Fluid flow and heat transfer investigation of blood with nanoparticles through porous vessels in the presence of magnetic field

Abstract

In this study, the blood flow was investigated involving nanoparticles through porous blood vessels in the presence of magnetic field by means of collocation and least squares techniques. Blood was assumed to be non-Newtonian fluid having nanoparticles in which different models were used to determine the viscosity of the nanofluids. Vogel’s, Reynolds, variable, and constant viscosity models were discussed by using the two aforementioned techniques. Hence, we compared our results with a numerical technique RK-4 and already existing results to show the credibility of our solutions. Further, some physical parameters and their effects are also stated in this research. Increase in the thermophoresis parameter and pressure gradient along with the decrease in the Brownian motion parameter provide a rapid change in the velocity profile, which has been disclosed by results. Additionally, a dramatic change in the velocity has been realized by using Vogel’s model.

Introduction

Blood, which is made of plasma, white and red platelets, blood cells, etc., can be deliberated as a standout amongst the most vital multi-segment blends occurring in nature. Because of the wide range of applications, the study on non-Newtonian fluids is of great significance. Late advances in nanotechnology have prompted another inventive class of heat transfer called nanofluids made by the scattering of nanoparticles (10–50 nm) in usual heat transfer fluids. These fluids are broadly involved in many applications of technological, industrial, engineering, and biological sciences, such as paints, glues, biological solutions, asphalts, melts of polymers, and tars. Due to lesser attention by researchers, this area has not been studied much but, on the other hand, medical application problems regarding non-Newtonian fluids are widely involved. By using Navier–Stokes equations, the blood flow in the cardiovascular system was modeled by Ogulu and Amos ¹ where it was stated that the porosity of the medium and wall shear stress were directly proportional to each other. Kumar et al. ² experimentally examined the influence of gold nanoparticles in blood. Blood is considered as the third-grade non-Newtonian fluid, therefore, by using this assumption Hatami et al. ³ analytically investigated hollow vessel and blood conveying gold nanoparticles in a porous medium. In a tube, blood flow and modeling was studied by Moyers-Gonzalez et al. ⁴ and their work witnessed that the peak values of velocity and frequency of the (constant amplitude) pressure gradient oscillations were inversely proportional to each other. In the presence of magnetic field, Misra et al. ⁵ studied electrically conducting and viscoelastic blood in a channel in which walls are stretched and oscillatory. Mathematical model for the modified second-grade blood fluid was developed by Phuoc and Massoudi ⁶ in which normal stress coefficient and the viscosity were dependent upon the shear rate. They discussed fluid behavior by considering Fahraeus–Lindqvist effect where blood behaves as a non-Newtonian fluid in the core and behaves as a Newtonian fluid near the wall. By considering blood as a third-grade fluid, in Majhi and Nair, ⁷ mathematical model of pulsatile blood flow subject to externally imposed periodic body acceleration was given. As discussed, blood can be considered as a non-Newtonian fluid and so many researchers gave attention to second- and third-grade non-Newtonian fluids. Aziz et al. ⁸ investigated the unsteady flow of an incompressible third-grade fluid over a porous plate within a porous medium. Moreover, work related to nanofluids has a significant role in biosciences and many researchers paid attention to the field of nanofluids. Impact of exterior magnetic field for divergent/convergent channels and models of such type have a vital role in biosciences. Ellahi et al.^9,10 investigated nanofluids of blood. Recently, in the presence of magnetic field, Rahbari et al. ¹¹ studied heat transfer and fluid flow of blood with nanoparticles. Mohyud-Din and Usman ¹² formulated the problem of nanofluids by the use of Buongiorno’s model and successfully obtained its solution. Umar et al. ¹³ numerically investigated the impact of heat transfer and velocity slip boundaries of nanofluids in a channel. In stretchable converging/diverging channels, heat transfer for multihydrodynamic (MHD) nanofluid flow was studied by Mohyud-Din et al. ¹⁴ Axisymmetric squeezing and two-dimensional flow of Cu–kerosene and Cu–water nanofluids and some of their effects were discussed in Khan et al. ¹⁵ Khan et al. ¹⁶ also studied the MHD nanofluids flow and effects of radiation. Later, Umar and Naveed ¹⁷ examined first of its kind Stoke’s problem for nanofluids in the presence of boundary slip condition. Usman et al. ¹⁸ used Adomian’s decomposition method to examine the solution between two parallel disks for squeezing flow and heat transfer. Further, Usman et al. ¹⁹ developed a perturbation solution for MHD incompressible viscous fluid flow. And, Mohyud-Din et al. ²⁰ carried out an investigation on rotating channel for three-dimensional nanofluids squeezing flow, which were stretched by carbon nanotubes. In porous channels, numerical and optimal solutions for MHD flow of radiative micropolar nanofluid and later an analysis about mass and heat transfer for MHD flow of nanofluids was given by Mohyud-Din et al. ²¹ and Khan et al. ²² Here, we are presenting the solution through collocation method (CM) and least squares method (LSM). Least squares technique was used by Eason ²³ for solving partial differential equations. LSM have been implemented for solving singularly perturbed two-point boundary value problems by Evrenosoglu and Somali. ²⁴ Later on, Wu ²⁵ successfully applied the LSM for solving partial differential equations by using Bézier control points. Recently, Hoshyar et al. ²⁶ used the LSM to find the numerical solutions of porous fin in the presence of uniform magnetic field and unsteady motion of vertically falling spherical particles in non-Newtonian fluid. ²⁷ In 1971, Newman ²⁸ developed a technique to analyze stress for crake plates and named it as a method of collocation. Zhu and Atluri ²⁹ carried out modification in element-free Galerkin method and titled it as modified collocation method. Without input data, solution of elliptic partial differential equations via stochastic CM has been discussed by Babuska et al. ³⁰ Recently, Rahimi-Gorji et al. ²⁷ used this technique to investigate the unsteady motion of vertically falling spherical particles in non-Newtonian fluid. In this research, we considered blood as a third-grade non-Newtonian fluid involving nanoparticles. Determination of viscosity has been done by using different viscosity models. A detailed evaluation of results obtained by LSM, CM, and RK-4 has been made to show the efficiency of suggested algorithms. Comparison of results for LSM, CM, and numerical technique (RK-4) has also been asserted graphically. In this work, we also emphasize on the impacts of some physical parameters for instance thermophoresis parameter, temperature, Brownian motion parameter, velocity, pressure gradient, and nanoparticles concentration profiles.

Mathematical analysis of the problem

In the presence of magnetic field, a porous artery comprising non-Newtonian, steady, and incompressible nanofluid has been assumed. Graphical representation and physics of the problem is discussed in Rahbari et al. ¹¹ Here, we study the modeling of blood flow in one dimension since one-dimensional blood flow study is convenient for the analysis of the dynamics along with cardiovascular diseases and arterial hemodynamics. Additionally, it has been understood that the wavelengths of the pressure-flow waves created by heart have a high order of magnitude and, thus, there has been substantial speculation to accept it to be quasi-one-dimensional. Nanofluid density $\rho$ and structure can be modeled ³¹ as

$$\rho =\phi {\rho }_P+{\rho }_{f_0}\left(1-\phi \right)\cong \phi {\rho }_P\left(1-\phi \right)\left\{{\rho }_f\left(1-{\beta }_T\left(\theta -{\theta }_w\right)\right)\right\}$$

(1)

Here, ${\rho }_P,{\theta }_w,{\beta }_T,{\rho }_f,$ and ${\rho }_{f_0}$ are density of nanoparticles, reference temperature, volumetric expansion coefficient, base fluid density at ${\theta }_w$ , and base fluid density, respectively. It is obvious that a viscous fluid is represented by coherence and Navier–Stokes conditions and when the fluid is supposed to be total mass, the conservation of momentum, thermal energy, incompressible, and nanoparticles are as follows

$${\rho }_f\left(\frac{\partial V}{\partial t}+V.\nabla V\right)=\text{div}\left(T\right)-\frac{\mu \varphi }{k}\left(1+{\lambda }_r\frac{\partial }{\partial t}\right)V+\left[\phi {\rho }_P+\left(1-\phi \right)\left\{1-{\beta }_T\left(\theta -{\theta }_w\right)\right\}\right]\text{g}+\text{J}\times \text{B}$$

(2)

$${\left(\rho c\right)}_f\left(\frac{\partial \theta }{\partial t}+V.\nabla \theta \right)=k{\nabla }^2\theta +{\left(\rho c\right)}_P\left[D_b\nabla \phi +\frac{DT}{{\theta }_w}\nabla \theta .\nabla \theta \right]$$

(3)

$$\left(\frac{\partial \phi }{\partial t}+V.\nabla \phi \right)=D_b{\nabla }^2\phi +\frac{D_T}{{\theta }_w}{\nabla }^2\theta$$

(4)

$\varphi$ and $\phi$ are respectively the porosity of the medium and the nanoparticles mass concentration. Since electrically conducting fluids have been considered, by means of OHM’s law ³² the last term of equation (2) can be stated

$$\text{J}=\sigma \left(E+V\times B\right)$$

(5)

In equation (5), $B=B_0+b,E,\hspace{0.17em}J,\hspace{0.17em}V,\sigma$ are the total magnetic field, electric filed, electric current density, velocity vector, and electric conductivity, respectively. Origin for B can be found by using the same equation. Induced magnetic field can be neglected in the presence of small Reynolds number. Hence, the following can be stated ³²

$$J\times B=-\sigma B^{2}V$$

(6)

Following BCs, ⁹ have been used for this study

$$v\left(r\right)=v_0,\hspace{1em}\theta \left(r\right)={\theta }_w\phi \left(r\right)={\phi }_w,\hspace{1em}\text{at}\hspace{0.17em}\hspace{0.17em}{r}_{1}v\left(r\right)=\theta \left(r\right)=\phi \left(r\right)=0,\text{at}{r}_2$$

(7)

Stress in third-grade fluid can be expressed as

$$\begin{array}{c}T=-PI+\mu A_1+{\alpha }_1{A}_2+{\alpha }_2{A}_1^2+{\beta }_1{A}_3\\ +{\beta }_2\left(A_1{A}_2+A_2{A}_1\right)+{\beta }_3\left(tr{A}_1^2\right)A_1\end{array}$$

(8)

In equation (8), ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$ , and ${\beta }_3$ are supposed to be the functions of temperature in general and material modules. Factor $–PI$ shows spherical stress due to resistance of incompressibility and $A_1,A_2,$ and $A_3$ are kinematic tensors that can be expressed as ¹

$$A_1=\nabla V+{\left(\nabla V\right)}^t$$

(9)

$$A_n=\frac{\text{d}}{\text{d}t}{A}_{n-1}+A_{n-1}\left(\nabla V\right)+{\left(\nabla V\right)}^t{A}_{n-1}\text{For}n\text{=}2\text{ and }\text{3}$$

(10)

Locally, at rest the specific Helmholtz energy is minimum for fluid ³³

$$\begin{array}{l}\mu \ge 0,{\alpha }_1\ge 0,\left|{\alpha }_1+{\alpha }_2\right|\le \sqrt{24\mu {\beta }_3},{\beta }_1={\beta }_2=0,\\ {\beta }_3\ge 0\end{array}$$

(11)

Fluid was considered to be thermodynamically compatible by Ellahi et al., ⁹ and thus equation (8) can be summarized as

$$T=-PI+\mu A_1+{\alpha }_1{A}_2+{\alpha }_2{A}_1^2+{\beta }_3\left(tr{A}_1^2\right)A_1$$

(12)

So, equation (12) can be restated as

$$T=-PI+\left[\mu +{\beta }_3\left(tr{A}_1^2\right)\right]A_1+{\alpha }_1{A}_2+{\alpha }_2{A}_1^2$$

(13)

Some dimensionless parameters are presented in the following equations

$$\overline{v}=\frac{v}{V_0},\overline{r}=\frac{r}{R},\overline{\mu }=\frac{\mu }{{\mu }_0}$$

(14)

$$\overline{\theta }=\frac{\theta -{\theta }_w}{{\theta }_m-{\theta }_w},\overline{\phi }=\frac{\phi -{\phi }_w}{{\phi }_m-{\phi }_w}$$

(15)

In equations (14) and (15), $V_0$ is the reference velocity, ${\phi }_m$ is the mass concentration, ${\mu }_0$ is the viscosity, ${\theta }_m$ is the fluid temperature, and ${\theta }_w$ is the pipe temperature. Using equations (14) and (15) in equation (13), our system of equations (2) to (4) can be expressed as the following coupled system ¹⁰

$$N_b\frac{\text{d}\theta }{\text{d}r}\frac{\text{d}\phi }{\text{d}r}+\frac{1}{r}\frac{\text{d}\theta }{\text{d}r}+\alpha \frac{{\text{d}}^2\theta }{\text{d}{r}^2}+{\alpha }_1{N}_t{\left(\frac{\text{d}\theta }{\text{d}r}\right)}^2=0$$

(16)

$$N_b\left(\frac{{\text{d}}^2\theta }{\text{d}{r}^2}+\frac{1}{r}\frac{\text{d}\theta }{\text{d}r}\right)+N_t\left(\frac{{\text{d}}^2\phi }{\text{d}{r}^2}+\frac{1}{r}\frac{\text{d}\phi }{\text{d}r}\right)=0$$

(17)

$$\begin{array}{c}\frac{\text{d}\mu }{\text{d}r}\frac{\text{d}v}{\text{d}r}+\frac{\mu }{r}\frac{\text{d}v}{\text{d}r}+\mu \frac{{\text{d}}^2\theta }{\text{d}{r}^2}+\frac{\text{Λ}}{r}{\left(\frac{\text{d}v}{\text{d}r}\right)}^3+3\text{Λ}{\left(\frac{\text{d}v}{\text{d}r}\right)}^2\frac{{\text{d}}^{2}v}{\text{d}{r}^2}\\ =Pv+M^{2}v+c-G_r\theta -B_r\phi \end{array}$$

(18)

$M$ is MHD and $P$ is porosity parameter

$$v\left(r\right)=\theta \left(r\right)=\phi \left(r\right)=1\text{as}r\to 1$$

(19)

$$v\left(r\right)=\theta \left(r\right)=\phi \left(r\right)=0\text{as}r\to 2$$

(20)

Equations (19) and (20) are boundary conditions for the problem. Equations (16) to (18) are defined by allowing for the parameters given under

$$\text{Λ}=\frac{2{\beta }_3{v}_0^2}{{\mu }_0{R}^2},c=\frac{\frac{\partial p}{\partial z}{R}^2}{v_0{\mu }_0},G_r=\frac{\left({\theta }_m-{\theta }_w\right){\rho }_{f,w}{R}^2(1-{\phi }_w)\text{g}}{v_0{\mu }_0}$$

(21)

$$M^2=\frac{\sigma B_0^2{R}^2}{{\mu }_0},N_t=\frac{D_T({\theta }_m-{\theta }_w)}{{\theta }_w},N_b=D_b\left({\phi }_m-{\phi }_w\right)$$

(22)

$$B_r=\frac{\left({\rho }_P-{\rho }_w\right)R^2\left({\phi }_m-{\phi }_w\right)\text{g}}{v_0{\mu }_0},P=\frac{\phi }{k{R}_2^2}$$

(23)

In equations (21) to (23), $N_b,G_r,N_t,M,B_r,c,\text{Λ}$ are Brownian motion parameter, Grashof number, thermophoresis parameter, magnetic parameter, Brownian diffusion constant, pressure gradient parameter, and third-grade parameter, correspondingly.

Analysis of the problem for medical science

As mentioned earlier, nanofluids have a vital role in biological and medical sciences. Inhibition of growth or stimulation of blood vessels can be done using gold particles. ³ The growth of blood capillaries in certain specific diseases can be increased by consuming drugs but these treatments are effective only for a short period of time. Recently, scientists exposed that nanoparticles could solve some of the problems associated with the administration of drugs. The following models^9–11 are used to determine the viscosity of nanofluids.

Constant: This model can be expressed as $\mu =k,$ where $k=1.$

Variable: In this model, the value of viscosity changes with time. Mathematical expression for this model can be written as $\mu =r.$

Vogel’s: Vogel’s model can be stated as $\mu ={\mu }_0\exp \left(\frac{A}{B+\theta }{\theta }_0\right).$ Viscosity is an exponential function of viscosity and fluid temperature. O is the ambient condition and $A,B$ are constants.

Reynold’s: This model is defined as $\mu =\exp \left(-B\theta \right).$ In this model, viscosity is an exponential function of the total magnetic field and temperature.

Analysis of the applied methods

In this section, we obtain the solution of the above discussed problem by means of two techniques i.e. CM and LSM. In addition, we emphasize on the impact of various viscosity models stated above.

Collocation method

Collocation method, which finds the numerical solutions of the above-stated problem, has the following steps as follows:

Step 1: First consider equations (16) to (20) as $$\begin{array}{l}{N}_b\frac{\text{d}\theta }{\text{d}r}\frac{\text{d}\phi }{\text{d}r}+\frac{1}{r}\frac{\text{d}\theta }{\text{d}r}+\alpha \frac{{\text{d}}^2\theta }{\text{d}{r}^2}+{\alpha }_1{N}_t{\left(\frac{\text{d}\theta }{\text{d}r}\right)}^2=0\\ N_b\left(\frac{{\text{d}}^2\theta }{\text{d}{r}^2}+\frac{1}{r}\frac{\text{d}\theta }{\text{d}r}\right)+N_t\left(\frac{{\text{d}}^2\phi }{\text{d}{r}^2}+\frac{1}{r}\frac{\text{d}\phi }{\text{d}r}\right)=0\\ \frac{\text{d}\mu }{\text{d}r}\frac{\text{d}v}{\text{d}r}+\frac{\mu }{r}\frac{\text{d}v}{\text{d}r}+\mu \frac{{\text{d}}^{2}v}{\text{d}{r}^2}+\frac{\text{Λ}}{r}{\left(\frac{\text{d}v}{\text{d}r}\right)}^3+3\text{Λ}{\left(\frac{\text{d}v}{\text{d}r}\right)}^2\frac{{\text{d}}^{2}v}{\text{d}{r}^2}\\ \hspace{1em}=Pv+M^{2}v+c-G_r\theta -B_r\phi \end{array}$$ Step 2: In this method, the approximate solutions of $\theta \left(r\right)$ by $\tilde{\theta }\left(r\right)$ , $\phi \left(r\right)$ by $\tilde{\phi }\left(r\right)$ , and $v\left(r\right)$ by $\tilde{v}\left(r\right)$ are given as follows

$$\tilde{\theta }\left(r\right)=p_0+p_{1}r+p_2{r}^2+⃛+p_M{r}^M={\displaystyle \sum }_{k=0}^M{p}_k{r}^k$$

(24)

$$\tilde{\phi }\left(r\right)=p_{M+1}+p_{M+2}r+p_{M+3}{r}^2+⃛+p_{2M}{r}^M={\displaystyle \sum }_{k=1}^M{p}_{k+M}{r}^k$$

(25)

$$\tilde{v}\left(r\right)=p_{2M+1}+p_{2M+2}r+p_{2M+3}{r}^2+⃛+p_{3M}{r}^M={\displaystyle \sum }_{k=1}^M{p}_{k+2M}{r}^k$$

(26)

Here, $M$ signifies that the order of approximation for improved solution $M$ should be large. Since in the CM^28–30 the trial solutions (24) to (26) must satisfy the initial conditions, after application the trial solutions (24) to (26) shrink to

$$\tilde{\theta }\left(r\right)=2-r+{\displaystyle \sum }_{k=1}^M{p}_k\left[r^{k+1}-\left(2^{k+1}-1\right)r+\left(2^{k+1}-2\right)\right]$$

(27)

$$\tilde{\phi }\left(r\right)=2-r+{\displaystyle \sum }_{k=1}^M{p}_{k+M}\left[r^{k+1}-\left(2^{k+1}-1\right)r+\left(2^{k+1}-2\right)\right]$$

(28)

$$\tilde{v}\left(r\right)=2-r+{\displaystyle \sum }_{k=1}^M{p}_{k+2M}\left[r^{k+1}-\left(2^{k+1}-1\right)r+\left(2^{k+1}-2\right)\right]$$

(29)

It can be seen that $\tilde{\theta }\left(1\right)=\tilde{\phi }\left(1\right)=\tilde{v}\left(1\right)=1$ and $\tilde{\theta }\left(2\right)=\tilde{\phi }\left(2\right)=\tilde{v}\left(2\right)=0.$

Step 3: Residual of $\theta \left(r\right),\phi \left(r\right),$ and $v\left(r\right)$ obtained by substituting the trial solutions (27) to (29) into equations presented in step 1 can be given as follows

$$R_{\theta }\left(r,p_1,\dots ,p_{3M}\right)=N_b\frac{\text{d}\tilde{\theta }}{\text{d}r}\frac{\text{d}\tilde{\phi }}{\text{d}r}+\frac{1}{r}\frac{\text{d}\tilde{\theta }}{\text{d}r}+\alpha \frac{{\text{d}}^2\tilde{\theta }}{\text{d}{r}^2}+{\alpha }_1{N}_t{\left(\frac{\text{d}\tilde{\theta }}{\text{d}r}\right)}^2\ne 0$$

(30)

$$R_{\phi }\left(r,p_1,\dots ,p_{3M}\right)=N_b\left(\frac{{\text{d}}^2\tilde{\theta }}{\text{d}{r}^2}+\frac{1}{r}\frac{\text{d}\tilde{\theta }}{\text{d}r}\right)+N_t\left(\frac{{\text{d}}^2\tilde{\phi }}{\text{d}{r}^2}+\frac{1}{r}\frac{\text{d}\tilde{\phi }}{\text{d}r}\right)\ne 0$$

(31)

$$R_v\left(r,p_1,\dots ,p_{3M}\right)=\frac{\text{d}\mu }{\text{d}r}\frac{\text{d}\tilde{v}}{\text{d}r}+\frac{\mu }{r}\frac{\text{d}\tilde{v}}{\text{d}r}+\mu \frac{{\text{d}}^2\tilde{v}}{\text{d}{r}^2}+\frac{\text{Λ}}{r}{\left(\frac{\text{d}\tilde{v}}{\text{d}r}\right)}^3+3\text{Λ}{\left(\frac{\text{d}\tilde{v}}{\text{d}r}\right)}^2\frac{{\text{d}}^2\tilde{v}}{\text{d}{r}^2}=P\tilde{v}+M^2\tilde{v}+c-G_r\tilde{\theta }-B_r\tilde{\phi }\ne 0$$

(32)

Step 4: To find the unknowns i.e. $p_i$ ’s, we need to discretize the residuals of temperature, concentration, and velocity over the domain as

$$r_i=a+i\frac{1}{M},i=0,1,2,\dots ,M$$

Step 5: Solving the system of nonlinear algebraic equations obtained in step 4 and setting into equations (24) to (26) we obtain the approximate solutions of equations (16) to (18).

Least squares method

Step 4: In the least squares method we have to minimize

$$E_{\theta }={\int }_{\text{Γ}}{R}_{\theta }^2\left(r,p_1,\dots ,p_{3M}\right)\text{d}r{E}_{\phi }={\int }_{\text{Γ}}{R}_{\phi }^2\left(r,p_1,\dots ,p_{3M}\right)\text{d}r{E}_v={\int }_{\text{Γ}}{R}_v^2\left(r,p_1,\dots ,p_{3M}\right)\text{d}r$$

where $\text{Γ}$ represents the domain of the problem. In order to attain a minimum of functions $E_v,E_{\theta },$ and $E_{\phi }$ , putting derivatives of the functions w.r.t $p_i$ ’s equal zero. That is

$$\frac{\partial E_{\theta }}{\partial p_i}=2{{\displaystyle \int }}^{\text{}}{R}_{\theta }(r)\frac{\partial R_{\theta }}{\partial p_i}\text{d}r=0,i=1,2,3,\dots ,M$$

(33)

$$\frac{\partial E_{\phi }}{\partial p_i}=2{{\displaystyle \int }}^{\text{}}{R}_{\phi }(r)\frac{\partial R_{\phi }}{\partial p_i}\text{d}r=0,i=M+1,M+2,\dots ,2M$$

(34)

$$\frac{\partial E_F}{\partial p_i}=2{{\displaystyle \int }}^{\text{}}{R}_F(r)\frac{\partial R_F}{\partial p_i}\text{d}r=0,i=2M+1,2M+2,\dots ,3M$$

(35)

Nonetheless, the “2” coefficient can be dropped.

Step 5: Solving the above equations (33) to (35) for $p_i$ ’s and then putting back into the trial solutions we attain the estimated solutions of problems (16) to (18).

The solution obtained from CM for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1$ and for $M=12$ can be given as

$$\begin{array}{l}\tilde{\theta }\left(r\right)=1.3685876929-0.58058005979r\\ +2.711133429r^2-8.11392868230r^3\\ +14.73130402486r^4-19.67694042269r^5\\ +19.0820211212389r^6-13.3414462498r^7\\ +6.7362569683406r^8-2.444166456438r^9\\ +0.624057698072r^{10}-0.106810655492r^{11}\\ +0.011032694286r^{12}-0.00052110215533696r^{13}\end{array}$$

$$\begin{array}{l}\tilde{\phi }\left(r\right)=8.6787010612-24.742212113r\\ +48.3964019188r^2-75.64220576321r^3\\ +90.1939863699780r^4-80.9298622261405r^5\\ +55.10087668425312r^6-28.7140122433610r^7\\ +11.457224658735665r^8-3.45901627806015r^9\\ +0.7677146591605r^{10}-0.11844748910497r^{11}\\ +0.0113610685008r^{12}-0.000510307357337r^{13}\end{array}$$

$$\begin{array}{l}\tilde{v}\left(r\right)=2.99452374789906-3.5467541102674r\\ +3.833471532278367r^2-5.424589064457997r^3\\ +6.50519377054658r^4-6.1229250872581r^5\\ +4.49366364695392r^6-2.556614617283r^7\\ +1.1160753050992088r^8-0.36681430205858r^9\end{array}\begin{array}{l}+0.087838015073774r^{10}-0.0144642629369875r^{11}\\ +0.0014639879466256r^{12}\\ -0.000068561534593895568r^{13}\end{array}$$

Results and discussion

LSM and CM are successfully applied for the analytical solutions of the above presented problem for constant viscosity, variable viscosity, Reynold’s, and Vogel’s models.

In this section, we emphasize on the inspiration of the physical parameters i.e. third grade, pressure gradient, Brownian diffusion, magnetic, thermophoresis, Grashof, ${\alpha }_1$ , $\alpha$ , and Brownian motion parameters on the velocity, temperature, and concentration profile for constant viscosity case. Distinction in the temperature profile under the impact of Brownian motion parameter, thermophoresis parameter, $\alpha ,$ and ${\alpha }_1$ is discussed in Figures 1 to 4. The behavior of the temperature profile by varying $Nb$ is presented in Figure 1. Here, we see that as the Brownian parameters increase the values of the temperature distribution increase. The identical behavior of the temperature profile is examined in Figure 2 by increasing the thermophoresis parameter. Figure 3 illustrates the impact of $\alpha$ on the temperature profile. It is clear that $\theta \left(r\right)$ is a decreasing function in $\alpha$ . The reverse behavior of the temperature profile under the effect of ${\alpha }_1$ can be seen in Figure 4.

Figure 1.

Impact of $Nb$ on $\theta$ for $Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 2.

Impact of $Nt$ on $\theta$ for $Nb=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 3.

Impact of $\alpha$ on $\theta$ for $Nb=Nt=P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 4.

Impact of ${\alpha }_1$ on $\theta$ for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figures 5 to 8 demonstrate the impact of $Nb,Nt,\alpha ,$ and ${\alpha }_1$ on the concentration of the nanoparticles. Unlike Figure 1, the concentration profile decreases with the increase in the values of the Brownian parameters as presented in Figure 5. But similar behavior of the thermophoresis parameter is obtained in Figure 6, like in Figure 2. The concentration profile increases with the increase in $Nt$ . In Figure 7, reverse behavior obtained for $\alpha$ on the concentration profile is as seen in Figure 3. Similar behavior of ${\alpha }_1$ on $\phi$ is found in Figure 8 as has been seen for $\theta \left(r\right)$ .

Figure 5.

Impact of $Nb$ on $\phi$ for $Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 6.

Impact of $Nt$ on $\phi$ for $Nb=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 7.

Impact of $\alpha$ on $\phi$ for $Nb=Nt=P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 8.

Impact of ${\alpha }_1$ on $\phi$ for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

The concentration of nanoparticles increases with the increase in the values of ${\alpha }_1$ . A detailed study of how the velocity profile changes by varying the parameters $Nb,Nt,\text{ Λ},M,P,c$ , and $G_r$ is presented in Figures 9 to 15.

There is a progressive decrease in the velocity profile with the increase of $Nb$ and decrease of $Nt,$ which is shown in Figures 9 and 10. Figure 11 represents the impact of the third-grade parameter on $v\left(r\right)$ . It is obvious that an expansion of the $\text{Λ}$ number adds to an expansion of $v\left(r\right)$ .

Figure 9.

Impact of $Nb$ on $v$ for $Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 10.

Impact of $Nt$ on $v$ for $Nb=\alpha =P=M=c=\text{Λ}=G_r=B_r=1.$

Figure 11.

Impact of $\text{Λ}$ on $v$ for $Nb=Nt=\alpha =P=M=c=G_r=B_r=1.$

Variation of $v\left(r\right)$ due to the impact of $M$ is shown in Figure 12. It is observed that the velocity decreases at higher estimations of $M$ . The root cause of this pattern is that when $M$ is connected, the stream of blood is restricted as a consequence of the Lorentz force. In this way, the magnetic field presentation backs off the blood speed. Figure 13 shows that by expanding the estimations of porosity parameter, $v\left(r\right)$ is very little affected. Because at the point when $M$ y builds, liquid can, without much of a stretch, move between the pores furthermore, therefore there is an increase in $v\left(r\right)$ . Figure 14 represents the behavior of the velocity profile lessen the values of the pressure gradient that increase the velocity of blood in arteries.

Figure 12.

Impact of $M$ on $v$ for $Nb=Nt=\alpha =P=c=\text{Λ}=G_r=B_r=1.$

Figure 13.

Impact of $P$ on $v$ for $Nb=Nt=\alpha =M=c=\text{Λ}=G_r=B_r=1.$

Figure 14.

Impact of $c$ on $v$ for $Nb=Nt=\alpha =P=M=\text{Λ}=G_r=B_r=1.$

The increment in the Grashof number results in the increase in the velocity profile as shown in Figure 15. Comparison between the CM and RK-4 method is emphasized in Tables 1 and 2 for the velocity, temperature, and concentration profile for constant viscosity model and variable viscosity model, respectively.

Figure 15.

Impact of $Gr$ on $v$ for $Nb=Nt=\alpha =P=M=c=\text{Λ}=B_r=1.$

Table 1.

Comparison of the collocation method (CM) with Runge–Kutta method of order four for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,{\alpha }_1=0.05$ for constant viscosity model.

$r$	Solution $f\left(\eta \right)$		Solution $\theta \left(\eta \right)$		Solution $\phi \left(\eta \right)$
	CM	Numerical	CM	Numerical	CM	Numerical
1.0	1.000000000	1.000000000	1.000000000	1.000000000	1.000000000	1.000000000
1.1	0.918511301	0.918511300	0.806481652	0.806481652	0.871765980	0.871765980
1.2	0.829650077	0.829650077	0.644281111	0.644281111	0.750453061	0.750453061
1.3	0.734758110	0.734758110	0.508218643	0.508218644	0.635675851	0.635675851
1.4	0.635175417	0.635175416	0.393970929	0.393970930	0.527146103	0.527146103
1.5	0.532163357	0.532163357	0.297911641	0.297911642	0.424654562	0.424654562
1.6	0.426855790	0.426855790	0.217000400	0.217000400	0.328059900	0.328059900
1.7	0.320234781	0.320234781	0.148695727	0.148695726	0.237282933	0.237282933
1.8	0.213125016	0.213125015	0.090881171	0.090881171	0.152305275	0.152305274
1.9	0.106200747	0.106200745	0.041800416	0.041800417	0.073172188	0.073172188
2.0	0.000000000	0.000000000	0.000000000	0.000000000	0.000000000	0.000000000

Table 2.

Comparison of the collocation method (CM) with Runge–Kutta method of order four for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,{\alpha }_1=0.05$ for variable viscosity model.

$r$	Solution $f\left(\eta \right)$		Solution $\theta \left(\eta \right)$		Solution $\phi \left(\eta \right)$
	CM	Numerical	CM	Numerical	CM	Numerical
1.0	1.000000000	1.000000000	1.000000000	1.000000000	1.000000000	1.000000000
1.1	0.918511301	0.918511300	0.806481652	0.806481652	0.862128267	0.862128268
1.2	0.829650077	0.829650077	0.644281111	0.644281111	0.733314352	0.733314352
1.3	0.734758110	0.734758110	0.508218643	0.508218644	0.613210566	0.613210566
1.4	0.635175417	0.635175416	0.393970929	0.393970930	0.501553892	0.501553893
1.5	0.532163357	0.532163357	0.297911641	0.297911642	0.398142831	0.398142832
1.6	0.426855790	0.426855790	0.217000400	0.217000400	0.302819606	0.302819606
1.7	0.320234781	0.320234781	0.148695727	0.148695726	0.215455477	0.215455477
1.8	0.213125016	0.213125015	0.090881171	0.090881172	0.135937764	0.135937764
1.9	0.106200747	0.106200745	0.041800416	0.041800417	0.064157846	0.064157846
2.0	0.000000000	0.000000000	0.000000000	0.000000000	0.000000000	0.000000000

The results obtained using the CM agree well with the RK-4 results. Moreover, the error growth in LSM and CM for $\theta \left(r\right),\phi \left(r\right),$ and $v\left(r\right)$ by utilizing constant viscosity, variable viscosity, Reynold’s, and Vogel’s model is presented in Tables 3 to 6, respectively. It is detected that $\left\{\left|\theta -\tilde{\theta }\right|,\left|\phi -\tilde{\phi }\right|,\left|v-\tilde{v}\right|\right\}\to 0$ as the order of the approximation approaches infinity i.e. $M\to \infty$ .

Table 3.

Error analysis in results obtained from least squares method (LSM) and collocation method (CM) for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,{\alpha }_1=0.05$ for constant viscosity model.

$r$	$\left|\theta -\tilde{\theta }\right|$		$\left|\phi -\tilde{\phi }\right|$		$\left|v-\tilde{v}\right|$
	CM	LSM	CM	LSM	CM	LSM
1.0	0	0	0	0	0	0
1.1	2.214E-10	1.586E-10	1.238E-10	2.369E-09	8.842E-11	1.450E-10
1.2	4.597E-10	1.040E-10	4.263E-10	9.541E-10	9.160E-11	7.542E-11
1.3	7.850E-10	4.586E-10	6.754E-10	3.456E-10	3.822E-11	3.620E-11
1.4	1.004E-09	2.336E-10	8.818E-10	2.258E-10	6.274E-12	7.455E-11
1.5	2.647E-10	6.458E-10	3.414E-10	3.267E-10	1.866E-11	6.522E-11
1.6	3.592E-10	9.900E-10	3.397E-10	7.458E-10	9.253E-11	1.247E-11
1.7	5.037E-10	1.004E-10	3.443E-10	1.258E-10	1.165E-10	1.254E-11
1.8	9.009E-10	1.556E-09	5.714E-10	3.951E-09	1.689E-10	1.236E-10
1.9	1.992E-09	3.558E-09	1.295E-09	3.658E-09	2.862E-10	7.152E-10
2.0	0	0	0	0	0	0

Table 4.

Error analysis in results obtained from least squares method (LSM) and collocation method (CM) for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,{\alpha }_1=0.05$ for variable viscosity model.

$r$	$\left|\theta -\tilde{\theta }\right|$		$\left|\phi -\tilde{\phi }\right|$		$\left|v-\tilde{v}\right|$
	CM	LSM	CM	LSM	CM	LSM
1.0	0	0	0	0	0	0
1.1	2.277E-10	3.587E-10	1.283E-10	3.258E-10	1.941E-10	3.654E-10
1.2	4.716E-10	2.658E-10	4.356E-10	4.658E-10	3.129E-10	1.005E-10
1.3	7.987E-10	1.001E-10	6.872E-10	3.698E-10	3.696E-10	3.654E-10
1.4	1.029E-09	6.014E-09	9.011E-10	3.001E-10	4.440E-10	3.222E-10
1.5	2.961E-10	3.256E-10	3.682E-10	6.570E-10	5.330E-10	1.047E-10
1.6	3.890E-10	1.024E-10	3.638E-10	3.012E-10	5.111E-10	6.542E-10
1.7	4.565E-10	3.250E-10	3.070E-10	4.566E-10	5.062E-10	3.674E-10
1.8	9.286E-10	1.002E-10	5.904E-10	1.023E-10	2.760E-10	5.001E-10
1.9	2.055E-09	6.255E-09	1.340E-09	6.397E-09	1.424E-10	7.889E-10
2.0	0	0	0	0	0	0

Table 5.

Error analysis in results obtained from least squares method (LSM) and collocation method (CM) for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,{\alpha }_1=0.05$ for Reynold’s viscosity model.

$r$	$\left|\theta -\tilde{\theta }\right|$		$\left|\phi -\tilde{\phi }\right|$		$\left|v-\tilde{v}\right|$
	CM	LSM	CM	LSM	CM	LSM
1.0	0	0	0	0	0	0
1.1	2.624E-08	6.427E-07	3.179E-08	5.198E-07	2.761E-09	3.802E-08
1.2	2.484E-08	1.403E-08	2.291E-08	2.185E-08	2.638E-09	6.969E-08
1.3	2.368E-08	2.878E-07	1.672E-08	6.300E-08	2.574E-09	1.216E-08
1.4	2.168E-08	5.593E-07	1.160E-08	1.527E-08	2.415E-09	2.036E-08
1.5	1.822E-08	1.038E-07	8.271E-09	3.318E-08	2.159E-09	3.286E-08
1.6	1.516E-08	1.850E-07	5.251E-09	6.668E-08	1.905E-09	5.141E-08
1.7	1.083E-08	3.184E-08	3.603E-09	1.261E-08	1.550E-09	7.823E-08
1.8	8.528E-09	5.309E-08	9.570E-10	2.272E-08	1.161E-09	1.162E-09
1.9	5.724E-09	8.607E-08	1.184E-09	3.927E-09	8.312E-10	1.687E-09
2.0	0	0	0	0	0	0

Table 6.

Error analysis in results obtained from least squares method (LSM) and collocation method (CM) for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,{\alpha }_1=0.05$ for Vogel’s viscosity model.

$r$	$\left|\theta -\tilde{\theta }\right|$		$\left|\phi -\tilde{\phi }\right|$		$\left|v-\tilde{v}\right|$
	CM	LSM	CM	LSM	CM	LSM
1.0	0	0	0	0	0	0
1.1	6.364E-08	1.304E-07	1.567E-07	5.913E-07	5.499E-07	5.907E-07
1.2	8.203E-07	2.004E-07	1.660E-07	2.372E-06	6.868E-08	2.370E-07
1.3	8.176E-07	2.972E-06	1.233E-08	6.749E-06	6.261E-08	6.744E-07
1.4	7.420E-07	4.278E-06	7.828E-07	1.628E-06	5.225E-07	1.627E-07
1.5	6.327E-07	5.999E-06	5.070E-07	3.534E-06	4.699E-08	3.531E-07
1.6	5.024E-07	8.224E-06	4.143E-07	7.109E-06	5.012E-08	7.101E-07
1.7	3.621E-07	1.105E-06	4.124E-07	1.347E-06	5.884E-08	1.346E-07
1.8	2.268E-07	1.458E-06	3.817E-08	2.433E-07	6.429E-08	2.429E-06
1.9	1.080E-08	1.895E-06	2.400E-08	4.219E-06	5.161E-08	4.212E-07
2.0	0	0	0	0	0	0

Comparison of the already existing results with the proposed algorithms is also shown in Table 7. This comparison table shows that the proposed techniques are well-matched to finding the analytical solutions of the presented problems. In addition, the suggested algorithms can be extended to such type of nonlinear problem arising in biosciences.

Table 7.

Comparison results obtained from least squares method (LSM) and collocation method (CM) with the already existing technique ¹¹ for $Nb=Nt=\alpha =P=M=c=\text{Λ}=G_r=B_r=1,\mu \left(r\right)=1$ .

$r$	$\left|\theta -\tilde{\theta }\right|$			$\left|\phi -\tilde{\phi }\right|$			$\left|v-\tilde{v}\right|$
	CM	LSM	Rahbari et al. 11	CM	LSM	Rahbari et al.11	CM	LSM	Rahbari et al.11
1.0	0	0	0	0	0	0	0	0	0
1.1	6.6E-10	3.1E-09	4.0E-06	2.3E-09	4.0E-09	4.0E-06	4.1E-11	3.0E-10	5.0E-06
1.2	9.7E-10	1.2E-10	6.0E-06	1.5E-09	3.6E-09	1.3E-05	3.6E-11	1.2E-10	5.0E-06
1.3	1.3E-09	5.6E-09	5.0E-06	8.2E-10	7.5E-09	1.0E-05	8.5E-11	6.5E-10	6.0E-06
1.4	1.5E-09	3.6E-09	5.0E-06	2.6E-10	6.8E-09	7.0E-06	1.2E-10	5.2E-09	7.0E-06
1.5	7.7E-10	6.5E-09	4.0E-06	5.0E-10	6.4E-10	7.0E-06	1.2E-10	9.7E-09	4.0E-06
1.6	8.0E-10	7.1E-09	5.0E-06	2.5E-10	2.0E-09	7.0E-06	1.8E-10	6.5E-09	6.0E-06
1.7	1.4E-10	2.0E-10	5.0E-06	7.3E-10	1.0E-10	7.0E-06	1.8E-10	2.2E-10	5.0E-06
1.8	1.1E-09	8.0E-09	6.0E-06	3.5E-10	4.7E-09	8.0E-06	2.1E-10	1.0E-09	5.0E-06
1.9	2.1E-09	1.1E-09	7.0E-06	1.2E-09	9.0E-09	0.0E-00	3.1E-10	2.3E-09	5.0E-06
2.0	0	0	0	0	0	0	0	0	0

Concluded facts

In this research, we assumed blood to be a non-Newtonian third-grade fluid and successfully obtained the solution of the discussed problem by means of collocation and least squares techniques. The following remarks were noted from the study:

By controlling the behavior of nanoparticles we can improve the performance of biological system. This permits the balance of resistant framework connections, cooperation with target cells, thus helping the viable conveyance of payload inside cells or tissues.

Procedure depends on comprehension of the points of interest of blood moving through arteries.

Results obtained via LSM and CM disclose the worth of suggested algorithms for such type of nonlinear problems. Comparison with the numerical results via RK-4 shows the credibility of the proposed techniques. Different parameters and their effects have been illustrated by graphs.

The thermophoresis and Brownian movement perform an essential part on the temperature and nanoparticle focus conveyance.

The projected model gives an understanding into the part of red platelets in the appropriation of nanoparticles inside the permeable vein and can be utilized to recognize the impact of attractive field of various restorative treatments on human body.