Magnetohydrodynamic of Copper-Aluminium of Oxide Hybrid Nanoparticles Containing Gyrotactic Microorganisms over a Vertical Cylinder with Suction

Keywords: Magnetohydrodynamic, Gyrotactic Microorganisms, Hybrid Nanofluid, Suction, Stretching Cylinder

The boundary layer flow with heat and mass transfer are important since the quality of final product depends on factors such as the rate of cooling and stretching phenomenon. The pivotal aim of this research is to address magnetohydrodynamics (MHD) copperaluminium oxide hybrid nanoparticles containing gyrotactic microorganisms over a stretching vertical cylinder with suction. The mathematical model has been formulated based on Tiwari-Das nanofluid model. Two types of nanofluid containing Copper (Cu) and Aluminum Oxide (Al2O3) immersed in water is considered in this study. In the analysis, the governing partial differential equations (PDEs) are transformed into a set of ordinary differential equations (ODEs) by a similarity transformation. Corresponding boundary conditions are analysed numerically along with these equations and are programmed in MATLAB software through the bvp4c method to obtain the solutions. The numerical solutions are obtained for the skin friction coefficient, the local Nusselt number, local Sherwood number and local density of motile microorganism as well as velocity, temperature, concentration, and microorganism profiles. The present analysis is validated by comparing with previously published work and found to be in good agreement. The effects of the parameter are analysed and discussed. According to the findings, suction increases shear stress, heat transport rate, mass transfer, and mass diffusivity. Moreover, hybrid nanofluid was discovered to be faster than nanofluid in terms of transit rate. Furthermore, the local density motile microorganism bioconvection Peclet number and bioconvection Lewis number increased. 

Introduction

In modern industrial and technical practice, boundary layer flow and heat transfer with stretching boundaries have drawn a lot of attention. The characteristics of the ultimate product are heavily reliant on stretching and heat transfer rate during the final step of processing. Because of its practical significance, scientists and engineers have developed an interest in understanding this phenomenon. Extrusion of metals into cooling liquids, food, and plastic items are common examples, as is the reprocessing of material in the molten state at high temperatures [1]. Double diffusive convection was studied in a porous medium by Hafizah et al., [1] to enhance the heat transfer. The study of cylinders has recently bought interest among researchers. Mukhopadhyay and Ishak [2] explored the impact of curvature on the skin friction coefficient. They found that when a cylinder is compared to a flat plate, the skin friction coefficient is shown to be higher for the cylinder. Furthermore, as the curvature parameter is increased, the radius of the cylinder shrinks, causing the fluid's motion to accelerate. This is due to the reduced contact surface area of the cylinder with the fluid, which provides less resistance to fluid flow [4]. Additional application regarding cylinder may be found in these literature [5,0. MHD mixed convective flow along a stretching vertical cylinder has many applications in areas such as heat exchangers, geothermal power generation, nuclear reactors, drilling operations, insulation system and plastic products formation, polymer processing units, etc. Mukhopadhyay 0 stated that heat and mass transfer flow past a stretching surface is mostly investigated. Sharma et al., 0 have investigated heat generation/absorption on MHD mixed convective stagnation point flow in the presence of external magnetic field. Sinha and Yadav 0 did research on steady MHD mixed convection Newtonian fluid flow along a vertical stretching cylinder embedded in porous medium. Lorentz force works as a resistance and retards the fluid motion, due to this resistance some amount of energy transforms into heat and increases the temperature. Further, preliminary investigation by Fatimah et al., 0 on the energy harvesting results in a higher output voltage with the presence of the magnets due to the action of nonlinear magnetic forces. 
A hybrid nanofluid is a type of nanofluid that contains two or more different nanoparticles. According to Yang et al., 0 hybrid nanofluid is produced by combining two different types of nanoparticles in the equivalent base fluid to improve thermophysical, optical, rheological, and morphological qualities. Hybrid nanofluids are expected to replace simple nanofluids for a variety of reasons, including a wider absorption range, lower extinction, good thermal conductivity, low pressure-drop, and fewer frictional losses and pumping power when compared to mono nanofluids. Solar collectors, electronic component heat management, machine cutting, engine applications, and automobile cooling have all been tested with hybrid nanofluids. Waini et al., 0 discovered that the heat transfer rate enhanced in the presence of hybrid nanoparticles when compared to standard nanofluid. It has been demonstrated that hybrid nanofluids have greater thermal conductivity than pure nanofluids. Additional hybrid nanofluid references in the direction of MHD may be found in the current literature [12-0. Microorganisms colonise all ecological niches, including hostile places where few other living things can survive such as the inside engine fuel lines, coal mines and hot springs. Bioconvection happens when microorganisms are swimming in upward direction of fluid and yield instability and amorphous pattern. Due to upward swimming, the gyrotactic microorganisms such as algae tends to concentrate on the upper portion of fluid and produces dense stratification that usually develops wobbly. Bioconvection flows appear in a variety of exciting applications, including environmental systems (ocean algae), fuel cells, and biological polymer synthesis [17]. Hosseinzadeh et al., 0 investigated cross-fluid flow with gyrotactic microorganisms and nanoparticles on a horizontal and three-dimensional cylinder while taking viscous dissipation and magnetic field into account. From the study, the change of bioconvection Lewis number from 0.2 to 0.5, it was observed that the concentration of the microorganisms reduced about 78.38%. We extended Devi and Devi, 0 by adding nanoparticle concentration equation and density of gyrotactic microorganism (bioconvection flow) equation. The pivotal aim of this research is to address and as nanoparticles with water as a base fluid containing gyrotactic microorganism swimming into it with suction and governed by a Al O boundary layer problem formulation over a stretching vertical cylinder using bvp4c method in MATLAB software. 

Methodology

This study considers a MHD of copper-aluminium oxide hybrid nanoparticles containing gyrotactic microorganisms over a stretching vertical cylinder with suction and a two-dimensional incompressible hybrid nanofluids embedded with water as base fluid through a stretching vertical cylinder. The fluid is a homogeneous mixture of nanoparticles, and gyrotactic microorganisms are also swimming in it. Microorganisms stabilize the suspension of nanoparticles by bio-convective flow which is generated by the combined simultaneous effects of nanoparticles and buoyancy. The linear stretching velocity is denoted by which is given by where reference length, and stretching rate, . The axis of the cylinder is treated along the x-axis and radial direction along the r-axis. The externally applied magnetic field, in the transverse direction is taken. The wall is permeable with . The viscous and Joule dissipation effects are negligible. ( ) 
w U x ( ) o w U x U x L = L o U 0 B ( ) 0 0 0 f v v v = - > ( ) Cu ( ) 2 3
Al O The subscript 1 and 2 indicate the physical properties of titanium alloy and aluminium alloy respectively. The total volume concentration of both nanoparticles is denoted by given as . 
1 j 2 j ( ) r ( ) C r r ( ) µ ( ) k ( ) s
The governing equations are: 
( 
(2) 
( ) 2 Cu j ( ) 2 3 1 Al O j ( ) 3 kg m r ( ) p C J kgK ( ) k W mK ( ) s m s 6 59.6 10 ´6 35 10 ´6 5.5 10 - ( ) r ( ) ( ) 2 1 1 1 2 2 1 1 hnf f y y r j j r jr jr é ù = - - + + ë û ( ) C r r ( ) ( ) ( )( ) ( ) ( ) 2 1 1 2 1 2 1 1 p p p p hnf f y y C C C C r j j r j r j r é ù = - - + + ë û ( ) µ ( ) ( ) 2.5 2.5 1 2 1 1 f hnf µ µ j j = - - ( ) k ( ) ( ) 2 2 2 2 2 2 2 2 2 y bf bf y hnf bf y bf bf y k k k k k k k k k k j j + - - = + + - ( ) (
) 
1 1 1 1 1 1 2 2 2 y f f y bf f y f f y k k k k k k k k k k j j + - - = + + - ( ) s ( ) ( ) 2 
+ - - = + + - ( ) ( ) 1 2 1 1 2 1 2 2 2 y f f y bf f y f f y s s j s s s s s s j s s + - - = + + - j 1 2 j j j = + ( ) ( ) 0 ru rv x y ¶ ¶ + = ¶ ¶ ( )( ) ( )( ) ( )( ) 2 2 0 2 1 1 1 0 hnf hnf hnf hnf hnf f n f m f B u u u u u u v x r r r r g C T T g C C g H H µ s r r r b r r r g r r ¥ ¥ ¥ ¥ ae ö ae ö ¶ ¶ ¶ ¶ + = + - + ç ÷ ç ÷ ç ÷ ¶ ¶ ¶ ¶ è ø è ø é ù - - - - - - - - = ë û (3) (4) (5) 
where is the component velocity along x-axis, is the component velocity along the r-axis, is the kinematic viscosity of the hybrid nanofluid, is the coefficient of fluid viscosity of the hybrid nanofluid, is the fluid density of hybrid nanofluids, is the acceleration due to gravity, is the coefficient of thermal expansion of the hybrid nanofluid, is electrical conductivity of hybrid nanoparticles, is the strength of magnetic field, is the thermal expansion coefficient, is the concentration of nanoparticles, is the fluid temperature , is the thermal conductivity of the hybrid nanofluid, is the heat capacity of the hybrid nanofluid, is the density of nanoparticles, is the density of microorganisms particle, is the diffusivity of nanoparticles, is the concentration of microorganisms, and is the diffusivity of microorganisms, is the constant velocity of the plate and is the maximum cell swimming speed. 
The governing equations (1) to (5) are subjected to the boundary conditions given as follows, 
Where is the suction velocity where , is the surface stretching velocities, is the surface temperature, is the surface concentration, is the surface microorganism, is the constant ambient temperature. is the constant ambient concentration of nanoparticles and is the constant ambient density of microorganism. 
The governing equations which are in the form of partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by using similarity transformation method. The similarity variables used are as shown below, , 
where is the similarity variable, is the kinematic viscosity of the base fluid, is the dimensionless temperature, is the radius of the cylinder, is the characteristic length of the cylinder, is the dimensionless nanoparticle fraction function and is the dimensionless microorganisms fraction function. By using the similarity variables (7), the PDEs are transformed into ODEs as follows, ( ) 
2 2 1 hnf p hnf k T T u u u v x r r r r C r ae ö ¶ ¶ ¶ ¶ + = + ç ÷ ¶ ¶ ¶ ¶ è ø 2 2 1 B C C C C u v D x r r r r ae ö ¶ ¶ ¶ ¶ + = + ç ÷ ¶ ¶ ¶ ¶ è ø 2 2 1 C M bw H H H H H u v H D x r C r r r r r ae ö ¶ ¶ é ¶ ¶ ù ¶ ¶ ae ö + + = + ç ÷ ç ÷ ê ú ¶ ¶ D ¶ ¶ ¶ ¶ è ø ë û è ø u v hnf hnf hnf u µ r = hnf µ hnf r g hnf b hnf s 0 B b C T hnf k ( ) hnf C r r n r m r B D H M D b C w 0 , , , , a t w w w w v v u U T T C C H H r R = = = = = = 0, , , at u T T C C H H r ¥ ¥ ¥ ® ® ® ® ®¥ 0 v ( ) 0 0 v > w U w T w C w H T ¥ C ¥ H ¥ 2 2 0 2 f U r R R L h u - = ( ) 0 ' U x u f L h = ( ) 0 1 f U v R f r L u h = - ( ) w T T T T q h ¥ ¥ - = - ( ) w C C C C f h ¥ ¥ - = - ( ) w H H H H c h ¥ ¥ - = - h f u q R L f c (8) (9) 
(10) (11) Along with the boundary conditions, (12) where , , , 
Using the similarity variables (7) where . Then, , , , 
( ) [ ] 2 1 2 2 2 1 2 ' 0 hnf f f f f ff Mf Nr Rb s e hk k e e l q f c s ae ö ¢¢¢ ¢¢ ¢¢ ¢ é ù + + - - - + - - = é ù ç ÷ ë û ë û ç ÷ è ø ( ) 4 2 1 2 0 Pr hnf f k f k e hk q kq q ae ö ¢¢ ¢ ¢ + + + = é ù ç ÷ ë û ç ÷ è ø ( ) 1 2 1 2 0 f Sc f hk f k f ¢ ¢ ¢ ¢ + + + = é ù ë û ( ) ( ) ( ) ( )( ) ( ) ( ) 2 1 2 2 1 2 1 0 Lb f Pe hk c kc c hk c w f hk c wk kc f é ù ¢¢ ¢ ¢ ¢¢ ¢ ¢ + + + - + + + + + + = ë û ( ) ( ) ( ) ( ) ( ) , 1 , 1 , = 1 , = 1 a t 0 f S f h h q h fh c h h ¢ = = = = ( ) ( ) ( ) ( ) 0, 0, 0, 0 as f h q h f h c h h ¢ ® ® ® ® ®¥ (14) 
f L R U k u = 2 0 0 f f M B L U s r = 2 Re x x Gr l = ( )( ) 3 2 1 x w f Gr g C T T x b u ¥ ¥ = - - Re x w f U x u = ( )( ) ( )( ) 1 n f w f w Nr C C C T T r r br ¥ ¥ ¥ = - - - - ( )( ) ( )( ) 
1 m f w f w Rb H H C T T g r r br ¥ ¥ ¥ = - - - - ( ) Pr p f f f C k r u = f B Sc D n = f M Lb D n = c M Pe bw D = w H H H w ¥ ¥ = - 0 0 f U S v L u = ( ) f C ( ) x Nu ( ) x Sh ( ) x Nn 2 hnf f f w r R u C U r µ r = ¶ ae ö = ç ÷ ¶ è ø ( ) hnf x f w r R k x T Nu k T T r ¥ = ¶ ae ö = - ç ÷ - ¶ è ø ( ) x w r R x C Sh C C r ¥ = ¶ ae ö = - ç ÷ - ¶ è ø ( ) x w r R x H Nn H H r ¥ = ¶ ae ö = - ç ÷ - ¶ è ø Re x w f U x u = ( ) ( ) ( ) 1 2 2.5 2.5 1 2 0 Re 1 1 f x f C j j ¢¢ = - - ( ) 1 2 Re 0 hnf x x f k Nu k q - ¢ = - ( ) 1 2 Re 0 x x Sh f - ¢ = - ( ) 1 2 Re 0 x x Nn c - ¢ = -

Results

The transformed governing boundary layer equations are non-linear and are solved numerically by using bvp4c in MATLAB. The flow and the heat transfer characteristics are obtained using hybrid nanofluid and nanofluid for the hydromagnetic flow. Figure 2 shows the effect for velocity profile when the magnetic increases in the presence of suction. shows the significance of mixed convection parameter on the velocity profile . 
0 f ¢¢ ( ) 0 q¢ - ( ) 0 f¢ - ( ) 0 c¢ - ( ) 0 f ¢¢ ( ) 0 q¢ - ( ) 0 q¢ - 1 0 j = 2 0 j = 0 k = Pr ( ) 0 q¢ - Pr 1 2 0 j j = = Pr ( ) 0 f ¢¢ / water Cu 1 0 j = 2 3 / water Cu Al O - 1 0.1 j = 0 k = 2 j ( ) 0 f ¢¢ 2 j / water Cu 2 j
Clearly, dimensionless velocity of fluid boosts up for growing mixed convective number . 
Larger corresponds to an increase in temperature difference leading to enhancement of fluid velocity. For , an additional buoyancy induced flow appears. Therefore, fluid velocity enhances and an increment in momentum layer thickness is seen. 
Figure 3 shows the effect on temperature profile mixed convection parameter increases in the presence of suction. Here distribution profile shows decreasing significance for mixed convection parameter . Larger leads to an increment in temperature difference which produces decay in temperature profile. Concentration profile is presented in Figure 4 for varies values of the mixed convection parameter . It has been observed that the concentration of nanoparticles is depreciated by growing mixed convective number. Higher corresponds to an increment in temperature difference which creates reduction in concentration of nanoparticles. Microorganism profile is presented in Figure 5 for varies values of the mixed convection parameter in the presence of suction. For higher values of mixed convective number , microorganism field diminishes. Basically, mixed convective number is in direct relation with gravitational acceleration. Therefore, larger estimations of mixed convective number lead to higher gravitational acceleration which shows decay in motile microorganism's concentration. 
( ) , it signifies the cylinder's outer surface behaves like a flat surface. Temperature distribution is increasing as the value of curvature parameter, increasing. Kelvin temperature has to do with the kinetic energy of molecules and called as the average kinetic energy. When the velocity of the fluid increases, kinetic energy also will increase and cause temperature to increase. It shows that with an increase in , the radius of the cylinder decreases which in turn leads to an increase in temperature, near the surface of the cylinder. 
0 f ¢¢ 2 j 2 3 / water Cu Al O - 2 j l ( ) f h ¢ ( ) f h ¢ l l 0 l > l ( ) q h l l l l l ( ) f h l l l ( ) c h
Concentration profile is presented in Figure 7 for varies values of the Schmidt number, . The results present the deceleration of nanoparticle concentration with increasing Schmidt number. The Schmidt number is a fraction of the momentum diffusion rate to the mass diffusion rate. This decrement in the magnitude of nanoparticle concentration is due to the momentum diffusivity increment. This explains about the relation between the concentration of microorganisms and the Schmidt number. The density of the microorganisms decreases with the raising values of the Schmidt number. As Schmidt number increases dynamic viscosity of the fluid increases but the density and mass diffusivity of the nanoparticles decreases hence concentration of the microorganisms decreases. Microorganism profile is presented in Figure 8 and 9 for varies values of the bioconvection Lewis number, and Peclet number, in the presence of suction. An augmentation in the values of both parameters leads to a less spread of microorganism and physically results a decline of motile boundary layer thickness of nanofluid. The Lewis number causes a decrease in the thickness of the motile microbe layer. This occurs because as the Lewis number grows, so does the viscous diffusion rate, which lowers the velocity distribution at the surface and, as a result, the density of the microorganisms decreases. The bioconvection Peclet number helps in increasing the speed of the microorganisms in relation to the fluid and resulting in a reduction in microbe density at the surface. The outcomes of local Sherwood number for various variations of involved parameters are tabulated in Tables 6. From Table 6, local Sherwood number increases for higher values of Schmidt number parameter . Hence, the rate of mass transfer also increases. An upsurge in is associated to thinner concentration boundary layer. Physically, connects the relative thickness of the mass transfer layer and the hydrodynamic boundary layer. 

Conclusions

This present result is extended from the work of Devi and Devi 0 by considering hybrid nanofluid in vertical cylinder and adding term of mixed convection parameter, nanoparticle concentration equation and density of gyrotactic microorganism. The similarity transformation has been used in order to transform the partial differential equation of boundary layer into the form of ordinary differential equation, and then solved by bvp4c solver in MATLAB software. Some of parameters are being considered in this study to determine the behavior of the skin friction coefficient, local Nusselt number, local Sherwood number and local density of motile microorganism, also the velocity, temperature, concentration and microorganism profiles respectively. The results obtained are as follow: 
a. The velocity profile increases when curvature parameter, mixed convection parameter, buoyancy ratio parameter, and bioconvection Rayleigh number increases. However, the profile decreases when volume fraction of Copper, suction, and magnetic field parameter decreases. b. The temperature profile increases when volume fraction of Copper, and curvature parameter increases. Furthermore, the profile decreases when suction, and Prandtl parameter decreases. c. The concentration profile increases when volume fraction of Copper, and curvature parameter increases. 

Cu

  


Fig. 1 .

 1 Fig. 1. Physical model of the problem 


Fig. 2 . 3 .

 23 Fig. 2. Effects of Velocity profile for various Fig. 3. Effects of Temperature profile for various 


Fig. 4 .

 4 Fig. 4. Effects of Concentration profile for various 


Fig. 6 .Fig. 5 .Fig. 7 .

 657 Fig. 6. Effects of Temperature profile for various 


Fig. 9 .Fig. 8 .

 98 Fig. 9. Effects of Microorganism profile for various Pe 


In this study, two types of nanoparticles used with water as base fluid are considered. The

  
	nanoparticles are copper	and aluminium oxide	termed as	and	respectively. The
	equations for evaluating the thermophysical properties of hybrid nanofluids which includes density
	, heat capacity	, dynamic viscosity	, thermal conductivity	and electrical
	conductivity	are presented in Table 1 [19]. The thermophysical properties of the base fluid and
	nanoparticles are presented in Table 2 [19].		

Table 1

 1 Thermophysical Properties of nanoparticles and water 0 


	Thermophysical
	properties

Table 3

 3 
	Comparison of values of	for several values of	when
		W. A. Khan and Pop 0	Devi and Devi 0	Ahmad et al., 0	Present
						results
	2	0.9113	0.91135		0.91045	0.9113
	7	1.8954	1.89540		1.89083	1.8954
	20	3.3539	3.35390		3.35271	3.3539
	70	6.4621	-		6.47814	6.4622
	The numerical value of Skin Friction Coefficient,	are obtained by comparing the numerical
	results of present study with Devi and Devi 0, and Ahmad et al., 0 which are portrayed in Table 4 for
	(pure nanofluid) in the absence of solid volume fraction	and Table 5 for
		(hybrid nanofluid) with	is fixed throughout the study, curvature
	parameter	for different values of , and it is noted that our results are found to be in excellent
	agreement with them. Therefore, this verifies the reliability and accuracy of the method used for this
	study.				

Table 4

 4 
	Comparison of values of	for several values of	for	(Pure Nanofluid)
		Devi & Devi 0	Ahmad et al. 0	Present results
	0.005	-2.54730	-2.54599	-2.54730
	0.02	-2.69147	-2.68993	-2.69147
	0.04	-2.88932	-2.88749	-2.88932
	0.06	-3.09419	-3.09209	-3.09419

Table 5

 5 
	Comparison of values of	for several values of	for	(Hybrid Nanofluid)
		Devi & Devi 0	Ahmad et al., 0	Present results
	0.005	-3.33526	-3.33339	-3.33526
	0.02	-3.50377	-3.50168	-3.50377
	0.04	-3.73592	-3.73354	-3.73592
	0.06	-3.97734	-3.97469	-3.97734

Table 6

 6 Local Sherwood number for and 


	0.005



  The profile decreases when Schmidt number, and suction decreases. d. The microorganism profile increases when volume fraction of Copper, curvature parameter, bioconvection Lewis number, bioconvection Peclet number, and Schmidt number increases. However, the profile decreases when suction, and bioconvection parameter decrease. e. The skin friction coefficient increases when volume fraction of Copper, curvature parameter, suction and magnetic field increase. It is decrease when mixed convection parameter, bioconvection ratio parameter, and bioconvection Rayleigh number decrease. f. The local Nusselt number increase when volume fraction of Copper, curvature parameter, suction and Prandtl number increase. g. The local Sherwood number increase when curvature parameter, and suction increases. Furthermore, it is decrease when volume fraction of Copper and Schmidt number decrease. h. The local motile microorganism increases when curvature parameter, suction, bioconvection Lewis number, and bioconvection parameter increase. However, it is decrease when volume fraction of Copper, bioconvection Peclet number, and Schmidt number decreases. Acknowledgement This research was funded by a grant from Ministry of Higher Education of Malaysia (FRGS/1/2021/STG06/UITM/02/11) and Universiti Teknologi MARA Shah Alam is grateful acknowledged. 


	f¢ -	( ) 0
	Sc	Sc
		Sc
Siti Khuzaimah Soid, Afiqah Athirah Durahman, Nur Hazirah Adilla Norzawary, Mohd Rijal Ilias, Amirah Mohamad Sahar
Magnetohydrodynamic of Copper-Aluminium of Oxide Hybrid Nanoparticles Containing Gyrotactic Microorganisms over a Vertical Cylinder with Suction