Numerical analysis of bioconvection-MHD flow of Williamson nanofluid with gyrotactic microbes and thermal radiation: New iterative method

Abbreviations

A

stagnation quantity

A ^∗

coefficient of heat source/sink for space-dependency

B

magnetic field

B ^∗

coefficient of heat source/sink for temperature-dependency

B ₀

constant magnetic field

Cp

specific heat

_{D T}

thermophoretic diffusion-coefficient

f

dimensionless stream function

K

heat velocity slip factor

k

thermal conductivity [W(m k)⁻¹]

k ^*

Rosseland mean absorption coefficient 1 m

Le = α D B

Lewis number

M = 2 σ B 0 2 U 0 ρ ( 1 + m )

elucidated magnetic field

m

flat sheet flow

N

velocity slip

N c ˆ = ρ ρ c ˆ ρ ρ c ˆ ( C ˆ w − C ˆ ∞ )

nanofluid heat capacity

Nb = B T ˆ T ˆ ∞ D T ˆ ( C ˆ w − C ˆ ∞ ) ( T ˆ w − T ˆ ∞ )

Brownian diffusivity/thermophoretic diffusivity

Nt

thermophoresis parameter

Pe = b w c D m

Peclet numbers

Pr = v α

Prandtl number

Q

non-Fourier heat flux

q _r

radiative heat flux

q′″

variable heat source/sink

Rd = 4 σ ⁎ T ˆ ∞ 3 k ∞ k ⁎

radiation parameter

Sh x ˆ = X q w k ( T ˆ w − T ˆ ∞ )

physical quantities

S c = v D B

Schmidt number

T

fluid temperature

T _w

fluid temperature

T _∞

ambient temperature

u _w = aX

velocity

V

fluidic velocity

V _w

mass transfer at wall

α = A U 0 ( 1 + m ) 2 v

wall thickness parameter

ε

elucidated magnetic field

ρ

fluid density

u ˜ , v ˜

velocity components

u ˜ ∞ = a 0 X

velocity of free stream

λ = T ˆ U 0 3 ( x ˆ + b ) 3 m − 1 ( m + 1 ) v

Williamson fluid parameter and thermal conductivity parameter

Ω = N ∞ N w − N ∞

difference in microorganism concentration

1 Introduction

In recent years, researchers have focused on the importance of promoting thermal suspension (size less than 100 nm) of pure liquids such as oil, water (H₂O), glycol, and other nanofluids and small particles. These nanoparticles, such as silver, alumina (Al₂O₃), copper oxide, etc., conduct and regulate heat transfer in freshwater, so these liquids play a vital role in the type of heat carrier in heat exchangers. Choi [1] proposed enhancing the thermal conductivity of fluids with nanoparticles in developments and applications of non-Newtonian flows.

These fluids play important roles in industrial and engineering applications like microelectronics, biomedical equipment, automotive cooling, and energy production. Hay et al. [2] discussed the structures of nanofluids in different forms and conditions decline in the production of entropy in nanofluids passing through a thin end under the influence of thermal radiation and the development of entropy is studied under the thermodynamic second law. Salleh et al. [3] investigated the the stabilization of the depth axis of the movement of a hydromagnetic fluid on the then narrow and Buongiorno mode is used for Brownian study and thermophoretic interference power in this system. Waini et al. [4] discussed uniform heat emission of mixed nanofluids on a flat tip. In this position, the sample measurements were converted into a draught form using sets of other variables and then adjusted the resulting numbers using Mathematica Programming. Gul et al. [5] discussed the flow of nanofluids on fine needles for convective nanotubes of fractional order. In the last few years, the debate on the thinning of needles has been very successful. According to Lee [6] and Narain and Uberoi [7], the strength as well as the absence of temperature as the nanofluids pass through a thin vertical needle of cold water is measured. In this expanded work, the authors consider the solution to the problems of metal flow locally and systematically. Chen and Smith [8] discussed heat transfer to a fine needle. In this article, the temperature properties, and the number and volume of the effect of Prandtl to accelerate the uniform flow of water are studied. Van [9] studied the combined flow of a hot needle plate placed on the handles. Finally, various studies have been conducted on water flowing through a good needle using different flow methods [10,11,12,13,14,15]. Ramesh et al. [16] discussed the therapeutic study of the hybrid nanofluid flow in a thin needle using the Darcy-Forchheimer characteristic of the upper package and the outer heat source.

The researcher of this article changed the modeling equation to that of the measurement model and thus changed the response rate to this set of measurement equations. Hamid et al. [17] discussed the properties of thermal electrophoresis of a continuously running nanofluid.

The behavior of driving in a nanofluid is explained in two types. One way is recommended by Buongiorno [18], and the other way is recommended by Das and Tiwari [19]. The Buongiorno model consists of more than one inhomogeneous substance in which the flow of pure fluid and nanoparticles is separated from zero. This mode showed more than six different ways to slide the flow of nano drinks. Magnetic fields, Magnus force, Brownian conduction, inertia, drainage, diffusion phosphorylation, and thermophoretic effects are examples of these processes. It has also been mentioned in the work of Buongiorno [18] that of the seven fine methods, only thermophoresis and Brown’s diffusion are important for liquids containing nanoparticles. Considering the value of the Buongiorno model, several researchers have successfully applied this model using different geometry and different flow conditions. Over the years, Nield and Kuznetsov [20,21] increased Buongiorno’s work. Both authors used the terms “thermophoretic” and “Brownian” and went to great lengths with electrical equations to study the effects of Brownian movement and thermophoresis on these equations by proposing several new scenarios that are on the verge of trouble. Finally, some researchers used the morning image model for flow to apply different types of flow patterns and states. Khan et al. [22,23] discussed the Buongiorno’s design for the flow of nanoparticles in a series of logs using different water methods. The authors of these studies addressed the hypotheses arising from the results by using appropriate equations and shapeless and partial analysis (homotopy analytical method, HAM) methods. The reader can continue to study the Buongiorno water cycle using a variety of geometry and different flow states in refs. [24,25,26,27,28].

Bioconvection is another fascinating area that has many physical and real-world forms. The movement of an object due to its density gradient on a small, microscopic step is defined as a dynamic change. This density gradient instability is caused by the swimming of microorganisms. This phenomenon usually occurs at the height of the fluid due to the condensation of the fluid in the affected area. Many physicians and procedures require these physical phenomena, such as biofuels, enzymes, micro-systems, biological cells, bacteria, and biotechnology, among others. Biological infection processes fall into different types, such as gyroscopic microorganisms and chemotaxis. and microbes found on land.

This classification depends on the dynamics of many microorganisms. Kuznetsov [29,30] studied biological convection using a large number of nanoparticles. Marikarjuna et al. [31] discussed in a vertical kiln, continuous biomarkers of nanoliquids and contracted microorganisms, transforming experimental problems into non-invasive models using a dimensionless variation, and the difference between numerical methods and finite numbers. Uddin et al. [32] discussed a mathematical structure for examining the effects of secondary velocity slides on horizontal filter plates. Chebyshev’s method was used in this study to get an approximate answer to the problem. The reader can search more for convection oil with different types of water and stream provisions [33,34,35,36,37].

The magnetic field plays an important role in the field of hydrodynamics. The magnetic fields are very important. The term magnetohydrodynamics (MHD) was coined by the Swedish physician Noble Laurat Hannes Alfven. The thermal action and the large amount of MHD over a long period find useful applications in the fields of fiberglass manufacturing, polymer technology, and metallurgy. MHD nano contributes greatly to fluid flow and heat transfer. Akbar et al. [38] studied and investigated MHD nanofluidic flow caused by surface tension, compression, and slip effect. Several other studies have addressed the flow of MHD nanowires and recommended effective solutions [39,40,41,42,43,44]. Nadeem and Hussein [45] observed Williamson’s liquid on a rapidly rising surface and examined the heat transfer. Waheed [46] discussed mixed sensory heat transfer in rectangular containers guided by continuous horizontal plates. Acharya et al. [47] proposed unsteady bioconvective squeezing flow with higher‐order chemical reaction and second‐order slip effects. Acharya [48] studied spectral quasi linearization simulation on the radiative nanofluid spraying over a permeable inclined spinning disk considering the existence of a heat source/sink. Acharya et al. [49] studied spectral quasi linearization simulation of radiative nanofluidic transport over a bent surface considering the effects of multiple convective conditions. Shah et al. [50] investigated numerical modeling on hybrid nanofluid (Fe₃O₄ + MWCNT/H₂O) migration considering MHD effect over a porous cylinder. Shah et al. [51] discussed the simulation of entropy optimization and thermal behavior of nanofluid through the porous media. Shah et al. [52] studied the impact of nanoparticle shape and radiation on the behavior of nanofluids under the Lorentz forces. Bilal et al. [53] investigated unsteady hybrid-nanofluid flow comprising ferrous oxide and CNTs through a porous horizontal channel with dilating/squeezing walls. Marzougui et al. [54] discussed entropy generation on the magneto-convective flow of copper–water nanofluid in a cavity with chamfers.

In 2006, a new method was developed called the new iterative method (NIM) for linear and nonlinear functional equations by Daftardar-Gejji and Jafari [55]. The fixed method operates linear and non-linear classifications in a straightforward polynomial way in nonlinear terms such as ADM. The need for the Lagrange function increases with its algorithm, such as VIM, and the need for route choice of route numbers. The judiciary is free from a small intellectual parliament, in contrast to the judicial tribunal. Since NIM is an iterative method, it needs the first condition to become its main value. Subsequently, the method was developed by many analysts for different problems, including algebraic equations, change equations, common and varied equations, and a system of non-linear measurement equations. Daftardar-Gejji and Bhalekar [56] discussed that lines tested with non-linear separation equilibrium equations terminate domains with the Dirichlet boundary conditions using a new repeating method. String and non-linear separation-wave equations were analyzed using the iterative method developed by Daftardar-Gejji and Bhalekar [57]. Bhalekar and Daftardar-Gejji [58] evaluated a separate component of effective use using a new iterative method. Bhalekar and Daftardar-Gejji [59] used the hypothetical method of control, which is complete in the context of the conversion of that method of decay. The new iterative approach was used to address the nth-command lines and non-linear integral-differentiation equations by Hemeda [60]. The Newell–Whitehead–Segel equation was investigated using NIM by Yaseen and Samraiz [61]. Patade and Bhalekar [62] showed an iterative method for correcting unequal equations that include only the first design function. A family of repeated regression models in which the system of linear equations was presented by Noor and Noor [63]. Noor et al. [64] promoted and evaluated alternative design relationships that create different classes of iterative behaviors for the informal investigation of problems using a combined simulation system. Shah and Noor [65] discussed the higher order iteration method for non-linear equations with decomposition technique.

2 Problem formulation

Considered a two dimensional, incompressible, hydromagnetic flow of Williamson nanofluid present across a horizontal stretching surface. A slit point is assumed in the center from which the surface is drawn in a fluid medium. The flow is assumed under the effect of thermal radiative flux and bioconvection with gyrotactic microorganisms. The Cartesian coordinate x-axis is in the stretchable sheet direction with the velocity u ˆ w = u ˆ 0 ( x ˆ + b ) m whereas y-axis lies in the normal direction of the stretchable sheet surface. Assume that the thickness of the surface is variable according to y ˆ = A ( x ˆ + b ) 1 − m 2 , where A ≥ 0 indicates the constant of the surface and m is the velocity power index. Assume T _w and C _w are the wall temperature and concentration which are higher than the ambient temperature T _∞ and concentration C _∞ far away. The nanofluid is considered to not influence the way of swimming of microorganisms and their velocity. Magnetic field is exerted in y-axis direction of strength B with a low Reynolds number. Figure 1 shows the flow model configuration.

Figure 1

Geometry of microorganism nanofluids model.

The mathematical expressions for mass, momentum, energy, nanofluid, and microorganism concentration are as follows:

The corresponding boundary conditions are:

Roseland approximation for radiation is defined by

where extending the Taylor series at temperature T ˆ ∞ with ignoring higher order of T ˆ ∞ as,

By utilizing Eqs. (7) and (8), we set:

3 Numerical solution of ODE systems

A suitable numerical method is too much necessary for the computational of fluidic problems. Here we use a new iterative technique to solve the ODE systems in Eqs. (11–15).

3.2 Comparison of NIM with HAM

Computational outcomes of Eqs. (11–15) is achieved via a new iterative technique and HAM. Excellent agreement between numerical and analytical solutions reveals the significance mode of NIM. Both methods’ comparison and their residual errors are expressed in Tables 1–4 and Figures 2–5. A falling tendency in average squared residual error is detected for higher-order deformations. Average residual errors for all computational outcomes ranging from 10⁻³ to 10⁻⁶ show that NIM is reliable and asymptotically convergent for fluidic problems.

Table 1

Numerical technique vs analytical solution for velocity profile

η	NIM	HAM	Absolute error
0.0	1.00000	1.00000	1.25608 × 10–6
0.5	0.616065	0.616119	0.0000534592
1.0	0.376676	0.376915	0.000239218
1.5	0.229501	0.229725	0.000223707
2.0	0.139570	0.139729	0.000159905
2.5	0.0847875	0.0848917	0.000104138
3.0	0.0514753	0.0515406	0.0000653048
3.5	0.0312394	0.0312797	0.0000402754
4.0	0.0189543	0.0189789	0.0000246443
4.5	0.0114988	0.0115138	0.0000150203
5.0	0.00697525	0.00698438	9.13554 × 10–6

Table 2

Numerical technique vs analytical solution for temperature profile

η	NIM	HAM	Absolute error
0.0	1.00628	1.00	0.00628196
0.5	0.52617	0.606531	0.0803606
1.0	0.300561	0.367879	0.0673187
1.5	0.178216	0.22313	0.0449137
2.0	0.107204	0.135335	0.0281313
2.5	0.0648315	0.082085	0.0172535
3.0	0.0392825	0.0497871	0.0105046
3.5	0.0238181	0.0301974	0.00637925
4.0	0.014445	0.0183156	0.0038706
4.5	0.00876119	0.011109	0.00234781
5.0	0.00531394	0.00673795	0.00142401

Table 3

Numerical technique vs analytical solution for nanofluid concentration profile

η	NIM	HAM	Absolute error
0.0	1.00268	1.00002	0.002656
0.5	0.599198	0.606542	0.00734398
1.0	0.360821	0.367885	0.00706385
1.5	0.218048	0.223133	0.00508493
2.0	0.131995	0.135337	0.00334216
2.5	0.0799723	0.0820861	0.00211377
3.0	0.0484756	0.0497877	0.00131208
3.5	0.0293913	0.0301978	0.000806442
4.0	0.0178229	0.0183159	0.000492947
4.5	0.0108088	0.0111091	0.00030037
5.0	0.00655534	0.00673803	0.000182688

Table 4

Numerical technique vs analytical solution for nanofluid concentration profile

η	NIM	HAM	Absolute error
0.0	1.00884	0.999987	0.00885519
0.5	0.611377	0.606524	0.00485344
1.0	0.370617	0.367876	0.00274073
1.5	0.224751	0.223129	0.00162255
2.0	0.136317	0.135335	0.000982388
2.5	0.0826835	0.0820848	0.000598777
3.0	0.0501521	0.049787	0.000365143
3.5	0.0304197	0.0301973	0.000222405
4.0	0.0184509	0.0183156	0.000135288
4.5	0.0111912	0.011109	0.0000822121
5.0	0.00678786	0.00673794	0.0000499239

Figure 2

Absolute errors for the velocity, temperature, and concentration profile.

Figure 3

Consequences of magnetic parameter on: (a) velocity profile, (b) temperature gradient, (c) nanofluid concentration, and (d) motile microorganism concentration.

Figure 4

Consequences of velocity power index on: (a) velocity profile, (b) temperature gradient, (c) nanofluid concentration, and (d) motile microorganism concentration.

Figure 5

Consequences of Williamson parameter on: (a) velocity profile and (b) temperature gradient.

4 Results and discussion (interpretation of variations)

Demonstrates characteristics of variations such as magnetic field M, power index velocity m, Williamson parameter λ, wall thickness parameter α, thermal conductivity variation ε, Prandtl number Pr, thermal radiation Rd, Brownian motion Nb, Peclet number Pe, and different concentration parameter Ω against velocity profile f′(η), temperature distribution θ(η), solutal nanoparticles distribution ϕ(η), and concentration of motile microorganism distribution χ(η) of Eqs. (11–14) with boundary conditions, Eq. (15) is analyzed in Figures 3–8. Properties of magnetic field effects M on the velocity, temperature, solutal, and microorganism concentration fields are described in Figure 3. Lorentz force is the force that a magnetic field exerts on a moving electric charge. The fluid viscosity increases due to the Lorentz force characteristic in the magnetic field, thereby reducing the velocity profile and temperature gradient as well, as shown in Figure 3(a and b), whereas Figure 3(c and d) dispatch the strong concentration behavior of nanofluid and microorganisms when the viscosity created by magnetic force increases. As the name shows, velocity power index m speeds up the momentum boundary layer thickness and thermal boundary layer thickness when the values of m vary. This causes the stability in diluted distribution of nanofluid and swimming microorganisms (Figure 4). A non-Newtonian fluid does not obey Newton’s law of viscosity, which states that viscosity should remain constant regardless of stress. In non-Newtonian fluids, viscosity can alter when subjected to force, becoming either more liquid or more solid. When shaken, ketchup, for example, gets runnier, making it a non-Newtonian fluid. Williamson parameter possesses the viscoelastic shear thinning characteristic of non-Newtonian fluid. The higher the non-Newtonian Williamson parameter, the greater the thickness of the boundary layer. In Figure 5, the flow rate and temperature distribution upsurge for a greater value of Williamson quantity. Thermal radiation refers to electromagnetic radiation that generates energy, which is emitted by a heated surface in all dimensions. Enhancing the thermal radiation parameter enhances the energy field but the higher electromagnetic radiation decreased the diluted concentration of Williamson nanofluid (Figure 6). Prandtl quantity is essential for the fluidic problem as it sets the fluid viscosity in correlation with thermal conductivity. Figure 7 displays the opposite impact of Prandtl number against the temperature field, concentration, and swimming microorganism profile. It controls the momentum thickness and thermal boundary layer; hence, the greater the Pr, the lesser the thermal boundary layer thickness. Figure 8(a) shows the Schmidt quantity (in a chemical process, it describes the hydrodynamic impact of the flowing fluid’s inertia and friction forces) against the microorganism concentration profile. It is the ratio of kinematic viscosity to microorganism molecular diffusivity. Elevation in kinematic viscosity (a value that represents a fluid’s dynamic viscosity per unit density) restricts the movement of motile microbes. To slow down this movement, Schmidt parameter is increased. Figure 8(b) shows the decreasing behavior of the density of microorganisms against the Peclet number, which is the study of transportation in the swimming continuum of microorganisms defined as maximum motile microbes’ swimming speed vs microbes’ diffusion rate (the quantity of diffusing molecules inside cell changes throughout time). Microbe’s diffusion is the movement of motile from a highly concentrated area to a lower concentration in the light of gyrotactic.

Figure 6

Consequences of thermal radiation parameter on: (a) velocity profile and (b) temperature gradient.

Figure 7

Consequences of Prandtl number on: (a) temperature gradient, (b) nanofluid concentration, and (c) motile microorganism concentration.

Figure 8

Consequences of (a) Schmidt number, (b) Peclet number on motile microorganism concentration profile.

5 Conclusion

This study makes various aspects of heat transfer and radiation effects on hydromagnetic boundary layer Williamson nanofluid flow via a new iterative technique. With few iterations, the correlation with the HAM is ranging from 10⁻⁶ to 10⁻³, which proves that the proposed technique is reliable, accurate, and rapid convergent for fluidic problems.

Additionally, the effects of the variations are highlighted as:

• Flow rate upsurges for Hall current parameter and wall thickness quantity whereas it slows down for the increase in the range of the magnetic parameter and Williamson quantity.

• Temperature distribution rises for Hall current parameter, wall thickness quantity, thermal conductivity parameter, and thermal radiation quantity but it has opposite fluctuation against the Prandtl number, magnetic force, Williamson quantity, and Lewis number.

• Nanofluidic concentration has direct correlation with magnetic quantity, Hall current quantity, Williamson parameter, and thermal conductivity whereas the behavior of Lewis number and thermal radiation is seen to oppose this profile.

• The Bioconvection microorganism concentration field goes fast in the presence of the magnetic field, Hall current parameter, and wall thickness quantity but it becomes slow for the higher value of Prandtl number, Schmidt number and Peclet number.