Influence of viscous dissipation and Joule heating on MHD bioconvection flow over a porous wedge in the presence of nanoparticles and gyrotactic microorganisms
 Umar Khan^{1},
 Naveed Ahmed^{2} and
 Syed Tauseef MohyudDin^{2}Email author
Received: 16 July 2016
Accepted: 21 November 2016
Published: 30 November 2016
Abstract
Background
The flow over a porous wedge, in the presence of viscous dissipation and Joule heating, has been investigated. The wedge is assumed to be saturated with nanofluid containing gyrotactic microorganisms. For the flow, magnetohydrodynamic effects are also taken into consideration. The problem is formulated by using the passive control model. The partial differential equations, governing the flow, are transformed into a set of ordinary differential equations by employing some suitable similarity transformations.
Results
A numerical scheme, called Runge–Kutta–Fehlberg method, has been used to obtain the local similarity solutions for the system. Variations in the velocity, temperature, concentration and motile microorganisms density profiles are highlighted with the help of graphs. The expressions for skin friction coefficient, Nusselt number, Sherwood number and motile microorganisms density number are obtained and plotted accordingly. For the validity of the obtained results, a comparison with already existing results (special cases) is also presented.
Conclusion
The magnetic field increases the velocity of the fluid. Injection at the walls can be used to reduce the velocity boundary layer thickness. Thermal boundary layer thickness can be reduced by using the magnetic field and the suction at the wall. The motile microorganisms density profile is an increasing function of the bioconvection Pecket number and bioconvection constant. The same is a decreasing function of m, M and Le. The skin friction coefficient increases with increasing m and \( M \). Nusselt number and the density number of motile microorganisms are higher for the case of suction as compared to the injection case. The density number of motile microorganisms is an increasing function for all the involved parameters.
Keywords
Background
In recent times, scientists and researchers are keenly working on the ways to improve the heat transfer characteristics of the fluids used in everyday life. For this purpose, many theoretical as well as practical studies have been presented over the years. Choi (1995), in one of the important studies in this regard, presented a useful and important model. The proposed model uses nanoparticles to improve the heat transfer characteristics of the fluids like water, kerosene and the other traditional fluids. He proved that the thermal properties of these fluids (termed as base fluids) can be enhanced by the addition of nano particles (Choi et al. 2001). After this benchmark study, many researchers dedicated their time to work in the field of nanofluids. In another study, Buongiorno (2006), suggested a model that incorporates the Brownian motion and thermophoresis effects in energy and concentration equations. Working on the idea of Buongiorno, Khan and Pop (2010) studied the boundary layer flow of nanofluid over a stretching surface. Makinde and Aziz (2011) extended the same idea for the case of convective boundary conditions. Several studies on this topic have been presented over the years. Some of the most relevant and useful ones can be seen in (Sheikholeslami and Ellahi 2015a, b; Ellahi et al. 2015; Khan et al. 2015; MohyudDin et al. 2015a, b; Gul et al. 2015a, b) and the references therein.
Flow over a wedge has gained interest of many researchers due to the practical applications it has in polymer processes, cooling or heating of films/sheets, insulating materials, conveyor belts, cylinders and metallic plates. The seminal work regarding the flow over a wedge has been carried out by Falkner and Skan (1931). Their study considers a fixed wedge and the absence of any external forces. Hartree (1937) and Koh and Hartnett (1961), extended the idea of Falkner and Skan by considering the various factors involved, and, provided an extended solution to the traditional wedge problem. Suction/injection and variable wall temperature were the major factors considered by them. Magnetohydrodynamic effects in the flow over a wedge were considered by Thakar and Pop (1984). Khan and Pop (2013) presented the boundary layer flow past a wedge moving in a nanofluid. Khan et al. (2015) used the Xue model to analyze the flow of carbon nanotubes suspended nanofluid over a static/moving wedge. Khan et al. (2015) used the nonlinear form of thermal radiation to study the flow properties in a porous wedge under the influence of magnetic field.
Bioconvection is due to the macroscopic convective motion of fluid caused by the density gradient. The collective swimming of the motile microorganisms creates the bioconvection. The swimming of these selfpropelled motile microorganisms results in increased values for the density that cause bioconvection. Studies related to bioconvection can be seen in Kuznetsov (2010), Khan and Makinde (2014), Nield and Kuznetsov (2006), Avramenko and Kuznetsov (2004), Makinde and Animasaun (2016a, b), Mutuku and Makinde (2014), Khan et al. (2014) and the references therein. All these researchers considered the flow by taking nanofluids and concluded that motion due to selfpropelled microorganisms result in enhancement in mixing and thus preventing nanoparticle cluster.
A careful literature survey reveals that to date, no study is available which considers the boundary layer flow of a nanofluid over a wedge in presence of microorganisms. To fill up this gap, we present here a mathematical study analyzing the flow of a nanofluid over a porous wedge in the presence of gyrotactic microorganisms. MHD along with the Joule heating effects for the flow are also taken into consideration. The flow analysis is carried out after reducing the equations governing the flow into a set of ordinary differential equations. The solution of the problem is obtained numerically. The graphs are plotted to highlight the effects of various emerging parameters. A comprehensive discussion over those graphs is also presented.
Problem formulation
Methods
Results and discussion
A graphical description of the effects of m and magnetic number \( M \) on Nusselt number is presented in Fig. 20. An interesting behavior is seen. With an increase in \( m \), the value of Nusselt number decreases for the case of suction at the wall; while for the injection case, the same gets a rise. The influence of \( M \) on Nusselt number is alike for both suction and injection cases, i.e. an increase in Nusselt number is observed. In Fig. 21, the influence of Ec on Nusselt number, due to the increasing values of \( m \), is plotted. With an increase in \( Ec \), there is a drop in the rate of heat transfer. Since Ec raises the temperature of the fluid, due to that the rate of heat transfer drops significantly. This behavior is same for both suction and the injection cases. Figures 22 and 23 give a description of effects of \( m \), Nb and Le on Nusselt number. The Brownian motion decreases the temperature of the fluid, in a result, the rate of heat transfer at the wall increases (Fig. 22). An opposite behavior for the suction and the injection cases is also evident. Almost alike behavior of Nusselt number is observed for the increasing values of \( Le \). For injection and suction at the wall, the rate of heat transfer is seen to be increasing with increasing values of \( Le \). All these figures also show that the values of Nusselt number for the injection at the wall are on a higher side than the case of suction.
The next set of figures gives a description of the variations in density number of the motile microorganisms caused by the varying values of involved parameters. Figures 24, 25, 26, and 27 are plotted for the said purpose. The density number increases with increasing values of \( m \) for both suction and the injection cases. For increasing M, the bioconvection Lewis number \( Lb \), bioconvection number \( Pe \) and the bioconvection constant σ give a rise in the density number of the motile microorganisms.
Comparison of current results with already existing ones in the literature when \( \varvec{ Pr} = 0.73 \)
m  \( f^{{\prime \prime }} (0) \)  \(  \theta^{{\prime }} (0) \)  

Khan and Pop ( 2013)  Present  Khan and Pop (2013)  Present  
0  0.4697  0.4690  0.4207  0.4201 
0.0141  0.5047  0.5046  0.4263  0.4257 
0.0435  0.5690  0.5689  0.4359  0.4354 
0.0909  0.6550  0.6549  0.4477  0.4473 
0.1429  0.7320  0.7320  0.4572  0.4569 
0.2000  0.8021  0.8021  0.4653  0.4650 
0.3333  0.9276  0.9276  0.4780  0.4781 
Conclusions

The magnetic field increases the velocity of the fluid.

Injection at the walls can be used to reduce the velocity boundary layer thickness.

Thermal boundary layer thickness can be reduced by using the magnetic field and the suction at the wall.
The motile microorganisms density profile is an increasing function of the bioconvection Pecket number and bioconvection constant. The same is a decreasing function of m, M and Le.

Nusselt number and the density number of motile microorganisms are higher for the case of suction as compared to the injection case.

The density number of motile microorganisms is an increasing function for all the involved parameters.
Declarations
Authors’ contributions
Author STMD developed the problem. First author UK, in collaboration with NA, did the literature review, developed and implemented the computer code, and interpreted the subsequently obtained results. All authors read and approved the final manuscript.
Acknowledgements
Authors are thankful to the anonymous reviewers for their comments that really helped to improve the quality of the presented work.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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