$divV=0$ (1)

$\rho \frac{\text{d}V}{\text{d}t}=\nabla \cdot T_0+\rho b$ (2)

where $\frac{\text{d}V}{\text{d}t}$ is the material time derivative. The Cauchy stress tensor τ is given

as:

$\tau =\left[{\mu }_{\infty }+\frac{{\mu }_0-{\mu }_{\infty }}{1-{\Gamma }_{\stackrel{\dot{}}{\gamma }}}\right]A_1$ (3)

The Cauchy stress tensor τ for an incompressible flow is $-pI+S$ in which $I$ is the identity tensor, Here, ${\mu }_{\infty },{\mu }_0,\Gamma ,A_1,\stackrel{\dot{}}{\gamma }$ corresponds to the viscosity at infinity, viscosity at zero, positive time constant, first Rivlin-Erickson tensor and shear rate. The shear rate is defined as:

$\stackrel{\dot{}}{\gamma }=\sqrt{\frac{\chi }{2}}$, (4)

where

$\chi =\frac{\text{trace}\left(A_1^2\right)}{2}$ (5)

$A_1$ = Rivlin-Erickson tensor = $A_1=L+L^T$;

L = Gradient of the velocity vector $L^T$ = is the transpose of L.

Here we consider only the case for ${\mu }_{\infty }=0$ and ${\Gamma }_{\stackrel{\dot{}}{\gamma }}<1$. ⁠Now Cauchy stress tensor, τ is reduced to

$\tau =\left[\frac{{\mu }_0}{1-{\Gamma }_{\stackrel{\dot{}}{\gamma }}}\right]A_1$ (6)

Or, by using binomial expansion to Equation (6), we get

$\tau ={\mu }_0\left[1+{\Gamma }_{\stackrel{\dot{}}{\gamma }}\right]A_1$ (7)

$A_1=L+L^{T\ast }$, (8)

$L=\nabla v$ Gradient of the velocity vector $L^{T\ast }={\left(\nabla v\right)}^{T\ast }$, the transpose of L;

$T^{\ast }$ Represents transpose.

$\nabla v=\left(\begin{array}{ccc}\frac{\partial u}{\partial r}& \frac{1}{r}\frac{\partial u}{\partial \vartheta }-\frac{v}{r}& \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial r}& \frac{1}{r}\frac{\partial v}{\partial \vartheta }+\frac{u}{r}& \frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial r}& \frac{1}{r}\frac{\partial w}{\partial \vartheta }& \frac{\partial w}{\partial z}\end{array}\right)\nabla \cdot \tau =\left(\begin{array}{c}\frac{\partial {\tau }_{rr}}{\partial r}+\frac{1}{r}\frac{\partial {\tau }_{r\vartheta }}{\partial \vartheta }+\frac{\partial {\tau }_{zr}}{\partial z}+\frac{{\tau }_{rr}-{\tau }_{\vartheta \vartheta }}{r}\\ \frac{\partial {\tau }_{r\vartheta }}{\partial r}+\frac{1}{r}\frac{\partial {\tau }_{\vartheta \vartheta }}{\partial \vartheta }+\frac{\partial {\tau }_{z\vartheta }}{\partial z}+\frac{2{\tau }_{\vartheta r}}{r}\\ \frac{\partial {\tau }_{rz}}{\partial r}+\frac{1}{r}\frac{\partial {\tau }_{\vartheta z}}{\partial \vartheta }+\frac{\partial {\tau }_{zz}}{\partial z}+\frac{{\tau }_{rz}}{r}\end{array}\right)$ (9)

In the absence of pressure gradient T = τ. The boundary layer equations that govern the present flow, as per the above assumptions, are listed below as stated by Manjunatha [21] .

FLUID PHASE

$\frac{\partial \left(ru\right)}{\partial r}+\frac{\partial \left(rw\right)}{\partial z}=0$ (10)

$\begin{array}{c}u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}={\nu }_{nf}\left(\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial r^2}\right)+{\nu }_{nf}\left[\frac{\Gamma }{\sqrt{2}}{\left(\frac{\partial w}{\partial r}\right)}^2+\sqrt{2}\Gamma \frac{\partial w}{\partial r}\frac{{\partial }^2}{\partial r^2}\right]\\ +\frac{KN}{{\rho }_{nf}}\left(w_p-w\right)-\left(\frac{\sigma B_0^2}{{\rho }_{nf}}+\frac{{\upsilon }_{nf}}{k_p}\right)w\end{array}$ (11)

$\begin{array}{c}w\frac{\partial T}{\partial z}+u\frac{\partial T}{\partial r}=\frac{k}{\rho c_P}\left(\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial r^2}\right)+\tau \left\{D_B\frac{\partial C}{\partial r}\frac{\partial T}{\partial r}+\frac{D_r}{T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}^2\right\}\\ +\frac{Q_0}{\rho c_P}\left(T-T_{\infty }\right)\end{array}$ (12)

DUST PHASE

$\frac{\partial \left(r{u}_p\right)}{\partial r}+\frac{\partial \left(r{w}_p\right)}{\partial z}=0$, (13)

$w_p\frac{\partial w_p}{\partial r}+u_p\frac{\partial w_p}{\partial z}=\frac{K}{\rho }\left(w-w_p\right)$ (14)

$w_p\frac{\partial T_p}{\partial r}+u_p\frac{\partial T_p}{\partial z}=\frac{C{p}_f}{T_{t}Cp}\left(T-T_p\right)$ (15)

Equations (10)-(12) are continuity, momentum and energy equations for the fluid phase and Equations (13)-(15) are for the dust phase, where (w, u) and (w_pu_p) are the velocity components of the nanofluid and dust phases, in the z and r directions respectively. ν, ρ, N, m, K, and k₀ are the kinematic viscosity of the fluid, density of the fluid, number density of the particle phase, mass of the dust particle, Stoke’s resistance (drag co-efficient) and permeability of the porous medium respectively, in addition, α represent the thermal diffusivity, ρ refer to density of the fluid. A uniform magnetic field of strength B₀ is applied in the transverse direction of the flow, due to the small magnetic Reynolds number, it is not necessary to introduce the effect of the induced magnetic field.

The associated boundary conditions are as follows

$w=u_s+U_w\left(z\right),u=0,\text{at}r=a,$ (16)

$w\to {\mathcal{U}}_e\left(z\right),\text{as}r\to \infty$

$w\to 0,w_p\to 0,{\rho }_{\rho }\to {\omega }_{\rho },\text{at}r\to \infty .$ (17)

The following similarity variables are introduced in order to convert the governing equations into the relevant ordinary differential equations (ODEs).

$\begin{array}{l}\eta =\frac{r^2-a^2}{2a}\sqrt{\frac{{\mathcal{U}}_w}{\nu z}},\psi =a\sqrt{{\nu }_f{\mathcal{U}}_{w}z}f\left(\eta \right),\\ \theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},w=u_w\left(z\right)f^{\prime }\left(\eta \right),u=-\frac{a}{r}\sqrt{\frac{{\nu }_f{u}_w}{z}}f\left(\eta \right)\end{array}$ (18)

And dust phase:

$\begin{array}{l}{\psi }_P=\sqrt[a]{v_f{u}_{w}z}F\left(\eta \right),u_p=-\frac{a}{r}\sqrt{\frac{{\nu }_f{u}_w}{z}}F\left(\eta \right),\\ w_p=u_w\left(z\right)F^{\prime }\left(\eta \right),{\theta }_p\left(\eta \right)=\frac{T_p-T_{\infty }}{T_w-T_{\infty }}\end{array}$ (19)

ψ Is the stream function is introduced which satisfy the continuity equation,

such that $u=-\frac{1}{r}\frac{\partial \psi }{\partial r}$ and $w=-\frac{1}{r}\frac{\partial \psi }{\partial z}$. η, f and θ Aredimensionless transverse

distance, dimensionless stream function and dimensionless temperature of the

fluid respectively? The kinematic viscosity $\nu =\frac{{\mu }_{\infty }}{\rho }$.

It can be confirmed that Equation (10) satisfies the similarity transformations established above in an identical manner. By substituting (18) and (19) into (11)-(16), we obtain the following nonlinear ordinary differential equation

Fluid phase

$\begin{array}{l}{\varphi }_1\left\{\left(1+2\gamma \eta \right)f^{‴}+2\gamma f^{″}\right\}+{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\left[\gamma {f^{″}}^2+2\left\{\gamma f^{″}+\left(1+2\gamma \eta \right)f^{‴}\right\}\right]\\ +{\varphi }_2\left(f{f}^{″}-{f^{\prime }}^2\right)-\beta {\beta }_v\left(f^{\prime }-F^{\prime }\right)-\left({\varphi }_{5}Ha+{\varphi }_{1}Da\right)f^{\prime }=0\end{array}$

$\begin{array}{l}\frac{{\varphi }_1{\varphi }_4}{Pr}\left\{\left(1+2\gamma \eta \right){\theta }^{″}+2\gamma {\theta }^{\prime }\right\}+{\varphi }_1{\varphi }_3\left({\theta }^{\prime }f-2\theta f^{\prime }\right)\\ +{\varphi }_1{\varphi }_{2}Ha\cdot Ec{\left(f^{\prime }\right)}^2+\frac{{\varphi }_2{\varphi }_4}{Pr}\left(A{f}^{\prime }+B\theta \right)+{\varphi }_1{\beta }_T\beta \left({\theta }_p-\theta \right)=0\end{array}$ (20)

Dust phase

${F^{\prime }}^2-F{F}^{″}-{\beta }_v\left(f^{\prime }-F^{\prime }\right)=0$

${{\theta }^{\prime }}_{p}F-2{\theta }_p{F}^{\prime }+{\gamma }^{\ast }{\beta }_T\left(\theta -{\theta }_p\right)=0$

And the corresponding Dimensionless BCs for the prescribed surface temperature (PST)

$\begin{array}{l}\eta =0,f=f_w,f^{\prime }=1,\text{}\text{}\text{}\theta =1\\ \eta \to \infty ,f^{\prime }=0,F=f,F^{\prime }=0,\theta =0,{\theta }_p=0\end{array}$ (21)

Boundary conditions for the prescribed heat flux (PHF)

$\begin{array}{l}\eta =0,f=f_w,f^{\prime }=\zeta ,{\varphi }_4{\theta }^{\prime }=-1\\ \eta \to \infty ,f^{\prime }=0,F=f,F^{\prime }=0,\theta =0,{\theta }_p=0\end{array}$ (22)

The emerging parameters in the dimensionless equations are respectively listed below:

$P_r=\frac{\mu c_p}{k}$ , is Prandtl number. $Ha=\frac{{\sigma }_f{B}_{0}w}{{\rho }_f{U}_f}$ is Hartman/magnetic number;

$\gamma =\frac{1}{a}\sqrt{\frac{{\nu }_{f}z}{u_w}}$ is Curvature parameter, $\beta =\frac{z}{r{u}_w}$ , is porosity parameter;

$\delta =h_0{\left(\frac{2v}{U_{\infty }}\right)}^{-1/2}$ is the fluid particle interaction;

$\lambda =\Gamma \sqrt{\frac{2u_w}{v_{nf}z}}$ is the We is senbergparameter, $S=\frac{vz}{k_p{u}_w}$ is the permeability parameter, $Q=\frac{\sigma B_0^2}{{\rho }_{nf}{u}_w}$ is first order heat source/sink, $R_e=\frac{U_w^3\left(z\right)}{vL}$ is the

Reynold’s number;

$Ec=\frac{u_{w}z}{A{c}_p}$ is Eckert, $Da=\frac{u_w}{k_p{\rho }_{f}z}$ is Darcy/homogenous porous medium;

$N_t=\frac{\tau D_T}{\nu T_{\infty }}\left(T_w-T_{\infty }\right)$ is the thermophoresis parameter.

The Physical quantities of engineering interest are skin friction coefficient, (C_f) and Nusselt numbers (Nu), which physically indicate the wall shear stress, rate of heat transfer and rate of mass transfer, respectively. They are defined respectively as follows:

Skin friction coefficient, Nusselt number and defined as

$\begin{array}{l}{C}_f=\frac{{\tau }_w}{{\rho }_{nf}{W}_w},\\ N{u}_z=\frac{q_w}{k_f\left(T_w-T_{\infty }\right)}\end{array}\}$ (23)

${\tau }_w$ is the shearing stress and $q_w$ is the heat flux respectively and defined as

$\begin{array}{l}{\tau }_w={\mu }_{nf}{\left[\frac{\partial u}{\partial r}+\frac{\Gamma }{\sqrt{2}}{\left(\frac{\partial u}{\partial r}\right)}^2\right]}_{r=a}\\ q_w=-{k_{nf}\frac{\partial T}{\partial r}|}_{r=a}=-{k_{nf}\frac{\Delta Tr}{a}\sqrt{\frac{U_w}{{\nu }_{f}z}}{\theta }^{\prime }\left(\eta \right)|}_{r=a}\end{array}$ (24)

Using the dimensionless variables we obtain

$\begin{array}{l}R{e}_z^{\frac{1}{2}}{C}_{fz}=\frac{{\varphi }_1}{{\varphi }_2}\left[f^{″}\left(0\right)+We{f^{″}}^2\left(0\right)\right],\\ R{e}_z^{-\frac{1}{2}}N{u}_z=-{\varphi }_4{\theta }^{\prime }\left(0\right),\end{array}\}$ (25)

Local Reynold’s number $R{e}_z=\frac{U_{w}z}{{\nu }_f}$.

Here we employ the use of Homotopy Perturbation Method HPM to obtain the exact analytical solution, this is because the reduced set of similarity equations are coupled and nonlinear in nature, thus it is very difficult to get exact solutions. Here, choosing HPM is very important, because the HPM successfully couples the homotopy theory with the perturbation theory and the main idea is that a complicated problem is continuously deformed into a simple problem which is easy to solve in order to get an analytic exact or approximate solution.

Similarly, the governing equation of the model can be transformed to dimensionless forms using the similarity transformation defined (19 and 20) with a little modification in fluid temperature.

Consider a system of nonlinear ordinary differential equations given; there are four equations in total

Using homotopy perturbation method to solve velocity equation for the fluid phase mode

$\begin{array}{l}{\varphi }_1\left\{\left(1+2\gamma \eta \right)f^{‴}+2\gamma f^{″}\right\}+{\varphi }_1\lambda {\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\left[\gamma {f^{″}}^2+2\left\{\gamma f^{″}+\left(1+2\gamma \eta \right)f^{‴}\right\}\right]\\ +{\varphi }_2\left(f{f}^{″}-{f^{\prime }}^2\right)-\beta {\beta }_v\left(f^{\prime }-F^{\prime }\right)-\left({\varphi }_{5}Ha+{\varphi }_{1}Da\right)f^{\prime }=0\end{array}$ (26)

We rewrite the equations in form that is amendable for ease of analysis

$\begin{array}{l}{\varphi }_1{f}^{‴}+2{\varphi }_1\gamma \eta f^{‴}+2{\varphi }_1\gamma f^{″}+{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma {f^{″}}^2+2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma f^{″}\\ +2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{3}{2}}{f}^{‴}+4{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma \eta f^{‴}+{\varphi }_2\left(f{f}^{″}-{f^{\prime }}^2\right)\\ -\beta {\beta }_v\left(f^{\prime }-F^{\prime }\right)-\left({\varphi }_{5}Ha+{\varphi }_{1}Da\right)f^{\prime }=0\end{array}$ (27)

and the non linear velocity equations of fluid phase were separated into the Highest linear derivative (L), non linear (N) and remainder (R)

$L_f\left(f\right)={\varphi }_1{f}^{‴},$ (28)

$N_f\left(f\right)={\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma {f^{″}}^2+{\varphi }_2\left(f{f}^{″}-{f^{\prime }}^2\right)$ (29)

$\begin{array}{l}{R}_f\left(f\right)=2{\varphi }_1\gamma \eta f^{‴}+2{\varphi }_1\gamma f^{″}+2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma f^{″}+2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{3}{2}}{f}^{‴}\\ +4{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma \eta f^{‴}-\beta {\beta }_v\left(f^{\prime }-F^{\prime }\right)-\left({\varphi }_{5}Ha+{\varphi }_{1}Da\right)f^{\prime }\end{array}$ (30)

The solution of Equation (30) can be assumed to be written as a power series in p

$f=f_0+p{f}_1+p^2{f}_2+p^3{f}_3+$ (31)

We make assumption of approximating the solution and simplified we arrange the equation and the boundary conditions according to the power of the embedding parameter p, and by Decomposing into zero order, first order and second order.

Solution of Velocity Equation for Fluid Phase for the PST

$\begin{array}{c}{\varphi }_1{f^{‴}}_1=-2{\varphi }_1\gamma {f^{″}}_0-2{\varphi }_1\gamma \eta {f^{‴}}_1-{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma {\left({f^{″}}_0\right)}^2-2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma {f^{″}}_0\\ -2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{3}{2}}{f^{‴}}_0-4{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma \eta {f^{‴}}_0-{\varphi }_2\left(f_0{f^{″}}_0-{\left({f^{\prime }}_0\right)}^2\right)\\ +\beta {\beta }_v\left({f^{\prime }}_0-{F^{\prime }}_0\right)+\left({\varphi }_{5}Ha+{\varphi }_{1}Da\right){f^{\prime }}_0,\end{array}$ (32)

Solution of Velocity Equation for Fluid Phase for the PHF

$\begin{array}{c}{\varphi }_1{f^{‴}}_1=-2{\varphi }_1\gamma {f^{″}}_0-2{\varphi }_1\gamma \eta {f^{‴}}_1-{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma {\left({f^{″}}_0\right)}^2-2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma {f^{″}}_0\\ -2{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{3}{2}}{f^{‴}}_0-4{\varphi }_{1}We{\left(1+2\gamma \eta \right)}^{\frac{1}{2}}\gamma \eta {f^{‴}}_0-{\varphi }_2\left(f_0{f^{″}}_0-{\left({f^{\prime }}_0\right)}^2\right)\\ +\beta {\beta }_v\left({f^{\prime }}_0-{F^{\prime }}_0\right)+\left({\varphi }_{5}Ha+{\varphi }_{1}Da\right){f^{\prime }}_0.\end{array}$ (33)

Authentication of accuracy of the analytical procedure used was done by comparison, first computations for f'' and θ'' are carried out for Williamson fluid for various values of M and compared with the available published results as shown in Table 1 and Table 2 and they are establish an excellent agreement.

The accuracy of the result is tested and shown in Table 1 , when θ'(0) was compared with the results of Kumar et al. [16] and Manjunatha et al. [25] and for different values of Ha, the present results are found in a very good agreement with existing published results. Similarly in Table 2 when f''(0) was compared with the results of Ramesh et al. [4] , Giresha 22 and Chen [26] the present results are found in a very good agreement. This confirm that the HPM used in this work is perfect and accurate. This proves the validity of the present results, along with the accuracy of the present analytical technique. Therefore, we are confident that our results are highly accurate to analyze this boundary layer flow problem.

The exact analytical solution to the governing ODEs (21)-(23) with the boudary conditions (24)-(26) was achieved using the Homotopy Perturbation Method. Graphical illustrations were used to analyze the influences of the pertinent parameters, namely the Curvature parameter, the Magnetic parameter, Prandtl number, Porosity parameter, the Eckert number, Wesseinberg, Fluid particle interaction parameter and the Volume fraction parameter, on respective fluid- and dust-phase. The graphical representation are as follow:

Velocities (f') and (F') as well as, fluid- and dust-phase temperature (θ' and ${{\theta }^{\prime }}_p$) are shown in Figures 2-17 , Also, the nature of the velocities and temperatures of both phases is examined in this section and an appropriate explanation is provided here. For the analysis of the model Figures 2-17 are plotted for different values of physical parameters. Furthermore, the values of the skin friction coefficient C_fz and Nusselt number Nu_z in the presence of the pertinent parameter are shown in Table 3 . It is noted that the value of C_fz decreases as Wesseinberg λ, volume fraction and fluid particle interaction and increases with Ec. On the other hand, the value of Nu_z decreases with an increase in λ or a decrease in the remaining parameters, as displayed.

Figure 2 and Figure 3 illustrate the effect of the magnetic parameter M on the velocity and temperature distribution of the fluid phase and dust phase respectively. It is clear that higher values of the magnetic parameter M result in a reduction in the fluid and dust phase velocity, along with an elevation in the temperature profiles of both phases. This occurs due to the generation of a resistive drag force, referred to as the Lorentz force, when a transverse magnetic field is applied perpendicular to the flow direction. The force operates in the opposite direction of the flow. This has the effect of reducing the velocity boundary layer and increasing the thickness of the thermal boundary layer. Due to recent observations, we have noticed a rise in the temperature profiles of the flow fluid transport phenomena. With an increase in magnetic parameter M, the momentum boundary layer thickness decreases, leading to a higher velocity gradient.

Figure 4 and Figure 5 illustrate the influence of the curvature parameter, on the velocity and temperature profiles of both fluid and dust particles. With an increase in the curvature parameter, there is a notable enhancement in both the velocity of the fluid phase and the velocity of the dust phase. It is quite evident from Figure 4 that the velocity of the fluid as well as dust phase corresponding to (γ = 0) is minimum and the increase of (γ) is to increase the velocity within the boundary layer. Moreover (γ = 0), the problem reduces to flat surface, and hence, the velocity within the boundary layer in the case of cylinder is larger than the flat surface. The presence of a cylinder leads to an increase in the flow rate within the boundary layer compared to a flat surface. This indicates that the increasing speed within the boundary layer is due to expansion in the cylinder size… Consistent with expectations, there is a noticeable increase in both velocity and momentum as we approach the surface of the cylinder. On the other hand, as we move away from the surface, we see a decrease in velocity and momentum, accompanied by an increase in the thickness of the boundary layer. This is because the resistance exerted by the viscous forces near the surface of the cylinder is considerably higher compared to those located further away from it. Furthermore, as the curvature parameter increases, there is a corresponding decrease in the radius of the cylinder. As a result, there is a reduction in the resistance to fluid motion, leading to a subsequent increase in fluid velocity. In the dust phase, the rate of velocity gradient increase is significantly greater in comparison to the fluid phase. An increase in the curvature parameter leads to a significant reduction in the radius of the cylinder, thereby causing a decrease in the resistance to fluid motion. Consequently a decrease in the distribution of heat in both the fluid and dust phases is observed. The data clearly indicates that the temperature profiles of both the fluid and dust phases show an upward trend as the curvature parameter increases.

Figure 6 and Figure 7 show how the Eckert number affects the velocity and temperature distribution of the fluid phase and dust phase, respectively. Just like a mechanical engineer, the graphs clearly show that as the Eckert number Ec increases, the temperature profile also increases. Increases in the Eckert number enhance the intermolecular mobility and kinetic energy within liquids. The thermal variation, with its various stages and the accompanying boundary layer, is further amplified by the intermolecular tension. When the Eckert number (Ec) goes up, there is usually a corresponding increase in the dissipation of energy due to viscosity between particles. The main cause of this phenomenon is that the fluid’s viscosity absorbs energy from its motion and transforms it into internal energy, leading to an increase in the fluid’s temperature. As a result, the thickness of the thermal boundary layer increases, leading to a higher temperature distribution within the dusty Nanofluid.

The fluid particle interaction parameter β has a significant impact on the velocity and temperature profiles of the fluid and dust phases, Figure 8 and Figure 9 . It is worth noting that an increase in the value of β leads to an enhancement in the velocity of the dust phase and a reduction in the velocity of the fluid phase. This phenomenon occurs when there is a strong interaction between the fluid and particle phases, resulting in the particle phase exerting a force that opposes the fluid phase until the particle velocity equals the fluid velocity. It is evident that an increase in the fluid particle interaction parameter leads to an elevation in the temperature profiles of both the fluid and dust phases. This agrees with the general fact that the interaction between the fluid and the particle phase has a greater impact on enhancing the thermal conductivity of the flow than it might otherwise. From Figure 10 and Figure 11 , it is clear that an increase in the volume fraction of dust particles leads to a decrease in the velocity profiles of the fluid and dust phases, as well as the temperature profiles. The reason for this phenomenon is that the dust particles take up more space per unit volume of the mixture, resulting in an increase in the mass concentration of the dust phase and a decrease in velocity and thermal boundary layer thickness. The influence of Wesseinberg parameter, λ on velocity and energy profiles in both fluid and dust phases are seen in Figure 12 and Figure 13 . The Weissenberg parameter refers to a quantifiable correlation between the rate of fluid flow and a specific duration of the process. The increasing values of the λ lead to an increase in the relaxation time of fluid molecules, so elevating the significance of viscosity and generating a resistance to the flow of a fluid, ultimately leading to a decrease in velocity of the fluid. The rate of flow and the corresponding increase in boundary layer thickness decrease for the fluid phase and increase for the dust phase as the λ increases. However, the thickening of the temperature boundary layer increases for both the liquid and dust phases as the λ increases. The impact of the permeability parameter on velocity profiles in both the fluid and dust phases is illustrated in Figure 14 . The presence of porous medium is observed to cause increased restriction of fluid flow, resulting in slowing down the velocity. Hence, as the permeability parameter increases, there is a corresponding increase in the resistance to fluid motion. Figure 15 demonstrates the impact of increasing values of the permeability parameter on temperature distributions for both the fluid and dust phases. It is evident that higher values of the permeable parameter lead to the thickening of the thermal boundary layer. The influence of non-dimensional governing parameters on the skin friction coefficient for the fluid and dusty phases of Nanofluids is illustrated in Figure 16 . The data presented in the figures clearly indicate that an increase in the values of magnetic field parameter, Wesseinberg, mass concentration volume fraction of Nano particles, result in a rise in the skin friction coefficient, while the skin friction coefficient is depreciated as the value of the fluid particle interaction parameter increases. On the other hand the impact of non-dimensional governing parameters on the local Nusselt number can be observed in Figure 17 . From the figures, it can be seen that an increase in fluid particle interaction parameter leads to an enhancement in the heat transfer rate in both fluid and dust phases. However, an increase in the magnetic field parameter, Wesseinberg, parameter, mass concentration of dust particles and volume fraction of Nano particles, leads to a decrease in the heat transfer rate.