Maxwell Dusty Fluid Flow Past a Variable Thickness Porous Stretching Surface

Industrial applications like tap production, plastic and glass forming involve investigating flow across a stretching surface; some of these investigations focus on Newtonian fluids, while others look at non-Newtonian fluids. Many mathematical models simulate the non-Newtonian fluids in a computer simulation. Non-Newtonian fluid flow is more trustworthy from a technological standpoint than Newtonian fluid flow. For example, a viscoelastic non-Newtonian fluid can take on a variety of models, such as a third-grade model. A third-grade non-Newtonian fluid's shear thickening and shear thinning characteristics are illustrated in the study [1-3]. Hayat et al. [1] developed that in the effect of the chemical reaction. On the other hand, the impact of changeable viscosity due to temperature and thermal conductivity on the MHD flow past an extended cylinder by Salahuddin et al. [2]. In the presence of a magnetic field, Radi and colleagues [3] studied the convective flow induced by an extended sheet.

Viscous and elastic procedures are offered by viscoelastic non-Newtonian fluid. The study [4-6] covers a number of these tactics. Using velocity slip, cross-diffusion effects were investigated by Baag et al. [4], Jayachandra Babu and Sandeep [5] presented the flow of MHD Williamson fluid through a stretched surface of varied thickness. Unsteadiness and micropolarity were studied by Elbashbeshy et al. [6] over a flat extendable surface.

The study [7-16] shows that the fluid molecule connection has an influence on the study of dusty fluid. Researchers Vajravelu and Nayfeh [7] applied the hydro attractive dusty fluid advancement across an extended surface and found that molecular connection, stacking, and pull all affected the stream features. Dusty fluid was studied by Gireesha et al. [8] as it traveled across an upwards expanding surface. Extending the review, Gireesha et al. [9] discussed the unsteady dusty fluid flow across a sheet in the vicinity of a nonhomogeneous heat source/sink. Thoughts of the thick distribution occurred to Ramesh et al. [10] when they were studying a stream of dusty fluid.

Using a nonhomogeneous heat source and sink, Gireesha et al. [11] introduced the dusty MHD flow of liquid across a tilted level expanding surface. An unstable dusty fluid heat transfer and flow across an exponentially stretched sheet exposed to suction were investigated numerically by Pavithra and Gireesha [12, 13]. In the occurrence of radiation impacts, Ramesh and Gireesha [14] introduced a dusty fluid flow across a stretched sheet. When heat generation/absorption and thermal radiation were present as well as an unstable mixed convective dusty fluid flow. Gireesha et al. [15] studied this flow. Using a stretched surface and heat production inside the porous medium, Rauta and Mishra [16] analyzed two stage flow.

Countless studies have looked at the effect of nanoparticles on the dusty fluid. For examples, Sandeep et al. [17] studied the radiative flow and heat transfer of a dusty nanofluid across an exponentially stretched sheet in the existence of a volume proportion of nano particles and dust. When it comes to MHD flow and heat transmission, Manjunatha and Gireesha [18] focused on the impacts of thermal conductivity and viscosity. MHD nanofluid with conducting dust particles embedded in it was researched by Naramgari and Sulochana [19] for its momentum and heat transfer behavior as it passed an extending sheet with the existence of a volume proportion of dust particles. MHD flow of nanofluid toward a nonlinear expandable surface with changing thickness was introduced by Daniel et al. [20]. Bhatti et al. [21] showed how nonlinear thermal radiation affects the flow of a non-Newtonian dusty fluid with the inclusion of a solid substance past a porous channel. Utilizing the radiation effect, Reddy et al. [22] investigate the magneto-hydrodynamic boundary layer flow of Williamson nanofluid with heat and mass transfer across a stretched sheet with changing thickness. Powell-Eyring MHD nanofluid flow via non-directly extending surface of changeable thickness was presented by Hayat et al. [23]. Elbashbeshy et al. [24] have studied a Maxwell dusty fluid flow across an extended sheet using the RKBS-45 method.

The current research aims to investigate Elbashbeshy et al. [24] in the present of the porosity medium effect but using the effective Legendre-Galerkin technique. Legendre-Galerkin technique is broadly utilized in numerous applications. As of late, numerous analysts applied it to take care of their issues. Cai and Qi [25] introduced the arrangement of general Volterra useful necessary conditions utilizing a completely discrete Legendre-Galerkin technique. Klahn et al. [26] presented the arrangement the completely nonlinear potential stream issue for water waves in a solitary even measurement. Schmutzhard [27] concentrated on the assessment of the prolate spheroidal wave capacities utilizing Legendre-Galerkin strategy. For weakly singular Fredholm integral equations of the second type and their accompanying eigenvalue issue, Panigrahi and Nelakanti [28] study Galerkin's technique utilizing Legendre polynomial basis functions. Fathy et al. [29] utilized Galerkin strategy with Legendre basis for tackling direct Fredholm integro-differential. Legendre–Galerkin was presented the arrangement of the biharmonic conditions by Zhuang and Chen [30]. Sayed and Fathy [31] introduced how to obtain the flow of nanofluid through an upward cone.

The Legendre-Galerkin method is a strong and efficient numerical methodology for solving ordinary differential equations' linear and nonlinear boundary value issues. This numerical method's approach and primary definition have already been discussed in a number of publications. To make our computations easier, we should implement a suitable transformation to transfer the solution domain [0, ∞] to the domain [-1, 1]. Because the Legendre polynomials are in the computational interval [-1, 1], the transformation $\eta=\frac{\eta_{\infty}}{2}(\zeta+1)$ is employed to convert Eqns. (10)-(12) to the following forms in the first phase of this approach.

$\frac{8}{\eta_{\infty}^3} \frac{d^3 f}{d \zeta^3}+\frac{4}{\eta_{\infty}^2} f \frac{d^2 f}{d \zeta^2}-\frac{8 n}{\eta_{\infty}(n+1)}\left(\frac{d f}{d \zeta}\right)^2$$+\frac{8 \beta}{\eta_{\infty}^3}\left((3 n-1) f\left(\frac{d f}{d \zeta}\right)\left(\frac{d^2 f}{d \zeta^2}\right)-\frac{2 n(n-1)}{n+1}\left(\frac{d f}{d \zeta}\right)^3\right.$

$\left.+\frac{n-1}{2}(\xi+1)\left(\frac{d f}{d \zeta}\right)^2\left(\frac{d^2 f}{d \zeta^2}\right)-\frac{n+1}{2} f^2\left(\frac{d^3 f}{d \zeta^3}\right)\right)$$+\frac{4}{n+1} \frac{\ell \beta_v}{\eta_{\infty}}\left(\frac{d F}{d \zeta}-\frac{d f}{d \zeta}\right)-\frac{2 \gamma}{\eta_{\infty}} \frac{d f}{d \zeta}=0,$       (13)

$\frac{4 n}{\eta_{\infty}^2}\left(\frac{d F}{d \zeta}\right)^2-\frac{2(n+1)}{\eta_{\infty}^2} F\left(\frac{d^2 f}{d \zeta^2}\right)+\frac{2 \beta_v}{\eta_{\infty}}\left(\frac{d F}{d \zeta}-\frac{d f}{d \zeta}\right)=0,$      (14)

with boundary conditions:

$f(-1)=\alpha\left(\frac{1-n}{1+n}\right), \frac{d f(-1)}{d \zeta}=\frac{\eta_{\infty}}{2}, \frac{d f(1)}{d \zeta}=\frac{d F(1)}{d \zeta}=0$, $f(1)=F(1)$,     (15)

Assume that the nonlinear system's unknowns in (13) and (14) have an approximate solution:

$f(\xi) \approx \sum_{i=0}^M c_j P_j(\zeta) \quad$ and $F(\xi) \approx \sum_{i=0}^M d_i P_i(\zeta).$        (16)

By inserting into the preceding system and using the Galerkin approach, the following results may be obtained.

$\left(\frac{n-1}{2}\right)(\xi+1) \times \sum_{j=0}^M \sum_{i=0}^M \sum_{k=0}^M c_j c_i c_k\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta} \frac{d^3 P_k(\zeta)}{d \zeta^3}, P_r(\zeta)\right\rangle$

$+\frac{8 \beta}{\eta_{\infty}^3}\left(\sum_{j=0}^M c_j\left\langle\frac{d^3 P_j(\zeta)}{d \zeta^3}, P_r(\zeta)\right\rangle-\frac{2 n(n-1)}{n+1}\right.$

$\times \sum_{j=0}^M \sum_{i=0}^M \sum_{k=0}^M c_j c_i c_k\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta} \frac{d P_k(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle+(3 n-1)$

$\times \sum_{j=0}^M \sum_{i=0}^M \sum_{k=0}^M c_j c_i c_k\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta} \frac{d^2 P_k(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle+\frac{4 \ell \beta_v}{\eta_{\infty}^3(n+1)} \sum_{j=0}^M d_j\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle$

$+\frac{4}{\eta_{\infty}^2} \sum_{j=0}^M \sum_{i=0}^M c_j c_i\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d^2 P_i(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle-\left(\frac{n+1}{2}\right)$

$\left.\times \sum_{j=0}^M \sum_{i=0}^M \sum_{k=0}^M c_j c_i c_k\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta} \frac{d^3 P_k(\zeta)}{d \zeta^3}, P_r(\zeta)\right\rangle\right)$

$-\frac{8 n}{\eta_{\infty}(n+1)} \sum_{j=0}^M \sum_{i=0}^M c_j c_i\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle$

$-\frac{1}{\eta_{\infty}}\left(\frac{4 \ell \beta_v}{n+1}+2 \gamma\right) \sum_{j=0}^M c_j\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle=0,$        (17)

$\begin{aligned} \frac{4 n}{\eta_{\infty}^2} \sum_{j=0}^M \sum_{i=0}^M d_j d_i\left\langle\frac{d P_j(\zeta)}{d \zeta}\right. & \left.\frac{d P_i(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle-\frac{2(n+1)}{\eta_{\infty}^2} \sum_{j=0}^M \sum_{i=0}^M d_j d_i\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d^2 P_i(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle \\ & +\frac{2 \beta_v}{\eta_{\infty}} \sum_{j=0}^M\left(d_j-c_j\right)\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle=0\end{aligned}$     (18)

in terms of the boundaries:

$\sum_{j=0}^M(-1)^j c_j=\alpha\left(\frac{1-n}{1+n}\right), \sum_{j=0}^M(-1)^{j+1} j(j+1) c_j=\eta_{\infty},$

$\sum_{j=0}^M j(j+1) c_j=0, \sum_{j=0}^M j(j+1) d_j=0, \sum_{j=0}^M\left(c_j-d_j\right)=0,$     (19)

where, 〈⋅,⋅〉 is defined as the inner product:

$\langle\varphi(\zeta), \omega(\zeta)\rangle=\int_{-1}^1 \varphi(\zeta) \omega(\zeta) d \zeta$

The matrix form of the nonlinear system (17) and (18) with condition (19) is: 

$\mathbf{H} \mathbf{c}+\boldsymbol{\Psi} \tilde{\mathbf{c}}+\boldsymbol{\Theta} \overline{\mathbf{c}}=\mathbf{b}$     (20)

where,

$\mathbf{H}=\left(\begin{array}{ll}\mathbf{H}_{f_1} & \mathbf{H}_{F_1} \\ \mathbf{H}_{f_2} & \mathbf{H}_{F_2}\end{array}\right), \boldsymbol{\Psi}=\left(\begin{array}{cc}\boldsymbol{\Psi}_{f_1} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{\Psi}_{F_2}\end{array}\right), \boldsymbol{\Theta}=\left(\begin{array}{c}\boldsymbol{\Theta}_{f_1} \\ \boldsymbol{O}\end{array}\right), \mathbf{b}=\left(\begin{array}{l}\mathbf{b}_1 \\ \mathbf{b}_2\end{array}\right),$

and

${{\mathbf{H}}_{{{\mathbf{f}}_{\mathbf{1}}}}}=\left( \begin{align}  & \frac{{{\vartheta }_{j,r}}}{\begin{align}  & {{(-1)}^{j}} \\ & {{(-1)}^{j+1}}j(j+1) \\ & j(j+1) \\\end{align}} \\ &  \\\end{align} \right){{\mathbf{H}}_{{{\mathbf{f}}_{\mathbf{2}}}}}=\left( \begin{matrix}   \frac{{{\omega }_{j,r}}}{0}  \\   0  \\\end{matrix} \right)$

${{\mathbf{H}}_{{{\mathbf{F}}_{\mathbf{1}}}}}=\left( \begin{matrix}   \frac{{{\chi }_{j,r}}}{0}  \\   0  \\   0  \\\end{matrix} \right){{\mathbf{H}}_{{{F}_{2}}}}=\left( \begin{matrix}   \frac{{{\zeta }_{j,r}}}{j(j+1)}  \\   1  \\\end{matrix} \right)$

$c=\operatorname{span}\left\{c_j\right\}, \tilde{c}=\left\{\operatorname{span}\left\{c_i c_j\right\}, \operatorname{span}\left\{d_i d_j\right\}\right\},$

$\bar{c}=\left\{\operatorname{span}\left\{c_i c_j c_k\right\}, \operatorname{span}\left\{d_i d_j d_k\right\}\right\},$

$\left.b_1=\left(\begin{array}{llllllll}0 & 0 & 0 & 0 & 0 & 0 & \cdots & \alpha\left(\frac{1-n}{1+n}\right.\end{array}\right) \quad \eta_{\infty} \quad 0\right)^T,$

$b_2=\left(\begin{array}{llllllllll}0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0\end{array}\right)^T,$

$\vartheta_{j, r}=\frac{8}{\eta_{\infty}^3}\left\langle\frac{d^3 P_j(\zeta)}{d \zeta^3}, P_r(\zeta)\right\rangle$$-\left(\frac{4}{n+1} \frac{\ell \beta_v}{\eta_{\infty}}+\frac{2 \gamma}{\eta_{\infty}}\right)\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle,$    (21)

$\omega_{j, r}=-\frac{2 \beta_v}{\eta_{\infty}}\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle,$      (22)

$\chi_{j, r}=\frac{4}{n+1} \frac{\ell \beta_v}{\eta_{\infty}}\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle,$       (23)

$\varsigma_{j, r}=\frac{2 \beta_v}{\eta_{\infty}}\left\langle\frac{d P_j(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle,$       (24)

$\boldsymbol{\Psi}_{f_1}=\left\{\Lambda_{i, j, r}\right\}=-\frac{8 n}{\eta_{\infty}(n+1)}\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle$$+\frac{4}{\eta_{\infty}^2}\left\langle P_j(\zeta) \frac{d^2 P_i(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle$      (25)

$\begin{aligned} \boldsymbol{\Psi}_{\boldsymbol{F}_1}= & \left\{\kappa_{i, j, k, r}\right\}=\frac{4 n}{\eta_{\infty}^2}\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle \\ & -\frac{2(n+1)}{\eta_{\infty}^2}\left\langle P_j(\zeta) \frac{d^2 P_i(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle,\end{aligned}$

$\boldsymbol{\Psi}_{\boldsymbol{F}_{\mathbf{1}}}=\left\{\kappa_{i, j, k, r}\right\}=\frac{4 n}{\eta_{\infty}^2}\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta}, P_r(\zeta)\right\rangle$$-\frac{2(n+1)}{\eta_{\infty}^2}\left\langle P_j(\zeta) \frac{d^2 P_i(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle,$     (26)

$\boldsymbol{\Theta}_{f_1}=\left\{\Omega_{i, j, k, r}\right\}$$=\left(\frac{n-1}{2}\right)(\xi+1)\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta} \frac{d^2 P_k(\zeta)}{d \zeta^2}, P_r(\zeta)\right\rangle$$-\frac{2 n(n-1)}{n+1}\left\langle\frac{d P_j(\zeta)}{d \zeta} \frac{d P_i(\zeta)}{d \zeta} \frac{d P_k(\zeta)}{d \zeta}, P_r(\zeta), P_r(\xi)\right\rangle$

$+\frac{8 \beta}{\eta_{\infty}^3}\left((3 n-1)\left\langle P_j(\zeta) \frac{d P_i(\zeta)}{d \zeta} \frac{d^3 P_k(\zeta)}{d \zeta^3}, P_r(\zeta)\right\rangle-\frac{n+1}{2}\left\langle P_j(\zeta) P_i(\zeta) \frac{d^3 P_k(\zeta)}{d \zeta^3}, P_r(\zeta)\right\rangle\right).$      (27)

The preceding terms (21)-(27) are evaluated using theorems was discussed [34-37] by Fathy. Newton's iteration method used to obtain the solution of the nonlinear Eq. (20) of $3 M+3$ algebraic equations with $3 M+3$ unknowns $c_j$ and $d_j$. Solving the system of equations and substituting $c_j$ and $d_j$ in Eq. (16) may yield the asymptotic solutions. We need to use the inverse transformation $\zeta=\frac{2}{\eta_{\infty}} \eta-1$ in order to solve our problem. After that, we've identified an approximate solution to our problem.

The stretching surface profile (see Figure 1b) and stretching wall velocity are both defined by the index power $n$. The surface becomes thinner as the value of $n$ increases. The index power $n$ is also used to define the stretched wall velocity $U_w$. The parameter's impact on dimensionless velocity may be seen in Figure 2. By increasing the velocity index power $n$, the stretching surface velocity and the profile surface curvature is raised. As can be seen in Figure $2 \mathrm{~b}$, the momentum boundary layer gets thinner as the dimensionless velocity drops. This effect is reversed when $\eta>0.85$. Dusty particles' velocity drops in the same proportion. As a consequence, the amount of resistive force needed to keep the Maxwell dusty fluid from forming grows as this value is increased. For Newtonian dusty fluid, when the index power grows, the flow velocity decreases until $\eta=1.5$ then it rises with increasing the value of $n$, but the particle velocity falls as shown in Figure 2a.

2.png

Figure 2. (a) Effect of n for Newtonian dusty fluid and (b) Effect of n for Maxwell dusty fluid

Figure 3 discusses the results of an investigation into the relationship between the fluid flow velocity and the wall thickness parameter $\alpha$. As seen in Figure 3a, the boundary layer gets thinner and thinner as the velocity falls when $\alpha$ increases for $n<1$, but Figure 3b demonstrates that the boundary layer thickens as the velocity rises for $n>1$. Velocity of dusty particles has the similar pattern.

3.png

Figure 3. (a) Effect of α  for n<1 and (b) Effect of α for n>1

The elasticity parameter of the Deborah number is a dimensionless number that reflects the ratio of time scale flow to relaxation time flow. When determining the type of unstable flow, the time scale is used. A viscous fluid, for example, requires a certain amount of time to relax once shear tension has ceased to exert pressure on it. It can be shown in Figure 4 that increasing the elasticity parameter decreases the dimensionless velocity. This is due to the fact that increasing the viscosity of a fluid increases the fluid's resistance. Figure 4b shows that for n>1, the elasticity parameter has a somewhat larger effect than for n<1 (Figure 4a). Additionally, when the elasticity value is raised, dust particle velocity falls slightly.

4.png

Figure 4. (a) Effect of β for n<1 and (b) Effect of β for n>1

The dust phase's velocity rises while the fluid phase's velocity decreases as the value of $\beta_v$ increases in Figure 5. Because of the considerable contact between the fluid and particle phases, the particle phase may impose an opposing force on the fluid phase until the particle velocity is equal to the fluid velocity, allowing this effect to occur.

5.png

Figure 5. (a) Effect of $\beta_v$ for n<1 and (b) Effect of $\beta_v$ for n>1

Figures 6 demonstrate the impact of the porosity parameter on velocity profiles in the fluid and dust phases. Increasing the porosity parameter decreases the velocity profiles for both the fluid and dust phases in the figure. To explain this, raising the porosity parameter causes the pores in the porous material to enlarge, resulting in a decrease in velocity profiles.

6.png

Figure 6. (a) Effect of γ for n<1 and (b) Effect of γ for n>1

It is seen in Figure 7 that mass concentration of dust particles $\ell$ affects both the fluid and dust velocity profiles. The figure illustrates how increasing $\ell$ reduces the fluid and particle phase velocity profiles. As the mass concentration of dust particles rises, the velocity boundary layers are noticeably diminished.

7.png

Figure 7. (a) Effect of $\ell$ for n<1 and (b) Effect of $\ell$ for n>1