Flow of magnetohydrodynamic blood-based hybrid nanofluids with double diffusion in the presence of Riga plate for heat optimization and drug applications

Abstract

In a recent study, researchers investigated the flow behavior of Casson Hybrid nanofluids (HNFs) combination of single and multi-walled carbon nanotubes (SWCNTs), (MWCNTs) on a Riga plate for drug delivery applications. The study found that the Casson HNFs exhibited non-Newtonian behavior on the Riga plate, with the presence of nanoparticles causing an increase in viscosity and shear-thinning behavior. This rheological behavior is favorable for drug delivery applications as it improves the stability and dispersion of drug particles in the fluid. The similarity equations of the flow problem are easily tackled with the homotopy analysis method (HAM) built on fundamental homotopy mapping. In high-speed flows, Riga actuators are expected to achieve the requirements, since HNF is enhanced by modified Hartmann numbers. As the Eckert number, heat generation/absorption parameter, and thermal relaxation time parameter decrease the temperature, thermal transport increases. Furthermore, with the increments in paramount parameters, the skin friction coefficient and heat transfer rate are remarkably meliorated under higher modified Hartmann number. Furthermore, the study also found that the Casson Hybrid nanofluids showed enhanced heat transfer properties on the Riga plate, which is beneficial for localized drug delivery applications that require precise temperature control.

Introduction

Hybrid nanofluids (HNFs), which are a combination of nanoparticles (NPs) dispersed in a base fluid, play a significant role in drug delivery. These nanofluids have unique properties that make them ideal for delivering drugs to specific target areas in the body. One of the key advantages of using hybrid nanofluids for drug delivery is their ability to improve the solubility and stability of drugs. By dispersing nanoparticles in a base fluid, the surface area available for drug interactions is increased, allowing for enhanced drug solubility.^1,2 This helps to improve the bioavailability of the drug and ensure that it reaches its target site in the body more effectively. Hybrid nanofluids (HNFs) are engineered to have specific properties that make them ideal for targeted drug delivery. By modifying the size, shape, and surface chemistry of the NPs, researchers design nanofluids that selectively target specific tissues or cells in the body. ³ This helps to reduce the side effects of the drug and improve its therapeutic efficacy. HNFs are also used as carriers for drug delivery. By encapsulating drugs within NPs dispersed in a base fluid, researchers protect the drug from degradation and ensure that it is delivered to the target site intact. This helps to improve the stability of the drug and enhance its therapeutic effects.

Casson Hybrid nanofluids, in particular, have shown great potential for drug delivery applications due to their ability to transport drugs through biological tissues efficiently. Alnahdi et al. ⁴ and Arif et al. ⁵ investigated the flow behavior of Casson Hybrid nanofluids on a Riga plate for drug delivery applications.

The Riga plate is a device that is used to influence the flow of hybrid nanofluids. It is constructed with a specific configuration of electrodes and permanent magnets that have a direct impact on the movement and behavior of the nanofluid particles. When the nanofluid flows through the Riga plate, the electrodes and magnets interact with the particles, causing them to align or move in a certain direction. This is used to control and manipulate the flow of the nanofluid, allowing for more precise and efficient processes in various applications. The study found that the Casson Hybrid nanofluids exhibited non-Newtonian behavior on the Riga plate, with the presence of nanoparticles causing an increase in viscosity and shear-thinning behavior. The non-Newtonian fluids used in a variety of industrial and medical applications can be seen in Kocić et al., ⁶ Abbas et al., ⁷ Aslani et al., ⁸ Aslani and Sarris, ⁹ Bejawada and Nandeppanavar, ¹⁰ Wang et al. ¹¹ This rheological behavior is favorable for drug delivery applications as it improves the stability and dispersion of drug particles in the fluid.

Furthermore, the Casson Hybrid nanofluids showed enhanced heat transfer properties on the Riga plate, which could be beneficial for localized drug delivery applications that require precise temperature control.^12–15

(CNTs) have shown significant promise in various applications due to their unique properties. In the context of blood flow and heat transfer in the presence of a magnetic field, CNTs can potentially enhance these processes through their exceptional thermal and magnetic properties. CNTs have the potential and be used as drug carriers in blood flow. Similarly, therapeutics are delivered to specific locations in the body by functionalizing the sidewalls of CNTs with the right drug molecule.^16–21 Heat and mass transfer play a crucial role in drug delivery applications using carbon nanotubes (CNTs) hybrid nanofluids flow. CNTs have garnered significant attention in the field of drug delivery due to their unique properties, such as high surface area, thermal conductivity, and mechanical strength.^22,23 The incorporation of CNTs into nanofluids enhances the mass transfer properties of the fluid, enabling efficient and controlled drug delivery to targeted areas in the body. Heat transfer is essential in drug delivery applications as it affects the rate at which the drug is released from the carrier and delivered to the target site. The thermal conductivity of CNTs helps in improving the temperature distribution within the hybrid nanofluid, ensuring that the drug is released at the desired rate and temperature. Furthermore, heat transfer mechanisms are also utilized to trigger drug release through thermal stimuli, such as ultrasound or magnetic fields.^24,25

Double diffusion hybrid nanofluids flow on a Riga plate refers to a complex flow phenomenon where a mixture of nanoparticles and multiple fluids with different properties interact on a Riga plate.

In this case, the double diffusion hybrid nanofluids flow on the Riga plate involves the interaction of heat transfer, mass transfer, and fluid flow, making it a challenging and interesting research topic. Nayak et al. ²⁶ and Nadeem et al. ²⁷ studying this flow phenomenon aim to understand how the presence of nanoparticles and different fluid properties affect the heat transfer and flow characteristics on the surface of the Riga plate. This knowledge has applications in various fields such as thermal management systems, energy conversion, and heat exchangers. ²⁸

The effects on flow and heat transfer are illustrated graphically and statistically. The current work novelizes the published work of ²⁹ by extending it as follows:

Novelty

The newness of the present work is highlighted as:

• (SWCNTs) and (MWCNTs) Casson hybrid nanofluids are suggested for the applications of drug delivery.

• Riga plate in combination with the magnetic field is assumed for the proposed problem to strengthen the magnetic particles of the blood flow to work as medication.

• Heat and mass transfer properties are studied by using the Cattaneo-Christov thermal flux model, which incorporates double diffusion, instead of the conventional Fourier’s equation.

• The Darcy-Forchheimer medium is considered including the combination of thermophoretic and Brownian motion.

In Mathematica software, the homotopy analysis method (HAM) is used to solve the reduced ordinary differential system (ODS).

Mathematical formulation

We have envisioned simultaneous thermal and mass transport through a steady and incompressible MHD Casson hybrid nanofluid flow induced by a permeable Riga plate. The hybrid nanofluid is fabricated in the context of injecting the nanocomposites, made by hybridization of single and multi-walled carbon nanotubes, into blood. Moreover, the surface is continually set stretching/shrinking with constant velocity $u_w\left(x\right)=\mathrm{ax}$ (a is constant), isothermally heat with temperature $T_w$ , and constant amount of solute $C_w$ . At the ambient region implies the stagnant condition while the temperature and concentration of the surroundings are denoted by $T_{\infty }$ and $C_{\infty }$ , respctively. Despite of that, the consideration of thermally stratified medium a crucial factor since it is assumed that temperature fluctuates adjacent to the surface wall and away from it. The two-dimensional flow confines to the region $y\ge 0$ of $\mathrm{xy}$ Cartesian plane, which is geometrically illustrated in Figure 1.

Figure 1.

The flow in a Cartesian system with planar geometry.

Assumptions of the problem

• The fluid flow is considered steady,

• The fluid is assumed laminar,

• The fluid flow is two-dimensional,

• The external forces are considered negligible,

• The pressure is considered constant such that the pressure gradient is zero.

In the view of,^14,30,31 the governing model is thus given as:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=(1+\frac{1}{\beta }){\upsilon }_{\mathrm{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\pi j_0{M}^{*}}{8{\rho }_{\mathrm{hnf}}}\exp (-\frac{\pi }{a_1}y)}\\ -\frac{{\upsilon }_{\mathrm{hnf}}}{K^{*}}(1+\frac{1}{\beta })u-\frac{1}{{\rho }_{\mathrm{hnf}}}{F}_0{u}^2-\frac{{\sigma }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}{B}_0^{2}u,\hfill \end{array}$$

(2)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+{\tau }_e{\mathrm{\Lambda }}_T={\alpha }_{\mathrm{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\tau [\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2+D_B\left(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}\right)]}\\ +\frac{{\sigma }_{\mathrm{hnf}}}{{\left(\rho c_p\right)}_{\mathrm{hnf}}}{B}_0^2{u}^2+\frac{Q}{{\left(\rho c_p\right)}_{\mathrm{hnf}}}(T-T_{\infty }),\hfill \end{array}$$

(3)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+{\tau }_m{\mathrm{\Lambda }}_C=D_{\mathrm{hnf}}\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+\mathrm{Kr}(C-C_{\infty }),$$

(4)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Lambda }}_T=u\frac{\partial u}{\partial x}\frac{\partial T}{\partial x}+u\frac{\partial v}{\partial x}\frac{\partial T}{\partial y}+v\frac{\partial v}{\partial y}\frac{\partial T}{\partial y}+v\frac{\partial u}{\partial y}\frac{\partial T}{\partial x}+u^2\frac{{\partial }^{2}T}{\partial x^2}}\\ +2\mathrm{uv}\frac{{\partial }^{2}T}{\partial x\partial y}+v^2\frac{{\partial }^{2}T}{\partial y^2},\hfill \\ \multicolumn{1}{c}{{\mathrm{\Lambda }}_C=u\frac{\partial u}{\partial x}\frac{\partial C}{\partial x}+u\frac{\partial v}{\partial x}\frac{\partial C}{\partial y}+v\frac{\partial v}{\partial y}\frac{\partial C}{\partial y}+v\frac{\partial u}{\partial y}\frac{\partial C}{\partial x}+u^2\frac{{\partial }^{2}C}{\partial x^2}}\\ +2\mathrm{uv}\frac{{\partial }^{2}C}{\partial x\partial y}+v^2\frac{{\partial }^{2}C}{\partial y^2},\hfill \end{array}$$

(5)

along with the boundary conditions,

$$\begin{array}{l}u(0)=ax=U_w(x),v(0)=0,T(0)-T_w=0,C(0)-C_w=0,\\ u(\infty )\to 0,T(\infty )\to T_{\infty },\mathrm{C}(\infty )\to C_{\infty }.\end{array}$$

(6)

Here, $(u,v)$ denotes the velocity components along $(x,y)$ . Along with, $T$ , $C$ , $\rho$ , $K$ , $B_0$ , and $Q$ signify the temperature, concentration, density, Darcy permeability, uniform magnetic field strength, and heat source/sink, sequentially. The symbols $\mathrm{\Lambda }$ , $\nu$ , $\alpha$ , $D$ , $F_0$ , and $\tau$ symbolize the corresponding thermophoresis coefficient for diffusion, momentum diffusivity, thermal diffusivity, mass diffusivity, respectively. In addition to, the subscripts T and C denote the heat transfer and mass transfer properties, relatively.^30,31 have provided the thermophysical properties mentioned in Table 1, respectively, while the corresponding statistical data for concerned base fluid and nanomaterials are framed in Table 2 below.

Table 1.

Thermophysical relation for hybrid nanofluids (HNFs), see.^30,31

Property:	Relation:
Absolute viscosity	$\frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f}=\frac{1}{{(1-{\varphi }_{\mathrm{SWCNT}})}^{2.5}{(1-{\varphi }_{\mathrm{MWCNT}})}^{2.5}}$
Density	$\frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f}=(1-{\varphi }_{\mathrm{MWCNT}})[(1-{\varphi }_{\mathrm{SWCNT}})+{\varphi }_{\mathrm{SWCNT}}\left(\frac{{\rho }_{\mathrm{SWCNT}}}{{\rho }_f}\right)]+{\varphi }_{\mathrm{MWCNT}}\left(\frac{{\rho }_{\mathrm{MWCNT}}}{{\rho }_f}\right),$
Heat capacity	$\frac{{\left(\rho \mathrm{cp}\right)}_{\mathrm{hnf}}}{{\left(\rho \mathrm{cp}\right)}_f}=(1-{\varphi }_{\mathrm{MWCNT}})[(1-{\varphi }_{\mathrm{SWCNT}})+{\varphi }_{\mathrm{SWCNT}}\left(\frac{{\left(\rho \mathrm{cp}\right)}_{\mathrm{SWCNT}}}{{\left(\rho \mathrm{cp}\right)}_f}\right)]+{\varphi }_{\mathrm{MWCNT}}\left(\frac{{\left(\rho \mathrm{cp}\right)}_{\mathrm{MWCNT}}}{{\left(\rho \mathrm{cp}\right)}_f}\right),$
Thermal conductivity	$\begin{array}{c}\frac{k_{\mathrm{hnf}}}{k_{\mathrm{nf}}}=\frac{(1-{\varphi }_{\mathrm{MWCNT}})+2{\varphi }_{\mathrm{MWCNT}}\left(\frac{k_{\mathrm{MWCNT}}}{k_{\mathrm{MWCNT}}-k_{\mathrm{nf}}}\right)\ln \left(\frac{k_{\mathrm{MWCNT}}+k_{\mathrm{nf}}}{k_{\mathrm{nf}}}\right)}{(1-{\varphi }_{\mathrm{MWCNT}})+2{\varphi }_{\mathrm{MWCNT}}\left(\frac{k_{\mathrm{nf}}}{k_{\mathrm{MWCNT}}-k_{\mathrm{nf}}}\right)\ln \left(\frac{k_{\mathrm{MWCNT}}+k_{\mathrm{nf}}}{k_{\mathrm{nf}}}\right)},\hfill \\ \frac{k_{\mathrm{nf}}}{k_f}=\frac{(1-{\varphi }_{\mathrm{SWCNT}})+2{\varphi }_{\mathrm{SWCNT}}\left(\frac{k_{\mathrm{SWCNT}}}{k_{\mathrm{SWCNT}}-k_f}\right)\ln \left(\frac{k_{\mathrm{SWCNT}}+k_f}{k_f}\right)}{(1-{\varphi }_{\mathrm{SWCNT}})+2{\varphi }_{\mathrm{SWCNT}}\left(\frac{k_f}{k_{\mathrm{SWCNT}}-k_f}\right)\ln \left(\frac{k_{\mathrm{SWCNT}}+k_f}{k_f}\right)},\hfill \end{array}$
Electric conductivity	$\begin{array}{c}\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_f}=\frac{{\sigma }_{\mathrm{MWCNT}}+2{\sigma }_{\mathrm{nf}}-2({\sigma }_{\mathrm{nf}}-{\sigma }_{\mathrm{MWCNT}}){\varphi }_{\mathrm{MWCNT}}}{{\sigma }_{\mathrm{MWCNT}}+2{\sigma }_{\mathrm{nf}}+({\sigma }_{\mathrm{nf}}-{\sigma }_{\mathrm{MWCNT}}){\varphi }_{\mathrm{MWCNT}}},\hfill \\ \frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=\frac{{\sigma }_{\mathrm{SWCNT}}+2{\sigma }_f-2({\sigma }_f-{\sigma }_{\mathrm{SWCNT}}){\varphi }_{\mathrm{SWCNT}}}{{\sigma }_{\mathrm{SWCNT}}+2{\sigma }_f+({\sigma }_f-{\sigma }_{\mathrm{SWCNT}}){\varphi }_{\mathrm{SWCNT}}},\hfill \end{array}$

Table 2.

The experimental values of SWCNTs, MWCNTs, and Blood.^1,2,25

	Density	Heat capacity	Thermal conductivity	Electrical conductivity
Material	$\rho (\mathrm{kg}/m^3)$	$\mathrm{cp}(j/\mathrm{kgK})$	$k(W/\mathrm{mK})$	$\sigma (S/m)$
Blood	1053	3594	0.492	$5.5\times {10}^{-6}$
SWCNTs	2600	425	3.7	$1.1\times {10}^7$
MWCNTs	1600	796	3000	$59.6\times {10}^6$

In Table 1, ${\varphi }_1$ and ${\varphi }_2$ imply the concentration of SWCNTs and MWCNTs, accordingly. Besides, the subscripts nf and hnf are used to show the corresponding properties of NF and HNF, respectively. Introducing the similarity variables,

$$\begin{array}{c}\eta =y\sqrt{\frac{a}{{\upsilon }_f}},u=\mathrm{axf}\prime \left(\eta \right),v=-\sqrt{a{\upsilon }_f}f\left(\eta \right),\hfill \\ \mathrm{\Theta }\left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\phi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\hfill \end{array}$$

(7)

Equations (1)−(7) transform as:

$$\begin{array}{c}\frac{{\upsilon }_{\mathrm{hnf}}}{{\upsilon }_f}(1+\frac{1}{\beta })f″\prime \left(\eta \right)+f\left(\eta \right)f″\left(\eta \right)-f\prime {\left(\eta \right)}^2-\mathrm{Fr}\frac{{\rho }_f}{{\rho }_{\mathrm{hnf}}}f\prime {\left(\eta \right)}^2\hfill \\ -\lambda \frac{{\nu }_{\mathrm{hnf}}}{{\nu }_f}(1+\frac{1}{\beta })f\prime \left(\eta \right)-\frac{{\sigma }_{\mathrm{hnf}}/{\sigma }_f}{{\rho }_{\mathrm{hnf}}/{\rho }_f}\mathrm{Mf}\prime \left(\eta \right)+\frac{{\rho }_f}{{\rho }_{\mathrm{hnf}}}\alpha e^{(-B\eta )}=0,\hfill \end{array}$$

(8)

$$\begin{array}{c}\frac{{\alpha }_{\mathrm{hnf}}}{{\alpha }_f}\frac{1}{\mathrm{Pr}}\mathrm{\Theta }″\left(\eta \right)+f\left(\eta \right)\mathrm{\Theta }\prime \left(\eta \right)+\frac{{\sigma }_{\mathrm{hnf}}/{\sigma }_f}{{\left(\rho c_p\right)}_{\mathrm{hnf}}/{\left(\rho c_p\right)}_f}\mathrm{EcMf}\prime {\left(\eta \right)}^2\hfill \\ +\frac{{\left(\rho c_p\right)}_f}{{\left(\rho c_p\right)}_{\mathrm{hnf}}}\chi \mathrm{\Theta }\left(\eta \right)+\mathrm{Nt}\mathrm{\Theta }\prime {\left(\eta \right)}^2+\mathrm{Nb}\mathrm{\Theta }\prime \left(\eta \right)\phi \prime \left(\eta \right)-{\gamma }_e{\chi }_T=0,\hfill \end{array}$$

(9)

$$\begin{array}{c}\frac{(1-{\varphi }_1)(1-{\varphi }_2)}{\mathrm{Sc}}\phi ″\left(\eta \right)+f\left(\eta \right)\phi \prime \left(\eta \right)+\frac{\mathrm{Nt}}{\mathrm{Nb}}\mathrm{\Theta }″\left(\eta \right)\hfill \\ -k_1\phi \left(\eta \right)-{\gamma }_m{\chi }_C=0,\hfill \end{array}$$

(10)

where

$$\{\begin{array}{c}{\chi }_T=\frac{{\left(\rho c_p\right)}_f}{{\left(\rho c_p\right)}_{\mathrm{hnf}}}\chi \mathrm{\Theta }\prime \left(\eta \right)-\frac{{\sigma }_{\mathrm{hnf}}/{\sigma }_f}{{\left(\rho c_p\right)}_{\mathrm{hnf}}/{\left(\rho c_p\right)}_f}\mathrm{EcM}(f\prime {\left(\eta \right)}^2+f\left(\eta \right)f″\left(\eta \right))\hfill \\ +f\left(\eta \right)f\prime \left(\eta \right)\mathrm{\Theta }\prime \left(\eta \right)+f{\left(\eta \right)}^2\mathrm{\Theta }″\left(\eta \right),\hfill \\ {\chi }_C=f\left(\eta \right)f\prime \left(\eta \right)\phi \prime \left(\eta \right)+f{\left(\eta \right)}^2\phi ″\left(\eta \right)-k_{1}f\left(\eta \right)\phi \prime \left(\eta \right)f\left(\eta \right)f\prime \left(\eta \right)\phi \prime \left(\eta \right)\hfill \\ +f{\left(\eta \right)}^2\phi ″\left(\eta \right)-k_{1}f\left(\eta \right)\phi \prime \left(\eta \right),\hfill \end{array}$$

(11)

and

$$\begin{array}{c}f\prime \left(0\right)=1,f\left(0\right)=0,f\prime \left(\infty \right)=0,\mathrm{\Theta }\left(0\right)=1,\mathrm{\Theta }\left(\infty \right)=0,\hfill \\ \phi \left(0\right)=1,\phi \left(\infty \right)=0.\hfill \end{array}$$

(12)

After transformation the obtained dimensionless parameters are the Forchheimer number ( $\mathrm{Fr}$ ), porosity parameter ( $\lambda$ ), magnetic parameter ( $M$ ), modified Hatmann number ( $\alpha$ ), Casson fluid parameter ( $\beta$ ), Prandtl number (Pr), Eckert number ( $\mathrm{Ec}$ ), heat generation/absorption parameter ( $\chi$ ), thermal relaxation time parameter ( ${\gamma }_e$ ), concentration relaxation time parameter ( ${\gamma }_m$ ), thermophoresis parameter ( $Nt$ ), Brownian diffusivity parameter ( $\mathrm{Nb}$ ), Schimdt number ( $Sc$ ), and chemical reaction parameter ( $k_1$ ), which are defined as:

$$\{\begin{array}{c}\mathrm{Fr}=\frac{x{F}_0}{{\rho }_f},\lambda =\frac{{\nu }_f}{a{k}^{*}},M=\frac{{\sigma }_f{B}_0^2}{{\rho }_{f}a},\alpha =\frac{j_0{M}^{*}\pi }{8a^{2}x{\rho }_f},\beta =\frac{\pi }{a_1}\sqrt{\frac{{\upsilon }_f}{a}},\hfill \\ \mathrm{Pr}=\frac{{\upsilon }_f}{{\alpha }_f},\mathrm{Ec}=\frac{a^2{x}^2}{{\left(c_p\right)}_f(T_w-T_{\infty })},\chi =\frac{Q}{{\left(\rho c_p\right)}_{f}a},{\gamma }_e=a{\tau }_e,{\gamma }_m=a{\tau }_m,\hfill \\ \mathrm{Nt}=\frac{\tau D_T(T_w-T_{\infty })}{{\upsilon }_f{T}_{\infty }},\mathrm{Nb}=\frac{\tau D_B(C_w-C_{\infty })}{{\upsilon }_f},\mathrm{Sc}=\frac{{\upsilon }_f}{D_B},k_1=\frac{\mathrm{kr}}{a},\hfill \end{array}$$

(13)

respectively.

Besides, the physical quantities associated with the behaviors of drag force, heat transfer, and mass transfer are skin friction coefficient ( $C_f$ ), Nusselt number ( $\mathrm{Nu}$ ), and Sherwood number ( $\mathrm{Sh}$ ). With the use of equation (7), the above-mentioned physical quantities shape as (see^31,32):

$$\begin{array}{c}{\mathrm{Re}}_x^{1/2}{C}_f=\frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f}f″\left(0\right),{\mathrm{Re}}_x^{-1/2}\mathrm{Nu}=-\frac{k_{\mathrm{hnf}}}{k_f}\mathrm{\Theta }\prime \left(0\right),\hfill \\ {\mathrm{Re}}_x^{-1/2}\mathrm{Sh}=\phi \prime \left(0\right)\hfill \end{array}$$

(14)

where ${\mathrm{Re}}_x$ is the local Reynolds number.

Results and discussions

The unique properties of CNTs have made them highly promising in various applications. CNTs have exceptional thermal and magnetic properties that can potentially enhance blood flow and heat transfer in the presence of a magnetic field. For example, therapeutics are delivered to specific locations in the body by functionalizing the sidewalls of CNTs with the appropriate drug molecules. The bloodstream’s movement is guided and controlled by the magnetic properties of CNTs. So, in view of the above importance of CNTs the parameter discussion is pointed out as.

Through the use of schematic plot, this section presents the fluctuations in dimensionless outlook of various quantities based on the variations in sundry parameters. The impacts on non-dimensional velocity field ( $f\prime \left(\eta \right)$ ), temperature field ( $\mathrm{\Theta }\left(\eta \right)$ ), and concentration field ( $\phi \left(\eta \right)$ ) of the hybrid nanofluid (SWCNTs + MWCNTs/Blood) with respect to the concerned parameters are described through Figures 2 to 19, accordingly, while in contrast, the impressions regarding the skin friction coefficient ( ${\mathrm{Re}}_x^{1/2}{C}_f$ ), Nusselt number ( ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ ), and Sherwood number ( ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ ) against the influencing parameters are shown by Figures 21 to 35, relatively. Additionally, the sketched plots show the comparison between blood-based HNF and NF as well.

Figure 2.

$\alpha$ versus $f\prime \left(\eta \right)$ .

Figure 3.

$\beta$ versus $f\prime \left(\eta \right)$ .

Figure 4.

$\mathrm{Fr}$ versus $f\prime \left(\eta \right)$ .

Figure 5.

$\lambda$ versus $f\prime \left(\eta \right)$ .

Figure 6.

$M$ versus $f\prime \left(\eta \right)$ .

Figure 7.

$\mathrm{Ec}$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 8.

I $\chi$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 9.

$\mathrm{Nt}$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 10.

$\mathrm{Nb}$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 11.

$\mathrm{Fr}$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 12.

$\lambda$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 13.

$M$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 14.

${\gamma }_e$ versus $\mathrm{\Theta }\left(\eta \right)$ .

Figure 15.

$k_1$ versus $\phi \left(\eta \right)$ .

Figure 16.

$\mathrm{Sc}$ versus $\phi \left(\eta \right)$ .

Figure 17.

$\mathrm{Nt}$ versus $\phi \left(\eta \right)$ .

Figure 18.

$\mathrm{Nb}$ versus $\phi \left(\eta \right)$ .

Figure 19.

${\gamma }_m$ versus $\phi \left(\eta \right)$ .

Velocity profile f′(η)

Figures 2 to 6 demonstrate the crucial influences of parameters $\alpha$ , $\beta$ , $\mathrm{Fr}$ , $\lambda$ , and $M$ on dimensionless velocity $f\prime \left(\eta \right)$ , respectively. Figure 2 illustrates how the increments in modified Hartman number $\alpha$ leads to increasing patterns in term of flow rates $f\prime \left(\eta \right)$ , while the thickness of momentum boundary layer is adversely affected. The modified Hartman number, or mass Hartman number, expresses the relationship between the gravitational forces acting on a dispersed particle and the viscous forces constraining it. The behavior of particles in fluid flows is commonly characterized using this method. CNTs nanofluid derived from blood can be significantly affected by the modified Hartman number. A higher modified Hartman number implies stronger gravitational forces acting on the CNTs. This could lead to increased sedimentation of the CNTs, causing them to settle at the bottom of the fluid. In other words, high intensity in modified Hartman number indicates a strong external electromagnetic field, which is responsible for generating the span-wise Lorentz force to the surface wall that restricts the motion and recommending further momentum losses. Blood flow dynamics: In the context of blood-based nanofluids, the modified Hartman number can affect the flow behavior of blood containing CNTs. Blood is a complex fluid, and the addition of CNTs alters its rheological properties. Understanding the modified Hartman number can provide insights into how the CNTs may interact with blood flow, potentially impacting blood circulation and related physiological processes.

From Figure 3, it is observed that intensified Casson parameter $\beta$ diminishes the fluidity of the media since it starts behaving like shear-thickening non-Newtonian liquid. The Casson parameter is a rheological parameter used to describe non-Newtonian fluids, including nanofluids. It quantifies the yield stress or the resistance of the fluid to start flowing. The flow of blood-based carbon nanotubes (CNTs) nanofluid is influenced by the Casson parameter in terms of the yield stress. A higher Casson parameter would indicate a higher yield stress, meaning that the fluid requires more force to start flowing. This would result in a slower initial flow rate of the CNTs nanofluid.

In terms of the momentum boundary layer thickness, it minimizes with the incremental change in $\beta$ . Figure 4 depicts the slowing down of fluid flow rate against the increasing values of Forchheimer number $\mathrm{Fr}$ . Physically, higher $\mathrm{Fr}$ opposes the motion and cause the drag forces to increase, preventing fluid from moving easily. Accordingly, the velocity field diminishes. Figure 5 narrates that as the porosity parameter $\lambda$ is increased, the fluid faces retardation. Physically the statement can be justified because the higher porosity intend to augment the damping characteristics in the flow. This, thus, results is an improvement in frictional measures of fluid, though decelerates it. In Figure 6, $f\prime \left(\eta \right)$ is observed to decrease against increasing range of the magnetic parameter $M$ . Due to the opposing Lorentz force, a magnetic field tends to reduce the velocity field and create a notable resistance when applied to electrically conducting fluids. Besides, the Casson hybrid nanofluid exhibits higher results as compared to Casson nanofluid when it comes to their velocities. The physical explanation of the fact is that SWCNTs are usually denser than MWCNTs, see Table 2. However, the difference is insignificant on the basis of the results shown in all plots. Furthermore, in regard to the impacts on thickness of the momentum boundary layer (MBL), it is concluded that Figures 2 and 3 that the momentum losses occurring with the each upsurge in the measure of $\alpha$ and $\beta$ are confined to relatively larger region as compared to the previous one, while in case of $\mathrm{Fr}$ , $\lambda$ , and $M$ the viscous layer correspondingly becomes thinner, as demonstrated in Figures 4 to 6.

Temperature profile Θ (η)

Physical significance of $\mathrm{Ec}$ , $\chi$ , $\mathrm{Nt}$ , $\mathrm{Nb}$ , $\mathrm{Fr}$ , $\lambda$ , $M$ , and ${\gamma }_e$ on dimensionless temperature $\mathrm{\Theta }\left(\eta \right)$ can be visualized from Figures 7 to 14, sequentially. Figure 7 implies that the temperature field monotonically increases against rising Eckert number. The Eckert number is a dimensionless number that relates the kinetic energy to the thermal energy of a fluid flow. It is often used to study the heat transfer characteristics of nanofluids, including carbon nanotubes (CNTs) nanofluids.

As the Eckert number increases, the impact on the blood-based CNTs nanofluids can be significant in terms of heat transfer. A higher Eckert number suggests a higher kinetic energy compared to the thermal energy of the fluid flow. This can lead to enhanced heat transfer and better cooling capability in the nanofluid. By enhancing $\mathrm{Ec}$ , the internal friction within the adjacent layers causes the fluid mechanical energy to convert into thermal energy. As a result, the fluid temperature rises. Figure 8 shows the consequences faced by $\mathrm{\Theta }\left(\eta \right)$ associated with the heat generation/absorption parameter $\chi$ . By gradually increasing the heat source parameter, the thickness of the thermal boundary layer increases, which physically indicates that an increase in $\chi$ leads to an increase in heat generated in the boundary layer, which in turn increases the temperature. From Figure 9, it is clear that the temperature fluxes are increasing function of thermophoresis parameter $\mathrm{Nt}$ . According to the graph, $\mathrm{\Theta }\left(\eta \right)$ rises under the action of thermophoretic force, which compels the nanoparticles to diffuse into ambient streams by the virtue of temperature gradients. On the other hand, $\mathrm{\Theta }\left(\eta \right)$ fluctuates in up-and-down (non-monotonic) patterns regarding $\mathrm{Nb}$ , which is successfully illustrated in Figure 10. In terms of physics, Brownian motion describes the disorderly motion of the fluid particles, and an increase in $\mathrm{Nb}$ indicates that the molecular agitations improve, which is why a greater amount of heat is produced. Thus, the caloric range in the vicinity of surface wall enhances as the-said parameter increases. Figures 11 and 12 emphasize on the variations in temporal measure based on the varying values of Forchheimer number $\mathrm{Fr}$ and porosity parameter $\lambda$ , sequentially. Similar to $\mathrm{Nb}$ , the graphs follow upsurge-to-downsurge for increments in either of $\mathrm{Fr}$ or $\lambda$ . However, the enhancement in the case of $\mathrm{Fr}$ are little more intense. In Figure 13, it is found that climbing values of Magnetic field $M$ have intensifying impressions on temperature, which can be taken into the corresponding with Figure 6. Figure 14 frame the physical significance of thermal relaxation parameter ${\gamma }_e$ on the temperature profile. There is a decreasing relationship between ${\gamma }_e$ and $\mathrm{\Theta }\left(\eta \right)$ . The analysis also shows a decline in thermal boundary layer thickness. It is because physical particles require extra time to convert thermal energy to their nearby units when the thermal relaxation parameter is increased. Alternatively, we may say that materials with greater values of relaxation parameter have a non-conducting nature, thereby reducing their temperature distribution. Just like the case of $f\prime \left(\eta \right)$ , same observations are made for temperature distribution as well. It is noticed that the Casson hybrid nanofluid shows a bit intense results in contrast to the Casson nanofluid since the thermal conducting performance of SWCNTs is much higher than the MWCNTs, see Table 2. In the context of thermal boundary layer (TBL) thickness, it can be clearly seen in Figures 7, 9 and 11 to 14 that the temperature gradients define a large region for rising values of $\mathrm{Ec}$ , $\mathrm{Nt}$ , $\mathrm{Fr}$ , $\lambda$ , $M$ , and ${\gamma }_e$ , while the heat transfer is concluded to happen quickly throughout in the case of $\chi$ and $\mathrm{Nb}$ , referred to Figure 8 and Figure 10, relatively.

Concentration profile φ (η)

Despite of dimensionless velocity $f\prime \left(\eta \right)$ and temperature $\mathrm{\Theta }\left(\eta \right)$ , this paper also examines the behaviors of mass transfer mechanism within the concentration boundary layer with respect to the altering $k_1$ , $\mathrm{Sc}$ , $\mathrm{Nt}$ , $\mathrm{Nb}$ , and ${\gamma }_m$ . These impact are accordingly shown in Figures 15 to 19, sequentially. Figure 15 illustrates the reverse relation of chemical reaction parameter $k_1$ with the nanoparticles’ concentration $\phi \left(\eta \right)$ . Chemical reaction parameter value increases with decreasing nanoparticle concentration profiles. There is a significant delay in hybrid nanofluid concentration profile due to a large $k_1$ , which indicates a high chemical conversion rate between the molecules. An increased chemical reaction parameter indicates a high rate of chemical conversion, which causes a significant delay in the profile of $\phi \left(\eta \right)$ . In Figure 16, the Schmidt number $\mathrm{Sc}$ is depicted as having an impact on the concentration factor. In general, concentration profiles tend to diminish with increasing $\mathrm{Sc}$ because of the inverse relationship between the Schmidt number and diffusion coefficient. An increase in nanoparticles’ concentration $\phi \left(\eta \right)$ is reported on the behalf of a cumulative action of thermophoresis parameter $\mathrm{Nt}$ , as shown in Figure 17. Due to the larger impact of the thermophoresis, the nanoparticles move closer to the hot disk on their way to a cold fluid at ambient, resulting in higher penetration depth. On contrary basis, it can be seen in Figure 18 that $\phi \left(\eta \right)$ is a decreasing function of escalating Brownian parameter $\mathrm{Nb}$ . This happens because the rapid Brownian motion causes higher rate of interaction among particles. As a matter of fact, the dependency of concentration relaxation time to the mass relaxation time parameter ${\gamma }_m$ is direct. A sketch of the effect of ${\gamma }_m$ on the concentration field is shown in Figure 19. Greater solute viscosity and $\phi \left(\eta \right)$ level within the boundary layer ensue it closer toward the stretching surface. In the view of all above plots, the influencing nature of $k_1$ , $\mathrm{Sc}$ , $\mathrm{Nt}$ and $\mathrm{Nb}$ are stronger in comparison to ${\gamma }_m$ . Moreover, Figures 15, 16, and 19 accordingly indicate that the increasing measures of $k_1$ , $\mathrm{Sc}$ , and ${\gamma }_m$ cause the curves of concentration boundary layer (CBL) to migrate closer to the surface.

The skin friction coefficient ${\mathrm{Re}}_x^{1/2}{C}_f$ , Nusselt number ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ , and Sherwood number ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$

Among the important characteristics associated with heat and mass transfer through corresponding boundary layers, evolved in viscoelastic fluids, can be depicted by the behaviors of skin friction coefficient ${\mathrm{Re}}_x^{1/2}{C}_f$ , Nusselt number ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ , and Sherwood number ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ with respect to some specified conditions. Related to the preceding physical quantities, the variations regarding ${\mathrm{Re}}_x^{1/2}{C}_f$ , ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ , and ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ are discussed against magnetic parameter $M$ , thermopherosis parameter $\mathrm{Nt}$ , and chemical reaction parameter $k_1$ , accordingly. Figures 20 to 23 elaborates the variations in ${\mathrm{Re}}_x^{1/2}{C}_f$ while increasing values of $\alpha$ , $\mathrm{Fr}$ , $\lambda$ , and ${\varphi }_1,{\varphi }_2$ , respectively. Clearly, ${\mathrm{Re}}_x^{1/2}{C}_f$ increases with $\mathrm{Fr}$ , $\lambda$ , and ${\varphi }_1,{\varphi }_2$ , whereas it decreases with increasing $\alpha$ . On the other hand, Figures 24 to 31 reports the increasing-or-decreasing impression of ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ against proliferating $\alpha$ , $\mathrm{Ec}$ , $\chi$ , $\mathrm{Nb}$ , $\mathrm{Fr}$ , $\lambda$ , $M$ , and ${\gamma }_e$ , sequentially. A significant shoot-up in heat transfer coefficient are observed as opposed to inflaming $\alpha$ , while ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ undergoes some notable shoot-downs with increasing $\mathrm{Ec}$ , $\chi$ , $\mathrm{Nb}$ , $\mathrm{Fr}$ , $\lambda$ , $M$ , and ${\gamma }_e$ . Likewise, the etiquette of ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ are recorded with respect to the vacillating values of $\mathrm{Sc}$ , $\mathrm{Nt}$ , $\mathrm{Nb}$ , and ${\gamma }_m$ , correspondingly. As can be seen, the transport rate of solid nanoparticles pile-ups by reinforcing $\mathrm{Nt}$ , although it reduces for augmentations in $\mathrm{Sc}$ , $\mathrm{Nb}$ , and ${\gamma }_m$ . Besides, the plots in Figures 20 to 35 also display that the hybrid nanofluid exhibits higher skin friction coefficients, heat transfer rate, and nanoparticles transport rate at the surface wall, as compared to nanofluid.

Figure 20.

$M$ and $\alpha$ versus ${\mathrm{Re}}_x^{1/2}{C}_f$ .

Figure 21.

$M$ and $\mathrm{Fr}$ versus ${\mathrm{Re}}_x^{1/2}{C}_f$ .

Figure 22.

$M$ and $\lambda$ versus ${\mathrm{Re}}_x^{1/2}{C}_f$ .

Figure 23.

$M$ and ${\varphi }_1,{\varphi }_2$ versus ${\mathrm{Re}}_x^{1/2}{C}_f$ .

Figure 24.

$\mathrm{Nt}$ and $\alpha$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 25.

$\mathrm{Nt}$ and $\mathrm{Ec}$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 26.

$\mathrm{Nt}$ and $\chi$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 27.

$\mathrm{Nt}$ and $\mathrm{Nb}$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 28.

$\mathrm{Nt}$ and $\mathrm{Fr}$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 29.

$\mathrm{Nt}$ and $\lambda$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 30.

$\mathrm{Nt}$ and $M$ versus ${\mathrm{Re}}_x^{-1/2}Nu$ .

Figure 31.

$\mathrm{Nt}$ and ${\gamma }_e$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Nu}$ .

Figure 32.

$k_1$ and $\mathrm{Sc}$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ .

Figure 33.

$k_1$ and $\mathrm{Nt}$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ .

Figure 34.

$k_1$ and $\mathrm{Nb}$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ .

Figure 35.

$k_1$ and ${\gamma }_m$ versus ${\mathrm{Re}}_x^{-1/2}\mathrm{Sh}$ .

The present study on fluid flow over a Riga plate focuses specifically on the analysis of skin friction, which is the force per unit area acting parallel to the surface of the plate. This is a significant parameter in fluid flow analysis as it represents the resistance experienced by the fluid as it moves along the surface of the plate. In comparison to existing literature, ¹⁴ the present study offers a detailed investigation into the skin friction characteristics over a Riga plate as shown in Table 3. The common parameters $\alpha =Q$ of the Riga plate are considered avoiding the rest of the parameters. The results obtained from the study are found close in agreement.

Table 3.

Common parameters are taken into account in comparison with existing literature.

$\alpha =Q$	Anjum et al. 14	Present
	${\mathrm{Re}}^{0.5}{C}_f$	${\mathrm{Re}}^{0.5}{C}_f$
0.3	0.3244287961	0.32453278197
0.5	0.352675189	0.3527234152
0.7	0.37286819	0.37293452645
0.9	0.397628071	0.397782345167

Conclusions

Current study focuses on the theoretical study of the blood-based hybrid nanofluid over a Riga device. Here, a number of terms are introduced throughout the governing equations due to the effects of Darcy-Forchheimer medium, uniform magnetic field, heat generation/absorption, thermal expansion, joule heating, viscous dissipation, and chemical reaction. Two types of carbon nanotubes, that is, SWCNTs and MWCNTs, are incorporated into blood (shear-thinning non-Newtonian fluid). A set of similarity variables shrink the describing PDEs to a system of ODEs, that are analyzed and numerically solved using the HAM. Based on above discussions, the prime conclusions are as follows:

• In the case of blood-based CNTs nanofluids, an increase in the Eckert number can result in more efficient removal of heat from the system. This is particularly important in medical applications where efficient cooling is necessary, such as during hyperthermia treatment or in cooling systems used in medical devices.

• The flow rate of the blood-based hybrid nanofluid improvise against $\alpha$ , while become worsen for rising $\mathrm{Fr}$ , $\beta$ , $\lambda$ , and $M$ . Henceforth, the results regarding modified Hartmann number, which corresponds to Riga actuator, can be efficiently applicable where the high-speed flows are required.

• A higher Casson parameter would indicate a higher viscosity, which means that the fluid would have a thicker consistency. This higher viscosity would result in a slower flow rate of the CNTs nanofluid.

• The temperature field enhances for $\mathrm{Nt}$ , $\mathrm{Nb}$ , $\mathrm{Fr}$ , $M$ , and ${\gamma }_e$ , though the behaviors are recorded vice-versa in the case of $\mathrm{Ec}$ , $\chi$ , and $\lambda$ . The latter situation implies the rapid heat transfer.

• The concentration field is seen to occupy accumulation of nanoparticles with respect to $\mathrm{Nt}$ . However, the scenario reverses under escalating measure of $k_1$ , $\mathrm{Sc}$ , $\mathrm{Nb}$ , and ${\gamma }_m$ , which represent the uniform and stable distribution of nanostructures/solute.

• Based on the increments in paramount parameters, the skin friction coefficient and heat transfer rate is remarkably ameliorated under higher $\alpha$ , while the mass transfer improves for greater selection of $\mathrm{Nt}$ .

• The thickness of MBL is revamped for an upsurge in $\mathrm{Fr}$ , $\lambda$ , and $M$ , contrary to $\alpha$ and $\beta$ that have adverse affects on displacement thickness.