Regression analysis of squeezing-induced hybrid nanofluid flow in Darcy-Forchheimer porous medium

doi:10.1007/s10483-025-3202-9

Abstract

Abstract:

This article presents a mathematical model addressing a scenario involving a hybrid nanofluid flow between two infinite parallel plates. One plate remains stationary, while the other moves downward at a squeezing velocity. The space between these plates contains a Darcy-Forchheimer porous medium. A mixture of water-based fluid with gold (Au) and silicon dioxide (SiO₂) nanoparticles is formulated. In contrast to the conventional Fourier's heat flux equation, this study employs the Cattaneo-Christov heat flux equation. A uniform magnetic field is applied perpendicular to the flow direction, invoking magnetohydrodynamic (MHD) effects. Further, the model accounts for Joule heating, which is the heat generated when an electric current passes through the fluid. The problem is solved via NDSolve in MATHEMATICA. Numerical and statistical analyses are conducted to provide insights into the behavior of the nanomaterials between the parallel plates with respect to the flow, energy transport, and skin friction. The findings of this study have potential applications in enhancing cooling systems and optimizing thermal management strategies. It is observed that the squeezing motion generates additional pressure gradients within the fluid, which enhances the flow rate but reduces the frictional drag. Consequently, the fluid is pushed more vigorously between the plates, increasing the flow velocity. As the fluid experiences higher flow rates due to the increased squeezing effect, it spends less time in the region between the plates. The thermal relaxation, however, abruptly changes the temperature, leading to a decrease in the temperature fluctuations.

Key words: convective boundary condition, Darcy-Forchheimer medium, hybrid nanofluid, Joule heating, magnetohydrodynamic (MHD), numerical solution, squeezing flow, regression analysis

2010 MSC Number:

O361

K. MUHAMMAD, M. SARFRAZ. Regression analysis of squeezing-induced hybrid nanofluid flow in Darcy-Forchheimer porous medium. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 193-208.

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Figures/Tables 10

Fig. 1

Table 1

Thermal properties of nanoparticles (Au+SiO2) and the base fluid (H2O) at 20 °C (following Refs. [27] and [30])"

Material	$ρ / (kg ⋅ m − 3)$	$κ / (W ⋅ m − 1 ⋅ K − 1)$	$σ / (S ⋅ m − 1)$	$c p / (J ⋅ kg − 1 ⋅ K − 1)$	$P r$
H₂O	997	0.613	$5.5 × 10 − 6$	4 179	6.2
SiO₂	2 650	1.5	$1.0 × 10 − 18$	730	–
Au	19 300	318	$4.1 × 106$	129	–

Table 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 2

Response of the skin friction (CFRe) to variations in involved variables"

$M *$	$r sq$	$λ *$	$ϕ np1$	$F r *$	$ϕ np2$	$m *$	$− C F R e$
$M *$	$r sq$	$λ *$	$ϕ np1$	$F r *$	$ϕ np2$	$m *$	Au+SiO₂+water	Au+water
0.1	0.6	0.3	0.05	0.2	0.05	0.3	3.245 50	3.378 44
0.2	0.6	0.3	0.05	0.2	0.05	0.3	3.256 53	3.389 21
0.3	0.6	0.3	0.05	0.2	0.05	0.3	3.267 53	3.399 95
0.5	0.1	0.3	0.05	0.2	0.05	0.3	5.154 18	5.243 46
0.5	0.2	0.3	0.05	0.2	0.05	0.3	4.788 35	4.889 26
0.5	0.3	0.3	0.05	0.2	0.05	0.3	4.419 01	4.530 01
0.5	0.6	0.1	0.05	0.2	0.05	0.3	3.264 63	3.397 11
0.5	0.6	0.2	0.05	0.2	0.05	0.3	3.277 06	3.409 25
0.5	0.6	0.3	0.05	0.2	0.05	0.3	3.289 46	3.421 35
0.5	0.6	0.3	0.01	0.2	0.05	0.3	2.866 88	2.530 58
0.5	0.6	0.3	0.02	0.2	0.05	0.3	2.968 45	2.620 28
0.5	0.6	0.3	0.03	0.2	0.05	0.3	3.072 63	2.712 01
0.5	0.6	0.3	0.05	0.1	0.05	0.3	3.279 71	3.407 87
0.5	0.6	0.3	0.05	0.2	0.05	0.3	3.289 46	3.421 35
0.5	0.6	0.3	0.05	0.3	0.05	0.3	3.299 20	3.434 78
0.5	0.6	0.3	0.05	0.2	0.01	0.3	2.884 36	3.395 72
0.5	0.6	0.3	0.05	0.2	0.02	0.3	2.885 29	3.397 07
0.5	0.6	0.3	0.05	0.2	0.03	0.3	2.886 22	3.398 42
0.5	0.6	0.3	0.05	0.2	0.05	0.1	3.297 10	3.428 80
0.5	0.6	0.3	0.05	0.2	0.05	0.2	3.294 00	3.425 78
0.5	0.6	0.3	0.05	0.2	0.05	0.3	3.289 46	3.421 35

Table 2

Table 3

Regression analysis of the skin friction (CFRe) for rsq∈[0.1,0.5] and m∗∈[0.1,0.3] with varying Fr∗ for the hybrid nanofluid (Au+SiO2+water)"

$F r ∗$	$C$	$c 1$	$c 2$	$c 3$	$c 4$	$c 5$
0.1	5.611 53	$− 2.358 93 (T = − 0.555 1)$	$− 2.434 29 (T = − 0.285 6)$	2.873 530	11.147 400	$− 11.748 90$
0.2	5.591 80	$− 2.961 42 (T = − 0.733 5)$	$− 1.283 93 (T = − 0.158 5)$	0.622 552	1.158 860	$− 2.098 71$
0.3	5.612 53	$− 2.870 69 (T = − 0.713 5)$	$− 1.537 32 (T = − 0.190 4)$	$− 0.580 813$	$− 0.592 987$	1.081 71

Table 3

Table 4

Regression analysis of the skin friction (CFRe) for rsq∈[0.1,0.5] and m∗∈[0.1,0.3] with varying Fr∗ for the single-component nanofluid (Au+water)"

$F r ∗$	$C$	$c 1$	$c 2$	$c 3$	$c 4$	$c 5$
0.1	5.618 33	$− 3.384 40 (T = − 0.845 4)$	$− 0.066 195 5 (T = − 0.008 2)$	$− 2.132 360$	$− 9.567 37$	8.477 14
0.2	5.605 86	$− 2.455 43 (T = − 5.057 9)$	$− 1.515 370 0 (T = − 1.479 3)$	$− 0.617 931$	$− 2.146 15$	1.855 09
0.3	5.618 33	$− 3.384 40 (T = − 0.845 4)$	$− 0.066 195 5 (T = − 0.008 2)$	$− 2.132 360$	$− 9.567 37$	8.477 14

Table 4

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