“…Moreover, the uniform magnetic field of strength is applied normal to a shrinking sheet. The governance model with all assumptions are as follows 20 – 22 : with boundary conditions 20 , 21 : Figure 1 Physical model of the problem. …”
Section: Mathematical Description Of Problemmentioning
confidence: 99%
“…In this study the thermophysical properties of nanomaterials, base fluid, and hybrid nanofluid are used. In relation to, Tables 1 and 2 are presented 20 , 21 .…”
Section: Mathematical Description Of Problemmentioning
confidence: 99%
“…The following similarity transformation variables will be adopted to convert the system of equations into ODEs 20 , 21 . where represents the stream function while the velocities are as and .…”
Section: Mathematical Description Of Problemmentioning
confidence: 99%
“…Haq et al 19 depicted a theoretical thermal visualization of hybrid nanofluid problem with permeable cylinder. The fractional based mathematical model for hybrid nanofluid has been worked out by Wang et al 20 . Lund et al 21 performed a stability measurement of nanofluid via shrinking surface.…”
Following to improved thermal impact of hybrid nanomaterials, wide range applications of such materials is observed in the thermal engineering, extrusion systems, solar energy, power generation, heat transfer devices etc. The hybrid nanofluid is a modified form of nanofluid which is beneficial for improving energy transfer efficiency. In current analysis, the solid nanoparticles aluminium ($$\phi_{{{\text{Al}}_{2} {\text{O}}_{3} }}$$
ϕ
Al
2
O
3
) and copper ($$\phi_{{{\text{Cu}}}}$$
ϕ
Cu
) have been mixed with water to produce a new hybrid nanofluid. The investigation of a steady two-dimensional mixed convection boundary layer flow of the resultant hybrid nanofluid on a vertical exponential shrunk surface in the existence of porous, magnetic, thermal radiation, velocity, and thermal slip conditions is carried out. Exponential similarity variables are adopted to transform the nonlinear partial differential equation into a system of ordinary differential equations which has been then solved by employing the shooting method in Maple software. The obtained numerical results such as coefficient of skin friction $$f^{\prime \prime } \left( 0 \right)$$
f
″
0
, heat transfer rate $$- \theta^{\prime } \left( 0 \right)$$
-
θ
′
0
, velocity $$f^{\prime } \left( \eta \right)$$
f
′
η
and temperature $$\left( {\theta \left( \eta \right)} \right)$$
θ
η
distributions are presented in the form of different graphs. The results revealed that duality exists in solution when the suction parameter $$S \ge S_{ci}$$
S
≥
S
ci
in assisting flow case. Due to non-uniqueness of solutions, a temporal stability analysis needs to be performed and the result indicates that the upper branch is stable and realizable compared to the lower branch.
“…Moreover, the uniform magnetic field of strength is applied normal to a shrinking sheet. The governance model with all assumptions are as follows 20 – 22 : with boundary conditions 20 , 21 : Figure 1 Physical model of the problem. …”
Section: Mathematical Description Of Problemmentioning
confidence: 99%
“…In this study the thermophysical properties of nanomaterials, base fluid, and hybrid nanofluid are used. In relation to, Tables 1 and 2 are presented 20 , 21 .…”
Section: Mathematical Description Of Problemmentioning
confidence: 99%
“…The following similarity transformation variables will be adopted to convert the system of equations into ODEs 20 , 21 . where represents the stream function while the velocities are as and .…”
Section: Mathematical Description Of Problemmentioning
confidence: 99%
“…Haq et al 19 depicted a theoretical thermal visualization of hybrid nanofluid problem with permeable cylinder. The fractional based mathematical model for hybrid nanofluid has been worked out by Wang et al 20 . Lund et al 21 performed a stability measurement of nanofluid via shrinking surface.…”
Following to improved thermal impact of hybrid nanomaterials, wide range applications of such materials is observed in the thermal engineering, extrusion systems, solar energy, power generation, heat transfer devices etc. The hybrid nanofluid is a modified form of nanofluid which is beneficial for improving energy transfer efficiency. In current analysis, the solid nanoparticles aluminium ($$\phi_{{{\text{Al}}_{2} {\text{O}}_{3} }}$$
ϕ
Al
2
O
3
) and copper ($$\phi_{{{\text{Cu}}}}$$
ϕ
Cu
) have been mixed with water to produce a new hybrid nanofluid. The investigation of a steady two-dimensional mixed convection boundary layer flow of the resultant hybrid nanofluid on a vertical exponential shrunk surface in the existence of porous, magnetic, thermal radiation, velocity, and thermal slip conditions is carried out. Exponential similarity variables are adopted to transform the nonlinear partial differential equation into a system of ordinary differential equations which has been then solved by employing the shooting method in Maple software. The obtained numerical results such as coefficient of skin friction $$f^{\prime \prime } \left( 0 \right)$$
f
″
0
, heat transfer rate $$- \theta^{\prime } \left( 0 \right)$$
-
θ
′
0
, velocity $$f^{\prime } \left( \eta \right)$$
f
′
η
and temperature $$\left( {\theta \left( \eta \right)} \right)$$
θ
η
distributions are presented in the form of different graphs. The results revealed that duality exists in solution when the suction parameter $$S \ge S_{ci}$$
S
≥
S
ci
in assisting flow case. Due to non-uniqueness of solutions, a temporal stability analysis needs to be performed and the result indicates that the upper branch is stable and realizable compared to the lower branch.
“…The theory based on Prabhakar fractional calculus [23] is expansively considered in current years. It takes differential equations modeled through Prabhakar operators, which are fascinating for their applied and pure mathematical characteristics [24][25][26] and due to applications in different fields such as anomalous dielectrics, viscoelasticity and options pricing [27][28][29]. It is pointed out that the operators of Prabhakar fractional calculus may be observed as exceptional cases of at least two classes of fractional operators: i.e., the class of operators along with analytic and Sonine kernels [30,31].…”
The functional implications of substances, such as retardation and relaxation, can be studied for magnetized diffusion coefficient based on the relative increase throughout magnetization is a well-known realization. In this context, we have explored the Oldroyd-B hybrid nanofluid flowing through a pored oscillating plate along with an inclined applied magnetics effect. The slipping effect and sinusoidal heating conditions are also supposed to be under consideration. An innovative and current classification of fractional derivatives, i.e., Prabhakar fractional derivative and Laplace transform, are implemented for the result of transformed leading equations. The graphical representation is also described to understand the physical implementation of all effecting parameters. In order to justify and physically examine the considered problem, some limiting cases, the rate of heat and mass transfer, and friction factors are also analyzed. As a result, we have concluded that the thermal enhancement can be improved more progressively with the interaction of silver-water-based nanofluid suspension compared to copper-nanoparticles mixed nanofluid. Furthermore, It has examined the impact of both parameters, i.e., time relaxation and retardation is opposite of the momentum field.
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