Computers in Biology and Medicine 44 (2014) 44–56

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Network and Nakamura tridiagonal computational simulation of electrically-conducting biopolymer micro-morphic transport phenomena O. Anwar Bég a,n, J. Zueco b, M. Norouzi c, M. Davoodi c, A.A. Joneidi d, Assma F. Elsayed e a

Gort Engovation (Propulsion and Biophysics), Southmere Avenue, Bradford, BD7 3NU, UK Departamento de Ingeniería Térmica y Fluidos, Universidad Politécnica de Cartagena, Murcia, Spain c Mechanical Engineering Department, Shahrood University of Technology, Shahrood, Iran d Mechanical-Polymer Technology Group, Eindhoven University of Technology, Eindhoven, Netherlands e Mathematics Department, Faculty of Education, Ain shams, University, Heliopolis, Cairo, Egypt b

art ic l e i nf o

a b s t r a c t

Article history: Received 2 October 2013 Accepted 26 October 2013

Magnetic fields have been shown to achieve excellent fabrication control and manipulation of conductive bio-polymer characteristics. To simulate magnetohydrodynamic effects on non-Newtonian electroconductive bio-polymers (ECBPs) we present herein a theoretical and numerical simulation of free convection magneto-micropolar biopolymer flow over a horizontal circular cylinder (an “enrobing” problem). Eringen's robust micropolar model (a special case of the more general micro-morphic or “microfluid” model) is implemented. The transformed partial differential conservation equations are solved numerically with a powerful and new code based on NSM (Network Simulation Method) i.e. PSPICE. An extensive range of Hartmann numbers, Grashof numbers, micropolar parameters and Prandtl numbers are considered. Excellent validation is also achieved with earlier non-magnetic studies. Furthermore the present PSPICE code is also benchmarked with an implicit tridiagonal solver based on Nakamura's method (BIONAK) again achieving close correlation. The study highlights the excellent potential of both numerical methods described in simulating nonlinear biopolymer micro-structural flows. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Electro-conductive biolpolymers (ECBPs) Magnetohydrodynamics (MHD) Micropolar fluids Buoyancy Thermal convection Hartmann number Boundary layers Vortex viscosity Grashof number Network simulation Nakamura method Micro-rotation

1. Introduction Magnetic fields are deployed in many biomedical engineering systems owing to their excellent facility in manipulating properties and characteristics of materials and fluids. These include separation processes [1], smart biopolymer synthesis [2] and micro-channel magnetofluid devices [3]. In next generation “smart bio-engineered” polymers, magnetic manipulation in biomolecular separation from a sample, is achievable via initially utilizing adhesion to minute magnetized particles, followed thereafter, via detachment by virtue of carefully orientated external magnetic fields. In contrast to electric manipulation, magnetic interactions are generally not affected by surface charges, pH, ionic concentrations or temperature. This makes magneto-hydrodynamic processing of biolpolymers particularly attractive to bioengineers. To exploit this technology, magneto-hydrodynamics (MHD) must

n

Corresponding author. Tel.: þ 44 1274504653. E-mail address: [email protected] (O. Anwar Bég).

0010-4825/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compbiomed.2013.10.026

be used judiciously and engineers frequently conduct computational simulations leading to optimized designs. Magnetic forces have also been exploited in a vast range of modern biophysical applications including hemodynamic control [4,5], magnetic drug targeting [6–8], smart bio-lubrication [9–11], magnetic resonance imaging of the brain [12], hydromagnetic peristaltic pumps [13,14] and magnetic tweezers [15]. In the present study we simulate the enrobing process of a magnetic rheological biopolymer flow. These complex materials exhibit a number of advantages over synthetic polymers. They are commercially cheaper and non-toxic compared to the vast majority of synthetic polymers [16]. Such materials include methanesulfonic acid doped polyaniline (PANI) conductive blends which possess high electrical conductivity which has been experimentally verified with X-ray Photoelectron Spectroscopy. Mallick and Sakar [17] have also reported in detail on magnetohydrodynamic properties of conductive biopolymers, highlighting enhanced electrical conductivity even at room temperature. They have further emphasized the importance of optimizing fluid mechanical models of micro-structural rheological aspects of such polymers. As early as 1964, Eringen [18] pioneered a new branch

O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

of fluid mechanics- microcontinuum theory- which to date has provided the most elegant and “computable” model for complex liquids exhibiting microstructure. The micro-morophic framework simulates very accurately microscopic rheological (and vortex) effects which arise from rotary motions of the fluid microelements. The micropolar model, also introduced by Eringen [19] provides a simpler (but equally useful) theory for simulating couple stress and gryration effects and infact generalizes the Stokesian polar (couple stress) and other micro-structural model, by incorporating local rotary inertia. As such the micropolar model is a comprehensive and accurate framework which can describe a wide variety of complex non-Newtonian flows including bubbly liquids, liquid crystals, suspension polymers, paints, gels, physiological fluids (blood), synovial lubricants and even contaminated air in the human lungs. Thus far however this model has not been implemented for conducting biopolymers. Bég et al. [20] have reviewed progress in micropolar flow modeling in biomechanics in addition to aeronautics, chemical engineering and materials processing. Transport phenomena in micropolar flows have also been considered with multi-physical effects in [20]. In the context of biopolymeric enrobing flows, thermal convection boundary layer flows from cylindrical bodies are particularly relevant. Hassanien [21] investigated transverse curvature and viscosity effects numerically for the steady state mixed convective boundary layer flow of a micropolar fluid along vertical slender cylinders, showing that micropolar fluids display drag reduction as well as heat transfer rate reduction when compared to Newtonian fluids. Mahfouz [22] studied computationally the Rayleigh number and spin viscosity effects on unsteady natural convection from an isothermal cylinder placed horizontally in a micropolar fluid is investigated, confirming that in comparison with Newtonian fluids, micropolar fluids display a reduction in heat transfer rate at the cylinder surface. Bhargava et al. [23] obtained numerical solutions for the mixed convection flow of an incompressible micropolar fluid near a stagnation point on a horizontal cylinder, with wall transpiration effects showing that micropolar viscosity reduces drag forces and also acts as a cooling agent i.e. reduces surface heat transfer rates. Kumar et al. [24] used the finite element method to investigate the mixed convection on a moving vertical cylinder with suction in a moving micropolar fluid medium showing that temperature distribution is affected moderately by the motion of the cylinder as well with the buoyancy parameter. Gorla et al. [25] have also studied free and foced micropolar thermal boundary layers from a vertical cylinder. The above studies have all been restricted to electrically nonconducting fluid regimes i.e. magnetohydrodynamic effects have been ignored. Dunn [26] presented a simplified analytical model of the magnetohydrodynamic natural-convection heat transfer from a finite cylinder at various orientations with respect to an applied magnetic field, showing that the flow regime can be expressed solely as a function of Lykoudis number (magnetic parameter) and a constant. Aleksandrova [27] analyzed the hydromagnetic flow around an infinitely long elliptical cylinder. Aldos and Ali [28] studied the effect of suction on convection from a horizontal cylinder in a cross field hydromagnetic flow using both local numerical nonsimilarity and coordinate perturbation methods. They showed that blowing decreases Nusselt number and suction increases it both for free and forced convection flows. Rao et al. [29] studied numerically the two-dimensional hydromagnetic flow past a circular cylinder, describing the effects of a strong magnetic field on wake formation and stagnation point characteristics. El-Amin [30] studied porous drag force, Joule heating and viscous dissipation effects on hydromagnetic forced convection flow from a horizontal circular cylinder under transverse magnetic field, with variable wall temperature conditions using a second-level local non-similarity numerical method. Ganesan and

45

Loganathan [31] presented finite difference numerical solutions for the unsteady magnetohydrodynamic convection from a moving vertical cylinder with constant heat flux showing that an increase in the magnetic field decelerates the flow and elevates thermal boundary thickness. El-Kabeir et al. [32] employed the group theoretic method to simulate coupled magnetohydrodynamic heat and mass transfer from an impermeable horizontal cylinder to porous medium saturated with a nonNewtonian power law fluid. Several researchers have also discussed magneto-micropolar flows with heat transfer, where the nonlinearity of the differential flow conservation equations has generally necessitated numerical analysis. For example Mansour et al. [33] investigated hydromagnetic mixed thermal convection from a horizontal cylindrical body using numerical methods. In the present study we extend the non-magnetic study by Nazar et al. [34] to consider hydromagnetic effects on micropolar free convection boundary layer from a constant heat flux horizontal circular cylinder. Such a study has important applications in the simulation of magnetic field control of ECBP flows with heat transfer and has to the authors' knowledge, not appeared thus far in the technical literature. We study the effects of Hartmann number (Ha), Grashof number (Gr), micropolar vortex viscosity parameter (K) and Prandtl number (Pr) on the linear velocity, micro-rotation and temperature profiles using Network Simulation Methodology (NSM) [35]. Validation of numerical solutions is achieved with the Nakamura tridiagonal implicit finite difference scheme (NTS) [36]. The present study, as elaborated earlier, is motivated by further investigating enrobing hydrodynamics of magnetic nonNewtonian biopolymers. We employ micromorphic magnetofluid mechanics to investigate the industrial manufacturing processes for biologically-orientated applications of complex fluids including magnetic hydrogels (for bioassays), [37] rheological magnetic directed particle assembly [38], magnetorheological thin film biopolymers [39,40], magnetic “barcoded hydrogel microparticle” suspensions (of interest in multiplexed detection) [41] and microfluidic-based fabrication of magnetohydrodynamic hydrogel compositions [42]. These applications, many of which have been pioneered at the Massachusetts Institute of Technology (MIT), in recent years, have stimulated interest in how such materials can be magnetically manipulated during large-scale manufacture (industrial) for mass production and deployment in the biotechnology and magnetic medicine markets. It is envisaged that the present study will therefore stimulate some interest among researchers in further addressing this intriguing branch of computational biofluid mechanics.

2. Dynamics of magneto-thermo-micropolar biopolymeric fluids In this study the electrically conducting thermal micropolar constitutive model i.e. magneto-thermo-micropolar non-Newtonian fluid model is implemented to simulate microstructural chacateristics of conducting biopolymers. Such a fluid is a special subclass of the much more complex electrically-conducting thermomicromorphic fluid. In thermo-micropolar fluid mechanics, the classical continuum and thermodynamics laws are extended with additional equations which account for the conservation of microinertia moments and the balance of first stress moments which arise due to the consideration of micro-structure in a fluid. Hence new kinematic variables (gyration tensor, microinertia moment tensor), and concepts of body moments, stress moments and microstress are combined with classical continuum fluid dynamics theory. Thermo-micropolar fluids can accurately simulate liquids consisting of randomly orientated particles suspended in a viscous medium and offer an excellent framework to study advanced

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O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

gyration vector disappears and Eq. (3) due to Eringen [43] vanishes. Eq. (2) also reduces in this special case to the classical Navier-Stokes equations i.e. Newtonian flow. We also note that for the case of zero vortex viscosity (κ) only, the velocity vector V and the micro-rotation vector ω n are decoupled and the global motion is unaffected by the micro-rotations. Maxwell’s generalized electromagnetic field equations, following Sutton and Sherman [44] may also be presented as follows: ∞

∇B ¼ μe J ðAmpe`re0 slawÞ ∇E ¼  Fig. 1. Geometry of magnetic biopolymer enrobing flow regime.

geophysical and environmental pollution flows. The governing equations for magneto-thermo-micropolar fluids in terms of vector fields may be presented following Eringen [43] as follows:

∂B 0 ðFaraday s lawÞ ∂t

ð7Þ ð8Þ

∇B ¼ 0 ðMaxwell equation i:e: magnetic field continuityÞ

ð9Þ

∇J ¼ 0 ðGauss0 s Law i:e: conservation of electric chargeÞ

ð10Þ

where μe is the magnetic permeability of the fluid, E electrical field vector, and t denotes time. A simplified version of these equations is utilized in the mathematical transport model presented next.

Conservation of mass ∂ρ ¼  ρ∇  V ∂t

ð1Þ

Conservation of translational momentum ∂V ¼  ∇p þ κ ∇ω n  ðμ þ κ Þ∇  ∇  V þ ðλ þ 2μ þ κ Þ∇∇  V þ ρf þ JB ∂t

ð2Þ

Conservation of angular momentum (micro-rotation) ∂Ω

ρ jn

∂t

¼ κ ∇V  2κω n  γ ∇  ∇  ω n þ ðαv þ βv þ γ Þ∇ð∇  ω nÞ þ ρl

ð3Þ

Conservation of energy (heat)

ρ

∂E ¼ p∇  V þ ρΦ  ∇  q ∂t

ð4Þ

Dissipation function of mechanical energy per unit mass ∇ 2 þ γ ∇ω n : ∇ω n þ β v ∇ω n : ð∇ω nÞT

ρΦ ¼ λð∇  V Þ2 þ 2μD : D þ 4κ ð V  ω nÞ2 þ αv ð∇  ω nÞ2

3. Mathematical biopolymeric magnetohydrodynamic model The regime to be studied comprises a circular cylinder of radius, a, which is heated with a constant surface heat flux, qw and immersed in a viscous, incompressible, electrically-conducting micropolar biopolymer of ambient temperature, Tw. A magnetic field, B is applied perpendicular to the cylinder curved surface i.e. along the negative Y axis, as shown in Fig. 1. The X-direction is along the curved surface starting from the lower stagnation point of the cylinder i.e. X ¼0. Magnetic Reynolds number is assumed to be small enough to neglect magnetic induction effects. Hall current and ionslip effects are also neglected since the magnetic field is weak. We also assume that the Boussineq approximation holds i.e. that density variation is only experienced in the buoyancy term in the linear momentum equation. Additionally the electron pressure (for weakly conducting micropolar fluid), and the thermoelectric pressure are negligible. The steady free convection micropolar boundary layer regime may then be described under these assumptions, by the following equations in which the Lorentz force is included in the linear momentum equation.

ð5Þ Mass conservation

Deformation tensor 1 D ¼ ½V ij þ V ji  2

ð6Þ

where E is the specific internal energy, q the heat flux, Φ is the viscous dissipation function of mechanical energy per unit mass, ρ denotes the mass density of magneto-thermo-micropolar fluid, V is translational velocity vector, B is the applied magnetic field vector, J is current density vector, ω n is angular velocity (microrotation or gyration) vector, jn is microinertia, p is the thermodynamic pressure, f is the body force per unit mass vector, l is the body couple per unit mass vector, μ is the Newtonian dynamic viscosity, λ is the Eringen second order viscosity coefficient, κ is the vortex (microrotation) viscosity coefficient, and αv, βv and γ are spin gradient viscosity coefficients for thermo-micropolar fluids. In the special case where the fluid has constant physical properties, no external body forces exist and for steady state flow, the conservation equations can be greatly simplified. Additionally for the case where κ ¼ α ¼ β ¼ γ ¼0 and with vanishing l and f, the

∂U ∂V þ ¼0 ∂X ∂Y

ð11Þ

Linear momentum conservation    2  ∂U ∂U ∂ U ¼ ðμ þ κ Þ ρ U þV ∂X ∂Y ∂Y 2 X ∂H þ ρg βðT  T 1 Þ sin ð Þ  sBo 2 U þ κ a ∂Y Angular momentum (micro-rotation) conservation      2  ∂H ∂H ∂U ∂ H ¼  κ 2H þ þγ ρj U þ V ∂X ∂Y ∂Y ∂Y 2

ð12Þ

ð13Þ

Energy (heat) conservation  U

∂T ∂T þV ∂X ∂Y

 ¼

  k ∂2 T ρcp ∂Y 2

ð14Þ

O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

The spin gradient viscosity, γ, takes the following form: μ þ κ  γ¼ 2

ð15Þ

The appropriate surface (wall) and free stream boundary conditions are respectively:   ∂U ∂T q ; ¼ w ð16aÞ At Y ¼ 0 : U ¼ 0; V ¼ 0; H ¼  n ∂Y ∂Y k As Y-1 : U-0; H-0; T-T 1

ð16bÞ

where U is the velocity in the X-direction, V is the velocity in the Ydirection, H is the angular velocity (micro-rotation) of the microelements of the magneto-micropolar fluid, T is temperature, k is thermal conductivity of the magneto-micropolar fluid, μ is the Newtonian dynamic viscosity of the magneto-micropolar fluid, ρ is the density of the magneto-micropolar fluid, κ is the Eringen vortex viscosity of the magneto-micropolar fluid, s is the electrical conductivity the magneto-micropolar fluid, g is gravitational acceleration, β is the coefficient of thermal expansion, j is the micro-inertia density of the magneto-micropolar fluid and cp is the specific heat capacity of the magneto-micropolar fluid. In the micro-rotation boundary condition (16a), the parameter, n, is based on the study by Arafa and Gorla [45] and relates the angular velocity to the surface shear stress. Values of n may lie between 0 rn r1; for n ¼0 micro-rotation vanishes at the cylinder surface i.e. the micropolar fluid in this case has high particle concentration preventing micro-elements from rotating close to the wall. For n ¼0.5, the antisymmetric component of the stress tensor vanishes and this case simulates weak micro-element concentration at the wall. This is the case studied in this paper. For such a scenario, Ahmadi [46] has shown that micro-element rotation and fluid vorticity at the wall are equal for fine biopolymer particle suspensions. This may simulate accurately for example weakly concentrated bio-polymeric suspensions. Further details are given in Bég et al. [47].

47

Implementing the variables defined in (17) and (18) into the conservation Eqs. (11)–(14) we arrive at the following dimensionless equations: Mass conservation ∂u ∂v þ ¼0 ∂x ∂y

ð19Þ

Linear momentum conservation    2  ∂u ∂u ∂ u Ha2 ∂h ¼ ð1 þ KÞ þ Θ sin ðxÞ  2=5 u þ K u þv ∂x ∂y ∂y ∂y2 Gr Angular momentum (micro-rotation) conservation       ∂h ∂h ∂u K ∂2 h ¼  K 2hþ þ 1þ u þv ∂x ∂y ∂y 2 ∂y2 Energy (heat) conservation     ∂Θ ∂Θ 1 ∂2 Θ þv ¼ u Pr ∂y2 ∂x ∂y

ð20Þ

ð21Þ

ð22Þ

where x is dimensionless X-coordinate, y is dimensionless Y-coordinate, K is the Eringen vortex viscosity parameter (¼ κ/μ), u and v are the non-dimensional x- and y-direction velocities, Ha is the Hartmann (magnetohydrodynamic) number (¼ [sB02a2/μ]1/2), h denotes dimensionless micro-rotation, Gr is the Grashof (free convection) number (¼[gβa4qw/kν2] where ν is the kinematic Newtonian viscosity), Θ is the non-dimensional temperature function and Pr is the Prandtl number (¼ μcp/k) and T1 denotes the free stream temperature i.e. far from the cylinder curved surface. The nondimensionalized boundary conditions (16a) and (16b) take the form:   1 ∂u ∂Θ ; ¼ 1 ð23aÞ At y ¼ 0 : u ¼ v ¼ 0; h ¼  2 ∂y ∂y

4. Transformation of model

As y-1 : u-0; h-0; Θ-0

The steady boundary layer equations presented in (11)–(14) under boundary conditions (16a) and (16b) may be solved by finite difference or other computational methods. However to facilitate numerical solutions and circumvent the need for actual physical material data in the computations (e.g. viscosities, density etc.) we transform the model into dimensionless form as follows. Introducing the non-dimensional variables:

The further facilitate numerical solutions, we define the following transformed functions:



X a

ð17aÞ



Gr 1=5 Y a

ð17bÞ

a u ¼ Gr  2=5 U

ð17cÞ

ν

a

v ¼ Gr  1=5 V

ν



a2

ν

Gr  3=5 H

Θ ¼ Gr

1=5

  T  T1 aqw =k

ð17dÞ ð17eÞ ð17f Þ

Following Nazar et al. [34] we additionally define the micro-inertia density as: j¼

a2

ν

Gr  2=5

ð18Þ

ψ ðx; yÞ ¼ xFðx; yÞ; Θ ¼ Θðx; yÞ; h ¼ xGðx; yÞ where the stream function, tions:

ð23bÞ

ð24Þ

ψ satisfies the Cauchy–Riemann equa-



∂ψ ∂y

ð25aÞ



∂ψ ∂x

ð25bÞ

The dimensionless Eqs. (19)–(22) effectively reduce to the following trio of coupled, nonlinear, nonsimilar partial differential equations: Linear momentum  2      3  ∂ F ∂2 F ∂F sin x ∂G Ha2 ∂F ð1 þ KÞ   þ Θ þ K þ F ∂y x ∂y ∂y3 ∂y2 Gr 2=5 ∂y  ∂F ∂2 F ∂F ∂2 F   ¼x ∂y ∂x∂y ∂x ∂y2

ð26Þ

Angular momentum       1 þ K ∂2 G ∂G ∂F ∂2 F ∂F ∂G ∂F ∂G  þ F  G K 2G þ 2 ¼ x 2 2 ∂y ∂y ∂y ∂x ∂x ∂y ∂y ∂y

ð27Þ

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O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

Ha¼0. Eqs. (26)–(28) then reduce exactly to those considered by Merkin and Pop [48], with Eq. (27) vanishing.

Energy     1 ∂2 Θ ∂θ ∂F ∂Θ ∂F ∂Θ ¼ x  þ F Pr ∂y2 ∂y ∂x ∂x ∂y ∂y

ð28Þ

The transformed boundary conditions finally take the form: At y ¼ 0 :

As y-1 :



∂F ¼ 0; ∂y

∂F -0; ∂y

G¼ 

G-0;

1 ∂2 F ; 2 ∂y2

∂Θ ¼ 1 ∂y

Θ-0

ð29aÞ

Θðx; 0Þ ¼ Θw ðxÞ

ð33Þ

The transformed boundary conditions also reduce for the Merkin-Pop case [48] to: At y ¼ 0 : As y-1 :

ð30aÞ

ð30bÞ

The wall couple stress i.e. at surface of the cylinder is defined by:   ∂G ð30cÞ M wf ¼ γ ∂y y ¼ 0 The local Nusselt number is defined by:   hx Nux ¼ k

Energy     1 ∂2 Θ ∂θ ∂F ∂Θ ∂F ∂Θ ¼ x  þF Pr ∂y2 ∂y ∂x ∂x ∂y ∂y

ð32Þ

ð29bÞ

In chemical engineering applications, the gradients of the velocity, micro-rotation and temperature fields and also the cylinder surface temperature are of importance. Following Nazar et al. [34] we define a skin friction (non-dimensional surface shear stress) and cylinder temperature as, respectively:    2  K ∂ F x Cf ¼ 1 þ 2 ∂y2 y ¼ 0

Linear momentum  2  3    ∂ F ∂2 F ∂F sin x ∂F ∂2 F ∂F ∂2 F   þ Θ ¼ x þ F ∂y x ∂y ∂x∂y ∂x ∂y2 ∂y3 ∂y2

ð30dÞ



∂F ¼ 0; ∂y

∂F -0; ∂y

∂Θ ¼ 1 ∂y

ð34aÞ

Θ-0

ð34bÞ

Case III. Lower stagnation point hydromagnetic micropolar free convection from a cylinder At x  0, we have the special situation of lower stagnation point flow. Eqs. (26)–(28) in this case reduce to the following ordinary differential equations, in terms of the single spatial independent variable, y, viz: Linear momentum !  2     3 2 d F d F dF dG Ha2 dF ð1 þKÞ  2=5 ¼0 ð35Þ þK þF 2  3 dy dy dy dy dy Gr

where hx is the local heat transfer convection coefficient, defined by:   qw ðxÞx hx ¼ ð30eÞ Tw T1

Angular momentum ! ! 2 2 1þK d G dG dF d F  G  K 2G þ þF ¼0 2 dy dy dy2 dy2

The two-point boundary value problem defined by Eqs. (26)–(28) subject to the surface and free stream conditions (29a) and (29b) will not admit analytical solutions. As such we seek a numerical solution. Prior to this let us examine some important special cases.

Energy

! 2 1 d Θ dθ þF ¼ 0 Pr dy2 dy

ð36Þ

ð37Þ

subject to boundary conditions: 5. Special cases of the model

Case I. Non-magnetic micropolar free convection from a cylinder When magnetic field effects are negated, Ha¼0. Eqs. (26)–(28) then reduce exactly to those considered by Nazar et al. [34], in which only the linear momentum equation is affected, for which the micropolar fluid is electrically non-conducting:  2    3    ∂ F ∂2 F ∂F sin x ∂G ∂F ∂2 F ∂F ∂2 F ¼x  þF 2  þ ΘþK ð1 þ KÞ 3 2 ∂y x ∂y ∂y ∂x∂y ∂x ∂y ∂y ∂y ð31Þ Eqs. (27) and (28) and the boundary conditions (29a) and (29b) are identical to those considered in [34]. Case II. Non-magnetic Newtonian free convection from a cylinder. Merkin and Pop [48] studied the case where the fluid is both Newtonian and electrically non-conducting. This is extracted from our general model by setting K ¼ 0 (micropolar effects vanish) and

At y ¼ 0 :

As y-1 :

Fð0Þ ¼

∂Fð0Þ 1 ∂2 Fð0Þ ¼ 0; Gð0Þ ¼  ; ∂y 2 ∂y2

∂Fð1Þ -0; ∂y

Gð1Þ-0;

Θð1Þ-0

∂Θð0Þ ¼ 1 ∂y ð38aÞ ð39bÞ

In the present study we shall consider biopolymer flows for which Pr «1, since the thermal conductivity of such liquids is very high. Also we are interested in the case where a cylinder is being thermally controlled by an applied magnetic field, as encountered in enrobing flows. We shall present NSM solutions to the governing equations to provide a thorough perspective of the physics of the flow regime. Validation is achieved with a second order accurate tridiagonal finite difference method.

6. NSM computational solutions The governing Eqs. (26)–(28) subject to the surface and free stream conditions (29a) and (29b) constitute a seventh order set of

O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

nonlinear, coupled partial differential equations with seven corresponding boundary conditions. The Network Simulation Methodology (NSM) approach is an excellent technique which can robustly solve such a system. NSM is based on the thermoelectrical analogy and has been implemented in many diverse areas of fluid dynamics, being equally adept at solving linear and non-linear, steady or transient, hydrodynamic or coupled transport problems. Network simulation methodology is founded on the theory of network thermodynamics, in which flux–force relationships in dynamical systems are modeled using electric networks. Network thermodynamics exploits the formal similarities between the mathematical structure underlying different phenomena with the same balance and constitutive equations. Effectively NSM aims to design an electric circuit which possesses the same balance and constitutive equations as the physical problem of interest. NSM was introduced by Nagel [49] originally for semi-conductor problems. It was further refined for biological membrane science flows in the 1990s and has been applied to nonstationary diffusion through heterogeneous membranes [50], ionic transport in membranes [51], magnetic bio-tribology [52] and pulsatile blood flows [53]. NSM is extremely adaptable and equally adept at simulating both steady and transient flow regimes. In the NSM technique, discretization of the differential equations is founded on the finite-difference methodology, where only a discretization of the spatial co-ordinates is necessary. Numerical differentiation is implicit in such methods and some expertise is required in avoiding numerical diffusion, instability and convergence problems. NSM simulates the electrical variable of voltage as being equivalent to the velocities (u,v), micro-rotation (G) and temperature (Θ), while the electrical current is equivalent to the velocity flux (∂u/∂x, ∂u/∂y, ∂v/∂y), micro-rotation flux (∂G/∂x, ∂G/∂y) and temperature flux (∂Θ/∂x, ∂Θ/∂y). A network electrical model for each volume element is designed so that its electrical equations are formally equivalent to the spatial discretized equation. The whole network model, including the devices associated with the boundary conditions, is solved by the numerical computer code Pspice [54]. Fourier's law is utilized in the spatial discretization of the dimensionless transport equations. The electrical analogy is applied to the discretized equations together with Kirchhoff's law for electrical currents. To implement the boundary conditions, constant voltage sources are employed for both velocities. The principal advantage of the NSM approach is that it negates the requirement in standard numerical finite difference schemes of manipulation of difference equations and the constraints of specified yardsticks around the convergence of numerical solutions. Details of the discretization and electronic network diagram construction have been provided in previous studies by Zueco and Bég et al. [52,53], and are therefore omitted here for brevity. This code is designated the “electric circuits simulator”. Time remains as a real continuous variable. The user does not need to manipulate the finite difference differential equations to be solved nor expend effort in convergence problems. Nagel [55] has elucidated in detail the local truncation errors present in the SPICE algorithm. A necessary criterion for using SPICE effectively is a familiarity with electrical circuit theory. Momentum and angular momentum (micro-rotation) balance “currents” are defined systematically for each of the discretized equations and errors can be quantified in terms of the quantity of control volumes. The use needs to code a special protocol file, (file “Network.cir”). This program rapidly generates the file for execution in Pspice, and the program permits the reading of the solutions provided by Pspice (file “Network. out”). Following the PSPICE simulation, the code plots waveform results so the designer can visualize circuit behavior and determine design validity. Graphical results of each simulation are presented in Pspice's “Probe window waveform viewer” and analyzer, where it is possible to see the axial and azimuthal

49

Diagram 1. NSM methodology.

Table 1 NSM Comparison with Nazar et al.'s [34] implicit finite difference solutions for Θw(x) with Pr¼ 0.72 and Ha¼ 0. x ↓, K-

Nazar et al. [34]

NSM solution

0.5

1.5

2.5

0.5

1.5

2.5

0 0.4 0.8 1.0 1.2 1.6 2.0 2.4 3.0

2.2887 2.2975 2.3229 2.3424 2.3669 2.4329 2.5281 2.6665 3.0601

2.4200 2.4295 2.4568 2.4778 2.5041 2.5754 2.6782 2.8285 3.2623

2.5168 2.5267 2.5554 2.5774 2.6050 2.6799 2.7881 2.9465 3.4075

2.2563 2.2622 2.3026 2.3128 2.3356 2.4109 2.4998 2.6348 2.9867

2.3967 2.4008 2.4238 2.4438 2.4689 2.5469 2.6298 2.8001 3.2324

2.4643 2.5008 2.5255 2.5367 2.5698 2.6358 2.7537 2.9129 3.3788

π

3.2420

4.4682

3.6262

3.2009

4.4269

3.5976

components of the velocity and magnetic field any point of the medium. NSM implements the most recent advances in software in the resolution of electrical networks to solve diverse types of partial differential equations which may be elliptical, hyperbolical, parabolic, linear, non-linear and 1-, 2- or 3-dimensional. The intrinsic procedures to NSM are summarized in Diagram 1. Table 1 shows the comparison of NSM solutions for the variation of cylinder wall temperature, Θw(x) with streamwise coordinate, x with the non-magnetic solutions of Nazar et al. [34] obtained with the Keller box method. Excellent agreement is obtained for all three cases of micropolar vortex viscosity, K ¼0.5, 1.5 and 2.5. We further note that Table 1 shows the results of Nazar et al. [34] for Pr¼ 0.72 (air). The equivalent non-magnetic results are generated using Ha¼0 in our model, which are much too low for biopolymers. We therefore consider higher Prandtl numbers (e.g. 10). As with our computations, those by Nazar et al. [34] commence at the lower stagnation point of the cylinder (x  0) and proceed around the curved surface with increasing x up to the upper stagnation point (x¼ π).

7. Nakamura triadiagonal solutions and NSM validation The system of Eqs. (26)–(28) with boundary conditions (29a) and (29b) is a well-posed nonlinear two-point boundary value problem which has also been solved with the efficient implicit Nakamura Tridiagonal finite difference Scheme (NTS) [36]. We provide a second validation of NSM here since no available data exists for comparing the magnetic solutions. The BIONAK code has been developed to implement the Nakamura method for general

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O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

Table 2 NSM and NTS solutions for skin friction, Cf, and various streamwise coordinate values (x) with various Hartmann numbers (Ha) and Pr ¼10, Gr ¼10, K¼ 1. Ha ¼1

x

0 π/4 π/2 3π/4 π

Ha¼3

Ha¼ 7

Cf (NSM)

Cf (NTM)

Cf (NSM)

Cf (NTM)

Cf (NSM)

Cf (NTM)

0.0000 0.3245 0.5124 0.5428 0.2013

0.0000 0.3247 0.5131 0.5430 0.2016

0.0000 0.2403 0.3905 0.3878 0.1243

0.0000 0.2404 0.3904 0.3876 0.1241

0.0000 0.1231 0.2134 0.2005 0.0583

0.0000 0.1233 0.2135 0.2007 0.0584

Table 3 NSM and NTS solutions for wall couple stress function (micro-rotation gradient) ∂G/ ∂y, at x ¼0 (lower stagnation point) with various micropolar vortex viscosity parameters (K) and Pr ¼ 10, Gr ¼10, Ha¼ 1. y K¼ 0 (Newtonian) ∂G/∂y (NSM)

∂G/∂y (NTM)

0 0.14403 2 0.01075 4  0.0034 6  0.0004 12 0.0000

K¼2 ∂G/∂y (NSM)

K¼ 5 ∂G/∂y (NTM)

0.11951 0.01043  0.0042  0.0003 0.00000

∂G/∂y (NSM)

∂G/∂y (NTM)

0.0800 0.01032  0.0053  0.0002 0.0000

0.0802 0.01031  0.0054  0.0019 0.0000

Table 4 NSM and NTS solutions for temperature (Θ) at x¼ π (upper stagnation point) with various Hartmann numbers (Ha) and Pr ¼10, Gr¼ 10, K¼ 1.

Fig. 2. Cylinder temperature Θw(x) distribution versus x, for Gr ¼10 (strong buoyancy), K¼ 1, Pr ¼10 with various Hartmann numbers (Ha).

For the angular momentum and energy Eqs. (27) and (28) which are second order equations, only a direct substitution is needed. However a reduction is required for the linear momentum Eq. (26). Setting: P ¼ F=

ð40aÞ

Q ¼G

ð40bÞ

R¼Θ

ð40cÞ

The Eqs. (26)–(28) then assume the form: Nakamura linear momentum equation:

y

Ha¼1

0 5 10 20 30

Ha¼ 3

Ha¼ 7

A1 P== þ B1 P= þ C1 P ¼ S1

Θ (NSM)

Θ (NTM)

Θ (NSM)

Θ (NTM)

Θ (NSM)

Θ (NTM)

2.5134 0.0235 0.0002 0.0000 0.0000

2.5133 0.02034 0.0002 0.0000 0.0000

3.5024 1.4063 0.7035 0.2046 0.0817

3.5023 1.4062 0.7034 0.2045 0.0816

5.3027 3.0128 2.1052 1.2065 0.8064

5.3026 3.1027 2.1051 1.2064 0.8063

Nakamura angular momentum equation: A2 Q == þ B2 Q = þ C2 Q ¼ S2

The flow domain for the convection field is discretized using an equi-spaced finite difference mesh in the x, y-directions. The partial derivatives for F, G, Θ with respect to x, y are evaluated by central difference approximations. A single iteration loop based on the method of successive substitution is utilized due to the high nonlinearity of the momentum, angular momentum and energy conservation equations. The finite difference discretized equations are solved as a linear second order boundary value problem of the ordinary differential equation type on the x, y-domain.

ð42Þ

Nakamura energy equation: A3 R== þ B3 R= þ C3 R ¼ S3

biofluid dynamics problems. As with other difference schemes, a reduction in the higher order differential equations, is also fundamental to this method. It is also particularly effective at simulating highly nonlinear flows as characterized by biopolymers and rheological biofluids. Interesting applications include power-law fluid heat transfer [56], nanofluid bioconvection micro-organism flows in microbial fuel cells [57] and viscoelastic polymer wedge processing flows [58]. In micropolar flows Nakamura's method has been successfully used by Gorla et al. [59] and Gorla and Nakamura [60]. NTS works well for both one-dimensional (ordinary differential systems) and two-dimensional (partial differential systems) non-similar flows. NTS entails a combination of the following aspects.

ð41Þ

ð43Þ

where Ai ¼ 1…3, Bi ¼ 1…3, Ci ¼ 1…3 are the Nakamura matrix coefficients, Si ¼ 1…3 are the Nakamura source terms containing a mixture of variables and derivatives associated with the lead variable. The Nakamura Eqs. (41)–(43) are transformed to finite difference equations and these are orchestrated to form a tridiagonal system which is solved iteratively.Tables 2–4 compare the Nakamura solution with the magnetic NSM solution for skin friction with various Hartmann numbers (Table 2), micro-rotation gradient (wall couple stress function) with micropolar vortex viscosity parameter (Table 3) and temperature variation with Hartmann number (Table 4). In all cases very close correlation is achieved. Confidence in the present NSM solutions is therefore very high.

8. Results and discussion The flow regime is dictated by four rheological and thermophysical parameters: Eringen vortex viscosity parameter (K), Hartmann (magnetohydrodynamic) number (Ha), Grashof (free convection) number (Gr) and the Prandtl number (Pr). Selected computations are presented in Figs. 2–20, where all data is documented.

O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

Fig. 3. Skin friction Cf distribution versus x, for Gr ¼10 (strong buoyancy), K¼1, Pr ¼10 with various Hartmann numbers (Ha).

51

Fig. 5. Micro-rotation (angular velocity, G) distribution versus y for Gr ¼10 (strong buoyancy), K¼ 1, Pr ¼10 with various Hartmann numbers (Ha) at x ¼0 (lower stagnation point), x¼ π/2 (along horizontal diameter) and x ¼ π (upper stagnation point).

Fig. 4. Temperature (Θ) distribution versus y for Gr¼ 10 (strong buoyancy), K¼1, Pr ¼10 with various Hartmann numbers (Ha) at x ¼0 (lower stagnation point), x ¼π/2 (along horizontal diameter) and x¼ π (upper stagnation point).

Fig. 2 shows the distribution of cylinder wall temperature

Θw(x) versus streamwise coordinate x, for Gr ¼10 (strong buoy-

ancy), K ¼1, Pr¼10 (biopolymer) with various Hartmann numbers (Ha). With an increase in Hartmann number from 0 (non-conducting case) through 1, 3, 5 to 7, the wall temperature is found to be enhanced considerably. Hartmann number (Ha)¼[sB02a2/μ]1/2 represents the ratio of the magneto-hydrodynamic (Lorentzian) drag force to the viscous hydrodynamic force in the regime. The transverse magnetic field generates a Lorentzian magnetic drag force in the opposite direction to the flow. We note that for the case of Ha¼ 1 both magnetic retarding force and viscous hydrodynamic force will be of the same order. For Hao1, magnetic forceoviscous force. For Ha41, magnetic force4viscous force. The work done by the micropolar biopolymer in overcoming this drag force is converted into thermal energy i.e. dissipated as heat. This causes an accentuation in wall temperature of the cylinder. While all profiles are gentle slopes from x¼0 (lower stagnation point) through x ¼ π/2 (cylinder mid-point i.e. on periphery of a horizontal diameter in Fig. 1), as we approach the upper stagnation point there is a marked increase in gradients of the profiles. As such transverse magnetic field effects are maximized at the upper stagnation point and minimized at the lower stagnation point. In Fig. 3 the distribution of skin friction,    2  K ∂ F Cf ¼ 1 þ x 2 ∂y2 y ¼ 0

with streamwise coordinate, x, is shown. Increasing Hartmann number clearly significantly reduces Cf values i.e. retards the flow along the surface of the cylinder. Profiles however ascend from zero at the lower stagnation point (x ¼0) and peak at some distance beyond x¼ π/2 (cylinder mid-point i.e. on periphery of a horizontal diameter in Fig. 1), thereafter decaying considerably as

Fig. 6. Micro-rotation gradient (∂G/∂y) i.e. wall couple stress function distribution versus y for Gr¼ 10 (strong buoyancy), K¼ 1, Pr ¼10 with various Hartmann numbers (Ha) at x ¼ 0 (lower stagnation point), x¼ π/2 (along horizontal diameter) and x¼ π (upper stagnation point).

Fig. 7. Cylinder temperature Θw(x) distribution versus x, for Ha ¼1 (weak magnetic field), K¼1, Pr¼ 10 with various Grashof numbers (Gr).

we approach the upper stagnation point (x ¼ π). The flow at any given Hartmann number is therefore accelerated to downstream of the mid-point but then decelerated further along the cylinder. There is also a very sharp descent close to the upper stagnation point. In Fig. 4 the temperature (Θ) profiles versus coordinate normal to the cylinder surface (y) are plotted for three distinct locations along the cylinder i.e. x ¼0 (lower stagnation point), x¼ π/2 (cylinder mid-point) and x ¼ π (upper stagnation point), with different Hartmann number (Ha). Increasing Ha clearly serves to enhance temperatures which decay from a maximum at the cylinder surface (y¼0) to zero in the free stream (very large y). Stronger transverse magnetic field clearly acts to heat the fluid

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O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

Fig. 8. Skin friction Cf distribution versus x, for Ha ¼1 (weak magnetic field), K ¼1, Pr ¼10 with various Grashof numbers (Gr).

Fig. 9. Temperature (Θ) distribution versus y for Ha¼1 (weak magnetic field), K ¼1, Pr ¼10 with various Grashof numbers (Gr) at x¼ 0 (lower stagnation point), x ¼ π/2 (along horizontal diameter) and x¼ π (upper stagnation point).

Fig. 11. Micro-rotation gradient (∂G/∂y) i.e. wall couple stress function distribution versus y for Ha ¼1 (weak magnetic field), K¼ 1, Pr¼ 10 with various Grashof numbers (Gr) at x¼ 0 (lower stagnation point), x ¼π/2 (along horizontal diameter) and x¼ π (upper stagnation point).

Fig. 12. Cylinder temperature Θw(x) distribution versus x, for Gr¼ 10, Ha¼ 1 (weak magnetic field), Pr ¼10 with various micropolar parameter values (K).

Fig. 10. Micro-rotation (angular velocity, G) distribution versus y for Ha¼1 (weak magnetic field), K¼1, Pr¼ 10 with various Grashof numbers (Gr) at x ¼0 (lower stagnation point), x ¼π/2 (along horizontal diameter) and x¼ π (upper stagnation point).

Fig. 13. Skin friction Cf distribution versus x, for Gr¼ i0, Ha ¼1 (weak magnetic field), Pr¼ 10 with various micropolar parameter values (K).

regime i.e. generates more thermal energy in the boundary layer regime owing to dissipation of the work done in dragging the fluid against the imposition of a magnetic field. Maximum temperature therefore corresponds to maximum Hartmann number, a feature of some importance in magnetic materials processing. Temperatures are also respectively greater at x¼ π, then they are for x ¼ π/2 and x ¼0 i.e. the fluid is increasingly heated with progression from the lower stagnation point to the upper stagnation point. Micro-rotation profiles (G) versus y for various Ha values again at three discrete locations along the cylinder i.e. x ¼0, x¼ π/2 and x ¼ π are depicted in Fig. 5. The micro-rotation of micro-elements at the cylinder surface is dictated by the surface boundary

condition G ¼ ð1=2Þð∂2 F=Þ∂y2 in (29a). This physically implies that weak micro-element concentrations arise at the cylinder surface and can represent fine particle suspensions, as encountered in materials processing operations. We observe that microrotation at the surface is always negative i.e. micro-elements rotate in the reverse sense; this characteristic is however suppressed with increasing magnetic field. The least negative G values at the cylinder surface correspond to the upper stagnation point (x ¼ π) and the greatest negative values to the lowest stagnation point (x ¼0). All profiles ascend and peak at y  2 thereafter decaying smoothly to zero in the free stream. Downstream of y  2, increasing Hartmann number has a reverse effect on micro-rotation

O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

Fig. 14. Temperature (Θ) distribution versus y for Gr¼ 10, Ha ¼1 (weak magnetic field), Pr¼ 10 with various micropolar parameter values (K) at x¼ 0 (lower stagnation point) and x¼ π (upper stagnation point).

Fig. 15. Micro-rotation (angular velocity, G) distribution versus y for Gr ¼10, Ha¼1 (weak magnetic field), Pr ¼10 with various micropolar parameter values (K) at x ¼0 (lower stagnation point) and x¼ π (upper stagnation point).

53

Fig. 17. Cylinder temperature Θw(x) distribution versus x, for K¼ 1, Ha ¼1 (weak magnetic field), Gr ¼10 with various Prandtl numbers (Pr).

Fig. 18. Skin friction Cf distribution versus x, for K¼ 1, Ha¼1 (weak magnetic field), Gr¼ 10 with various Prandtl numbers (Pr).

Fig. 16. Micro-rotation gradient (∂G/∂y) i.e. wall couple stress function distribution versus y for Gr¼ 10, Ha ¼1 (weak magnetic field), Pr ¼10 with various micropolar parameter values (K) at x ¼0 (lower stagnation point) and x ¼π (upper stagnation point).

Fig. 19. Temperature (Θ) distribution versus y for K¼ 1, Ha ¼1 (weak magnetic field), Gr¼ 10 with various Prandtl numbers (Pr) at x ¼0 (lower stagnation point) and x ¼ π (upper stagnation point).

and acts to decrease G values. Therefore while increasing magnetic field suppresses reverse microelement rotary motions near the surface it also depresses rotary motions in the x–y plane further into the boundary layer transverse to the cylinder surface. The distribution of micro-rotation gradient (∂G/∂y) i.e. wall couple stress function with transverse coordinate, y is shown in Fig. 6. Increasing Hartmann number acts to decrease substantially ∂G/∂y values which rise momentarily peaking near the cylinder surface and then plummet sharply from the maxima at the cylinder surface, for the lower stagnation point case (x ¼0). At some distance from the wall, the values become negative and then vanish closer to the free stream. For the upper stagnation point

however, while increasing magnetic field also serves to decrease ∂G/∂y values, these remain positive from the cylinder surface throughout the boundary layer regime. Peak values also arise further from the wall for this case compared with the lower stagnation point. The variations of Θw(x), Cf, Θ, G and ∂G/∂y distributions with different Grashof numbers (Gr) are shown in Figs. 7–11. Grashof number¼ [gβa4qw/kν2] and represents the ratio of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer regime. Strong free convection effects will arise with very high Gr values. For Gr ¼1 the viscous and thermal buoyancy forces are equal. Wall cylinder temperature Θw(x) as shown in Fig. 7,

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O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

Fig. 20. Micro-rotation (angular velocity, G) distribution versus y for K¼ 1, Ha ¼1 (weak magnetic field), Gr¼ 10 with various Prandtl numbers (Pr) at x ¼0 (lower stagnation point) and x ¼π (upper stagnation point).

is observed to be strongly decreased with increasing Grashof number. Profiles ascend from a minimum at the wall to maximum magnitudes at the upper stagnation point. Cylinder surface shear stress (skin friction), as depicted in Fig. 8, is conversely found to be increased considerably with an increase in Grashof number. The flow is accelerated along the cylinder surface from the lower stagnation point, climaxes beyond x¼ π/2 and then decays towards the upper stagnation point. Free convection currents therefore aid the flow in the boundary layer. With increasing Gr values, temperature, Θ, is strongly reduced in the boundary layer regime, as illustrated in Fig. 9. Values are highest for the upper stagnation point and minimized at the lower stagnation point. Increasing thermal buoyancy therefore serves to cool the flow in boundary layer regime with a maximum cooling influence at the lowest point of the cylinder surface. Fig. 10 shows the micro-rotation profiles for various Grashof number again with a high Prandtl number (Pr¼ 1) and relatively weak transverse magnetic field (Ha¼ 1). Increasing thermal buoyancy induces significant microrotation reversal at the cylinder surface (y¼0); for Gr ¼ 100 (maximum), the most negative G value is observed at the wall and this also corresponds to the lower stagnation point (x¼ 0). Progressively weaker micro-rotation at the wall is associated with the cylinder mid-point (x ¼ π/2) and the upper stagnation point. In the latter case, micro-rotation further from the wall becomes positive but values remain very low, indicating that at the uppermost point of the cylinder geometry only very small rotary motions of the micro-elements are achieved. Peak positive micro-rotation arises at y  2, for the maximum Grashof number (100) and at the lower stagnation point. A very different response is exhibited by the micro-rotation gradient, ∂G/∂y, for various Gr values, as shown in Fig. 11. At the upper stagnation point, angular velocity gradient is generally positive although magnitudes are small, especially near the cylinder surface. Values are however accentuated with an increase in thermal buoyancy force i.e. increasing Grashof number. At the lower stagnation point, maximum micro-rotation gradient is achieved with highest Grashof number. A similar trend is apparent for the ∂G/∂y, profilers at the cylinder mid-point (x ¼ π/2), although magnitudes are noticeable smaller than for the lower stagnation point. Far from the cylinder surface micro-rotation gradient vanishes in the free stream. Figs. 12–16 depict the distributions of Θw(x), Cf, Θ, G and ∂G/∂y with various micropolar parameters, K. This parameter expresses the ratio of micropolar vortex viscosity to the Newtonian dynamic viscosity i.e. κ/μ,. For K ¼1 both viscosities are identical. For K ¼ 0 the flow is Newtonian. With increasing K, micropolarity i.e. micro-element concentration is enhanced. Cylinder wall temperature, Θw(x),is strongly enhanced with increasing K values, as shown in Fig. 12. Hence the lowest surface temperature computed

corresponds to the Newtonian case. These results concur with the non-conducting case studied by Nazar et al. [34]. A sharp ascent in Θw(x) versus x profiles is identified as we arrive at the furthest point along the cylinder periphery i.e. the upper stagnation point (x¼ π). Prior to this location the profiles grow more gently with streamwise coordinate, x. Skin friction, Cf, (Fig. 13) is also shown to be enhanced significantly with an increase in K values. Increasing micropolarity therefore accelerates the flow along the cylinder surface i.e. reduces drag, a characteristic identified by many other researchers including Mansour and Gorla [61], Nazar et al. [34], Ahmadi [46] and Arafa and Gorla [45]. Newtonian fluids therefore sustain greater drag and retardation then do micropolar fluids. As indicated earlier, generally skin friction (surface shear stress) profiles will grow with distance along the cylinder from the lower stagnation point, peak beyond the cylinder midpoint (i.e. in the range π/2 ox o π) and then descend towards the upper stagnation point. These result indicate the attractive nature of micropolar fluids in for example lubrication flows and external flow enhancement in materials processing. Fig. 14 shows the influence of K on temperature profiles (Θ) with transverse coordinate (y) at the lower (x ¼0) and upper stagnation point (x ¼ π/2) locations. Θ values are markedly greater at the upper stagnation point, in particular at the cylinder surface (y¼0). With increasing K, temperatures are generally increased in the boundary layer regime. Increasing micropolarity of the biopolymer therefore serves to enhance thermal energy distribution in the flow regime and leads to a more even distribution across the boundary layer than for Newtonian fluids (K ¼0) for which the descent is very sharp from the wall. In Fig. 15, the evolution of angular velocity (G) profiles with y-coordinate for various micropolar parameters is shown. Significant micro-element reversal arises at the cylinder surface (y¼ 0) at the lower stagnation point, with a much weaker reversal (negative G values) for the upper stagnation point. With increasing K i.e. stronger micropolar vortex viscosity, this reversal is suppressed. At the upper stagnation point, micro-rotation, G attains positive spin much faster than at the lower stagnation point, G values remaining negative in the latter case for longer distances transverse into the boundary layer. Peak positive micro-rotation is associated with K¼ 5 i.e. maximum vortex viscosity and occurs at y  1.7 i.e. in close proximity to the cylinder surface. With further separation from the surface, microelements possess greater volume to rotate within and this augments rotary motions. With increasing micropolarity (K), microrotation gradients (∂G/∂y) are observed in Fig. 16 to generally decrease considerably. Much higher values are computed for the lower stagnation point and these remain positive for some distance into the boundary layer. For the upper stagnation point (x ¼ π) very low gradients values are apparent for all y, although magnitudes are marginally higher for weak micropolarity (K ¼1,2). Finally in Figs. 17–20 the influence of Prandtl number on Θw(x), Cf, Θ and G distributions are shown. We have considered a wider range of Pr values then Nazar et al. [34]. Pr defines the ratio of momentum diffusivity (ν) to thermal diffusivity. It also expresses the ratio of the product of dynamic viscosity and specific heat capacity to the thermal conductivity of the micropolar fluid. Pr o1 physically implies that heat will diffuse faster than momentum in the fluid. For Pr¼ 1 the diffusion rates will be the same for heat and momentum i.e. thermal and velocity boundary layer thicknesses will be equal. For Pr 41 momentum will diffuse faster than heat. Cylinder surface temperature, Θw(x), is generally decreased strongly in Fig. 17, with a rise in Prandtl number. All profiles grow slowly from the wall (cylinder surface) with streamwise coordinate, x, and reach a climax at the upper stagnation point. Therefore for any given Prandtl number the temperature at the cylinder increases from the lower stagnation point to the upper stagnation location. Skin friction i.e. dimensional shear stress at the cylinder

O. Anwar Bég et al. / Computers in Biology and Medicine 44 (2014) 44–56

surface, Cf, is also suppressed considerably with increasing Prandtl number, as observed in Fig. 18. For low Pr values profiles are parabolic and become increasingly flattened with increasing Pr. Peak skin friction arises in all cases downstream of the cylinder mid-point. Generally the flow along the cylinder is therefore decelerated with an increase in Prandtl number. For Pr o1, the thermal (heat) diffusion rate is much greater than the momentum diffusion rate since such fluids have higher thermal conductivities. For Pr¼ 1 the linear velocity and thermal boundary layers will have the same thicknesses. In Fig. 19 the temperature, Θ, is also significantly reduced with an increase in Prandtl number. Smaller Pr values (i.e. 0.1) cause a thinner thermal boundary layer thickness and more uniform temperature distributions across the boundary layer. The profile therefore descends very gradually from the maximum at the wall to the free stream. Values are generally also greater at the upper stagnation point (x ¼ π) compared with the lower stagnation point (x ¼0). For very high Prandtl number (e.g. polymeric suspensions, lubricants, Pr¼ 100), and temperature decays extremely quickly from the wall. Smaller Pr fluids possess higher thermal conductivities so that heat can diffuse away from the cylinder surface (wall) faster than for higher Pr fluids (thicker boundary layers). Micro-rotation (G) distributions with transverse coordinate (y) for various Pr values, as shown in Fig. 20, indicate that maximum micro-element reversal arises for low Prandtl numbers i.e. Pr ¼0.1 at the lower stagnation point (x ¼0). With an increase in Pr, angular velocity (G) at the wall is increased i.e. values are progressively less negative. At the upper stagnation point (x ¼ π), the micro-rotation reversal is much less prominent at the wall, although again the greatest effect is with the minimum Prandtl number. For Pr¼ 100 micro-rotation practically vanishes at the wall and for some distance transverse to it. Weak positive micro-rotation is achieved further from the cylinder surface at the lower stagnation point, a trend sustained thereafter into the boundary layer towards the free stream. Generally positive microrotation is only achieved at the upper stagnation point for Pr ¼100.

9. Conclusions We have presented a mathematical model for the steady hydromagnetic viscous incompressible micropolar biopolymer flow and heat transfer from a horizontal cylinder under the action of a uniform transverse magnetic field. The two-dimensional momentum and thermal boundary layer equations have been transformed to a set of coupled nonlinear partial differential equations subject to appropriate boundary conditions. Several special cases have been considered. NSM solutions have been obtained for the cylinder wall temperature, skin friction, temperature, micro-rotation and micro-rotation gradient distributions with variation in Hartmann number, Grashof number, Prandtl number and the micropolar parameter. NSM solutions have also been validated for the non-magnetic case with previous studies in the literature and for the conducting case with Nakamura tridiagonal method (NTM) implicit finite difference solutions. The NSM computations have shown: (I) Increasing magnetic field parameter i.e. Hartmann number (Ha), strongly increases strongly wall temperature of the cylinder, reduces skin friction, increases temperatures through the boundary layer transverse to the cylinder surface and suppresses negative micro-rotation (reverse spin) at the cylinder surface. (II) Increasing buoyancy parameter i.e. Grashof number (Gr), decreases cylinder wall temperature, enhances cylinder surface shear stress (skin friction) i.e. accelerates the flow from the lower stagnation point to a peak just downstream of the

55

cylinder mid-point, reduces temperatures in the boundary layer transverse to the cylinder surface (maximum values are recorded at the upper stagnation point and the least values at the lower stagnation point) and causes significant microrotation reversal at the cylinder surface. (III) Increasing rheological parameter i.e. micropolar vortex viscosity parameter (K), generally elevates cylinder surface (wall) temperature, boosts skin friction (i.e. reduces drag in the biopolymer flow), increases temperatures in the boundary layer transverse to the cylinder surface, stifles angular velocity (micro-rotation) reversal at the cylinder surface (much weaker reversal occurs at the upper stagnation point) and reduces micro-rotation gradients. (IV) Increasing the Prandtl number (Pr) strongly depresses cylinder surface temperature, skin friction i.e. dimensional shear stress at the cylinder surface and temperatures in the boundary layer regime and causes positive micro-rotation at the upper stagnation point (for Pr ¼100 i.e. dense biopolymers). The present study finds important applications in biopolymeric magnetic materials processing. Future investigations will consider nanofluid [62] and viscoelastic models [63] for conducting biopolymers under variable magnetic field strength effects, motivated by the works in [38] and [39], and will be communicated imminently.

Conflict of interest There is no conflict of interest declared.

Acknowledgments This work was partially supported (11021/EE1/09) by the Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia (Spain). The first author also wishes to acknowledge several fruitful discussions with Professor R.S.R. Gorla of Cleveland State University (USA) and Professor H.S. Takhar (Manchester, UK) regarding boundary conditions for micropolar transport phenomena simulation in Spring 2013. References [1] J.W. Choi, T.M. Liakopoulos, C.H. Ahn, An on-chip magnetic bead separator using spiral electromagnets with semi-encapsulated permalloy, Biosens. Bioelectron 16 (2001) 409–416. [2] A. Kumara, A. Srivastavaa, I.Y. Galaev, B. Mattiasson, Smart polymers: physical forms and bioengineering applications, Prog. Polym. Sci. 32 (2007) 1205–1237. [3] N. Pamme, Magnetism and microfluidics, Lab Chip 36 (2006) 24–38. [4] A. Basiri Parsa, O. Anwar Bég, M.M. Rashidi, S.M. Sadri, Semi-computational simulation of magneto-hemodynamic flow in a semi-porous channel using optimal homotopy and differential transform methods, Comput. Biol. Med. 43 (2013) 1142–1153. [5] M.M. Hoque, M.M. Alam, M. Ferdows, O. Anwar Bég, Numerical simulation of Dean number and curvature effects on magneto-biofluid flow through a curved conduit, Proc. IMechE—Part H- J. Eng. Med. (2013), http://dx.doi.org/ 10.1177/095441 191349 38 44. [6] J.B. Freund, B. Shapiro, Transport of particles by magnetic forces and cellular blood flow in a model microvessel, Phys. Fluids 24 (2012) 051904. [7] C.F. Driscoll, R.M. Morris, A.E. Senyei, K.J. Widder, G.S. Heller, Magnetic targeting of microspheres in blood flow, Microvasc. Res. 27 (1984) 353. [8] Q.A. Pankhurst, N.K.T. Thanh, S.K. Jones, J. Dobson, Progress in applications of magnetic nanoparticles in biomedicine, J. Phys. D 42 (2009) 224001. [9] M. Zaki, A. Aljinaidi, M. Hamed, Tribological behavior of artificial hip joint under the effects of magnetic field in dry and lubricated sliding, Bio-Med. Mater. Eng. 13 (2003) 205–221. [10] J. Zueco, O. Anwar Bég, Network numerical analysis of hydromagnetic squeeze film flow dynamics between two parallel rotating disks with induced magnetic field effects, Tribol. Int. 43 (3) (2010) 532–543. [11] O. Anwar Bég, M.M. Rashidi, Tasveer A. Bég, M. Asadi, Homotopy analysis of transient magneto-bio-fluid dynamics of micropolar squeeze film in a porous

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[12]

[13]

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Network and Nakamura tridiagonal computational simulation of electrically-conducting biopolymer micro-morphic transport phenomena.

Magnetic fields have been shown to achieve excellent fabrication control and manipulation of conductive bio-polymer characteristics. To simulate magne...
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