Abstract
Heat transfer is a critical function in many technical, industrial, home, and commercial structures. As a result, the purpose of this study is to investigate the effects of slip velocity and variable fluid characteristics on Casson bionanofluid flow across a stretching sheet that has been saturated by gyrotactic microorganisms. The suggested system will be converted to a computationally tractable form using the Galerkin method. The shifted Vieta-Lucas polynomials are then used as basis functions on the provided domain to solve the nonlinear system of ordinary differential equations that has been constructed (ODEs). The results are presented in the form of graphs and tables to assess the impact of the problem’s governing parameters. The estimated solutions produced by using the proposed techniques were physically acceptable and accurate. The current outcomes are confirmed by comparing them to the available literature. It appears that the temperature distribution is enhanced whereas the velocity distribution declines, caused by rising values of the magnetic parameter, slip parameter, and Casson parameter. Also, the local Nusselt number escalates with the strength of the viscosity parameter while the friction drag decays with the same parameter. In addition, the effectiveness and accuracy of the proposed method are satisfied by computing and the residual error function.
1. Introduction
Many technological and industrial operations, such as the manufacture of glass fibers, polymer refining, and the extrusion process of aerodynamics, rely heavily on nanofluid flow through stretched sheets [1, 2]. Nanofluids are also very useful in biomedical research. Cancer therapies, nanocryosurgery, and magnetic resonance imaging are just a few examples [3]. The suspended stability of nanoparticles is commonly found to be greatly increased when gyrotactic bacteria are present. As a result of their numerous practical applications in assorted disciplines of biotechnology and applicable science, the innovative research of nanofluid flow encompassing gyrotactic microorganisms as well as heat (mass) transmission across a stretching surface has gotten a lot of attention from the research community and scientists. Various forms of microorganisms exist, including gravitaxis, oxytocic, chemotaxis, and gyrotactic. Microorganisms have been employed to make a variety of industrial and commercial products, including fertilizers, medicine delivery, and waste-derived biofuel [4]. In recent years, many researchers have written about these remarkable phenomena [5–8]. On a stretched surface, recently, Hosseinzadeh et al. [9] investigated the MHD nanofluid flow in conjunction with thermal radiation, microorganisms, and nanoparticles. Sankad et al. [10] very recently examined a semi-intuitive and numerical solution for the system governing the fluid flow problem containing gyrotactic microorganisms across a linearly expanding sheet.
In fact, most nonlinear ordinary differential equations (ODEs) have no exact solutions, so numerical and approximate techniques are the semiunique way to solve these types of ODEs. One of the most useful tools to simulate the ODEs is the spectral method [11]. The most famous advantage of these methods is their capability to generate accurate outcomes with a very small degree of error of freedom. The orthogonality property of some important polynomials as the shifted Vieta-Lucas polynomials is used to approximate functions on their domain. These polynomials have a main and important role in these methods for ODEs [12]. Many researchers used and implemented these polynomials to solve numerically many problems such as in [13] to solve the 1D and 2D nonlinear generalized Benjamin-Bona-Mahony-Burgers equation, in [14] to solve the 1D and 2D sinh-Gordon equation, and in [15] to get the approximate solution for certain types of fourth-order BVPs. In the same field of study, many kinds of research are also introduced [16].
The major objective of this study is to use the Galerkin method to numerically investigate the results acquired while investigating a non-Newtonian Casson dissipative nanofluid flow containing gyrotactic microorganisms and influenced by a magnetic field and slip velocity phenomenon. The novelty of the proposed method is that it gets a more accurate numerical solution than methods that appear in the literature. For the first time, the Galerkin method and the shifted Vieta-Lucas polynomials are combined to take advantage of the properties of each of them in approximation on one hand and in increasing the applicability and accuracy of the proposed technique on the other hand, which in turn gives good and closest solutions for the real solutions for the problem under study. As a result of applying the proposed technique to the Casson bionanofluid flow caused by a stretched sheet saturated by gyrotactic microorganisms in the presence of slip velocity, the provided study is novel according to the literature review.
2. Description of the Problem
A two-dimensional non-Newtonian Casson bionanofluid saturated by gyrotactic microorganisms is addressed with slip velocity, magnetic field, and viscous dissipation. The fluid flow is created by the stretching of an elastic impermeable sheet at a velocity of . During this examination, the axis is focused on the direction of flow, but the axis is perpendicular to it. Furthermore, the microbe concentration, fluid concentration, and temperature at the sheet surface are represented by , , and (Figure 1).
In the case of varying properties, the equations that regulate the physical model can be written as [10]where the velocity components in the and directions are and , respectively. Likewise, the magnetic field’s strength is denoted by , is the temperature of the nanofluid, is the thermophoretic diffusion coefficient, is the Casson parameter, is the density of the nanofluid at the ambient, and is the diffusivity of microorganisms. Also, is the Brownian diffusion coefficient, is the chemotaxis constant, is the ratio of the nanoparticle’s effective heat capacitance to that of the base fluid, and represents the maximum speed at which a cell can swim. Further, the microbe concentration, fluid concentration, and the temperature all at the ambient of the surface are represented by , and . On the other hand, the following are the pertinent physical boundary conditions:where is the rate of stretching for the sheet, is the velocity slip coefficient, and is the Casson nanofluid viscosity at the ambient. Furthermore, the fluctuation in viscosity, thermal conductivity, and nanoparticle and microorganism diffusivities are represented with the corresponding formulae functions, based on works by Noghrehabadi and Behseresht [17] and Megahed [18].where is the viscosity parameter, is the thermal conductivity parameter, is the mass diffusivity parameter, and is the microorganism diffusivity parameter. Now we will use the dimensionless variables listed below [10]:
To get the nonlinear ordinary differential equations given below, we plug (8)–(11) into (2)–(7).
The converted boundary conditions are as follows:where the parameters present in equations (12)–(17) can be defined as follows: is the magnetic parameter, is the slip velocity parameter, is the Prandtl number, is the thermophoresis parameter, is the Brownian motion parameter, is the Eckert number, is the Schmidt number, is the bioconvection Schmidt number, is the bioconvection Peclet number, and is the bioconvection constant. Further, the local skin friction factor , local Nusselt number , local Sherwood number , and local density number of motile microorganism are all important engineering factors. They are formulated as where is the local Reynolds number.
3. Basic Concepts of the Proposed Methods
Based on Sturm-Liouville’s theory, some special polynomials are created through well-known linear algebra methods. These polynomials are often found to use the power series method to solve the ODEs. It is usually demonstrated that they may be constructed using a generating function, the Rodrigues formula, for each specific polynomial and, ultimately, a contour integral. These polynomials are of great importance in the theory of approximation, mathematical physics, engineering, the theory of mechanical quadrature, and so forth. This section shows a brief description of the most important applications for one special polynomial, namely, shifted Vieta-Lucas polynomials. In addition, we will introduce some concepts about the Galerkin method.
3.1. The Galerkin Method
This subsection introduces the Galerkin method which is based on the approximation concept of the exact solution utilizing a set of basis polynomials. The mathematical theory behind the Galerkin technique proves to be immensely useful in laying the groundwork for both qualitative and quantitative analysis of the ODEs, which have been studied extensively since the mid-nineteenth century. Many researchers used and implemented these polynomials to solve numerically many problems such as in [19] to get the approximate solution for linear second-order ordinary differential equations with mixed boundary conditions, and in [20] they were used to solve the wave propagation problems. To clarify this method, we consider the following operator equation [20]:
is usually a differential operator acting on . The approximate solution for (19) is as follows:where the coordinate function is the approximating function and is a natural number. Note that the weights should satisfy the criteria that the difference (the residual of the approximation which is denoted by ) is minimal (i.e., nonzero):
The parameters ’s in the weighted residual technique are derived by requiring the residual to disappear in the weighted-integral sense; that is:
For more details about the Galerkin method, the reader is urged to examine [19–21].
3.2. Vieta-Lucas and the Shifted Vieta-Lucas Polynomials
In this part of the paper, we are presenting the basic definitions of the Vieta-Lucas and the shifted Vieta-Lucas polynomials and their notations and properties that we will use in our study and they are necessary to reach our goal [22].
We are researching for a class of orthogonal polynomials which lies at the heart of our research. Using the recurrence relations and analytical formula of these polynomials, they can be generated to construct a new family of orthogonal polynomials that will be known as Vieta-Lucas polynomials.
The Vieta-Lucas polynomials of degree are defined by the following relation [22]:
It is easy like other famous functions; we can prove that these Vieta-Lucas polynomials satisfy the following recurrence formula:
Using the transformation , we can generate from the family of Vieta-Lucas polynomials a new class of orthogonal polynomials on the interval , which are the orthogonal family of the shifted Vieta-Lucas polynomials, and it will be denoted by and can be obtained as follows:
The shifted Vieta-Lucas polynomials satisfy the following recurrence relation:where . Also, we find and .
For more details about the shifted Vieta-Lucas polynomials, see [22, 23].
4. The Proposed Numerical Scheme
In this section, we are going to implement the Galerkin method associated with the shifted Vieta-Lucas polynomials to solve numerically the proposed model (12)–(15) with its boundary conditions (16) and (17).
In the Galerkin method, we take a finite space solution’s domain [24].
We will apply the proposed numerical method with the following steps:
Step 1. We use the auxiliary parameter to reduce (12) of the second order. So we can rewrite system (12)–(15) as follows:The associated transformed boundary conditions become
Step 2. The idea behind the proposed method assumes that the approximate solution for (21)–(33) can be provided as a linear combination of shifted the Vieta-Lucas polynomials as follows:
Step 3. The values of the constants (weights) can be given by applying the boundary conditions (32) and (33) and the Galerkin approach to solve system (27)–(31) as follows:This is equivalent to
Step 4. After considerable rearrangement, with the result of substituting (36) with (as a special case to show the method) into the system of equations (36)–(38) and in the associated boundary conditions (32) and (33), we can write the nonlinear equations (36)–(40) in a matrix notation by using the Newton iteration method (NIM) as follows:where , is the vector which represents the nonlinear equations, and is the inverse of the Jacobian matrix [25].
Note that, here in constructing the nonlinear system of equations, we take into account the facts and , in the right hand side of the system of equations (36)–(40).
Step 5. The values of all coefficients are acquired using the direct approach of solving the obtained nonlinear system of equations by the NIM and by substituting these values into formulae (34), the approximate solution for the proposed model (12)–(17) in the specified range is generated.
5. Results and Discussion
Table 1 compares the skin friction values to those previously published. It should be emphasized that the obtained findings are in line with the previously reported outcomes. This lends credibility to the current mathematical formulation’s authenticity and precision, as well as the numerical solutions.
Numerical values are sketched in Figures 2–11 to illustrate the findings, and this part includes a full explanation of the impacts of the pertinent parameters on the velocity, temperature, concentration, and motile bacteria density profiles. The skin friction coefficient, local Nusselt number, local Sherwood number, and local density number of motile microorganisms are also calculated in this part with the same output responses. The implications of the dimensionless velocity , dimensionless temperature , dimensionless concentration , and dimensionless microorganism due to the magnetic number are depicted in Figure 2. Clearly, when the magnetic number is enhanced, the velocity declines rapidly, whereas for dimensionless temperature, dimensionless concentration, and dimensionless microorganisms, the opposite pattern is observed. Physically, when the magnetic number is present, a reduction in the boundary layer thickness and a diminished behavior for the velocity of the nanofluid along the sheet occur. Physically, because the Lorentz force is proportional to the magnetic field, the larger the Lorentz force, the lower the fluid velocity and the higher the temperature.
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Figure 3 conveys the impacts of slip velocity parameter on velocity , temperature , concentration , and dimensionless microorganism distributions, respectively. The dimensionless microorganism, dimensionless temperature, and dimensionless concentration of the Casson nanofluid rise as the slip velocity parameter is increased, whereas the contrary trend occurs for the dimensionless velocity with the same parameter. The resistance force that was created to act towards raising the fluid temperature distribution, but acts against the velocity field, demonstrates the physical importance of slip velocity parameter on the fluid velocity characteristics.
Figure 4 depicts the behavior of the viscosity parameter for velocities, temperature, concentration, and microbe field in Casson nanofluid motion. The viscosity parameter has a strong connection with the shear stress, and as a result, with greater values of the viscosity parameter, velocity increases besides the sheet, while the contrary trend happens away from the sheet.
Furthermore, when the same parameter is improved, the temperature, concentration, and microbe field all increase because a boost in viscosity creates higher resistance in the nanofluid layers which raises the temperature field, as well as concentration and microbe field. To assess the impact of the Casson parameter on the flow and heat mass transfer as well as the microbe field, Figure 5 is depicted. The velocity field drops as the Casson parameter is increased, whereas the temperature, concentration, and microbe fields show the opposite pattern. Physically, when the fluid reduces to the Newtonian type , it is possible to accomplish a robust warming process throughout the thermal boundary layer.
Figure 6 shows the temperature distribution as a function of the Eckert number and the thermal conductivity parameter . It is obvious from this diagram that increasing the value of both parameters enhances the thickness of the thermal boundary layer as well as the temperature distribution. Physically, when the dissipation impact is addressed, the distribution of the temperature is at a higher level due to internal heating related to the dissipation phenomenon. Physically, increasing the Eckert number reduces the temperature difference between the surface and the ambient temperature , improving temperature dispersion. Figure 7 depicts the distributions of dimensionless concentration and dimensionless microorganisms for a horizontal sheet with various thermophoresis parameter values .
As can be observed in this graph, as the thermophoresis parameter is increased, both the dimensionless concentration and the dimensionless microorganisms rise. Figure 8 also depicts the temperature profiles for the thermophoresis and Brownian motion parameters, respectively. When both parameters are increased, the fluid temperature in the boundary layer medium is observed to rise. Physically, as the fluid dynamic viscosity coefficient grows, the magnitude of both the Brownian motion and thermophoresis parameters rises, and therefore the fluid temperature rises.
When the Brownian motion parameter is adjusted for Casson nanofluid flow, Figure 9 shows the variation in concentration function and dimensionless microorganisms function plotted against independent variable . Increasing suppresses both the concentration distribution and the spread of dimensionless microbes.
Figure 10 depicts the behavior of the concentration function and the dimensionless microorganisms function against for the mass diffusivity parameter , respectively. It may be concluded that the mass diffusivity parameter has a greater impact on the concentration function than on the function of dimensionless microorganisms. As can be seen, raising the mass diffusivity parameter resulted in more bacterial spreading and a more even distribution of concentrations.
Figure 11(a) depicts the changing of the dimensionless temperature for various values of the mass diffusivity parameter . For a constant amount of , the dimensionless temperature increases slightly with the mass diffusivity parameter. This occurs since boosting the mass diffusivity parameter causes inductive heat to be generated within the boundary layer, resulting in a higher temperature profile. Figure 11(b), on the other hand, shows how the microbe diffusivity parameter affects the dimensionless microorganisms profile . It is worth noting that the dimensionless microorganisms profile in the Casson nanofluid is heavily influenced by the microorganism diffusivity parameter . We discovered that raising raises the profile of nanofluid bacteria.
To validate the accuracy of the approximate method, we evaluated through Table 2 the residual error function (REF) [27] of the current method with the values of parameters , , with . These values show the thoroughness of the proposed method in this article and confirm that the current method gives better accuracy. In addition, we studied the effect of the order of the approximation by the Galerkin method associated with the shifted Vieta-Lucas polynomials on the skin friction and the Nusselt number through Table 3. From this table, we can get more accurate values of these parameters by increasing the order of the approximation.
The drag force, heat transfer coefficient, mass transfer coefficient, and microorganism transfer coefficient are now represented in nondimensional form as , , , and , respectively, versus various values of the controlling parameters in Table 4. The heat transfer coefficient improves as the magnetic number, slip velocity parameter, and viscosity parameter increase; however the mass transfer coefficient and microorganism transfer coefficient exhibit the reverse trend for the same parameters. Both the thermal conductivity and mass diffusivity parameters appear to have a little effect on the local skin friction coefficient, as well as a gradual decrease in the local Sherwood number, local Nusselt number, and microorganism transfer rate at the wall. Furthermore, it is undeniable that raising the Brownian motion parameter of Eckert number lowers the local Nusselt number, whereas the local skin friction coefficient, the local Sherwood number, and the local microbe transfer rate at the wall show the opposite trend. What is more, in the same table, it is indicated that the local skin friction coefficient, the local Sherwood number, the local Nusselt, and the microorganism transfer coefficient all are rapidly dropping for the increment of the Casson parameter.
6. Conclusions
The nondimensionality governing equations of MHD non-Newtonian Casson dissipative bionanofluid flow saturated by gyrotactic microorganisms with variable transport characteristics and slip velocity phenomenon are solved numerically using the Galerkin technique and shifted Vieta-Lucas polynomials. Under the controller parameters, the temperature function, velocity function, concentration function, and microbe concentration function are evaluated using graphical representations. The efficiency and accuracy of the results were ascertained by comparing the approximate solutions with the solution obtained from the special case of the problem or by evaluating the REF. The numerical simulation work is performed with the aid of the Mathematica software package.
The following are the most crucial decisive points based on the numerical results:(1)As the magnetic parameter, slip velocity parameter, viscosity parameter, and Casson parameter are increased, the dimensionless microorganisms increase; however, as the Brownian motion parameter is increased, the opposite trend is observed.(2)When the magnetic parameter, slip velocity parameter, viscosity parameter, and Eckert number are increased, the temperature function displays an increasing trend.(3)For increasing levels of the magnetic number, slip velocity parameter, Casson parameter, and viscosity parameter, the nanofluid’s velocity collapses.(4)The skin friction factor shrinks as the viscosity, slip velocity, and Casson parameters swell, whereas the magnetic parameter swells in the opposite direction.(5)Nusselt number values rise with cumulative increases in thermal conductivity quantity, slip, and viscosity parameters but fall with increases in magnetic factor and Casson parameter.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through project number IFP-IMSIU202213.