Oblique Stagnation Point Flow of Nanofluids over Stretching/Shrinking Sheet with Cattaneo–Christov Heat Flux Model: Existence of Dual Solution
Abstract
:1. Introduction
2. Basic Equations
3. Results and Discussion
4. Conclusions
- Increasing the values of suction parameter gradually decreases the rate of heat transfer in a fluid both for first and second solutions. This rate is maximum when the sheet is shrunken and minimum when sheet is stretched.
- Increasing the concentration of nanoparticles increases the rate of heat transfer in a fluid. Here has same effect both on first and second solution and is maximum for shrinking sheet as compared to stretching sheet.
- It is notified that magnifying values of thermal relaxation parameter , only leads to a decrease in the rate of heat transfer and vice versa. Since we have one way coupling of momentum equation and temperature equation, therefore, thermal relaxation parameter γ, does not influence the skin friction coefficient, also it has no effect on the critical values of stretching/shrinking parameter, i.e., .
- The increasing values of and leads to an increase of .
- The local Nusselt number decreases with positive values of (stretching sheet), however it increases with the negative values of (shrinking sheet).
- decreases with high values of mass suction .
- The streamlines pattern for three different values of free stream parameter over a shrinking surface shows that both positive and negative values of increases the obliquity of flow toward the left or right of the origin, but for orthogonal stagnation flow has been realized.
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Symbols | Meaning and Dimensions | Dimensionless |
x, y | Spatial coordinates (L) | |
u, v | Velocity components (L/T) | , |
p | Pressure field (ML/ | p |
Stream function (/ | ||
Normal component of the flow | ||
Shear component of flow | ||
Density of nanofluids | _ | |
Density of base fluid and solid fraction ( | _ | |
Dynamic viscosity of nanofluid | ||
Dynamic viscosity of base fluid and solid fraction ( | _ | |
Thermal relaxation time | ||
Kinematic viscosity of nanofluid | _ | |
Kinematic viscosity of base fluid and solid fraction ( | _ | |
Thermal conductivity of nanofluids | _ | |
Thermal conductivity of nanoparticles and base fluid () | _ | |
Heat capacity of nanofluids | _ | |
_ | Prandtl number | |
Heat capacity of nanoparticles and base fluid | _ | |
Thermal diffusivity of nanofluids | _ | |
Thermal diffusivity of nanoparticle and base fluid ( | _ | |
Electrical conductivity of nanofluids | _ | |
Electrical conductivity solid fraction and base fluid ( | _ | |
_ | Skin friction coefficient | |
_ | Nusselt’s number | Nu |
Boundary layer control parameters (L) | , | |
Reference and ambient temperature | _ | |
_ | Reynolds number | |
_ | Stretching/shrinking parameter | |
_ | Nanoparticles concentration |
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Thermophysical Properties | Cu | Pure Water |
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Li, X.; Khan, A.U.; Khan, M.R.; Nadeem, S.; Khan, S.U. Oblique Stagnation Point Flow of Nanofluids over Stretching/Shrinking Sheet with Cattaneo–Christov Heat Flux Model: Existence of Dual Solution. Symmetry 2019, 11, 1070. https://doi.org/10.3390/sym11091070
Li X, Khan AU, Khan MR, Nadeem S, Khan SU. Oblique Stagnation Point Flow of Nanofluids over Stretching/Shrinking Sheet with Cattaneo–Christov Heat Flux Model: Existence of Dual Solution. Symmetry. 2019; 11(9):1070. https://doi.org/10.3390/sym11091070
Chicago/Turabian StyleLi, Xiangling, Arif Ullah Khan, Muhammad Riaz Khan, Sohail Nadeem, and Sami Ullah Khan. 2019. "Oblique Stagnation Point Flow of Nanofluids over Stretching/Shrinking Sheet with Cattaneo–Christov Heat Flux Model: Existence of Dual Solution" Symmetry 11, no. 9: 1070. https://doi.org/10.3390/sym11091070
APA StyleLi, X., Khan, A. U., Khan, M. R., Nadeem, S., & Khan, S. U. (2019). Oblique Stagnation Point Flow of Nanofluids over Stretching/Shrinking Sheet with Cattaneo–Christov Heat Flux Model: Existence of Dual Solution. Symmetry, 11(9), 1070. https://doi.org/10.3390/sym11091070