Mixed Convection Flow Along Vertical Thin Needles Containing Gyrotactic Microorganism with Variable Heat, Mass and Motile Microorganism Flux

The problem of steady laminar mixed convection boundary layer flow along vertical thin needle with variable surface heat, mass and motile microorganism flux in the presence of gyrotactic microorganism is considered in this study. The dimensionless leading equations of continuity, momentum, concentraton and motile microorganism conservation are reduced to ordinary differential equations with the help of similarity transformations. The transformed governing equations are then numerically solved by using MATLAB BVP4C function. The research is reached to excellent argument by comparison in few cases between the results obtained from MATLAB and Maple algorithm with the help of dsolve command. Numerical calculations are carried out for various values of the dimensionless parameters of the problem which includes mixed convection parameter λ, power law index m, buoyancy parameters 𝑁 1 , 𝑁 2 , Lewis parameter Le, bioconvection lewis parameter Lb, Bioconvection peclet number Pe and also the parameter a representing the needle size. It is also shown from the results that the surface (wall) temperature, surface fluid concentration, surface motile microorganism concentration and the corresponding velocity, temperature, concentration and motile microorganism profiles are significantly induced by these parameters. The results are pictured and discussed in detail.


Introduction
The study of mixed convection which is the combination of free and forced convection flow has become the great interest of many researchers over the last few decades because of its wide range of technological and industrial applications that have been reviewed in Refs. [1][2][3] such as heat exchanges placed in a low velocity environment, solar collectors exposed to wind currents, atmospheric boundary layer flows, nuclear reactors cooled during emergency shutdown and various electronic equipments. Several studies were done on mixed convection flow through different surfaces by many researchers such as Refs. [4][5][6][7]. Heat flux is another important phenomenon that can actually affect the heat process mechanism. Several studies in Refs. [8][9][10][11][12][13] are introduced on these topics and it has been shown that the fluid flow together with the uniform/ variable heat flux is very important in some of the manufacturing processes. Very recently Ibrahim [14] studied mixed convection nanofluid flow over non isothermal solid sphere with uniform heat flux and Ahmed [15] observed Magnetohyrodynamic flow for unsteady stretching sheet with variable surface heat flux.
The study of microorganism has become greater attention of researchers to observe the development of microorganisms in bioconvection. Because of their mixing property and ability to improve mass transport, bioconvection phenomena have major applications in many biological systems and biotechnology. There are several notable applications in the fields of biomedicine (nanodrug delivery, cancer therapeutics), bio-microsystems (enzyme biosensors, biotechnology). Bioconvection is a development process in the field of fluid flow that works with the steps of selfpropelled up swimming microorganisms such as algae and bacteria that contain oxytaxis, gyrotaxis or gravitaxis organisms. Since motile microorganisms are denser than the liquid they are surrounded by, they swim upwards, resulting in different flow profiles in the system according to Refs. [16][17][18][19][20]. Gyrotaxis is most commonly found in a bottom-heavy microorganism. Kuznetsov [21] came up with the concept of putting motile microorganisms in nanofluid. Adding motile microorganisms to the suspension has many advantages, including better mass transfer, microscale mixing, and fluid stability. Significant papers on the application of bioconvection in thermobioconvection, microbial enhancement, bio-Microsystems, biofuels, and other bioengineering systems have been written in Refs. [22][23][24][25][26][27][28][29].Very recently Mahdy [30,31] studied time-mixed convection nanofluid flow for rotating sphere and also Eyring -Powell nanofluid flow containing motile microorganisms. Waqas [32] observed Magneto-Burgers Nanofluid Stratified Flow with Swimming Motile Microorganisms and also bionanoconvective flow past a needle in the presence of Stefan blowing and gyrotactic microorganisms were observed by Amirsom [33] and Beg [34].
The flow over a thin needle has become a topic of interest for the researchers because of its importance in biomedical engineering where it arises in, for example cancer therapy and dermal administration of drugs. In order to measure the momentum and heat transfer behavior, flow analysis over such slender needles is also a major concern in experimental work. This occurs as a result of the needle's motion, which disrupts the free stream flow. As a result, this subject is of considerable practical importance, especially in the context of hot anemometers for finding wind's velocity, transportations, geothermal power generation, fiber technology, lubrications, aerospace, wire coating according to [35] and also in metal spinning and in micro/nanoscale equipment [36] including the removal of subterranean nuclear waste [37]. The thin needle is viewed as a slender body with the geometry having paraboloid of revolution where its thickness is smaller compared to the boundary layer thickness. Many researchers such as [38][39][40][41][42][43][44] have studied steady combined free and mixed convection boundary layer flow over a vertical thin needle. Grosan and Pop [45] pioneered the influence of nanoparticles on flow over a thin needle, which was later applied to a moving needle in a nanofluid by Soid et al. [46]. Hayat et al. [47] looked at the flow in a nanoliquid with a variable surface heat flux caused by a slender needle near a stagnation point. The Buongiorno mathematical model of nanofluid due to moving slender needle was used by Ahmad et al. [48]. Salleh et al. [49] then studied stability analysis to understand the additional impact of the heat source and chemical reaction in a nanofluid flow towards a slender needle. Very recently Waini et al. [50] and Khan et al. [51] studied hybrid nanofluid flow through the thin needle.
All studies mentioned above on free or mixed convection boundary layer flow over vertical thin needles refer to fixed needles immersed in a viscous and incompressible fluid. And have generally considered heat/mass transfer along either isothermal/non-isothermal or constant/variable wall heat flux needles. However, to the authors' best knowledge, the mixed convection boundary layer flow past a vertical thin needle containing gyrotactic microorganism with variable surface heat, mass and motile microorganism flux has not been reported in the literature, which motivates the present work. In view of the fact that this type of variable surface heat flux condition is commonly encountered in engineering practice, it merits separate attention. To the best of my knowledge, the results obtained from the present study are new and original and have not been published elsewhere. Present study could be used in food and pharmaceutical industries, chemical processing equipment, fuel cell technology, enhanced oil recovery, etc.
, where x and r are the axial and radial coordinates, respectively, with the x -axis measures from the needle leading edge. The needle is considered thin when its thickness does not exceed that of boundary layer over it. u and v are the velocity components along the x and r axes.
 is the velocity of vertical thin needle and g is the acceleration due to gravity. Following [33,34,52,53] under the Boussinesq approximations the appropriate dimensionless equations in cylindrical coordinates are, Following variables were used for non-dimensionalizing Where T, C, n are temperature, concentration and volume fraction of motile microorganism. w q is the variable surface heat flux, m q is the variable surface mass flux and n q is the variable surface motile microorganism flux. And also, Rayleigh number , Mixed convection parameter Where ψ is the stream function, which is defined in the usual way as The transformed boundary conditions become The surface temperature , surface fluid concentration and surface motile microorganism concentration are defined as

Numerical Method
Simulation of the transformed Eqns. (6)(7)(8)(9) subject to the boundary conditions (10)-(11) are found with the help of Matlab BVP4C numerical method for various values of the flow controlling parameters. In the context of bvp4c function described above we need to transform the governing equations into first order differential equation. At first Equations (6-9) can be rearranged in the following way Now we need to transform this above equation into first order differential equation. For this let = and 1 = , 2 = ′ 3 = , 4 = ′ , 5 = , 6 = ′ , 7 = , 8 The boundary conditions become considering be the left boundary, be the right boundary  Table 1. show good agreement and exactness of numerical calculations.

Result Discussion
Velocity profile distribution for different values of mixed convection parameter λ and buoyancy parameter 1 are shown in Fig (2-3). Velocity profile increases with the growing values of λ and 1 because of inciting attitude of buoyancy forces in case of both isothermal (m = 0.5) and nonisothermal (m = 1.0) needle scenarios shown in Fig (2a)-(3a) and also for different needle sizes shown in Fig (2b)-(3b). The velocity boundary layer thickness increases for isothermal needle comparing to non-isothermal needle. The velocity profile enhances with the increasing value of needle thickness. The larger contact area of needle surface which causes momentum diffusion enhances flow profile. And also physically, the surface of the needle in contact with the fluid particles decreases when the size of the needle is getting smaller. This causes the friction force that occurs on the needle surface and the fluid flow will be reduced. In Fig (4-5) Temperature profile reduces with the growing values of mixed convection parameter λ and Buoyancy parameter 2 . Higher reduction of temperature profiles is observed for nonisothermal needle (when m = 1.0) which is the opposite behavior to the velocity profile. The increment of the thickness of needle leads to a thicker thermal boundary layer which causes lower needle wall temperature and reduces corresponding temperature profiles are shown in Fig (4b)-5(b). And also according to [34] the ascension in momentum diffusion rate inhibits the thermal diffusion which induces cooling of the boundary layer and a minimization in thermal boundary layer thickness.  In Fig (8-10) it is observed that microorganism profiles decrease for higher values of mixed convection parameter λ and also for Bioconvection Lewis number Lb and Bioconvection peclet number Pe. Bioconvection peclet number Pe and Bioconvection Lewis number Lb rise the mobility of fluid causes the quantity of motile microorganism thickness reduces. It is observed in Fig (8(a)-10(a)) that the behavior of microorganism profiles are quite similar for the both isothermal (m =0.5) and non-isothermal (m= 1.0) cases. And also it is noticeable in Fig. 8 Variation of surface temperature, fluid concentration and also surface motile microorganism concentration with mixed convection parameter λ are shown in Fig (11-13). Surface temperature, concentration and motile microorganism density decrease with the increasing values of λ. In all cases surface temperature and concentrations decreases with the increasing values of m which states reduction is higher for non isothermal cases. On the other hand slender needle reduces wall temperature, fluid concentration and also motile microorganism concentration shown in Fig (11 -13).

Conclusion
A new mathematical model for steady two-dimensional mixed convection flow in the presence of gyrotactic microorganism with variable surface heat, mass and motile microorganism flux is studied theoretically and numerically. The influences of isothermal, non -isothermal needle with different sizes have been taken into consideration. The effects of governing parameter on the flow, heat, mass and microorganism characteristic are discussed. The key findings of this analysis can be summarized as follows:  Velocity profile increases with the increasing values of mixed convection parameter λ where temperature, concentration and microorganism profile decreases.  Buoyancy parameters enhance velocity profiles and reduce temperature profile.  Lewis parameter Le has great influence on concentration profile which reduces boundary layer thickness of concentration profile.  Boundary layer thickness of microorganism profile reduces with bioconvection Lewis number Lb and bioconvection peclet number Pe.  All the physical quantities reduce with non-isothermal power law index.  Velocity profile increases with the increasing value of needle thickness.
Temperature concentration and microorganism profiles decrease for slender needle.