Abstract
The article explores the threedimensional stream of silver (Ag), magnesium oxide (MgO), and motile microorganism waterbased hybrid nanofluids as independent of time through a circular cylinder with a sinusoidal radius. The goal of this research is to optimize the rate of energy and mass transfer through a circular cylinder having a periodic radius. The phenomena are simulated as a system of partial differential equations containing momentum, temperature, concentration, and the profile of motile microbes, which were then simplified to a dimensionless system of ordinal differential equations using the similarity technique. The problem is solved by using the parametric continuation method, which is a numerical methodology. From the analysis, it has been perceived that both the energy and velocity fields significantly enhance with the rising effect of hybrid nanoparticles (Ag–MgO). The effect of chemical reaction enhances the mass transition rate because chemical reaction parameter influence exercises the molecules inside the fluid. The motile microorganism outline is elevated with the increment of Lewis and Peclet number.
Nomenclature
c  noddle point 
u, v, w  velocity component 

thermal conductivity 

mass diffusivity 

volumetric heat capacity 
D _{ n }  microorganism diffusion 

motile microbes’ density 

Reynolds number 
Pe  Peclet number 

dimensionless energy profile 

concentration over the surface 

ambient concentration 

heat capacity 

dimensionless mass transition 
Sc  Schmidth number 

streamlines 

dynamic viscosity 

density 

temperature 
Pr  Prandtl number 

chemical reaction 
Le  Lewis number 

dimensionless motile microorganism 

dimensionless velocity profile 

volume friction 

ambient temperature 

Nusselt number 

kinematic viscosity of fluid 

similarity variable 

skin friction 

stream coefficients 

constant 
1 Introduction
Many engineering operations include flow through a circular cylinder, but far less research has been done on flow over a cylinder in a restricted domain. Several phenomena depend on wall upshots, such as blood flow through medical equipment in veins, flow over cylindrical objects near walls, and so on [1,2,3]. Moreover, the analysis of fluid flow across or over an irregular surface received enough attention. When regular sinusoidal ridges are present, a threedimensional (3D) printer, owing to desired vibrations throughout the printing procedure, prompts research of the resultant flow consequences [4]. Salahuddin et al. [5] documented the consequence of variously configured nanomaterials on the thermal characteristics and flow efficiency of ferrofluid across rigid and sinusoidal surfaces. Wu et al. [6] utilized several active mechanism wind tunnel to evaluate the aerodynamic workloads of a sinusoidal cylinder. Changing the amplitude and frequency produces a succession that is entirely coherent in the streamwise direction. Bilal et al. [7] described an extending cylinderinduced incompressible Maxwell nanofluid flow accompanied by an unfluctuating suction/injection effect. It was determined that the impacts of the thermophoresis considerably increased the velocity of mass transference, while the consequences of the viscosity factor’s expansion substantially decreased the velocity curve. Seo et al. [8] numerically estimated the free convection in a broad, diagonal domain with a sinusoidal cylinder. In order to outperform a circular cylinder in terms of overall heat exchange, the sinusoidal cylinders were investigated. Bilal et al. [9] addressed the hybrid nanoliquid’s Darcy–Forchhemier mixed convective flow across an angled, expanding cylinder. Alharbi et al. [10] documented the magnetic characteristics of nanoliquid flow with energy flux in a boundary layer incorporating nanocrystals.
Special prominence has been paid to the investigation of the hybrid nanoliquid with energy and mass transmission. Because of its critical significance in engineering and innovation, it has attracted the attention of numerous scientists and experts [11,12,13]. Propagation of the hybrid nanoliquid flow, as well as energy transference, plays crucial roles in biotechnology, nuclear sectors, paper manufacturing, geophysics, chemical plants, and unusual lubricants are only some of the uses in industry [13–15]. Commonly used fluids cannot fulfill the global demands, in the era of scientific and technological advancement. However, comparable base liquids with the deposition of small particles showed a significant enhancement in thermal characteristics [16]. Currently, we have employed the Ag and MgO nanoparticles (NPs). The compound MgO is composed of the ions Mg^{2+} and O^{2}, which are joined by a special interaction. It is more useful for metalworking and electrical procedures. Similar to this, the antiseptic capabilities of Ag NPs could be used to change bioactivity in a diverse range of settings, including dental procedures, surgery, wound care, and pharmaceutical equipment [17]. A 3D numerical estimation of Ag–MgObased nanofluid flow across a curved whirling disc, is investigated by Ahmadian et al. [18]. The hybrid composite was made by the addition of Ag–MgO NPs. By employing the numerical procedure parametric continuation method (PCM), the solution was found. Ag–MgO was considered to be more useful in addressing insufficient energy transport. Broadspectrum antibacterial activities in metal and metal oxide nanomaterials have been extensively demonstrated for silver and MgO [19]. Anuar et al. [20] added MgO and Ag nanoparticulates, to produce a hybrid nanoliquid to evaluate boundary layer flow and temperature distribution. The consequences exposed that growing the weight fraction of Ag nanomaterials in a base fluid reduces the Nusselt number. Gangadhar et al. [21] have conducted a quantitative analysis of the properties of a nanofluid combining MgO and Au nanoparticles for convective heating. The influence of increased slip condition massively enhances the energy conduction ratio in the saddle and nodal point regions, according to the conclusions. Hiba et al. [22] inspected the thermal performance and fluidity of a waterbased hybrid nanofluid made of MgO and Ag over a cylindrical, highly permeable region. Rasool et al. [15] examined numerical study of electromagnetohydrodynamic nanoliquid flows through a Riga pattern inserted diagonally in a Darcy–Forchheimer permeable media. Shah et al. [23] quantitatively investigated the convective fluxes of a remarkable nonhomogeneous nanofluids mixture across an impenetrable longitudinal electrostatic substrate. Ashraf et al. [24] documented the nanoliquid flow using the generalized numerical approach.
The analysis of gyrotactic microbes in free surface flows has recently gained the attention of the scientific community. A microorganism is a living organism that can reproduce, evolve, react to environmental stimuli, and maintain a structured order. It can be utilized to improve oil recovery, which involves adding micronutrients and microorganisms to fuel layers to balance out permeability discrepancies [25–27]. The benefits of motile microbe interruption include nanofluid stability [28]. Hydrodynamic convection is created by oxytactic microbes, which forms a flow system that moves cells and oxygenation from the highest to the lowest fluid regions. The nanostructure mobility is regulated by molecular diffusion. The motile bacteria’s motility seemed to be independent of nanomaterial gestures [29,30]. The working mechanism of gyrotactic microorganisms in nanofluid was first discussed in refs [29,31]. Kuznetsov [32] extended the idea of suspensions by including Buongiorno’s conceptualization of bioconvection in nanoliquids. Xu et al. [33] analyzed a ferrofluid flow, consisting of motile microbes that flowed across parallel surfaces and transmitted energy. The velocity curve enhances with the mounting upshot of bioconvection [34]. Sohail et al. [35] examined the varied thermophysical characteristics of the 3D flow of a liquid with mass and energy conveyance in the presence of peristaltic transport of motile microbes over a curved elongated substrate. Despite the fact that the earlier analyses have indeed focused on understanding nanoliquid convection, there has been no effort in the literature to inspect the mass and motile microorganism transition under the influence of chemical reaction through the cylinder with sinusoidal radius (Figure 1).
The objective of this analysis is to scrutinize the features of the 3D stagnation point flow of Ag–MgO based hybrid nanoliquids traveling through a spherical cylinder with a harmonic radius. The electrostatic source consequences are evaluated in the stream flow. Our second priority is to elaborate also the uses and applications of silver and magnesium particles for medical and industrial purposes. After depersonalization, the computational technique PCM is used to calculate the powerful nonlinear systems. Furthermore, a pictorial assessment of the outstanding characteristics is performed on the velocity, mass, temperature, and motile microbes’ profiles.
2 Mathematical formulation
We have presumed the 3D stream of Ag–MgO waterbased hybrid nanofluids through a circular cylinder of the periodic surface. It is important to note that there exists a motionlessness point at each point M, N, and O. The u, v, and w are the velocities terms in the path of x, y, and z, respectively. Where
Here,
Here,
The boundary conditions are:
The thermophysical properties of hybrid nanoliquid and model are [9]:









In order to diminish the system of partial differential equations (PDEs) to the system of nonlinear ordinal differential equations (ODEs), we defined the following variables:
Now, by using Eq. (8) in Eqs. (1)–(6) and (7), we get:
Where,
The renovated conditions are:
The physical interest quantities are:
Where,
The dimensionless form of Eq. (16) is:
3 Problem solution
Different steps, which are used during applying PCM to system of Eqs. (9)–(13) and (15), are [38–43]:
Step 1: Reduced BVPs to the first order ODEs
By employing Eq. (19) to Eqs. (9)–(13) and (15), we get:
boundary conditions are:
Step 2: Familiarizing the embedding constraint p to Eqs. (20)–(24):
4 Results and discussion
The results are exposed through Figures (2–4), and Tables and discussion on the obtained results are categorized as:
4.1 Velocity profile
Figure 2(a)–(d) highlights the presentation of axial
4.2 Energy profile
Figure 3(a)–(d) uncovers the behavior of energy
4.3 Mass and motile microorganism profile
Figure 4(a)–(d) emphasizes the performance of mass transition
Table 1 presents the experimental values of Ag, MgO, and water. Table 2 discovers the comparative valuation of bvp4c package and PCM with the arithmetical outcomes of skin friction and the local Nusselt number. The variation of both Ag and MgO nanoparticles boosts the drag force and energy transference rate. From Table 2, it can be noticed that the PCM is fast approaching technique than bvp4c. Tables 3 and 4 report the comparison of simple and hybrid nanofluid behavior for skin fraction and energy transition. As compared to nanofluid, hybrid fluid has tremendous tendency for energy propagation.






Pure water  997.1  4,179  0.613  21 
Magnesium oxide  3,560  955  45  1.80 
Silver  10,500  235  429  1.89 






PCM  bvp4c  PCM  bvp4c  PCM  bvp4c  
Ag  0.00  1.2683  1.2682  0.4994  0.4992  1.3304  1.3300 
0.01  1.9488  1.9385  0.7635  0.7629  1.6187  1.6183  
0.02  2.6974  2.6970  1.0621  1.0615  1.9296  1.9290  
MgO  0.00  1.2686  1.2682  0.4999  0.4991  1.3321  1.3299 
0.01  1.6667  1.6659  0.6565  0.6556  1.4977  1.4954  
0.02  2.1541  2.1535  0.8482  0.8474  1.7396  1.7383 


Ag–MgO/water  Ag/water  MgO/water  








0.01  0.01  1.300  0.450  1.231  0.423  1.231  0.424 
0.02  0.02  1.451  0.513  1.300  0.450  1.303  0.453 
0.03  0.03  1.647  0.591  1.380  0.482  1.388  0.485 
0.04  0.04  1.882  0.687  1.471  0.518  1.482  0.523 


Ag–MgO/water  Ag/water  MgO/water 





0.01  0.01  0.537  0.518  0.521 
0.02  0.02  0.573  0.537  0.541 
0.03  0.03  0.608  0.556  0.561 
0.04  0.04  0.643  0.576  0.581 
5 Conclusion
We have observed the features of 3D stagnation point flow of Ag–MgObased hybrid nanoliquids traveling through a spherical cylinder of sinusoidal radius. The facts have been formulated in the form of system of PDEs. After transformation, the computational technique PCM is used to estimate the nonlinear systems of differential equations. Furthermore, a graphical assessment of the physical characteristics is accomplished on the velocity, mass, temperature, and motile microbes’ profiles. The key conclusions are as follows:
Both axial and radial velocities significantly augment with the intensifying effect of hybrid nanoparticulates (Ag–MgO).
The energy transmission profile also boosts with the increment of nanoparticulate in the base liquid.
The energy transfer rate also improves at the noddle point.
The upshot of chemical reaction enhances the mass transition because the chemical reaction parameter influence exercises the molecules inside the fluid, whose encourages the mass transfer.
The motile microorganism outlines elevated with the increment of Lewis and Peclet number.
As compared to nanoliquid, hybrid nanoliquid is more convenient for heat and mass transmission.
The skin friction coefficient augments with the rising numbers of nanoparticulates in the base fluid.
The addition of MgO and Ag nanomaterials to the base fluid also enhances the Nusselt number.

Funding information: The author (Z. Raizah) extends her appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.1/334/43).

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
References
[1] Sohail M, Naz R. Modified heat and mass transmission models in the magnetohydrodynamic flow of Sutterby nanofluid in stretching cylinder. Phys A: Stat Mech Appl. 2020 Jul 1;549:124088.10.1016/j.physa.2019.124088Search in Google Scholar
[2] Wakif A. A novel numerical procedure for simulating steady MHD convective flows of radiative Casson fluids over a horizontal stretching sheet with irregular geometry under the combined influence of temperaturedependent viscosity and thermal conductivity. Math Probl Eng. 2020 May 5;2020.10.1155/2020/1675350Search in Google Scholar
[3] Wakif A, Chamkha A, Animasaun IL, Zaydan M, Waqas H, Sehaqui R. Novel physical insights into the thermodynamic irreversibilities within dissipative EMHD fluid flows past over a moving horizontal riga plate in the coexistence of wall suction and joule heating effects: a comprehensive numerical investigation. Arab J Sci Eng. 2020 Nov;45(11):9423–38.10.1007/s13369020047573Search in Google Scholar
[4] Cavanagh K, Wulandana R. 2D flow past a confined circular cylinder with sinusoidal ridges. In Proceedings of the 2019 COMSOL Conference in Boston. Boston, MA, USA; 2019 Oct. p. 2–4.Search in Google Scholar
[5] Salahuddin T, Bashir AM, Khan M, Xia WF. Multiple shaped nanoparticles influence on thermal conductivity of fluid flow between inflexible and sinusoidal walls. Case Stud Therm Eng. 2021 Jun 1;25:100930.10.1016/j.csite.2021.100930Search in Google Scholar
[6] Wu B, Li S, Zhang L, Li K. Experimental determination of the twodimensional aerodynamic admittances of a 5: 1 rectangular cylinder in streamwise sinusoidal flows. J Wind Eng Ind Aerodyn. 2021 Mar 1;210:104525.10.1016/j.jweia.2021.104525Search in Google Scholar
[7] Bilal M, Saeed A, Selim MM, Gul T, Ali I, Kumam P. Comparative numerical analysis of Maxwell’s timedependent thermodiffusive flow through a stretching cylinder. Case Stud Therm Eng. 2021 Oct 1;27:101301.10.1016/j.csite.2021.101301Search in Google Scholar
[8] Seo YM, Luo K, Ha MY, Park YG. Direct numerical simulation and artificial neural network modeling of heat transfer characteristics on natural convection with a sinusoidal cylinder in a long rectangular enclosure. Int J Heat Mass Transf. 2020 May 1;152:119564.10.1016/j.ijheatmasstransfer.2020.119564Search in Google Scholar
[9] Bilal M, Khan I, Gul T, Tassaddiq A, Alghamdi W, Mukhtar S, et al. Darcyforchheimer hybrid nano fluid flow with mixed convection past an inclined cylinder. Comput Mater Continua. 2021 Jan 1;66(2):2025–39.10.32604/cmc.2020.012677Search in Google Scholar
[10] Alharbi KA, Ahmed AE, Ould Sidi M, Ahammad NA, Mohamed A, ElShorbagy MA, et al. Computational valuation of darcy ternaryhybrid nanofluid flow across an extending cylinder with induction effects. Micromachines. 2022 Apr 9;13(4):588.10.3390/mi13040588Search in Google Scholar PubMed PubMed Central
[11] Sohail M, Naz R, Abdelsalam SI. Application of nonFourier double diffusions theories to the boundarylayer flow of a yield stress exhibiting fluid model. Phys A Stat Mech Appl. 2020 Jan 1;537:122753.10.1016/j.physa.2019.122753Search in Google Scholar
[12] Bilal S, Sohail M, Naz R. Heat transport in the convective Casson fluid flow with homogeneous‒heterogeneous reactions in Darcy‒Forchheimer medium. Multidiscip Model Mater Struct. 2019 Aug 1610.1108/MMMS1120180202Search in Google Scholar
[13] Algehyne EA, Wakif A, Rasool G, Saeed A, Ghouli Z. Significance of DarcyForchheimer and Lorentz forces on radiative aluminawater nanofluid flows over a slippery curved geometry under multiple convective constraints: a renovated Buongiorno’s model with validated thermophysical correlations. Waves Random Complex Media. 2022 May 14;1–30.10.1080/17455030.2022.2074570Search in Google Scholar
[14] Bilal M, Gul T, Alsubie A, Ali I. Axisymmetric hybrid nanofluid flow with heat and mass transfer amongst the two gyrating plates. ZAMM‐J Appl Math Mech/Zeitschrift für Angew Mathematik und Mechanik. 2021 Nov;101(11):e202000146.10.1002/zamm.202000146Search in Google Scholar
[15] Rasool G, Shah NA, ElZahar ER, Wakif A. Numerical investigation of EMHD nanofluid flows over a convectively heated riga pattern positioned horizontally in a DarcyForchheimer porous medium: application of passive control strategy and generalized transfer laws. Waves Random Complex Media. 2022 May 14;1–20.10.1080/17455030.2022.2074571Search in Google Scholar
[16] Tassaddiq A, Khan S, Bilal M, Gul T, Mukhtar S, Shah Z, et al. Heat and mass transfer together with hybrid nanofluid flow over a rotating disk. AIP Adv. 2020 May 1;10(5):055317.10.1063/5.0010181Search in Google Scholar
[17] Li YX, Muhammad T, Bilal M, Khan MA, Ahmadian A, Pansera BA. Fractional simulation for DarcyForchheimer hybrid nanoliquid flow with partial slip over a spinning disk. Alex Eng J. 2021 Oct 1;60(5):4787–96.10.1016/j.aej.2021.03.062Search in Google Scholar
[18] Ahmadian A, Bilal M, Khan MA, Asjad MI. Numerical analysis of thermal conductive hybrid nanofluid flow over the surface of a wavy spinning disk. Sci Rep. 2020 Nov 2;10(1):1–3.10.1038/s4159802075905wSearch in Google Scholar PubMed PubMed Central
[19] Zhang XH, Algehyne EA, Alshehri MG, Bilal M, Khan MA, Muhammad T. The parametric study of hybrid nanofluid flow with heat transition characteristics over a fluctuating spinning disk. PLoS One. 2021 Aug 16;16(8):e0254457.10.1371/journal.pone.0254457Search in Google Scholar PubMed PubMed Central
[20] Anuar NS, Bachok N, Pop I. Influence of buoyancy force on Ag–MgO/water hybrid nanofluid flow in an inclined permeable stretching/shrinking sheet. Int Commun Heat Mass Transf. 2021 Apr 1;123:105236.10.1016/j.icheatmasstransfer.2021.105236Search in Google Scholar
[21] Gangadhar K, Edukondala Nayak R, Venkata Subba Rao M, Kannan T. Nodal/Saddle stagnation point slip flow of an aqueous convectional magnesium oxide–gold hybrid nanofluid with viscous dissipation. Arab J Sci Eng. 2021 Mar;46(3):2701–10.10.1007/s1336902005195xSearch in Google Scholar
[22] Hiba B, Redouane F, Jamshed W, Saleel CA, Devi SS, Prakash M, et al. A novel case study of thermal and streamline analysis in a grooved enclosure filled with (Ag–MgO/Water) hybrid nanofluid: Galerkin FEM. Case Stud Therm Eng. 2021 Dec 1;28:101372.10.1016/j.csite.2021.101372Search in Google Scholar
[23] Shah NA, Wakif A, ElZahar ER, Ahmad S, Yook SJ. Numerical simulation of a thermally enhanced EMHD flow of a heterogeneous micropolar mixture comprising (60%)ethylene glycol (EG),(40%)water (W), and copper oxide nanomaterials (CuO). Case Stud Therm Eng. 2022 Apr 21;102046.10.1016/j.csite.2022.102046Search in Google Scholar
[24] Ashraf MU, Qasim M, Wakif A, Afridi MI, Animasaun IL. A generalized differential quadrature algorithm for simulating magnetohydrodynamic peristaltic flow of blood‐based nanofluid containing magnetite nanoparticles: a physiological application. Numer Methods Partial Differ Equ. 2022 May;38(3):666–92.10.1002/num.22676Search in Google Scholar
[25] Hamad NH, Wakif A, Alshehri A. Towards the dynamics of a radiativereactive magnetized viscoelastic nanofluid involving gyrotactic microorganisms and flowing over a vertical stretching sheet under multiple convective and stratification constraints. Waves Random Complex Media. 2022 Jul 20;1–31.10.1080/17455030.2022.2100944Search in Google Scholar
[26] Wakif A, Zaydan M, Alshomrani AS, Muhammad T, Sehaqui R. New insights into the dynamics of alumina(60% ethylene glycol + 40% water) over an isothermal stretching sheet using a renovated Buongiorno’s approach: A numerical GDQLLM analysis. Int Commun Heat Mass Transf. 2022 Apr 1;133:105937.10.1016/j.icheatmasstransfer.2022.105937Search in Google Scholar
[27] Naz R, Tariq S, Sohail M, Shah Z. Investigation of entropy generation in stratified MHD Carreau nanofluid with gyrotactic microorganisms under Von Neumann similarity transformations. Eur Phys J Plus. 2020 Feb;135(2):1–22.10.1140/epjp/s13360019000690Search in Google Scholar
[28] Kuznetsov AV. The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms. Int Commun Heat Mass Transf. 2010 Dec 1;37(10):1421–5.10.1016/j.icheatmasstransfer.2010.08.015Search in Google Scholar
[29] Geng P, Kuznetsov AV. Effect of small solid particles on the development of bioconvection plumes. Int Commun heat mass Transf. 2004 Jul 1;31(5):629–38.10.1016/S07351933(04)000508Search in Google Scholar
[30] Naz R, Noor M, Shah Z, Sohail M, Kumam P, Thounthong P. Entropy generation optimization in MHD pseudoplastic fluid comprising motile microorganisms with stratification effect. Alex Eng J. 2020 Feb 1;59(1):485–96.10.1016/j.aej.2020.01.018Search in Google Scholar
[31] Van Wijngaarden WK, Vermolen FJ, Van Meurs GA, Vuik C. Modelling biogrout: a new ground improvement method based on microbialinduced carbonate precipitation. Transp Porous Media. 2011 Mar;87(2):397–420.10.1007/s1124201096918Search in Google Scholar
[32] Kuznetsov AV. Nonoscillatory and oscillatory nanofluid biothermal convection in a horizontal layer of finite depth. Eur J MechanicsB/Fluids. 2011 Mar 1;30(2):156–65.10.1016/j.euromechflu.2010.10.007Search in Google Scholar
[33] Xu YJ, Bilal M, AlMdallal Q, Khan MA, Muhammad T. Gyrotactic microorganism flow of Maxwell nanofluid between two parallel plates. Sci Rep. 2021 Jul 26;11(1):1–3.10.1038/s41598021945434Search in Google Scholar PubMed PubMed Central
[34] Sohail M, Naz R, Abdelsalam SI. On the onset of entropy generation for a nanofluid with thermal radiation and gyrotactic microorganisms through 3D flows. Phys Scr. 2020 Feb 11;95(4):045206.10.1088/14024896/ab3c3fSearch in Google Scholar
[35] Sohail M, Naz R, Shah Z, Kumam P, Thounthong P. Exploration of temperature dependent thermophysical characteristics of yield exhibiting nonNewtonian fluid flow under gyrotactic microorganisms. AIP Adv. 2019 Dec 1;9(12):125016.10.1063/1.5118929Search in Google Scholar
[36] Gholinia M, Hosseinzadeh K, Ganji DD. Investigation of different base fluids suspend by CNTs hybrid nanoparticle over a vertical circular cylinder with sinusoidal radius. Case Stud Therm Eng. 2020 Oct 1;21:100666.10.1016/j.csite.2020.100666Search in Google Scholar
[37] Dinarvand S, Hosseini R, Damangir E, Pop I. Series solutions for steady threedimensional stagnation point flow of a nanofluid past a circular cylinder with sinusoidal radius variation. Meccanica. 2013 Apr;48(3):643–52.10.1007/s1101201296217Search in Google Scholar
[38] Shuaib M, Bilal M, Qaisar S. Numerical study of hydrodynamic molecular nanoliquid flow with heat and mass transmission between two spinning parallel plates. Phys Scr. 2020 Nov 30;96(2):025201.10.1088/14024896/abcce2Search in Google Scholar
[39] Shuaib M, Shah RA, Bilal M. Variable thickness flow over a rotating disk under the influence of variable magnetic field: An application to parametric continuation method. Adv Mech Eng. 2020 Jun;12(6):1687814020936385.10.1177/1687814020936385Search in Google Scholar
[40] Shuaib M, Shah RA, Durrani I, Bilal M. Electrokinetic viscous rotating disk flow of PoissonNernstPlanck equation for ion transport. J Mol Liq. 2020 Sep 1;313:113412.10.1016/j.molliq.2020.113412Search in Google Scholar
[41] Bilal M, Ayed H, Saeed A, Brahmia A, Gul T, Kumam P. The parametric computation of nonlinear convection magnetohydrodynamic nanofluid flow with internal heating across a fixed and spinning disk. Waves Random Complex Media. 2022 Mar 2;1–6.10.1080/17455030.2022.2042621Search in Google Scholar
[42] Algehyne EA, Areshi M, Saeed A, Bilal M, Kumam W, Kumam P. Numerical simulation of bioconvective Darcy Forchhemier nanofluid flow with energy transition over a permeable vertical plate. Sci Rep. 2022 Feb 25;12(1):1–2.10.1038/s41598022072549Search in Google Scholar PubMed PubMed Central
[43] Alrabaiah H, Bilal M, Khan MA, Muhammad T, Legas EY. Parametric estimation of gyrotactic microorganism hybrid nanofluid flow between the conical gap of spinning diskcone apparatus. Sci Rep. 2022 Jan 7;12(1):1–4.10.1038/s41598021030772Search in Google Scholar PubMed PubMed Central
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