Magnetized couple stress fluid flow past a vertical cylinder under thermal radiation and viscous dissipation effects

: Contemporary investigation studies the silent features of the dissipative free convection couple stress fluid flow over a cylinder under the action of magnetic field, thermal radiation and porous medium with chemical reaction effect. Present two-dimensional viscous in-compressiblephysicalmodelisdesignedbasedonthecon- sidered flow geometry. Present physical problem gives the highly complicated nonlinear coupled partial differential equations (PDE’s) which are not amenable to any of the known techniques. Thus, unconditionally stable, most accurate and speed converging with flexible finite difference implicit technique is utilized to simplify the dimensionless flow field equations. It is apparent from the current results that; the velocity profiles are diminished with enhancing values of magnetic field. Temperature profile increases with enhancing values of thermal radiation parameter. Velocity contours deviates away from the wall with enhancing magnetic parameter. Also, the effects of magnetic field, porous medium, thermal radiation, chemical reaction, buoyancy ratio parameter and Eckert number on couple stress flow velocity, temperature, and concentration profiles are studied. However, the present study has good number of applications in the various fields of engineering such as; polymer processing, solidification of liquid crystals, colloidal solutions, synovial joints, geophysics, chemical engineering, astrophysics and nuclear reactors etc. Finally, the current solutions are validated with the available results in the literature review and found to be in good agreement.


Introduction
Incompressible couple stress uid ow obeying Prandtl's theory with distinct e ects about a vertical cylinder kept in a porous medium is a signi cant study area related to many of the manufacturing process. The non-Newtonian uids have rich set of biomedical and industrial applications in the eld of science and engineering. Numerous non-Newtonian uids are available in nature which has signi cant number of applications in the various elds of science and engineering. Particularly, the time-dependent couple stress uid have important applications in numerous processes that are recognized in the industry like the extrusion of polymer uids, solidi cation of liquid crystals, and colloidal solutions. Application of couple stress uid is fairly found in synovial joints (shoulder, hip, knee, and ankle), geophysics, chemical engineering, and astrophysics. Based on the literature review it is observed that, many researchers modelled the synovial uids as couple stress uid and they investigated the thermodynamic behaviour. In the last few decades, the ow of couple stress liquid around a plate and cylinder with di erent physical e ects has acknowledged the consideration of various researchers and engineers.
Soundalgekar [1] investigated the natural convection ow over an in nite vertical plate by considering Stokes problem in accordance with the mass transfer applications. The action of variable temperature on uid ow about a moving plate was investigated by Murty and Soundalgekar [2]. Soundalgekar and Takher [3] employed the series solution for magnetized free convection ow past an in nite vertical plate under the action of uniform transverse magnetic eld with dissipation e ect. Their investigation demonstrates that, the rising magnetic eld parameter and the Prandtl number enhances the dissipation e ects in the ow regime. Ganesan and Rani [4,5] were disclosed the transient natural convection uid ow with heat and mass transfer process along the vertical cylinder by considering MHD e ects. Fathia et al. [6] studied the dissipative unsteady natural convective viscous uid ow with concentration di usion about an in nite vertical plate. Rani [7] has examined the time-dependent buoyancy motivated ow with thermal and species transfer e ects over a vertical cylinder. Unsteady magnetized natural convective viscous dissipative uid ow about a vertical plate was studied by Singh et al. [8] and they visualized that, the rising magnetic eld raises the temperature eld.
Palani [9] studied the viscous dissipative uid ow along a vertical plate with variable temperature eld. The research revealed that, the increasing dissipation e ect results in the rise of temperature. Rani and Kim studied the time-dependent natural convection ow over a vertical cylinder with variable viscosity and thermal conductivity along with temperature dependent viscosity e ect [10,11]. Rani and Kim examined the unsteady natural convection ow about an isothermal cylinder with Dufour and Soret e ects [12]. Magnetised uid ow with di usion of concentration eld about a vertical plate was studied by Palani and Srikanth [13]. They found that, the rising buoyancy force signi cantly raises the velocity eld. The e ects of variable viscosity and heat conduction on velocity, temperature and Nusselt number on vertical geometry were studied by Palani and Kim [14]. Rani and Kim studied the time-dependent free convective micro-polar uid ow about a cylinder with heat and mass transfer [15]. Rani et al. numerically examined the chemically reactive couple stress uid ow about a vertical cylinder [16,17]. In the same direction, the problem of uid moment about a moving cylinder with magnetic and thermal radiation impacts was discussed by Loganathan et al. [18].
Neog and Das investigated the unsteady free convective magnetised uid ow past a plate with varying temperature by employing Laplace transform method [19]. Rani et al. studied the buoyancy motivated unsteady couple stress uid ow over a cylinder under the action of Prandtl number [20]. Bella and Naikoti inspected the impacts of dissipation and thermal radiative on timedependent natural convective magnetized chemically reactive thermal and mass moment about a plumb plate using FEM method [21]. The application of viscosity on free convection ow of air around a vertical cylinder was studied by Hossain et al. [22]. Loganathan and Sivapoornapriya studied the free convective species concentration transfer past an in nite plate which is kept in porous medium [23]. Their analysis shows that, the rising permeability parameter enhances the velocity eld and rising Prandtl number diminishes the thermal eld.
Reddy and Raju analyzed the radiation and chemical reaction e ects on unsteady magnetized mixed convective ow around a vertical porous plate [24]. Palani et al. disclosed the impacts of MHD and viscous dissipation on buoyancy motivated uid moment about a plate with varying thermal eld [25]. Loganathan and Divya numerically discussed the e ects of chemical reaction and dissipation on Newtonian uid past a cylinder [26]. Reddy studied the impact thermal radiation and chemical application on time-dependent magnetized free convective parabolic uid ow over isothermal vertical plate with viscous dissipation e ect [27]. Basha et al. studied the impacts of free convection on supercritical water ow past a vertical using Redlich-Kwong equation of state approach [28]. Basha et al. studied the chemically reactive unsteady buoyancy motivated couple stress uid ow about a vertical plate [29]. Recently, Shekar and Kishan studied the impact of thermal radiation on natural convection thermal transfer in a square cavity lled with nano uid under the in uence of saturated porous medium with the di erent nanoparticles [30]. It is found from their investigation that, the heat transfer increases with increase in Rayleigh number. Haritha et al. investigated the impact of magnetic and viscous dissipation on free convection heat transfer process in a square cavity lled with nano uid under the e ect of saturated porous medium with the di erent nanoparticles [31]. It is observed from their investigation that, the enhancing magnetic number increases the ow eld. Balla et al. numerically studied the e ect of chemical reaction on bioconvective ow in a porous square cavity containing oxytactic microorganism [32]. Further, the bio-convection thermal and ow behavior is studied under the impact of Darcy model. It is recorded from their investigation that, chemical reaction increases the ow cell and causes the splitting of the cell in the uid medium.
Acharya et al. studied the impact of thermal radiation on laminar free convective two-dimensional couple stress uid ow over a stretching cylinder under magnetic eld [33]. It is recorded from their study that, the increasing magnetic number increases the entropy generation process. Basha et al. investigated supercritical free convection in couple stress and Newtonian uids under the in uence EOS approach [34]. It is found from their investigation that, the supercritical uid is sensible to thermal changes in the uid medium. Das et al. [35] numerically investigated the importance of Joule heating, viscous dissipation, magnetic eld and slip condition on the two-dimensional ow of an electrically conducting couple stress uid induced by an exponentially elongating sheet kept in a porous medium. It is recorded from their study that, the increasing radiation number increases the thermal eld in the ow regime. Ibrahim and Gadisa [36] studied the double strati ed mixed convection ow of couple stress nano uid about an inclined elongating cylinder under the impact of Cattaneo-Christov heat and mass ux model. It is found from their study that, the increasing Reynolds number increases the concentration distribution in the ow eld. The detailed literature on couple stress uid can be found in [37][38][39][40].
By observing the above published results, the combined e ects of chemical reaction with radiation, dissipation and porous medium on incompressible unsteady buoyancy driven couple stress uid ow around a vertical cylinder has obtained rare consideration in the literature. In the current work, authors made an attempt to examine the behaviour of time-dependent couple stress uid ow about a cylinder with magnetic force, porous medium, thermal radiation and chemical reaction. However, yet the exact solution is not available for the current physical problem with above mentioned physical e ects. The governing highly nonlinear physical system of oweld equations are simpli ed by employing the unconditionally stable nite di erence scheme.

Mathematical formulation and governing equations
Transient natural convective magnetised incompressible thermally radiative viscous couple stress uid about a vertical cylinder with viscous dissipation and chemical reaction impacts under the in uence of porous medium is studied. To de ne the problem well, a rectangular region of geometry is considered in which x-axis is measured along the axial path and r-coordinate is taken normal to cylinder. The schematic representation of the physical problem shown in Figure 1. The free stream thermal eld (T ∞ ) and species di usion eld (C ∞ ) are expected to be same as the thermal and mass di usion of the geometry which is maintained at the time t = . As time starts t > , the thermal eld and concentration di usion raised to T w > T ∞ and C w > C ∞ and this is same for all time t > .
With this thermal and mass di usion di erences within the boundary layer in the vicinity of the ow con guration density changes occurs and that in turn interacts with the gravity eld and produces the free convective couple stress uid ow about a cylinder. Based on the uid ow and geometry, the governing two-dimensional nonlinear coupled equations of the present problem are described below with Boussinesq's approximation [10][11][12][15][16][17].

Momentum equation:
Here η is the constant related to couple stress liquid ow. Energy equation:

Concentration equation:
(4) Based on the Rosseland theory of thermal radiation [41], the radiative heat ux (qr) can be approximated as: Stefan-Boltzmann constant is σs and k * is the mean absorption coe cient. If di erence of the temperature within the ow is too small, the term T can be written as a linear temperature function, then T can be written by utilizing Taylor's expansion about T ∞ with ignoring terms of greater order is given by: Using equations (5) and (6), equation (3) reduces to Stokes [16,17] given the two conditions for couple stress liquid moment in the boundary layer regime. However, second condition is employed for simplifying the current problem. Thus, the required boundary conditions of the current physical problem are [10-12, 15-17, 20, 34]: Utilized non-dimensional variables and parameters are as follows: . By introducing the above dimensionless quantities in the Eqs. (1)-(4) and (7), the reduced non-dimensional governing equations are obtained: The corresponding initial and boundary conditions in dimensionless form are given as follows: For all X and R, C = , Similarly, the Eq. (9) also changed to the following dimensionless form.

. Average momentum, heat transfer and mass transfer rates
In view of real-life engineering applications, the researchers required the good knowledge on the values average momentum, heat and mass transfer rates. When compared to Nusselt and Sherwood numbers, skin friction coe cient pays the key role in the ow phenomena. In many of the technical applications, the raising skin-friction coe cient is unwanted but even though the raised skin-friction can be utilised in some applications such as thermal and mass exchanger's systems. In the current investigation, the momentum, thermal and concentration ow rates are obtained from the literature Rani et al. [16]. Thus, the dimensionless average momentum transport coe cient (C f ), heat transfer (Nu) and mass transfer (Sh) rates are de ned as follows: The di erentials involved in Eqs. (17)- (19) are evaluated through a ve-point formula, later integrals are computed through a Newton-Cotes scheme. Simulated dimensionless momentum, thermal and species di usion rates for couple stress liquid are shown graphically and discussed below.

Numerical solution procedure
Governing non-dimensional unsteady uid ow equations Eqs. (11)- (14) are simpli ed by employing implicit nite di erence method. The associated nite di erence equations are listed as follows: The solutions of the Eqs. (20)- (23) are obtained in the rectangular region of the geometry limiting from X min = , Xmax = , R min = and Rmax = , here Rmax assuming as R = ∞ be located far from the momentum, heat and mass di usion boundary layers. Unsteady velocity, heat and mass di usion values are obtained using the mesh space of 100 × 500 varies 2 nd decimal with 50 × 250 and 50 × 250 di ering in 5 th decimal with 200 × 100. The mesh space of 100 × 500 has been nalised for all numerical computations with 0.03 and 0.01 as mesh sizes along R and X directions. Also, the step size of time dependence has veri ed with ∆t = 0.01 and provides good result. Further, for unsteady solutions of velocity, temperature and concentration, the absolute di erence between the ow eld at two successive times is smaller than − at all mesh nodes is maintained.
By solving the energy and concentration Eqs.
where Ψ represents the dependent variable velocity U and φ represents the dependent variables T and C. Therefore, the Eqs. (24) and (25) at each interior node on a speci c l-level creates a pentadiagonal and tridiagonal equations. These equations are simpli ed through pentadiagonal and Thomas schemes. Explicitly the velocity V is found from thenite di erence equation Eq. (20). This mechanism is continual for all successive l-levels with more time steps till the convergence criteria is attained − .

Results and discussions
Based on the present numerical analysis following results are demonstrated. To study the time-dependent behaviour of simulated patterns such as ow eld, thermal, concentration and their numerical values at various places are shown and which are neighbouring to heated surface of vertical cylinder. At X = . along R-coordinate, the steady mass di usion, thermal and ow elds are described.

. Validation of currents with existing solutions
The steady-state simulated mass di usion elds of couple stress liquid obtained in the current investigation is compared with the existing results of Rani et al. [17]. It is clear from Figure 2 that, the current ndings are showing accurate matching with the existing solutions.

. Analysis of numerical results
Current numerical solutions are expressed in terms of magnetic parameter (M), porous medium parameter (K ), radiation number (Nr), chemical reaction number (Kr) Buoyancy ratio parameter (Bu) and Eckert number and rest of the variables kept constant.

(i) E ect of magnetic eld (M) on ow pro les Velocity pro le:
The simulated velocity eld Uat position ( , . ) verses t is illustrated in Figure 3(a). It is determined that, the unsteady-state ow eld decayed with rising M. It is because of fact that, the increasing magnetic number ampli es the Lorenz forces in the ow region and hence velocity pro le is diminished. Also, the boundary layer thickness diminished with rising M. It is witnessed that, for the enhancing M, the maximum velocity peaks diminished. Also, the time required to reach the time-independent state increases with amplifying M. Initially (i.e., t ) velocity pro les are coinciding with each other due to convective thermal transfer process is dominated by the conduction heat transfer.
The computer generated ow eld U verses R at X = is demonstrated in the Figure 3(b) under the action of magnetic eld. Initially ow eld begun with zero near the cylindrical hot surface, touch its greatest values and monotonically decays to value zero along R-direction for all t. It is recorded from Figure 3(b) that, the steady-state velocity acts as a decaying function of magnetic parameter.             Figure 4(a). This plot shows that, the merged thermal curves indicates that, initially conduction process dominates the convection phenomena. Transient temperature pro le rises with time, reaches maximum values, shortly later, diminished and again gradually increases, at last asymptotically reached steady state. Unsteady temperature pro le rises with amplifying values of magnetic parameter.
At X = . along radial axis, the time-independent temperature pro le is shown in the Figure 4(b) for the giving di erent values of M. The temperature curves originate from the hot cylinder wall at T = and diminished monotonically to zero irrespective of time along the radial coordinate. However, steady state velocity pro le enhanced with increasing M.
Concentration pro le: The computer-generated unsteady concentration di usion eld for the various magnetic parameter M values at location ( , . ) is displayed in the Figure 5(a). Initially all the concentration di usion arcs are fully merged with one another and diverged after some time. Also, it is noticed that, the unsteady concentration eld increases, attains maximum as well as minimum and again increases, there after decreases, later asymptotically attains the stable state condition. Further, for the rising M, the mass di usion eld increases.
The time-independent simulated concentration prole at X = . verses R for various values of M is shown in the Figure 5(b). The concentration di usion curves are originated from the heated cylinder wall at C = and diminished monotonically to zero along R-axis irrespective of t. Steady-state concentration pro le rises with rising M values.
(ii) E ect of porous parameter (K ) on velocity and temperature pro le Velocity Pro le: The simulated time-dependent velocity eld for di erent values of porous parameter K is shown in the Figure 6(a) at the location ( , . ). It is found from this gure that, the enhancing porous parameter ampli es the velocity pro les. According to Darcy's law, the ow between the two points is directly related to the distance between points, pressure di erence between the points and connectivity of the ow between the points within the medium. Thus, an increment in porous number results the free ow of couple stress uid in the medium, hence velocity boundary layer thickness increases. It is shown that velocity pro le slowly rises with time, attained maximum around t = . and then decreases attain the temporal minima and again increases, decreases slightly, later attain the steady-state. The time-independent simulated velocity pro le at X = verses R is demonstrated in the Figure 6(b) for various K values. This graph shows that, the ow eld starts initially zero at heated wall of cylinder, reaches the highest value and shortly later diminished to zero monotonically in R-direction. Also, for the increasing values of K , the steady state velocity pro le are increases.
Temperature pro le: The Figure 7(a) at ( , . ) shows the simulated unsteady temperature pro le for var-ious K values. Temperature pro le diminished with rising K . It is due to fact that, the rising K raises the ow between the points and pressure di erence between the points decreases, then this decrease in the heat generation. Also, noticed that, initially all the thermal curves merged with each other, shortly later reaches temporal maxima as well as minima and again increases, later it attains the steady state. The Figure 7(b) shows the simulated steady state temperature pro le at X = . against R   ). It is observed that, rising Nr values raises the ow eld. It is because of the reason that, the bond holding tendency of components of the uid particles are easily broken when the intensity of heat produced through convicted heat is raised, hence by increasing the thermal radiation parameter, the translational velocity increases. From this gure it is noticed that, the transient velocity increases gradually, reached maxima as well as minima and slightly increases then shortly later attains the steady state. Figure 8(b) describes the computer generated ow eld at X = against Rfor distinct Nr. Initially all velocity curves are originated from the zero value and are merged with each other, after some distance they separate with each other and attains their maximum and gradually decreases to zero. Steady state ow eld magni es with rising Nr.
Temperature pro le: The Figure 9(a) provides the simulated time-dependent temperature pro le for distinct Nr values at position ( , . ). The parameter Nr de nes the comparative contribution of conduction heat transfer to convection heat transfer. It is seen from Figure 9(a) that, the rising value of Nr makes in increase in the temperature pro le. It is because of fact that, the increasing Nr enhances the heat generation process and hence increases the thermal boundary layer thickness. Also, it is perceived that, initially all the thermal curves are coinciding with each other, later gradually increases and scatter with respect to time, attain slightly temporal maxima as well as minima, and then increases slowly there after reaches the steady state. The time required to attain the steady state decays with rising values of Nr.
Steady state temperature pro le for varying values of Nr is shown in the Figure 9(b) at X = against R. Also, the temperature pro le generated from the heated cylinder wall at T = and diminished to zero in radial path irrespective of t. Further, the steady state thermal eld ampli es with enhancing Nr.
Concentration pro le: Figure 10(a) shows the timedependent mass di usion pro le at the location ( , . ) for various values of Nr. It is inspected that, the concentration eld decayed for rising values of Nr. Since, the thermal radiation process produces the heat by increasing Nr consequently the concentration eld decreases by rising the Nr. Further, the Figure 10(b) disclose the timeindependent mass di usion eld for various values of Nr at X = verses R. The concentration pro le begins from the hot wall of the cylinder at C = and diminished to zero in radial path. Also, the steady state concentration pro le decreases with increasing Nr values.
(iv) In uence of chemical reaction parameter (Kr) on velocity and concentration pro les Velocity pro le: Figure 11(a) indicates the impact of Kr on velocity pro le at location ( , . ). Initially all velocity curves rising with respect to time, attains its maximum, then declines, again enhances and there after accomplishes the stable state. Initially when t , the thermal ow is because of conduction e ect. Immediately after sometimes, there exists a convective heat transfer process and which dominates the conduction phenomena. It is observed that for rising the values of Kr, yields the diminished mass di usion process, which in turns decays the values of BuC in Eq.(12), due to this fact, the acceleration reduces. Hence, velocity pro le diminished for the rising values of Kr. Figure 11(b) discloses the time-independent behaviour of simulated velocity eld verses R at X = , at which ow curves begins with zero at cylinder, attains their highest value, then monotonically decayed to zero irrespective of time in R-path. Magnitude of the velocity eld decreases with rising Kr, since rising Kr results the reduced mass di usion process near the cylinder surface (ref. Eq. 14), and this fact decelerate the ow eld (ref. Eq. (12)). Hence, steady-state velocity diminished with rising Kr.
Concentration pro le: Figure 12(a) shows the e ect of Kr on transient simulated concentration eld at the location ( , . ). Where the concentration pro les initially near the wall dominated by conduction process, later attains temporal maxima as well as minima, then marginally enhances and shortly later touches to time-independent state. Temporal maxima attains early for smaller Kr values. Species di usion eld declined with amplifying Kr, since higher values of Kr decreases the mass di usion process in the ow domain (ref. Eq. (14)).
Simulated time-independent mass di usion pro le for various values of Kr at X = along R path is described in Figure 12(b). The mass di usion pro le begins from the heated cylinder surface at C = and monotonically diminished to zero. Further, as Kr rises, the species concentration pro le decreases. (

v) E ect of buoyancy ratio parameter (Bu) on velocity pro les
Figures 13(a) and 13(b) describes the in uence of buoyancy ratio parameter on velocity eld. It is recorded from these gures that, the both transient and steady velocity pro les increases with increasing values of Bu, this is due to the fact that, the increasing Bu enhances buoy-ancy forces and which in turn accelerates the ow eld in the ow regime. (

vi) E ect of Eckert number (Ec) on temperature pro les
Figures 14(a) and 14(b) describes the in uence of Eckert number on thermal eld. It is recorded from these gures that, the both transient and steady temperature elds enhanced with rising values of Ec, this is due to the fact that, the increasing Ec enhances frictional forces and which in turn accelerates the ow eld in the ow regime.The physical justi cation behind this fact is that the presence of frictional forces in the uid causes the release of heat energy into the uid which gives the intensied temperature eld in the ow region.
. Influence of magnetic parameter on physical quantities of interest Figure 15(a) reveals impact of magnetic parameter on transient averageC f . This gure discloses that, enhancing magnetic variable diminished the skin-friction coe cient in the ow domain. Figure 15(b) describes the action of M on average Nusselt number N u and it is known that,Nu is the ratio of heat transfer due to convection to the heat transfer due to conduction, if its value is one, it re ects the thermal transfer by pure conduction. Figure 15 is higher when compared to M = . , it is due to Lorentz's e ect. Further, it is observed that, as M increases, the velocity contours of couple stress uid moves away from the heated cylindrical surface. Figures 17(a) and 17(b) describe the stable state thermal contours for di erent M values. Thermal eld of couple stress uid for M = . is less when equated to M = . . Further, for the enhancing M, the thermal contours moves near to heated cylinder. Figure 18(a) and 18(b) shows the steady-state concentration contours for di er-ent values of M. It is noticed that, as M increases, mass di usion contours of couple stress uid moves far from the heated cylindrical wall.

Concluding remarks
In the current investigation, authors obtained the simulated results on the boundary layer ow of viscous incompressible non-Newtonian unsteady magnetized thermally radiative couple stress uid ow about a cylinder with viscous dissipation e ect under the in uence of porous medium and chemical reaction. The system of governing nonlinear non-dimensional equations with required conditions are simpli ed by employing the implicit FDM. Solutions are obtained for di erent values of magnetic eld, porous medium parameter, thermal radiation parameter and chemical reaction parameter. From the present study following observations are made in limiting sense: • Velocity pro le decreases, whereas temperature and concentration pro les increase for amplifying M. • Velocity eld increases and temperature pro le decreases with rising K values. • Velocity and temperature pro les increase whereas concentration pro le decreased with enhancing values of Nr. • Velocity and concentration pro les decreased with rising values of Kr. • Momentum, thermal and species transport rates decreased for amplifying values of M. • Velocity pro le increases with increase buoyancy ratio parameter. • Temperature enhanced with rising Eckert number. • Time-independent state ow and mass di usion contours deviates away from the cylinder surface, whereas, the temperature contours lies near to heated cylinder for the rising M.
However, it is anticipated that, the present numerical investigation provides the motivation for numerical modelling of two and three dimensional heat and mass transfer problems occurring in the eld of science and engineering. Further, the present paper gives a motivation for the modeling of couple stress uid ow problems past a horizontal cylinder, sphere, cone and stretching cylinder as well as sheets with various physical aspects. Also, the present problem can be extended to study the problems occurring in the eld of bioengineering and medicine, solar collectors, nonmaterial synthesis, thermal and chemical industries, etc.