COVID-19 has altered our way of life, causing us to become more aware of the limitations that we as humans confront. Scientists and clinicians around the world are scrambling to figure out what is causing the epidemic of the COVID-19 virus and what they can do to stop it. Scientists have concluded, as a result of recent investigations, that nanotechnology can be used to address a wide range of clinical difficulties, including the Coronavirus pandemic [1]. Nanotechnology has the potential to be used to prevent and identify diseases in their early stages. The nanoparticles, when introduced into human immune system, have the potential to target diseases at the cellular level. Nanoparticles have the extra advantage of delivering molecular adjuvants for loaded antigens, which can improve the efficacy and safety of a vaccine while also increasing its efficacy. In order to target delivery at the cellular level, a large number of vaccines are being produced on the basis of nanoparticles (Lipid nanoparticles). The BioNTech/Pfizer and Moderna mRNA Corona virus vaccines, for example, make use of Lipid nanoparticles to deliver treatments more efficiently to the patient. Aside from that, face masks constructed of nanomaterials are more capable of providing efficient air filtering and can also be reused for a longer period of time, giving them an added edge in overcoming the shortage of high-quality masks.

Nanotechnology operates at the nanometric scale, and as a result, it is the study of a wide range of systems and technologies that operate at this scale. Nanofluids, which are based on nanomaterials, are becoming increasingly used in a variety of industrial and technical operations. Yoo et al. [2] were the first to propose the concept of nanofluids in the year 1995, and since then, nanofluids have become a popular topic of discussion among scientists. Yoo et al. [2] investigated the prospect of increasing energy exchange among fluids by incorporating various nanoparticles into the mix. Fetecau et al. [3] developed a precise solution for the flow of nanoliquids by employing a fractional model. Meanwhile, Khan et al. [4] presented the challenge of thin film electrically guided nanoliquids sprayed over a stretching cylinder in their paper Tinny film electrically steered nanoliquids sprayed over a stretching cylinder in their paper. A study conducted by Mekheimer et al. [5] using gyrotactic suspended nanoparticles and micro organisms to investigate the discharge of Prandtl fluid flow by utilizing chemical reaction, radiation, and magnetic field effects has just been published. Meanwhile, Ahmed et al. [6] investigated the Fa lkner-Skan problem with double stratification in the flow of nanofluids by employing single/multiple wall carbon nanotubes in the flow of single/multiple wall carbon nanotubes [7]. We out a comparative investigation of ethylene glycol-based nanofluid flows, employing electro-osmotic and peristaltic pumping techniques. To examine Hybrid nanoliquid slip flows, Wahid et al. [8] used an exponentially stretching/shrinking permeable sheet that was stretched and shrunk exponentially. Because of the increasing demand for nanoliquid flows in numerous fields of research and development, we find a wealth of studies on nanoliquid flows conducted by a large number of other researchers [9,10].

Micropolar fluids are a class of non-Newtonian fluids in which we are interested in the micro-motion of fluid particles and are therefore named as such. Eringen [11] was the first to propose the concept of micro components in fluids, back in 1966. The formation of a boundary layer of micro-structured liquids, on the other hand, was predicted by Willson [12]. With the use of Lubrication approximation theory, Asghar et al. [13] were able to obtain a reduced version of the flow equation to represent Micropolar slime. In the contemporary age, Abbas et al. [14] investigated the prospect of increasing energy transformation among Micro-rotational nanoliquids by employing the Ota, Yamada, and Xue models of hybrid nanoliquids, which were developed by Ota, Yamada, and Xue. For their part, Ismail et al. [15] used generated magnetic field effects to analyze the flow of micro-structured fluids while conducting their experiments. With the help of a magnetic curved surface, Abbas et al. [16] investigated the unsteady flow of Micropolar fluid flows. Raza et al. [17] discovered dual solutions in their investigation of Micropolar fluid flows utilizing thermal radiations, which was published just a few months ago.

Casson fluid is yet another essential non-Newtonian liquid that is becoming increasingly important in our daily lives, according to the National Institute of Standards and Technology. Casson fluids include a variety of sauces, jellies, and soups, amongst other things. Casson [18] was the first to derive the constitutive equations for Casson fluid and to illustrate the properties of various polymers. He published his results in 1959. McDonald [19] pointed out in 1974 that the flow of blood is a good representation of the Casson fluid model. Under-discussed fluid classes find use in a diverse range of fields including medical, biology, the food industry, and several sorts of drilling procedures, just to name a few. Because of the wide range of applications for which this essential class of fluids is used, many researchers keep a close eye on it in their laboratories. Through the use of the homotopy approach, Nadeem et al. [20] found the optimal solutions of Casson nanoliquid flows. The authors of another paper [21] concluded that skin friction along the wall is lower in Newtonian fluids than in non-Newtonian Casson-nanofluids when comparing the two types of fluids. When it comes to the flow of Casson nanoliquids, Mustafa et al. [22] used general power law velocity dispersal to analyze the phenomenon. Sarkar et al. [23] investigated the evolution of Casson nanoliquids in the aggregation of wedge angle and melting processes, respectively. In a similar vein, Ibrar et al. [24] investigated the flow of Casson nanoliquids with magnetic current using single and multiple wall carbon nanotubes, respectively. Nayak et al. [25] used entangled motile gyrotactic micro-creatures to examine the flow of Casson nanoliquids utilizing chemical processes in order to better understand their behavior. Several researchers, including Khan et al. [26], have investigated the possibility of a dual solution in the evolution of Casson fluid flows across a Stretching/Shrinking sheet.

We believe that unsteady fluid mix is appropriate based on the literature research and to the best of our knowledge. Nanofluids flows across a curved Stretching/Shrinking surface using the Micropolar Casson-Buongiorno model have not been investigated previously. Based on the findings of the studies given in [27,28], we determined that it was reasonable to unleash the dual solution phenomena utilizing the fluid flows under consideration. Stability testing will be carried out in order to better understand and identify the most appropriate solutions that will be physically feasible. To achieve this goal, we will introduce a form of exponential function into our steady state system to represent perturbation to it. This results in the stable situation being turned into an eigen value problem, from which acceptable eigen values are selected and noted for future reference.

Consider the case of a liquid mixture. As illustrated in Fig. 1, a Casson Micropolar nanofluid is discharged across a bent stretching/shrinking surface that has been wrapped in a circle of radius R. For our two-dimensional frame of reference rs−axes , we will use the (r+R,s) Curvilinear Coordinates System as our starting point. The velocity components u and v are measured in the directions of the s and r axis, and correspondingly. It is proposed that a hydrodynamic force acting in the r direction of the surface will aid in the control of the flow of the fluid in question. When the normal assumptions for boundary layer approximations are taken into consideration, the set of equations that governs this particular flow is the following:

(1) ∂u∂s+1R∂∂r(rv+Rv)=0,

(2) ρu2=(r+R)∂p∂r,

(3) ∂u∂t+(Rur+R)∂u∂s+v∂u∂r+vu(r+R)=−(Rr+R)∂pρ∂r+ν(1β+K+1)[−u(r+R)2+∂2u∂r2+(1r+R)∂u∂r]−Kρ∂N∂r−σB0ρu,

(4) ∂∂t(N)+v∂∂r(N)+RNr+R∂∂s(N)=(1+0.5K)[∂2N∂r2+(1r+R)∂N∂r]−K[(1r+R)u+2N+∂N∂r],

(5) ∂T∂t+(Rur+R)∂T∂s+v∂T∂r=α1∗(∂2T∂r2+(1r+R)∂T∂r)+ρcpρcf[DB(∂T∂r)(∂C∂r)+(DTT∞)(∂T∂r)2],

(6) ∂C∂t+(Rur+R)∂C∂s+v∂C∂r=DB[(1r+R)∂C∂r+∂2C∂r2]+(DTT∞)[∂2T∂r2+(1r+R)∂T∂r].

When t>0 , the relevant boundary conditions are

u=bs,v=aν,N+n(∂u∂r)=0,DB(∂C∂r)=−DTT∞(∂T∂r),atr=0,

(7) u→0,∂u∂r→0N→0,C=C∞,T=T∞asr→∞.

Eqs. (1)–(4) represent velocity and angular velocity profile, Eqs. (5) and (6) represent temperature profile and concentration of nano particles.

To simplify the above equations, we will use these transformations:

v=−(Rr+R)aν(1−αt)f(η),u=asf′(η)(1−αt),

N=(as(1−αt))aν(1−αt)g(η),

φ(η)=(C−C∞)(Cw−C∞),p=(ρa2s2P(η))(1−αt)2,

(8) θ(η)=(T−T∞)(Tw−T∞),η=aν(1−αt)r.

After solving Eqs. (2)–(6) we get

(9) (1+1β+K)[fiv+2f′′′η+K∗−f′′(η+K∗)2+f′(η+K∗)3]−K∗η+K∗(f′f′′−ff′′′)−K∗(η+K∗)2(f′2−ff′′)−K∗(η+K∗)3ff′−K(g′′+g′η+K∗)−β∗2(3f′′+ηf′′′)−β∗η+K∗(f′+η2f′′)−M2(f′′+g′η+K∗)=0,

(10) (1+0.5K)(g′′+1η+K∗g′)+K∗η+K∗g′f−K∗η+K∗gf′′−K(2g+1η+K∗f′+f′′)−0.5β∗(3g+g′η)=0,

(11) θ″+1(η+K∗)θ′+Pr[K∗θ′fη+K∗+θ′2Nt+φ′θ′Nb−0.5ηβ∗θ′]=0,

(12) φ′′+1η+K∗φ′−Le[0.5ηβ∗φ′−K∗fη+K∗φ′]+NtNb(1η+K∗θ′+θ′′)=0.

The boundary conditions for above flow problem are:

f(0)=0,f′(0)=λ,g(0)+nf′′(0)=0,θ(0)=1,Nbφ′(0)=−Ntθ′(0),

(13) f′(η)→1,g(η)→0,f′′(η)→0,φ(η)→0,θ(η)→0,asη→∞.

Moreover, some of the important physical measures are Skin friction coefficient Cf , Nusselt number Ns , Sherwood number Shs and these are expressed as:

(14) Cf=τrsρuw2,Ns=sqwk(Tw−T∞),Shs=shmDB(Cw−C∞)

where, τrs , sq_w, sh_m are defined as:

(15) τrs=ρ(1+K+1β)(∂u∂r+ur+R+kN)r=0,τm=ρ(1+K/2)(∂N∂r+Nr+R)r=0,qw=−(∂T∂r)r=0,

Moreover, the dimensionless form these measures are:

Res1/2Cf=(1+K+1β)f′′(0)+f′(0)K*),Res−1/2Nus=−θ′(0),

(16) Res−12Shs=−φ′(0),

where Res=(as2νf) is the local Reynold number.

Under specified parametric combinations, the numerical simulation of Eqs. (9)–(13) yields dual solutions to the equations. It is therefore essential to choose the most appropriate solution that is both realistic and viable. The stability analysis will be conducted in accordance with the methodologies established by the authors Weidman et al. [27] and Rosca et al. [28], respectively. In order to accomplish this, we establish another variable τ=at , where is linked to the beginning value problem and thus provides us with the most likely replies between the two alternatives. We introduce some additional similarity variables:

u=as(1−αt)f′(η,t),v=−(Rr+R)a/ν(1−αt)f(η,t),

N=(as1−αt)a/ν(1−αt)g(η,t),φ(η,t)=(C−C∞)/(Cw−C∞),

p=[ρs2a2(1−αt)2]P(η,t),η=a/ν(1−αt)r,

(17) θ(η,t)=(T−T∞)(Tw−T∞),K∗=Raν(1−αt),α=aβ1.

Simplifying Eqs. (3)–(6) using transforming variables from (17) we have

(18) (1β+K+1)[−1(K∗+η)2∂2f∂η2+∂4f∂η4+1(K∗+η)3∂f∂η+2K∗+η∂3f∂η3]−K∗K∗+η(−f∂3f∂η3+∂2f∂η2∂f∂η)−K∗(K∗+η)2[−f∂2f∂η2+(∂f∂η)2]−K∗(K∗+η)3f∂f∂η−K[∂2g∂η2+1K∗+η∂g∂η]−β∗K∗+η(∂f∂η+0.5η∂2f∂η2)−0.5β∗(η∂3f∂η3+3∂2f∂η2)−M2(∂2f∂η2+1K∗+η∂f∂η)+(αt−1)∂3f∂τ∂η2+(αt−1)K∗+η∂2f∂τ∂η=0,

(19) (1+0.5K)(1K∗+η∂g∂η+∂2g∂η2)+(K∗K∗+η)[−g∂f∂η+f∂g∂η]−0.5β∗(η∂g∂η+3g)+(αt−1)∂g∂τ−K(2g+1K∗+η∂f∂η+∂2f∂η2)=0,

(20) ∂2θ∂η2+1(K∗+η)∂θ∂η+Pr[K∗K∗+ηf∂θ∂η+(∂θ∂η)2Nt+(∂φ∂η∂θ∂η)Nb−∂θ∂ηηβ∗2−(1−αt)∂θ∂τ]=0,

(21) ∂2φ∂η2+(1K∗+η)∂φ∂η−Le[(∂φ∂ηηβ∗2)+(1−αt)∂φ∂τ−(K∗fK∗+η)∂φ∂η]+[1K∗+η∂θ∂η+∂2θ∂η2]NtNb=0.

Subject to boundary conditions

f(0,τ)=0,n∂2f∂η2(0,τ)+g(0,τ)=0,∂f∂η(0,τ)−λ=0,

θ(0,τ)−1=0,∂φ∂η(0,τ)Nb=−∂θ∂η(0,τ)Nt.

Also, when η→∞ we have

∂f∂η(η,τ)→0,∂2f∂η2(η,τ)→0,θ(η,τ)→0,

(22) φ(η,τ)→0,g(η,τ)→0.

We have steady solution of the system of Eqs. (9)–(12) with boundary condition (13), in the form f(η)=f0(η),g(η)=g0(η),θ(η)=θ0(η) and φ(η)=φ0(η) .

We assume the following variables to generate the perturbation:

f(η,τ)=F(η,τ)e−γτ+f0(η),

g(η,τ)=G(η,τ)e−γτ+g0(η),

θ(η,τ)=H(η,τ)e−γτ+θ0(η),

(23) φ(η,τ)=Q(η,τ)e−γτ+φ0(η),

Here functions F(η,τ) , G(η,τ) , H(η,τ) and Q(η,τ) are very small as compared to f0(η),g0(η),θ0(η)andφ0(η) respectively with γ being the eigen value parameter. Simplifying Eqs. (18)–(22), by using Eq. (23) we have

(24) (1+1β+K)[(1(K∗+η)3)∂F∂η−(1(K∗+η)2)∂2F∂η2+(2K∗+η)∂3F∂η3+∂4F∂η4]−K∗K∗+η[(∂2F∂η2)f0′+(∂F∂η)f0′′−(∂3F∂η3)f0−Ff0‴]−K∗(K∗+η)2[−(∂2F∂η2)f0−Ff0′′+2(∂F∂η)f0′]−K∗(K∗+η)3(f0∂F∂η+f0′F)+(1−αt)K∗+η[−∂2F∂η∂τ+γ∂F∂η]−β∗K∗+η(0.5η∂2F∂η2+∂F∂η)−0.5β∗(η∂3F∂η3+3∂2F∂η2)−K(1K∗+η∂G∂η+∂2G∂η2)+(1−αt)[γ∂2F∂η2−∂3f∂η2∂τ]−M2(∂2F∂η2+1K∗+η∂F∂η)=0,

(25) (1+0.5K)[(1η+K∗)∂G∂η+∂2G∂η2]+K∗η+K∗[∂G∂ηf0+g0′F−f0′G−∂F∂ηg0]−β∗2(η∂G∂η+3G)−K[(1η+K∗)∂F∂η+2G+∂2F∂η2]−(1−αt)(−γG+∂G∂τ)=0,

(26) (1η+K∗)∂H∂η+∂2H∂η2−Pr[−K∗η+K∗(∂H∂ηf0+θ0′F)−(∂Q∂ηθ0′+∂H∂ηφ0′)Nb−2∂H∂ηθ0′Nt+(−Hγ+∂H∂τ)(1−αt)+0.5ηβ∗∂H∂η]=0,

(27) (1η+K∗)∂Q∂η+∂2Q∂η2−Le[−K∗η+K∗(∂Q∂ηf0+φ0′F)+(−Qγ+∂Q∂τ)(1−αt)+0.5ηβ∗∂Q∂η]+(1η+K∗)∂H∂ηNtNb+∂2H∂η2NtNb=0,

and the corresponding boundary conditions are

∂f∂η(0,τ)=0,F(0,τ)=0,G(0,τ)=−n∂2F∂η2(0,τ),

Nb∂Q∂η(0,τ)=−Nt∂H∂η(0,τ),θ(0,τ)=1.

For η→∞, we have

(28) ∂F∂η(η,τ)→0,∂2F∂η2(η,τ)→0,,H(η,τ)→0,G(η,τ)→0Q(η,τ)→0.

Keeping in view the work done by authors in [27,28], we investigate the steady solutions g0(η),θ0(η) , f0(η),φ0(η) in Eqs. (9)–(12), substituting τ=0 in the system of Eqs. (24)–(28). Thus G=G0(η) , F=F0(η) , H=H0(η) , and Q=Q0(η) is considered to be the initial rise and fall in Eq. (23). After simplifications we attained the following linearized eigen value system:

(29) (1β+K+1)[F0iv+(1(η+K∗)3)F0′−(1(η+K∗)2)F0′′+(1η+K∗)2F0‴]−(F0′′f0′+F0′f0′′−F0‴f0−F0f0‴)(K∗η+K∗)−K∗(η+K∗)2[−F0′′f0+2f0′F0′−F0f0′′]−K∗(η+K∗)3(f0′F0+F0′f0)−K[(1η+K∗)G0′+G0′′]−β∗η+K∗(F0′+0.5ηF0′′)+γη+K∗F0′−0.5β∗(3F0′′+ηF0‴)+γF0′′=0,

(30) (1+0.5K)[(1K∗+η)G0′+G0′′]+K∗η+K∗[f0G0′+F0g0′−G0f0′−g0F0′]−K∗(2G0+F0′′+1η+K∗F0′)−β∗2(ηG0′+3G0)+γG0=0,

(31) H0′′+1K∗+ηH0′−Pr[0.5ηβ∗H0′−K∗K∗+η(f0H0′+F0θ0′)−Nb(θ0′Q0′+φ0′H0′)−γH0−2Ntθ0′H0′]=0,

(32) Q0′′+Q0′(1K∗+η)−Le[−K∗K∗+η(Q0′f0+φ0′F0)−Q0γ+0.5ηβ∗Q0′]+[H0′′+(1K∗+η)H0′]NtNb=0.

The boundary conditions are as follows:

F0(0)=0,F0′(0)=0,G0(0)=−nF0′′(0),

θ0(0)=0,Q0′(0)Nb=−H0′(0)Nt.

For η→∞, we have,

(33) F0′(η)→0,F0′′(η)→0,H0(η,τ)→0,G0(η)→0,Q0(η)→0

As per findings of authors Weidman et al. [27] and Rosca et al. [28], the stability analysis will give smallest values for γ . We modify condition F0′(η)=1, to solve the system (29)–(33). Table 2 shows smallest eigen values obtained hence forth.

We will now investigate impacts of various embedded parameters on different profiles.

Some parametric options have been shown to have dual solutions in previous sections, as demonstrated by numerical simulations of Eqs. (9)–(13). For this reason, we used a numerical eigenvalue system of Eqs. (29)–(33) to determine the best solution. For positive lowest eigen values, we noticed that our solutions show an initial deterioration. Furthermore, we found that the solutions remained unstable when the eigen values were negative. Table 2 shows the least eigen values that were identified as a result of these solutions. Our findings show that the first options are more feasible than the second ones. According to [27,28], there were similar findings.

The performance of the Casson Micropolar nanoliquid flow mixer and energy transfer were examined in this study. We discovered that for some parametric values, our equations yield two solutions. As a result, we needed to do a stability analysis in order to select the best solution. Our steady state system was perturbed by exponential functions. Because exponential functions converge more quickly than some other function, this is why they are used. It was easy for us to identify the proper eigenvalues because they agreed so well with our possible solutions. The MATLAB solver bvp4c has stopped working.

We arrived at the following key observations:

We witnessed that the profile f′(η) increases if we enlarged the values of M . But the same profile is decreasing function of parameters K∗, β and K.