 st order equations Equations (6) subject to the boundary conditions (7) are first written as a system of first-order equations. For this purpose, we introduce new dependent variables, Therefore, we obtain the following seven first-order equations, the boundary conditions become: CHAPTER 1: contd.. 31 May 202124 (11a-g) (12)

25. vv Phase B: Finite Difference Discretization CHAPTER 1: contd.. 31 May 202125 (13)

26. vv We write the difference equations that are to approximate equations (11) by considering one mesh rectangle as shown in Figure 1. We start by writing the finite-difference approximations of the ordinary differential equations 11(a) to 11(d) for the midpoint (  n,  j-1/2 ) of the segment P 1 P 2, using centered-difference derivatives. This process is called “centering about (  n,  j-1/2 ). This gives: The finite-difference forms of the partial differential equations (11e) to (11g) are approximated by centering about the midpoint (  n-1/2,  j-1/2 ) of the rectangle P 1 P 2 P 3 P 4. This can be done in two steps. In the first step, we center equations (11e) to (11g) about the point (  n-1/2,  ) without specifying . Next we center equations (11e) to (11g) about the point (  n-1/2,  j-1/2 ) by using (13) CHAPTER 1: contd.. 31 May 202126 (14)

27. vv CHAPTER 1: contd.. 31 May 202127 (15)

28. vv Phase C: Quasilinearization of Non-Linear Keller Algebraic Equations To linearize the nonlinear system of equations (15) using Newton’s method, we introduce the following iterates: Then we substitute these expressions into equations (14) and (15) except for the term  n-1. Next we drop the terms that are quadratic in the following CHAPTER 1: contd.. 31 May 202128

29. vv CHAPTER 1: contd.. 31 May 202129 (16) the following linear tri-diagonal system of equations is obtained:

30. vv where CHAPTER 1: contd.. 31 May 202130

31. vv CHAPTER 1: contd.. 31 May 202131

32. vv CHAPTER 1: contd.. 31 May 202132

33. vv Phase D: Block-tridiagonal Elimination of Linear Keller Algebraic Equations The linearized difference equations of the system (16) have a block- tridiagonal structure. Commonly, the block-tridiagonal structure consists of variables or constants, but here, an interesting feature can be observed that is, for the Keller-box method, it consists of block matrices. Before we can proceed further with the block-elimination method, we will show how to get the elements of the block matrices from the linear system (16). We consider three cases, namely when j = 1, J-1 and J. When j = 1, the linear systems (16) become: CHAPTER 1: contd.. 31 May 202133

34. vv Designating CHAPTER 1: contd.. 31 May 202134

35. vv The corresponding matrix form assumes: For j = 1, we have CHAPTER 1: contd.. 31 May 202135

36. vv Similar procedures are followed at the different stations. Effectively the seven linearized finite difference equations have the matrix-vector form:   j =  j where  = Keller coefficient matrix of order 7 x 7,  j = seventh order vector for errors (perturbation) quantities and  j = seventh order vector for Keller residuals. This system is then recast as an expanded matrix-vector system, viz:  j  j -  j  j =  j where now  j = coefficient matrix of order 7 x 7,  j = coefficient matrix of order 7 x 7 and  j = seventh order vector of errors (iterates) at previous station on grid. Finally the complete linearized system is formulated as a block matrix system where each element in the coefficient matrix is a matrix itself. The numerical results are affected by the number of mesh points in both directions. Accurate results are produced by performing a mesh sensitivity analysis. After some trials in the η -direction a larger number of mesh points are selected whereas in the ξ direction significantly less mesh points are utilized. η max has been set at 30 and this defines an adequately large value at which the prescribed boundary conditions are satisfied. ξ max is set at 3.0 for this flow domain. CHAPTER 1: contd.. 31 May 202136

37. Staring conditions In the numerical computation, a proper step size and an appropriate (boundary layer thickness) value (an approximation to ) must be determined. This is done usually by a trial and error approach (Chen, 1988). In general, if the approximate value at a given is not known, the computation can be started by using a small value of and the successively increase the value of until a suitable value is obtained. In some cases, from trial and error experience, we found that too small or too large a value will give rise to convergence difficulties 31 May 202137

38. For most laminar boundary flows, the transformed boundary layer thickness is almost constant (Cebeci and Bradshaw, 1988). For instance, when, typically lies between 10 and 30. Once we obtain the proper value of, a reasonable choice of net spacing and should be determined. In most laminar boundary layer flows, a step size =0.02 to 0.04 is sufficient to provide accurate numerical results. In fact, for the present problem, we can even go up to =0.1 and still get accurate and comparable results. This particular value of =0.1 has also been used successfully by Merkin (1976, 1977a). Meanwhile, the step size can be arbitrary as it does not affect the converged results appreciably. A uniform grid across the boundary is quite satisfactory for most laminar flow calculations, especially in laminar boundary layer. However, the Keller-box method is unique in which various spacing in both directions can be used (Aldoss et al., 1996). 31 May 202138

39. In order to start and proceed with the numerical computation, it is necessary to make initial guesses for the functions f, u, v, g, p, s, and t across the boundary layer. The initial guesses can be obtained by a number of expressions. To start a solution at a given, it is necessary to assume distribution curves for u (velocity), s (temperature) and g (concentration) between and There are few possibilities in the selection of distribution curves, as long as they satisfy the boundary conditions (Chen, 1988). Once possibility is t assume that the initial velocity, temperature and concentration are given by (Began, 1984; Burmeister, 1993): 31 May 202139

40. 31 May 202140 Respectively. Therefore, integrating and differentiating expression (42a) with respect to, we get the following expressions for f and v, respectively:

41. 31 May 202141 Similarly, differentiating expression (42b) with respect to, we get

42. 31 May 202142 Finally, differentiation equation (42c) with respect to, gives The complete numerical results of this particular problem described in this chapter, which is the problem of free convection boundary layer flow over a permeable horizontal cylinder in a fluid- saturated non-Darcy porous regime with Soret/Dufour effects.

43. vv CHAPTER 1: contd.. 31 May 202143 FLOW CHART

44. vv RESULTS AND DISCUSSION Selected computations are presented in figures. The default values for the control parameters are selected as: Da = 0.1,  = 0.1, Pr = 0.7 (air), Sc = 0.25, Sr = 0.25, Du = 0.2 [i.e. Sr x Du = 0.05], f w, = 0.5, M = 1, N = 1, Gr = 10, these correspond to weak hydromagnetic convection in air (Pr = 0.71) flowing through a high permeability (Da = 0.1) with blowing (f w = 0.5) at the cylinder surface, weak inertial drag (Λ = 0.1 ), air diffusing in the electrolytic solution (Sc = 0.25) and a balance of thermal and species buoyancy forces (N = 1), at a general location along the cylinder curvature (  = 1). which represents physically buoyant non-Darcian case of hydrogen diffusing in a hydromagnetic boundary layer through porous media with suction (positive transpiration) and Soret and Dufour effects present. The values of Sr and Du have been selected to ensure that the product Sr and Du is constant, assuming that the mean temperature is constant. CHAPTER 1: contd.. 31 May 202144

45. vv RESULTS AND DISCUSSION CHAPTER 1: contd.. 31 May 202145  Decreasing Du clearly reduces the influence of species gradients on the temperature field, so that   values are clearly lowered and the boundary layer regime is cooled. With the exception of the profile corresponding to the lowest value of Du (= 0.01) and the highest value of Sr (= 5) there is a smooth decay of concentration distribution from the cylinder surface into the edge of the boundary layer. However for this exceptional case, a distinct concentration overshoot is caused in close proximity to the cylinder surface (  = 0). Effect of Sr and Du

46. vv RESULTS AND DISCUSSION CHAPTER 1: contd.. 31 May 202146 An increasing Da will serve to reduce the Darcian impedance since progressively less fibers will be present adjacent to the cylinder in the porous regime to inhibit the flow. The boundary layer flow will therefore be accelerated. Evidently lower permeability materials serve to decelerate the flow and this can be exploited in materials processing operation where the momentum transfer may require regulation. Diffusion of species is stifled with increasing Da i.e. concentration values decrease owing to an increase in permeability of the medium. Therefore lower permeability media aid in the diffusion of species in the boundary layer while higher permeability regimes oppose it. Effect of Da

47. vv RESULTS AND DISCUSSION CHAPTER 1: contd.. 31 May 202147 The Forchheimer effect is a second-order nonlinear porous medium inertial resistance. It arises also in the momentum Equation While the Darcy resistance dominates low Reynolds number viscous flows, with increasing inertial effects the Forchheimer drag will be dominant. Increasing  will evidently boost this Forchheimer drag which will decelerate the flow in the boundary layer as seen in Fig.5. A velocity peak is again witnessed close to the cylinder surface. At some distance from the wall all profiles tend to merge and the effect of increasing  is greatly diminished. Effect of Λ

48. vv RESULTS AND DISCUSSION CHAPTER 1: contd.. 31 May 202148 In magnetic materials processing, the flow can therefore be very effectively controlled with a magnetic field. Comparing with Fig. 5 we observe that smaller increases in M (from 0 through 10) have a more pronounced influence in decelerating the flow than very large changes in Forchheimer parameter (  changes from 0.001 to 100), despite the quadratic nature of the Forchheimer drag, indicating that magnetic field has a greater effect on flow retardation than nonlinear porous drag. A marked increase in temperature accompanies a rise in M i.e. temperatures are maximized with strong magnetic field. The supplementary work expended in dragging the fluid in the boundary layer against the action of the Lorentzian hydromagnetic drag is dissipated as thermal energy which heats the fluid. This induces a rise in temperatures. Effect of M

49. vv RESULTS AND DISCUSSION CHAPTER 1: contd.. 31 May 202149 Velocity is clearly decelerated with increasing migration from the leading edge i.e. larger  values for some distance into the boundary layer, transverse to the wall (  ~ 30). However closer to the free stream, this effect is reversed and the flow is accelerated with increasing distance along the cylinder surface. Conversely a very strong increase in temperature (  ) and concentration (  ), occurs with increasing  values. At various station ξ

50. vv RESULTS AND DISCUSSION CHAPTER 1: contd.. 31 May 202150 Aiding buoyancy forces (N >0) serve to accelerate the flow and this will increases skin friction at the cylinder wall; The species cross-diffusion term, in the energy Eq and the temperature cross-diffusion term, in the species Eq clearly exert a significant influence on both heat transfer and mass transfer rates at the cylinder surface in the porous media regime and should not be ignored in advanced studies of importance in materials processing. Skin friction, Nusselt, Sherwood numbers

51. vv CHAPTER 2 MHD FREE CONVECTION-RADIATION INTERACTION FROM A HORIZONTAL CYLINDER IN A NON-DARCY POROUS MEDIUM IN THE PRESENCE OF SORET/DUFOUR EFFECTS 31 May 202151

52. vv Governing Equations In the line with the approach of Yih (2000) and introducing the boundary layer approximations, the governing conservation equations can be written as follows: The boundary conditions are prescribed at the cylinder surface and the edge of the boundary layer regime, respectively as follows: 31 May 202152 CHAPTER 2: contd..

53. vv In The Rosseland diffusion flux model is used and is defined following Modest (1993) as follows: Where k* is the mean absorption coefficient and σ* is the Stefan-Boltzmann constant. Following Raptis and Perdikis (2004) we can express the quadratic temperature function in (6) as a linear function of temperature. The Taylor series for T 4 discarding higher order terms can be shown to give: Substituting of this expression into (5) and then the heat conservation equation (3), eventually leads to the following form of the energy equation: (6) (7) 31 May 202153 CHAPTER 2: contd..

54. vv The stream function is defined by Proceeding with the analysis we introduce the following dimensionless variables: (8) The coupled, nonlinear, dimensionless partial differential equations for momentum, energy and species conservation for the regime: The transformed dimensionless boundary conditions are: (9) (10) 31 May 202154 CHAPTER 2: contd..

55. vv The engineering design quantities of physical interest include the skin- friction coefficient, Nusselt number and Sherwood number, which are given by: (11) Where is the radiation parameter, 31 May 202155 CHAPTER 2: contd..

56. vv RESULTS AND DISCUSSION The present analysis integrates the system of Eqs. (9) with the boundary conditions Eq. (10) by the implicit finite difference approximation together with the modified Keller box method of Cebeci and Bradshaw. For the sake of brevity, the numerical method is not described. Computations were carried out with Δ  = 0.1; the first step size Δ η = 0.02. The requirement that the variation of the velocity, temperature and concentration distribution is less than 10 -5 between any two successive iterations is employed as the criterion convergence. A representative set of numerical results is presented graphically to illustrate the influence of hydromagnetic parameter ( M ), Forchheimer inertial drag parameter (  ), Darcy number ( Da ), Prandtl number ( Pr ), tangential coordinate (  ), Dufour number ( Du ), Soret number ( Sr ), Radiation parameter ( F ) and Schmidt number ( Sc ) on velocity, temperature, concentration, shear stress, local Nusselt number and Sherwood number profiles. 31 May 202156 CHAPTER 2: contd..

57. vv RESULTS AND DISCUSSION CHAPTER 2: contd.. 31 May 202157 It is observed from Fig. 2(a) that, a decrease in Du from 5.0 through to 0.01 (simultaneously Sr increases from 0.01 to 5.0, so that the product of Sr and Du remains constant i.e. 0.05) leads to a significant decrease in temperature values in the regime. From Fig. 2(b), it is notice that  (concentration function) in the boundary layer regime increases as Du decreases from 5.0 to 0.01 (and Sr simultaneously increases from 0.01 to 5.0). Mass diffusion is evidently enhanced in the domain as a result of the contribution of temperature gradients.

58. vv RESULTS AND DISCUSSION CHAPTER 2: contd.. 31 May 202158, and this defines the ratio of thermal conduction contribution relative to the thermal radiation. An increase in F from 0.1 (total thermal radiation dominance) through 0.5, 1.0, 3.0, 5.0 to 100, causes a significant decrease in velocity with distance into the boundary layer i.e. decelerates the flow. Velocities in all cases ascend from the cylinder surface, peak close to the wall and then decay smoothly to zero in the free stream. Thermal radiation flux therefore has a de-stabilizing effect on the flow regime. This is important in polymeric and other industrial flow processes since it shows that the presence of thermal radiation while decreasing temperature, will affect flow control from the cylinder surface into the boundary layer regime.

59. vv RESULTS AND DISCUSSION CHAPTER 2: contd.. 31 May 202159 Temperature values are also significantly reduced with an increase in F as there is a progressive decrease in thermal radiation contribution accompanying this. All profiles are monotonic decay from the wall to the free stream. The maximum reduction in temperature is witnessed relatively close to the cylinder surface since thermal conduction effects will be prominent closer to the cylinder surface, rather than further into the free stream. Concentration is conversely boosted with an increase in F i.e. decrease in thermal radiation contribution. The parameter F does not arise in the species conservation equation and therefore the concentration field is indirectly influenced by F via the coupling of the energy equation with the momentum equation, the latter also being coupled with the conservative acceleration terms in the species equation.

60. vv RESULTS AND DISCUSSION CHAPTER 2: contd.. 31 May 202160 f | An increasing N from -0.5 to 5, clearly accelerate the flow i.e. induces a strong escalation in stream wise velocity, f |, close to the wall; thereafter velocities decay to zero in the free stream. For all values of N there is a smooth decay in θ and  profiles from a maximum at the cylinder surface to the free stream. The buoyancy effect can clearly be exploited to control concentration distributions in laminar flow from a cylinder.

61. vv RESULTS AND DISCUSSION CHAPTER 2: contd.. 31 May 202161 Increasing suction (f w >0) causes the boundary layer to adhere closer to the flow and destroys momentum transfer; it is therefore an excellent control mechanism. Conversely with increased blowing i.e. injection of fluid via the cylinder surface into the porous medium regime, for which (f w < 0), the flow is accelerated i.e. velocities are increased. Temperature,  and concentration , are also markedly enhanced with increased blowing at the cylinder wall and depressed with increased suction.

62. vv RESULTS AND DISCUSSION CHAPTER 2: contd.. 31 May 202162 The effect of conduction radiation parameter, F on cylinder surface Shear stress, local Nusselt number and local Sherwood number variation are presented in Figs. 12(a)–12(c). With an increasing F, corresponding to progressively lower contributions of thermal radiation, wall shear stress is consistently reduced i.e. the flow is decelerated along the cylinder surface. With an increasing F, local Nusselt number is considerably increased. Also, with an increasing F, local Sherwood number is decreased.

63. vv CHAPTER 3: THERMO-DIFFUSION AND DIFFUSION-THERMO EFFECTS ON MAGNETO-HYDRODYNAMIC FREE CONVECTIVE HEAT AND MASS TRANSFER FROM A SPHERE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM 31 May 202163

64. vv MATHEMATICAL MODEL We consider the steady, laminar, two-dimensional, incompressible, electrically- conducting, buoyancy-driven convection heat and mass transfer flow form a permeable isothermal sphere embedded in a non- Darcy porous medium, under the action of an outwardly directed radial magnetic field. Fig illustrates the physical model and coordinate system. Here x is measured along the surface of the sphere, y is measured normal to the surface, respectively and r is the radial distance from symmetric axis to the surface. r = sin(x/a), a is the radius of the sphere. 31 May 202164 CHAPTER 3: contd..

65. vv The boundary conditions are prescribed at the cylinder surface and the edge of the boundary layer regime, respectively as follows: Governing Equations Under the usual Bousinessq and boundary layer approximations, the equations for mass continuity, momentum, energy and concentration, can be written as follows: 31 May 202165 CHAPTER 3: contd..

66. vv The stream function is defined by Proceeding with the analysis we introduce the following dimensionless variables: (5) We obtain the coupled, nonlinear, dimensionless partial differential equations for momentum, energy and species conservation for the regime: The transformed dimensionless boundary conditions are: (6) (7) 31 May 202166 CHAPTER 3: contd..

67. vv f w 0 ( the case of blowing), and f w > 0 for V w < 0 ( the case of suction). Of course the special case of a solid cylinder surface corresponds to f W = 0. The engineering design quantities of physical interest include the skin-friction coefficient, Nusselt number and Sherwood number, which are given by: (8) 31 May 202167 CHAPTER 3: contd..

68. vv RESULTS AND DISCUSSION CHAPTER 3: contd.. 31 May 202168 Forchheimer effects are associated with higher velocities in porous media transport. Forchheimer drag however is quadratic and the increase in this “form” drag swamps the momentum development, effectively decelerating the flow. With a dramatic increase in  there is a very slight elevation in temperatures and concentration in the regime. The deceleration in the flow results in thinner velocity boundary layers which serve to enhance energy and mass diffusions.

69. vv RESULTS AND DISCUSSION 31 May 202169 CHAPTER 3: contd.. Increasing M from 0 (non-conducting case), to 1.0 (magnetic body force and viscous force equal) through to 10.0 (very strong magnetic body force) induces a distinct reduction in velocities. Conversely with increasing M, temperature and concentration are observed to be markedly increased.

70. vv RESULTS AND DISCUSSION CHAPTER 3: contd.. 31 May 202170 Increasing Dufour number and decreasing Soret number strongly enhances the mass transfer rate at the wall i.e. boosts values. Generally with greater distance along the sphere surface i.e. with increasing  values, the local Sherwood number decreases. However for very low Dufour numbers (Du = 0.05, 0.01) and very high Soret numbers (Sr =1.0 and 5.0, respectively) there is a slight upturn in at large distances from the leading edge. The species cross-diffusion term in the energy equation and the temperature cross-diffusion term in the species equation clearly exert a significant influence on both heat transfer and mass transfer rates at the sphere surface in the porous media regime and should not be ignored in advanced studies of importance in materials processing.

71. vv CHAPTER 4 THERMO-DIFFUSION AND DIFFUSION-THERMO EFFECTS ON MHD FREE CONVECTION FLOW PAST A VERTICAL PLATE EMBEDDED IN A NON- DARCIAN POROUS MEDIUM The heat and mass transfer characteristics of MHD natural convection flow of a Newtonian fluid with uniform suction velocity along a uniformly heated vertical porous plate in the presence of diffusion-thermo (Dufour) and thermal-diffusion (Soret) effects. The dimensionless steady, coupled and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate Keller-Box implicit finite difference method 31 May 202171

72. vv MATHEMATICAL MODEL We consider the steady, laminar, two-dimensional, incompressible, electrically- conducting, buoyancy-driven convection heat and mass transfer flow form a permeable isothermal vertical plate embedded in a non-Darcy porous medium, under the action of diffusion-thermo and thermo-diffusion effects. Fig. 1 illustrates the physical model and coordinate system. Here x is measured along the surface of the plate, y is measured normal to the surface, respectively 31 May 202172 CHAPTER 4: contd..

73. vv The boundary conditions are prescribed at the cylinder surface and the edge of the boundary layer regime, respectively as follows: Governing Equations 31 May 202173 CHAPTER 4: contd..

74. vv Near the leading edge, the boundary layer is very much like that of the free convection boundary layer in the absence of suction. Therefore the following group of transformations are introduced: Substituting Eq.(5) into Eqs. (1) to (4), we obtain the coupled, nonlinear, dimensionless partial differential equations for momentum, energy and species conservation for the regime: The boundary conditions 31 May 202174 CHAPTER 4: contd..

75. vv 31 May 202175 where CHAPTER 4: contd..

76. RESULTS AND DISCUSSION CHAPTER 4: contd.. 31 May 202176 Values of and for different N, Sr, Du and ξ (Pr = 1.0, M = 1.0, Da = 0.1,  = 0.1, Sc = 0.25).,

77. RESULTS AND DISCUSSION CHAPTER 4: contd.. 31 May 202177 Values of and for different M, and  (Pr = 1.0, N = 1.0, Da = 0.1, ξ = 1.0, Sc = 0.25, Sr = 0.25, Du = 0.2)., Here we found that skin friction, local heat transfer and local mass transfer coefficients decrease with increasing (  ), because momentum, thermal and concentration boundary layers become thick due to the increase in the resistance to the motion. It is also seen that, the behavior remain same as increase in M.

78. vv RESULTS AND DISCUSSION CHAPTER 4: contd.. 31 May 202178 Increasing Da from 0.01 (extremely low permeability) through 0.05, 0.1, 0. 5, 1.0 to the maximum value of 10.0, clearly substantially enhances the flow velocity in the boundary layer. With higher Da values there will be a corresponding reduction in the Darcian drag force, and this will serve to effectively accelerate the flow in the medium adjacent to the plate. In all the velocity profiles the peak velocity is located close to the plate surface; with an increase in Da this peak is displaced progressively away from the plate surface. Temperature (  ) however, as shown in Fig. 3(b), is observed to progressively decrease with an increase in Darcy number (Da). With increasing Da there is a decrease in the density of solid matrix fibers in the regime

79. vv RESULTS AND DISCUSSION CHAPTER 4: contd.. 31 May 202179 It is clear from these figures that as suction parameter ξ increases, the maximum fluid velocity decreases; this is due to the fact that the effect of the suction is to take away the warm fluid on the vertical plate and thereby decrease the maximum velocity with a decrease in the intensity of the natural convection rate. Fig. 5(b) show the effect of the local suction parameter on the temperature profiles: We notice that the temperature profiles decrease with an increase in the suction effect and as the suction rate is increased, more warm fluid is taken away, and thus the thermal boundary layer thickness decreases.

80. vv RESULTS AND DISCUSSION CHAPTER 4: contd.. 31 May 202180 Conversely a strong decrease in concentration (  ), as shown in Fig. 5(c), occurs with increasing  values. It is also seen that as increase in  the impedance offered by the fibers of the porous medium will increase and this will effectively decelerate the flow in the regime, as testified to by the evident decrease in velocities shown in Fig. 5(a).

81. vv CHAPTER 5: COMPUTATIONAL ANALYSIS OF FREE CONVECTION HEAT AND MASS TRANSFER FROM AN ISOTHERMAL HORIZONTAL CIRCULAR CYLINDER IN A MICROPOLAR FLUID WITH SORET/DUFOUR EFFECTS 31 May 202181

82. vv INTRODUCTION  In recent years, the dynamics of micropolar fluids has been a popular area of research. As fluids consist of randomly oriented molecules and as each volume element of the fluid has translation as well as rotation motions, the analysis of physical problems in these fluids has revealed several interesting phenomena not found in Newtonian fluids.  Micropolar fluids are a special class of micro-morphic fluids, in which the elements are allowed to undergo rigid rotations only without stretch. The theory of micropolar fluids and thermo micropolar fluids developed by Eringen (1966).  In thermo micropolar fluid mechanics, the classical continuum and thermodynamics laws are extended with additional equations which account for the conservation of micro-inertia moments and the balance of first stress moments which arise due to the consideration of micro-structure in a fluid. CHAPTER 5: contd.. 31 May 202182

83. vv Schematic diagram of the physical system  The objective of the present work is to investigate the heat and mass transfer characteristics in free convection flow over a horizontal circular cylinder embedded in micropolar fluid with Soret/Dufour effects. CHAPTER 5: contd.. 31 May 202183

84. Under the usual Boussinesq and boundary layer approximation, the equations which govern the boundary layer flow are (Nazar et al. 2002a) (1) Governing Equations The boundary conditions appropriate for the regime corresponding to Eqs. (1) are:, (2) CHAPTER 5: contd.. 31 May 202184

85. Non-Dimensional variables (3) It is worth mentioning that in Equation (2) we have followed (Arafa and Gorla) by assigning a variable relation between microrotation and the surface skin friction, where n is a constant and. The value n = 0, which indicates, represents concentrated particle flows in which the particle density is sufficiently great that microelements close to the wall are unable to rotate (Jena and Mathur). This condition is also called “strong” interaction (Guram and Smith). The case corresponding to n = 1/2 results in the vanishing of antisymmetric part of the stress tensor and represents weak concentration (Ahmadi). In this case, the particle rotation is equal to fluid vorticity at the boundary for fine particle suspension. The case of n = 1/2 is considered in the present study When n = 1, we have flows which are representative of turbulent boundary layers (Peddieson). The case of n = 1/2 is considered in the present study. CHAPTER 5: contd.. 31 May 202185

86. The boundary conditions appropriate for the regime corresponding to Eqs. (1) are: (4) Governing Equations in Non- Dimensional Form The boundary conditions (5) CHAPTER 5: contd.. 31 May 202186

87. To solve the equations (5) - (9), subject to the boundary conditions (11), we assume the following variables where is the stream function defined in the usual way as Substituting (12) into equations (5) - (9) and after some algebraic manipulation we arrive at the following transformed equations: (6) (7) CHAPTER 5: contd.. 31 May 202187

88. Subject to the boundary conditions: The dimensionless parameters are defined as follows: The physical quantities of primary interest are the skin-friction coefficient; Nusselt number and Sherwood number are defined as (8) (9) (10) CHAPTER 5: contd.. 31 May 202188

89. RESULTS AND DISCUSSION A detailed parametric study has been performed for the influence of N, K, Du and Sr on dimensionless velocity (f / ), micro-rotation (g), temperature (  ), concentration (  ), skin friction coefficient (f ’’), Nusselt number (-  / ) and Sherwood number (-  / ). All data is provided as legends in figures 2a to 9. To validate the present numerical scheme we have compared solutions for the case of micropolar free convection with mass transfer Soret and Dufour effects neglected, a case studied earlier by Nazar et al. 2002a. Table 1 shows that excellent agreement is obtained over a wide range of x and K values. CHAPTER 5: contd.. 31 May 202189

90. CH 5. RESULTS AND DISCUSSION contd… 31 May 202190

91. Sr represents the effect of temperature gradients on mass diffusion. Du simulates the effect of concentration gradients on thermal energy flux in the flow domain. the product of Sr and Du must stay constant i.e. 0.05 Pr = 0.71, N = 1.0, Sc = 0.25, K = 2.0, x = 1.0. Figure 2a Decreasing Du (Sr values rise) clearly reduces the influence of species gradients on the temperature field, so that   values are clearly lowered and the boundary layer regime is cooled. Figure 2b,   in the boundary layer regime is increased as Du is decreased (and Sr simultaneously increased). Mass diffusion is evidently enhanced in the domain as a result of the contribution of temperature gradients. CH 5. RESULTS AND DISCUSSION contd… 31 May 202191

92. 1  Figures 2c and 2d show the temperature and concentration distributions with collective variation in Soret number Sr and Dufour number Du for the case of buoyancy opposition (opposing buoyancy force, N<0). The effects of temperature and concentration remain same as in the case of aiding buoyancy force.  The Sr & Du effects exert a dominant effect on concentration and temperature fields.  In micro-polar flows, although the linear velocity field is coupled to the angular velocity field, the influence of Soret and Dufour terms will be relatively weak on the velocity fields, and these are therefore not plotted. CH 5. RESULTS AND DISCUSSION contd… 31 May 202192

93. 1  Figures 3a to 3d depict the influence of the micropolar material parameter (K) on the velocity, micro-rotation, temperature and concentration profiles.  For K = 1 the micropolar and Newtonian dynamic viscosity are equivalent. For K = 0 micropolarity is neglected and the equations reduce to the Newtonian case.  We observe in figure 3a that initially an increase in K strongly decelerates the flow i.e depresses linear velocity (f / ); this trend is sustained until a certain distance normal to the cylinder surface after which a transition occurs and further towards the free stream increasing micropolar vortex viscosity serves to strongly accelerate the flow.  Nearer the cylinder surface, micro-elements are inhibited from spinning i.e. sustaining rotary motions; often to the extent that reverse spin is induced. 31 May 202193

94. 1  Further from the wall the micro-elements are able to spin more vigorously and this aids in accelerating the flow in the boundary layer.  Figure 3b indeed confirms the implications of figure 3a, as we note that strong reversal of micro-element rotation is induced near the cylinder surface with increasing K values; however with progressive distance into the boundary layer transverse to the cylinder surface, angular velocity (g) is indeed enhanced strongly.  Both temperature (  ) and species concentration (  ) are boosted throughout the boundary layer, with increasing micropolar vortex viscosity parameter, K. Clearly  and  distributions attain the least magnitude for K = 0 i.e. the Newtonian case. CH 5. RESULTS AND DISCUSSION contd… 31 May 202194

95. 1 Figures 4a to 4b illustrate the effect of the buoyancy ratio parameter, N, on linear velocity and micro-rotation distributions through the boundary layer regime. Initially for N 0 leads to a slight reduction in flow velocity (f / ) with the contrary for N < 0; however the influence of a large change in N is much less pronounced further from the wall. In figure 4b we observe that buoyancy- assistance (N>0) induces a significant reversal in angular velocity of the micro- elements (g) of the micropolar fluid in the near-wall region. CH 5. RESULTS AND DISCUSSION contd… 31 May 202195

96. 1  Figures 5 show the variation of flow variables with different locations along the cylinder surface i.e. x-coordinate for a weakly micropolar fluid.  We have computed the solutions from the lower stagnation point of the cylinder (x = 0) proceeding around the cylinder to x =  (i.e. 180 degrees) which corresponds to the upper stagnation point.  Near the lower stagnation point x ~ 0 and the governing pseudo-similar boundary layer equations reduce from the general case i.e. (13) to (16) to the following ordinary differential equations, viz:  Very low micropolar vortex viscosity the magnitudes of the micro-rotations are extremely small and consistently negative. With increasing x reverse micro-rotation is increased i.e. g values become increasingly more negative. CH 5. RESULTS AND DISCUSSION contd… 31 May 202196

97. 1  In figures 6a to 6d, the variation of velocity, angular velocity, temperature and concentration fields with different x values are shown.  Close to the cylinder surface, linear velocity (f / ) is found to be maximized closer to the lower stagnation point and minimized with progressive distance away from it, a marked acceleration in the flow is generated with greater distance from the lower stagnation point.  In figure 6b a very different response for the micro-rotation field is observed compared with the essentially monotonic profiles of figure 5.  Near the wall the micro-rotation is decreased i.e. reverse spin is enhanced, slightly with increasing x values. CH 5. RESULTS AND DISCUSSION contd… 31 May 202197

98. 1  Further form the wall however micro-rotation is positively affected the closer we approach the lower stagnation point. Yet further from the wall this trend is again altered and a considerable acceleration of angular velocity i.e. enhancement of micro-element spin is observed, a pattern which is sustained into the free stream of the boundary layer regime.  Temperature, , is found to noticeably increase through the boundary layer with increasing x values; as such the fluid regime is cooled most efficiently at the lower stagnation point and heated increasingly as we progress around the cylinder periphery upwards.  Concentration, is boosted throughout the boundary layer with increasing x values. CH 5. RESULTS AND DISCUSSION contd… 31 May 202198

99. 1  With strong micropolarity (K) the skin friction, is observed to be increased; increasing micropolar vortex viscosity accelerates the flow owing to increased spin of the micro- elements and this serves to increase skin friction. Effectively therefore micropolar fluids exhibit drag-reducing properties, a phenomenon identified by many other investigations including Peddieson and McNitt 1970. The lubricating properties of micropolar fluids i.e. liquids with micro-suspensions are greater therefore than Newtonian fluids (K = 0).  It is observed to strongly decrease with an increase in Sr and simultaneous rise in Du. The largest decrease arises near the lower stagnation point and the profiles tend to converge some distance thereafter. while increasing Sr and decreasing Du still inhibit surface mass transfer rate, the effect is less pronounced. CH 5. RESULTS AND DISCUSSION contd… 31 May 202199

100. 1  Local heat transfer rates are found to be enhanced substantially with an increase in Soret number and decrease in Dufour number. Again the maximum effect is sustained closer to the stagnation point and this influence is diminished with greater distances from this location. CONCLUSIONS  Buoyancy-assisted flow has been found to generally accelerate the linear flow field and also induced angular velocity acceleration of the micro-elements.  Increasing micropolarity of the fluid accelerates the flow and reduces drag effects compared with weaker micropolar and Newtonian fluids. Increasing Soret number has been shown to elevate surface heat transfer rates (Nu) but depress surface mass transfer rates (Sh) with the contrary response computed for increasing Dufour number.  The present numerical study it is hoped will stimulate further experimental investigations of micropolar diffusion flows, which require urgent attention in the research community. CH 5. RESULTS AND DISCUSSION contd… 31 May 2021100

101. vv CHAPTER 6: FREE CONVECTION HEAT AND MASS TRANSFER FROM AN ISOTHERMAL SPHERE TO A MICROPOLAR REGIME WITH SORET/DUFOUR EFFECTS 31 May 2021101

102. vv Schematic diagram of the physical system 31 May 2021102 CHAPTER 6: contd...

103. Under the usual Boussinesq and boundary layer approximation, the equations which govern the boundary layer flow are (see Nazar et al. 2002a) (1) (2) (3) (4) (5) Governing Equations The boundary conditions appropriate for the regime corresponding to Eqs. (1) - (5) are:, 31 May 2021103 CHAPTER 6: contd...

104. The boundary conditions appropriate for the regime corresponding to Eqs. (1) - (5) are: (9) (8) (7) (6) (5) Governing Equations in Non- Dimensional Form 31 May 2021104 CHAPTER 6: contd...

105. Non-Dimensional variables and Boundary conditions The dimensionless variables are defined as follows: The boundary conditions (11) 31 May 2021105 CHAPTER 6: contd...

106. To solve the equations (5) - (9), subject to the boundary conditions (11), we assume the following variables where is the stream function defined in the usual way as Substituting (12) into equations (5) - (9) and after some algebraic manipulation we arrive at the following transformed equations: (12) 31 May 2021106 CHAPTER 6: contd...

107. Subject to the boundary conditions: The dimensionless parameters are defined as follows: The physical quantities of primary interest are the skin-friction coefficient; Nusselt number and Sherwood number are defined as (17) (18) (19) 31 May 2021107 CHAPTER 6: contd...

108. RESULTS AND DISCUSSION A detailed parametric study has been performed for the influence of N, K, Du and Sr on dimensionless velocity (f / ), micro-rotation ( g ), temperature (  ), concentration (  ), skin friction coefficient (f’’), Nusselt number (-  / ) and Sherwood number (-  / ). All data is provided as legends in figures. To validate the present numerical scheme we have compared solutions for the case of micropolar forced convection with mass transfer (concentration equation) Soret and Dufour effects neglected, a case studied earlier by Nazar et al. Table 1 shows that excellent agreement is obtained over a wide range of x and K values. CHAPTER 6: contd.. 31 May 2021108

109. CH 6. RESULTS AND DISCUSSION contd… 31 May 2021109 Table1: Values of the Local Heat Transfer Coefficient (Nu  q w ) for various values of x and K with Pr = 0.7, Sr = Du = 0.0, N=0.0. (Nazar et al. [12] values are given in parenthesis)

110. CH 6. RESULTS AND DISCUSSION contd… 31 May 2021110 Decreasing Du clearly reduces the influence of species gradients on the temperature field, so that   values are clearly lowered and the boundary layer regime is cooled. The Sr values rise from 0.01 to 5.0 over this range (the product of Sr and Du must stay constant i.e. 0.05) Mass diffusion is evidently enhanced in the domain as a result of the contribution of temperature gradients. In micropolar flows, although the linear velocity field is coupled to the angular velocity field (micro-rotation), the influence of the Soret and Dufour terms will be relatively weak on the velocity fields, and these are therefore not plotted.

111. CH 6. RESULTS AND DISCUSSION contd… 31 May 2021111 For K = 1 the micropolar and Newtonian dynamic viscosity are equivalent. For K = 0 micropolarity is neglected and the equations reduce to the Newtonian (Navier-Stokes) case. We observe in Fig. 2(a) that initially an increase in K strongly decelerates the flow i.e depresses linear velocity this trend is sustained until a certain distance normal to the sphere surface after which a transition occurs and further towards the free stream increasing micropolar vortex viscosity serves to strongly accelerate the flow. Fig. 2(b) indeed confirms the implications of Fig. 2(a), as we note that strong reversal of micro- element rotation is induced near the wall (sphere surface) with increasing K values; however with progressive distance into the boundary layer transverse to the sphere surface, angular velocity (g) is indeed enhanced strongly.

112. CH 6. RESULTS AND DISCUSSION contd… 31 May 2021112 Both temperature (  ) and species concentration (  ) are boosted throughout the boundary layer, with increasing micropolar vortex viscosity parameter, K. Clearly  and  distributions attain the least magnitude for K = 0 i.e. the Newtonian case. Micropolar fluids therefore exhibit greater retardation compared with Newtonian fluids, which serves to suppress natural convection heat and mass transfer from the spherical body surface. Our results concur with the trends of Cheng [41], who has also shown that the greater viscosity of micropolar fluids acts to decelerate the flow and depress natural convection heat and mass transfer rates.

113. CH 6. RESULTS AND DISCUSSION contd… 31 May 2021113 Skin friction function response to micropolar parameter (K) values is illustrated in Fig. 7. With strong micropolarity the skin friction is observed to be increased; increasing micropolar vortex viscosity accelerates the flow owing to increased spin of the micro-elements and this serves to increase skin friction. strongly decrease with an increase in Sr and simultaneous rise in Du, the largest decrease arises near the lower stagnation point and the profiles tend to converge some distance thereafter.

114. vv CONCLUSIONS 31 May 2021114 Increasing Forchheimer inertial drag parameter (Λ) reduces velocity but elevates temperature and concentration. Increasing magnetic field parameter (M) reduces velocity but increases temperature and concentration. Increasing the radiation parameter (F) decreases velocity and temperature but increases concentration. Increasing Darcy number (Da) increases velocity but reduces temperature and concentration. Increasing Soret number and simultaneously decreasing Dufour number enhances the local heat transfer rate (local Nusselt number) at the cylinder surface with the opposite effect sustained for the mass transfer rate (local Sherwood number). Increasing positive buoyancy ratio parameter (aiding thermal and concentration buoyancy forces) acts to accelerate the flow with the contrary response computed for opposing buoyancy forces (negative buoyancy ratio parameters). The reverse effect is observed in the concentration field.

115. REFERENCES  Ali, A., N. Amin, and I. Pop, 2007, The unsteady boundary layer flow past a circular cylinder in micropolar fluids, Int. J. Numerical Methods Heat and Fluid Flow, 17, 692–714.  Bég, O. A., Bhargava, R., Rawat,S., Takhar, H. S. and Bég, T.A., 2007. Numerical study of Grashof and Darcy number effects on natural convection heat and species transfer past a stretching surface in micropolar saturated-porous medium with viscous heating. Int. J. Fluid.Mech. Res, 34(4), 287-307.  Bég, O.A., Tasveer A. Bég, A.Y. Bakier, Prasad, V., (2009). Chemically-reacting mixed convective heat and mass transfer along inclined and vertical plates with Soret and Dufour effects: Numerical solutions, Int. J. Applied Mathematics and Mechanics, 5, 2, 39-57.  Cebeci T., Bradshaw P., 1984. Physical and Computational Aspects of Convective Heat Transfer, Springer, New York.  Eringen, A.C., 1966.Theory of micropolar fluids. J. Math. Mech, 16, 1-18.  Mansour, M. A. and Gorla, R. S. R., 1999.MHD mixed convection boundary layer flow of a micropolar fluid from a horizontal cylinder, Applied mechanics and engineering, 4, 649-662. 31 May 2021115

116. 1  Merkin, J. H., 1976. Free convection boundary layer on an isothermal horizontal cylinder, ASMME/AIChE heat transfer conference, August 9-11, St. louis, USA.  Nazar, R., Amin, N, and Pop, I., 2002a. Free convection boundary layer flow on an isothermal horizontal circular cylinder in a micropolar fluid, proceedings of the 12 th international heat transfer conference, August 18-23, Paris: Elsevier, 2, 525-530.  Nazar, R., Amin, N, and Pop, I., 2002b. Free convection boundary layer flow on a horizontal circular cylinder with constant surface heat flux in a micropolar fluid, International journal of Applied Mechanics and Engineering, 7(2), 409 - 431.  Peddieson, J. and McNitt, R.P., 1970. Boundary layer theory for a micropolar fluid, Recent Adv. Engineering Science, 5, 405-476.  Seddeek, M.A., 2004. Thermal-diffusion and diffusion-thermo effects on mixed free- forced convective flow and mass transfer over accelerating surface with a heat source in the presence of suction and blowing in the case of variable viscosity, Acta Mech, 172, 83- 94.  Takhar, H.S., and Bég, O. A., 1997. Effects of transverse magnetic field, Prandtl number and Reynolds number on non-Darcy mixed convective flow of an incompressible viscous fluid past a porous vertical flat plate in saturated porous media, Int. J. Energy Research, 21, 87-100.  Willson, A.J., 1969.Basic flows of micropolar liquid, Appl. Sci. Res, 20, 338-355. 31 May 2021116

117. 1 31 May 2021117