Existence of the multiple exact solutions for nanofluid flow over a stretching/shrinking sheet embedded in a porous medium at the presence of magnetic field with electrical conductivity and thermal radiation effects

MHD flow and heat transfer over a permeable stretching/shrinking sheet in a hybrid nanofluid with a convective boundary condition

“…Comparison between the present results of Àf 00 0 ð Þ and Àu 0 0 ð Þ with those of Aly (2016), see Table III, when f 2 = 0, M = 0, Pr = 6.2, S = 0 and l = 1 (stretching sheet) 4.1 Studying the case of stretching sheet (l > 0) 4.1.1 Conditions to get the flow and temperature solutions. In this case, on studying the discriminant of equation ( 16) for any values of M and S, one can note that only the positive sign of the root that makes a 1 positive.…”

“…To solve the system ( 8) to ( 10), it should be mentioned that there are many numerical methods; see Kamyar et al (2012), Uddin et al (2014) and Aly (2016) and packages such as BVP4C in MATLAB or RFK45 in MAPLE. However, to the authors' knowledge, there is no software or technique to determine the critical values and regions on obtaining the solutions.…”

“…Further, Figure 8 indicates the temperature distributions of Al 2 O 3 /water vs Cu-Al 2 O 3 /water when l = 1 for various values of S. As discussed in Figure 5, increasing of S in suction/injection case slows down the velocity, which leads to decreasing the temperature also, as shown in Figure 8. The effects of Biot number (Bi) in the temperature distributions in suction and injection cases are On taking into account that the definition of Bi refers to a directly proportional to the heat transfer coefficient associated with the hot fluid h and the thermal resistance is inversely proportional to h, as a result, an increase in Biot number leads to increase efficiently u (h ), see Aly (2016). Further, the temperature of nanofluid is bigger than of hybrid nanofluid in both cases of S with remarkable difference in the injection case.…”

MHD Marangoni boundary layer problem for hybrid nanofluids with thermal radiation

“…It should be mentioned that in the present work, nanoparticles of alumina (Al 2 O 3 ) with fixed ϕ 1 = 0.1 are first added to the water to produce nanofluid (Al 2 O 3 /H 2 O). Then, copper (Cu) nanoparticles with 0 < ϕ 2 ≤ 0.1 are mixed to this nanofluid to get the HNF (Cu–Al 2 O 3 /H 2 O), such that 0 ≤ ( ϕ 1 + ϕ 2 ) ≤ 0.2 as introduced by Aly (2016, 2019) and Aly and Pop (2019). To validate the obtained results, Table 2 presents comparison between the results of Al-Mudhaf and Chamkha (2005) and the current ones for various values of M and S when ϕ 1 = ϕ 2 = 0 (regular fluid), K → ∞ and Pr = 0.78.…”

Flow and Heat Transfer Past a Stretching/Shrinking Sheet Using Modified Buongiorno Nanoliquid Model

“…Now, when λ > 0 (stretching sheet) and for any values of S, one can note that the positive sign of the second root makes β positive. This means that there exists only a unique solution for any combination of the considered parameters (see for e.g., Aly [36]). However, for the shrinking sheet (λ < 0), suppose that:…”