Pulsating flow of Casson fluid in a porous channel with thermal radiation , chemical reaction and applied magnetic field

Abstract. In the present analysis, the influence of thermal radiation, chemical reaction and thermaldiffusion on hydromagnetic pulsating flow of Casson fluid in a porous channel is investigated. The fluid is injected from the lower wall and sucked out from the upper wall with the same velocity. The governing flow equations are solved analytically by employing the perturbation technique. The influence of various emerging parameters on flow variables has been discussed. The obtained results show that the temperature distribution increases when there is an increase in heat source, whereas there is a decrease in temperature with an increase in radiation parameter. The concentration distribution decreases with an increase in chemical reaction parameter, while it increases for a given increase in Soret number. Further, the results reveal that, for both the Newtonian and non-Newtonian cases, Nusselt number distribution decreases at the upper wall with increasing Hartmann number and radiation parameter. The mass transfer rate decreases at the upper wall with increasing chemical reaction parameter and Soret number.


Introduction
Studies related to pulsating flow in a porous channel or pipe have attracted several researchers due to their applications in technological as well as biological flows such as transpiration cooling, gaseous diffusion, circulatory system and respiratory system, flow of blood, the process of dialysis of blood in an artificial kidney [18,32,41,47].Radhakrishnamacharya and Maiti [33] made an investigation of heat transfer to pulsatile flow in a porous channel.In their investigation, the fluid is injected through one wall c Vilnius University, 2018 and sucked out through the opposite wall at the same rate.Bestman [7] discussed that a combined forced and free convection flow through an inclined porous channel when a pulsatile pressure is applied across its ends.Datta et al. [10] studied the unsteady heat transfer pulsatile flow of a dusty fluid in a porous channel by employing the perturbation technique.Vajravelu et al. [46] has analyzed the pulsatile flow of viscous fluid between two permeable beds.Shit and Roy [38] appraised the flow and heat transfer characteristics of pulsating flow of magneto-micropolar fluid through a stenosed artery.
In 1959, Casson first introduced Casson fluid to analyze the flow behaviour of pigmentoil of printing ink, which shows yield stress in constitutive equation.Based on rheological characteristics, the Casson fluid is classified as the most popular non-Newtonian fluid.The Casson fluid model can be chosen to analyze the rheological behaviour of ingredients like tomato sauce, soup, honey, jelly and human blood [9,[21][22][23]26].Since the blood contains several substances such as fibrinogen, protein, globulin in aqueous base plasma and human red blood cells, Casson fluid can be preferred to analyze the flow characteristics of blood.Sankar [36] studied the pulsatile flow of blood through a catheterized artery by considering blood as Casson fluid.Abolbashari et al. [1] analytically analyzed the flow, heat and mass transfer characteristics of Casson nanofluid flow over a stretching surface.Sivaraj and Benazir [40] studied the unsteady hydromagnetic mixed convection flow of Casson fluid in a porous asymmetric wavy channel.
Studies pertaining to magnetohydrodynamic flow of non-Newtonian fluid in a porous medium have gained much attention of many researchers due to its applications in the geothermal sources investigation, the optimization of solidification processes of metals and metal alloys, and nuclear fuel debris treatment [3,11,20,29].Ali et al. [3] analyzed the hydromagnetic effects on blood flow in a horizontal circular tube by considering blood as Casson fluid.Khan et al. [15] examined the unsteady squeezing hydromagnetic flow of a Casson fluid between parallel plates.Vajravelu et al. [45] studied the mixed convective flow of a Casson fluid over a vertical stretching sheet by using optimal homotopy analysis method.Rashidi et al. [34] investigated laminar-free convective flow of a two-dimensional electrically conducting viscoelastic fluid over a moving stretching surface in the presence of a porous medium.Attia and Sayed-Ahmed [4] analyzed the unsteady magnetohydrodynamic flow of Casson fluid between two parallel non-conducting porous plates.The transient squeezing flow of a magneto-micropolar biofluid in a noncompressible porous medium intercalated between two parallel plates in the presence of a uniform strength transverse magnetic field is investigated by Beg et al. [5].The MHD boundary layer flow of a Casson fluid over an exponentially permeable shrinking sheet has been investigated by Nadeem et al. [25].The problem of heat transfer to hydromagnetic pulsatile flow of Oldroyd fluid was studied by Srinivas et al. [42].Prasad et al. [30] examined the flow and heat transfer of Casson nanofluid over a stretching sheet with variable thickness in the presence of a magnetic field.
Thermal radiation plays an imperative role in many engineering processes, which operate at high temperature such as gas turbines, nuclear plants and various propulsion devices.It is interesting to note that the thermal regulation in human blood flow by means of thermal radiation is most important in many medical treatments such as muscle spasm, chronic wide-spread pain, myalgia and thermal therapeutic [13,17,23,39,43].Sinha https://www.mii.vu.lt/NA and Shit [39] investigated the problem of electromagnetohydrodynamic flow and transfer of blood in a capillary with thermal radiation.Palani and Abbas [27] explored the free convection MHD flow from a impulsively saturated vertical plate with thermal radiation.The effect of thermal radiation and viscous dissipation on a combined free and forced convective flow in a vertical channel is investigated by Prasad et al. [28].Beg et al. [6] studied the combined heat and mass transfer by mixed magneto-convective flow of an electrically conducting flow along a moving radiating vertical flat plate with hydrodynamic slip and thermal convective boundary conditions.Mabood et al. [16] analyzed the influence of variable fluid properties on MHD flow of Casson fluid over a moving surface in the presence porous medium and radiation.Shehzad et al. [37] analytically studied the slip and thermal radiation effects on Casson fluid over a stretching surface by using homotopy analysis method.
The study related to heat and mass transfer with chemical reaction is of great practical importance in many engineering and industrial processes.Possible applications can be found in many industries like electric power industry, chemical industry and food processing chemical industry and so on.In many chemical engineering processes, a chemical reaction between a foreign mass and the fluid does occur.These processes occur in numerous industrial applications such as the polymer production, the manufacturing of ceramics or glassware, the food processing [8,12,23,35,44].Afify and Elgazery [2] studied the chemical reaction effect on MHD stagnation point flow towards a stretching sheet with suction or injection.Mythili et al. [24] carried out a numerical study to analyse the chemical reaction, cross diffusion effects on Casson fluid over a cone and flat plate.Hayat et al. [14] discussed the chemical reaction, Soret and Dufour effects on peristaltic flow of Casson fluid in a two-dimensional asymmetric channel.Prasad et al. [31] studied the effects of applied magnetic field, thermal and species concentration on flow, heat and mass transfer of Casson fluid over a vertical permeable stretching sheet.Recently, Malathy et al. [19] examined the chemical reaction and thermal radiation effects on MHD pulsating flow of an Oldroyd-B fluid in channel with slip and convective boundary conditions.
The literature reveals that no study related to MHD pulsatile flow of Casson fluid in a porous channel with thermal radiation and chemical reaction has been explored so far.The main aim of the present investigation is to explore the influence of thermal radiation, chemical reaction, Soret and Joule's heating effects on MHD pulsating flow of Casson fluid in a porous channel with porous medium.The dimensionless governing flow equations are solved analytically by using perturbation technique.The influence of various parameters on flow variables has been discussed.

Mathematical formulation
Consider the pulsatile flow of an incompressible Casson fluid in a porous channel under the influence of uniform transverse applied magnetic field.The flow is driven by an unsteady pressure gradient [33], where ρ is density of the fluid, p * is pressure, A is a known constant, 1 is a suitable chosen positive quantity, ω is the frequency and t * is time.As shown in Fig. 1, a Cartesian coordinate system is taken in such a way that the x * -axis is taken along lower wall and the y * -axis perpendicular to it.The lower wall maintains temperature T 0 and a concentration C 0 and the upper wall maintains temperature T 1 (T 1 > T 0 ) and concentration C 1 (C 1 > C 0 ).A uniform magnetic field B 0 is imposed along the direction normal to the flow.The fluid is injected into the channel from the lower wall with a velocity v 0 and is sucked out through the upper wall with the same velocity.Rheological equation for Casson fluid is defined as follows [9,21,22,26]: where τ ij is the (i, j)th component of the stress tensor, π = e ij e ij with e ij being the (i, j)th component of the deformation rate, π depicts the product of the component of the deformation rate with itself, π c is the critical value of this product based on the non-Newtonian model, µ B is the plastic dynamic viscosity of the non-Newtonian fluid, P y * is the yield stress of the fluid.The problem's governing equations as follows [9,21,22,25,26,33,41]: https://www.mii.vu.lt/NA The boundary conditions for the present analysis are where u * is dimensional velocity in x * direction, Φ and k are the porosity and permeability of porous medium, µ is the dynamic viscosity, σ is electrical conductivity, β = µ B √ 2π c /P y * is the Casson fluid parameter, ν is the kinematic viscosity, C p is the specific heat at constant pressure, Q 0 is the coefficient of heat source/sink, q r is the radiative heat flux, κ is the thermal conductivity, k 1 is the first-order chemical reaction rate (k 1 < 0 for generative reaction, k 1 > 0 for destructive reaction, k 1 = 0 for no reaction), T m is the mean temperature of the fluid, D is the coefficient of mass diffusivity, k T is the thermal diffusion ratio, T * , C * are the temperature and concentration of the fluid.By using Rosseland approximation for radiative heat flux, q r is defined as [13,16,17,43] where K * is the Rosseland mean absorption coefficient, σ * is the Stefan-Boltzmann constant.
We presume that the temperature variation within the flow are sufficiently small such that T * 4 may be expanded in a Taylor's series.Expanding T * 4 about T 0 and neglecting higher-order terms we obtain Substituting Eqs. ( 5) and ( 6) into Eq.( 3), we obtain We introduce the following dimensionless variables [10,33]: By using the above dimensionless variables, Eqs. ( 1), ( 7) and ( 4) become where Da = k/Φh 2 is the Darcy number of the porous media, ) is heat source/sink parameter (i.e., positive for heat source and negative for heat sink), ν is the chemical reaction parameter, which is positive for destructive chemical reaction and negative for generative reaction and So, the new boundary conditions are

Results and discussion
This section describes the influence of various physical parameters on the non-dimensional velocity, temperature, concentration, Nusselt number and sherwood number distributions graphically in Figs.2-13.In the present analysis, θ s , φ s ,u t , θ t , φ t represent steady temperature, steady concentration, unsteady velocity, unsteady temperature, unsteady concentration, respectively.8), i.e., −u/Da).An increase in permeability of porous regions, will increase Da, which will act as the Darcian drag force.Hence, there is an increase in u with increase in Da. Figure 2(d) shows that for a given increase in M , there is a decrease in velocity.This is due to fact that the retarding forces (called Lorentz forces) generated by the applied magnetic field act as resistive drag forces opposite to the flow direction, which results a decrease in velocity.The effect of t on unsteady velocity distribution is shown in Fig. 3.One can noticed that the unsteady velocity profiles oscillate with increasing t.
Figures 4(a) and 4(b) shows the influence of the cross flow Reynolds number R and heat source/sink parameter Q on the temperature distribution.Figure 4(a) depicts that the temperature distribution decreases with an increase of R. From Fig. 4(b) one can observed that there is a rise in temperature with an increase of heat source, while it is decreases with an increase of heat sink.The effects of radiation parameter (Rd ) Hartmann number (M ) and Eckert number (Ec) on steady and unsteady temperature distributions are shown in Figs.5-7. Figure 5 describes the variation of steady and unsteady temperature profiles for different values of radiation parameter.It is noticed that the steady temperature decreases for a given increase in radiation parameter (see Fig. 5(a)).From Fig. 5(b) one can see that the unsteady temperature oscillating with increasing Rd and the maximum is shifted to the near the walls.From Fig. 6 given increase in M towards the lower wall, while it increases towards the upper wall.From Fig. 6(b) one can infer that the unsteady temperature oscillates with increase M .Figure 7 presents the effect of Eckert number on steady and unsteady temperature distributions.From Fig. 7(a) one can noticed that the steady temperature increases with increasing Ec.This increase in temperature may be due to heat created by viscous distribution.From Fig. 7(b) it is observed that the unsteady temperature exhibits oscillating character with increasing Ec and the maximum is shifted to the boundary layers near the walls.The influence of time on unsteady temperature distribution is shown in Fig. 8.It is observed that the unsteady temperature profiles oscillate with increasing t.  and increases the mass transfer.Further, the unsteady concentration exhibits oscillating character.The opposite behaviour can be observed for the case of generative chemical reaction.Figure 10(a) shows that the steady concentration distribution decreases for given increase in Sc.From Fig. 10(b) one can see that the unsteady concentration profiles exhibits oscillation character.From Fig. 11(a), it is seen that there is a rise in steady concentration with an increase of Sr .Figure 11(b) presents the oscillating character of unsteady concentration distribution by varying Sr .
Figure 12 elucidates the effects of Rd and Q on Nusselt number distribution (Nu) against R. From Fig. 12(a) it is noticed that for a given increase in Radiation parameter, Nu increases at the lower wall, while it decreases at the upper wall.From Fig. 12(b) one can infer that, at the upper wall Nu increases with increasing heat source, while it decreases at the lower wall.The behaviour is reversed for the case of heat sink.The effects of Sc and Sr on Sherwood number distribution (Sh) against R is shown in Fig. 13.From Fig. 13(a), one can be noticed that Sh is an increasing function of Sc at the lower wall, while it is a decreasing function at the upper wall.From Fig. 13(b), the similar behaviour can be observed by varying Sr .Tables 1 and 2 show the variations of Nu and Sh for both the Newtonian and non-Newtonian cases.From Table 1 it is observed that for both the Newtonian and non-Newtonian cases, the Nusselt number distribution decreases at the upper wall with increasing M , R and Rd , while it decreases with increasing Q.However, this behaviour is reversed at lower wall.Table 2 depicts that the Sherwood number distribution is an increasing function of Sc, γ, and Sr for both the Newtonian and non-Newtonian cases at lower wall, while it is decreasing function at upper wall.

Conclusion
The present study deals with hydromagnetic pulsating flow of Casson fluid in a porous channel with thermal radiation and chemical reaction in the presence of heat source/sink.The problem considered is important as flow of Casson fluids (blood, drilling muds, https://www.mii.vu.lt/NA clay coating, certain oils, greases and many impulsions) in porous channel are used in modelling biological and industrial research.The governing flow equations are solved analytically by employing perturbation technique.The main findings of the present study are given below: • The velocity distribution increases with increasing Casson fluid parameter, frequency parameter and Darcy number, while it decreases with increasing Hartmann number.
• The temperature increases with increasing heat source whereas it decreases with increasing heat sink.• The steady temperature distribution is an increasing function of Ec, while it is a decreasing function of Rd .• The concentration distribution decreases with increasing destructive chemical reaction and Schmidt number, while it increases with increasing Soret number.• Nusselt number at upper wall increases with increasing heat source, while it decreases with increasing Rd .But this behaviour is reversed at the lower wall.
• Sherwood number at lower wall is a increasing function of Sc and Sr whereas it is a decreasing function at the upper wall.
• By taking M = 0, as a limiting case, the results corresponding to the problem for the hydrodynamic case can be captured.• The results of Radhakrishnamacharya and Maiti [33] (i.e., for the case of Newtonian fluid) can be captured from the present analysis by taking β → ∞, M = Q = Rd = γ = Sr = Sc = 0 in the absence of mass (concentration) diffusion.

Figure 1 .
Figure 1.Schematic diagram of the model.

Figure 2
presents the effects of Casson fluid parameter (β), Darcy number (Da), frequency parameter (H), Hartmann number (M ) on the velocity distribution.From Figs. 2(a) and 2(c), it is notice that the velocity increases with an increase in Casson fluid parameter and frequency parameter.

Figure 2 (
b) elucidates the variation of velocity for different values of Darcy number.It is noticed that the velocity is an increasing function of Da.Because the linear porous drag force, called the Darcian drag force, is inversely proportional to Darcy number (see the last term of Eq. (

Figures 9 -Figure 6 .Figure 7 .Figure 8 .Figure 9 .Figure 10 .
Figure7presents the effect of Eckert number on steady and unsteady temperature distributions.From Fig.7(a) one can noticed that the steady temperature increases with increasing Ec.This increase in temperature may be due to heat created by viscous distribution.From Fig.7(b) it is observed that the unsteady temperature exhibits oscillating character with increasing Ec and the maximum is shifted to the boundary layers near the walls.The influence of time on unsteady temperature distribution is shown in Fig.8.It is observed that the unsteady temperature profiles oscillate with increasing t.Figures 9-11 show the effects of the chemical reaction parameter (γ), Schmidt number (Sc) and Soret number on steady and unsteady concentration distribution φ.Figure9depicts the influence of γ on steady and unsteady concentration distributions.It is observed that the steady and unsteady concentration distributions decrease with an increase of destructive chemical reaction (γ > 0).This is due to fact that increasing destructive chemical reaction there is a decrease in the concentration boundary layer because the destructive chemical reaction reduces the solutal boundary layer thickness