Dynamical responses of inclined heated channel of MHD dusty fluids through porous media

Abstract

In this study, we analyze the unsteady laminar flow of two different layers of an incompressible, conducting Maxwell dusty fluids between two inclined parallel substrates, with heat transfer through porous media. Two constant but different temperatures are maintained at the lower and the upper planes. The dust particles are supposed to have a spherical form and uniform in size, and the number density of these is assumed to be constant during the flow process. The governing relations and the associated constraints give rise to the evaluation relations for the flow function and heat distribution, for the fluids and the dusty particles. In dealing with this issue, the long wave motion is considered for small wave numbers. The stability process is carried out, in the light of growth rate and neutral curves, in which stability and instability zones are identified. That is, numerical visualizations are carried out to graphically characterize the linear juncture of the interface progress. Depending on the selected physical parameters, it has been concluded that both the Hartmann number and the velocity relaxation time parameter have an effect depending on the order of the fluid layers. Some of the results of the investigation are the width of the channel plays an important and significant role in stabilizing the movement of flow, while it was found that the density ratio of the upper film to the lower layer helps stabilize the system unlike the effect of thermal conductivity ratio between top and bottom layers.

Appendix

I: The coefficients which appear in solution (28) are

$$\begin{aligned} a_1^{(1)}= & {} -{\mathbb {L}}^{(1)},\\ a_2^{(1)}= & {} \frac{1}{a_0}\Big \{ {\mathbb {L}}^{(1)} {\mathcal {H}}^{(1)} \sinh \big ( r{\mathcal {H}}^{(1)} \big ) \sinh \big ( (r-1){\mathcal {H}}^{(2)} \big )-\mu {\mathcal {H}}^{(2)} \cosh \big ( (r-1){\mathcal {H}}^{(2)} \big )\\&\times \big [ {\mathbb {L}}^{(1)} \big (\cosh ( r{\mathcal {H}}^{(1)})-1\big )+{\mathbb {L}}^{(2)}\big ] + \mu {\mathbb {L}}^{(2)} {\mathcal {H}}^{(2)} \Big \}, \\ {\mathbb {L}}^{(j)}= & {} \frac{\mu ^{1-j} \rho ^{j-1} Da^{(j)}}{1+Da^{(j)} {Ha^{(j)}}^2}, ~~~ {\mathcal {H}}^{(j)}=\sqrt{{Ha^{(j)}}^2 +\frac{1}{Da^{(j)}} },\\ a_0= & {} {\mathcal {H}}^{(1)} \cosh \big ( r{\mathcal {H}}^{(1)} \big ) \sinh \big ( (r-1){\mathcal {H}}^{(2)} \big )-\mu {\mathcal {H}}^{(2)}\cosh \big ( (r-1){\mathcal {H}}^{(2)} \big ) \sinh \big ( r{\mathcal {H}}^{(1)} \big ) ,\\ a_1^{(2)}= & {} \frac{1}{a_0}\Bigg \{ {\mathbb {L}}^{(2)}\Big [ {\mathcal {H}}^{(1)} \cosh (r {\mathcal {H}}^{(1)})(\sinh ( {\mathcal {H}}^{(2)})-\sinh (r {\mathcal {H}}^{(2)}))+\mu {\mathcal {H}}^{(2)}\cosh (r {\mathcal {H}}^{(2)})\sinh (r {\mathcal {H}}^{(1)}) \Big ] ,\\&-2{\mathbb {L}}^{(1)}{\mathcal {H}}^{(1)}\sinh ({\mathcal {H}}^{(2)}) \sinh ^2 \Bigg (\frac{r {\mathcal {H}}^{(1)}}{2}\Bigg )\Bigg \} ,\\ a_2^{(2)}= & {} \frac{1}{a_0}\Bigg \{ {\mathbb {L}}^{(2)}\Big [ {\mathcal {H}}^{(1)} \cosh (r {\mathcal {H}}^{(1)})\Big (\cosh (r {\mathcal {H}}^{(2)})-\cosh ( {\mathcal {H}}^{(2)})\Big )-\mu {\mathcal {H}}^{(2)}\sinh (r {\mathcal {H}}^{(1)})\sinh (r {\mathcal {H}}^{(2)}) \Big ] ,\\&+{\mathbb {L}}^{(1)}{\mathcal {H}}^{(1)}\cosh ({\mathcal {H}}^{(2)}) (\cosh (r{\mathcal {H}}^{(2)})-1)\Bigg \} . \end{aligned}$$

II: The parameters given in Eq. (81) are

$$\begin{aligned} {\mathbb {B}}_{10[0]}^{(j)}= & {} \frac{\text {i}}{8Da^{(j)}{{\mathcal {H}}^{(j)}}^3\ell _{m}^{(j)} }\Big \{10 {\mathcal {H}}^{(j)}\Big [ a_1^{(j)} Da^{(j)} {\mathbb {A}}_{1[0]}^{(j)}\big (\text {Re}^{(j)}\ell _{m}^{(j)}-{\mathfrak {N}}^{(j)} \big )+ {\mathbb {A}}_{3[0]}^{(j)}\big [ \ell _{m}^{(j)}Da^{(j)}\big (\text {Re}^{(j)}(c_{[0]}\\&-{\mathbb {L}}^{(j)} )+c_{[0]}De^{(j)} {Ha^{(j)}}^2 \big ) +(c_{[0]}-{\mathbb {L}}^{(j)} )\big (3+Da^{(j)}(3{Ha^{(j)}}^2+{\mathfrak {N}}^{(j)}-3{{\mathcal {H}}^{(j)}}^2)\big ) \big ] \Big ] \\&- a_2^{(j)}Da^{(j)}{\mathbb {A}}_{2[0]}^{(j)}\big (3 {\mathfrak {N}}^{(j)}+17\text {Re}^{(j)} \ell _{m}^{(j)} \big )\Big \},\\ {\mathbb {B}}_{11[0]}^{(j)}= & {} \frac{\text {i}}{4Da^{(j)}{{\mathcal {H}}^{(j)}}^2\ell _{m}^{(j)} }\Big \{2 {\mathcal {H}}^{(j)}\Big [ a_2^{(j)} Da^{(j)} {\mathbb {A}}_{1[0]}^{(j)}\big (\text {Re}^{(j)}\ell _{m}^{(j)}-{\mathfrak {N}}^{(j)} \big )+ {\mathbb {A}}_{4[0]}^{(j)}\big [ \ell _{m}^{(j)}Da^{(j)}\big (\text {Re}^{(j)}({\mathbb {L}}^{(j)}\\&- c_{[0]})+c_{[0]}De^{(j)} {Ha^{(j)}}^2 \big ) +(c_{[0]}-{\mathbb {L}}^{(j)} )\big (3+Da^{(j)}(3{Ha^{(j)}}^2+{\mathfrak {N}}^{(j)}-3{{\mathcal {H}}^{(j)}}^2)\big ) \big ] \Big ] \\&- a_1^{(j)}Da^{(j)}{\mathbb {A}}_{2[0]}^{(j)}\big ({\mathfrak {N}}^{(j)}-5\text {Re}^{(j)} \ell _{m}^{(j)} \big )\Big \},\\ {\mathbb {B}}_{12[0]}^{(j)}= & {} \frac{\text {i}a_2^{(j)}{\mathbb {A}}_{2[0]}^{(j)} \big ( {\mathfrak {N}}^{(j)}-\ell _{m}^{(j)} \text {Re}^{(j)} \big )}{4\ell _{m}^{(j)} {{\mathcal {H}}^{(j)}} },\\ {\mathbb {B}}_{20[0]}^{(j)}= & {} {\mathbb {B}}_{10[0]}^{(j)} ~ \text {at } ~ \{a_1^{(j)}\rightarrow a_2^{(j)}, {\mathbb {A}}_{3[0]}^{(j)}\rightarrow {\mathbb {A}}_{4[0]}^{(j)}, a_2^{(j)}\rightarrow a_1^{(j)} \},\\ {\mathbb {B}}_{21[0]}^{(j)}= & {} {\mathbb {B}}_{11[0]}^{(j)} ~ \text {at } ~ \{a_2^{(j)}\rightarrow a_1^{(j)}, {\mathbb {A}}_{4[0]}^{(j)}\rightarrow {\mathbb {A}}_{3[0]}^{(j)}, a_1^{(j)}\rightarrow a_2^{(j)}\},\\ {\mathbb {B}}_{22[0]}^{(j)}= & {} \frac{a_1^{(j)}}{a_2^{(j)} }{\mathbb {B}}_{12[0]}^{(j)},\\ {\mathbb {C}}_{1[0]}^{(j)}= & {} \frac{\text {i}\big (a_1^{(j)} {\mathbb {A}}_{3[0]}^{(j)}+a_2^{(j)} {\mathbb {A}}_{4[0]}^{(j)} \big )\big [ 3+Da^{(j)}\big ( 3{Ha^{(j)}}^2+4{\mathfrak {N}}^{(j)}-3 {{\mathcal {H}}^{(j)}}^2 \big ) \big ]}{24Da^{(j)}{{\mathcal {H}}^{(j)}}^2 \ell _{m}^{(j)} },\\ {\mathbb {C}}_{2[0]}^{(j)}= & {} \frac{\big (a_2^{(j)} {\mathbb {A}}_{3[0]}^{(j)}+a_1^{(j)} {\mathbb {A}}_{4[0]}^{(j)} \big )}{\big (a_1^{(j)} {\mathbb {A}}_{3[0]}^{(j)}+a_2^{(j)} {\mathbb {A}}_{4[0]}^{(j)} \big )}{\mathbb {C}}_{1[0]}^{(j)},\\ {\mathbb {C}}_{3[0]}^{(j)}= & {} \frac{3\text {i}\big (a_2^{(j)} {\mathbb {A}}_{4[0]}^{(j)}-a_1^{(j)} {\mathbb {A}}_{3[0]}^{(j)} \big )\big [ 1+Da^{(j)}\big ( {Ha^{(j)}}^2- {{\mathcal {H}}^{(j)}}^2 \big ) \big ]}{4Da^{(j)} \ell _{m}^{(j)} }. \end{aligned}$$

III: The calculations of the coefficients of Eq. (82) are

$$\begin{aligned} {\mathcal {B}}_{10[0]}^{(j)}= & {} \text {i}Pe^{(j)}{{\mathcal {H}}^{(j)}}^{-3}(1+D_c^{(j)} D_\rho ^{(j)})\Bigg [ {\mathcal {H}}^{(j)}\big ( a_1^{(1)}{\mathcal {B}}_2^{(j)}- b _1^{(j)}{\mathbb {A}}_{3[0]}^{(j)} \big )-2 a_2^{(1)} {\mathcal {B}}_1^{(j)}\Bigg ],\\ {\mathcal {B}}_{11[0]}^{(j)}= & {} \text {i}Pe^{(j)}{{\mathcal {H}}^{(j)}}^{-2}a_1^{(j)}{\mathcal {B}}_1^{(j)}\big (1+D_c^{(j)} D_\rho ^{(j)}\big ),\\ {\mathcal {B}}_{20[0]}^{(j)}= & {} {\mathcal {B}}_{10[0]}^{(j)}~\text {at}~\{a_1^{(j)}\rightarrow a_2^{(j)}, {\mathbb {A}}_{3[0]}^{(j)}\rightarrow {\mathbb {A}}_{4[0]}^{(j)}, a_2^{(j)}\rightarrow a_1^{(j)} \},\\ {\mathcal {B}}_{21[0]}^{(j)}= & {} \text {i}Pe^{(j)}{{\mathcal {H}}^{(j)}}^{-2}a_2^{(j)}{\mathcal {B}}_1^{(j)}\big (1+D_c^{(j)} D_\rho ^{(j)}\big ),\\ {\mathcal {C}}_{1[0]}^{(j)}= & {} \frac{\text {i}}{2}Pe^{(j)}\big (1+D_c^{(j)} D_\rho ^{(j)}\big ) \Bigg [{\mathcal {B}}_2^{(j)}({\mathbb {L}}^{(j)}-c_{[0]})- b _1^{(j)} {\mathbb {A}}_{1[0]}^{(j)}\Bigg ] ,\\ {\mathcal {C}}_{2[0]}^{(j)}= & {} \frac{\text {i}}{6}Pe^{(j)}\big (1+D_c^{(j)} D_\rho ^{(j)}\big ) \Bigg [{\mathcal {B}}_1^{(j)}({\mathbb {L}}^{(j)}-c_{[0]})- b _1^{(j)} {\mathbb {A}}_{2[0]}^{(j)}\Bigg ], \end{aligned}$$

where

$$\begin{aligned} b_1^{(1)}= & {} \frac{\kappa }{r (1-\kappa )-1}, \qquad b_1^{(2)} =b_1^{(1)}/\kappa , \qquad b_2^{(1)} =1, b_2^{(2)} =-b_1^{(2)}, \\ {\mathcal {B}}_1^{(1)}= & {} \frac{\kappa {\hat{h}}_{[0]} }{1+r (\kappa -1)}\left( \frac{\text {d}T_{0f}^{(2)}}{\text {d}y}-\frac{\text {d}T_{0f}^{(1)}}{\text {d}y}\right) \Big |_{y=r},\qquad {\mathcal {B}}_1^{(2)}={\mathcal {B}}_1^{(1)}/\kappa ,\qquad {\mathcal {B}}_2^{(1)}=0, \qquad {\mathcal {B}}_2^{(2)}=-{\mathcal {B}}_1^{(2)}. \end{aligned}$$