The new method of enhancing heat transfer through tri- hybrid nanofluid is discussed in the current study and represented in differential equation system.

2.  Presenter: Muhammad Waqar
MS Mathematics
Session 2021-23
 Supervisor: Dr. Muhammad Amjad
Department of Mathematics
COMSATS University Islamabad (Vehari Campus)

3.  Abstract
 Basic Definitions
 Literature Review
 Statement of Problems
 References

4. Abstract

5. The new method of enhancing heat transfer through tri- hybrid
nanofluid is discussed in the current study and represented in
differential equation system. Tri hybrid nanofluid is formed by
combining three different nanoparticles CuO , TiO2 and Al2O3
with water and CuO-TiO2-H2O tri hybrid nanofluid is formed by
that combination.
The properties of fluid like density ,thermal conductivity ,diffusion
and heat capacitance , etc. will be discussed mathematically .
The governing partial differential equations(PDEs) will transform
into ordinary differential equations(ODEs) and then solved by
using the Runge-Kutta-Fehlberg(RKF) method. The results will be
discussed graphically.

6. Basic Definitions

7. “The movement of heat across the border of the
system due to a difference in temperature between
the system and its surroundings.”
Conduction
Radiation
Convection

8. Fluid
A substance that has no fixed shape i.e. gas, plasma, or
liquid is called fluid. It adopts the shape of the
container in which it is contained. The shape of the
fluid can be changed by using outer stress.

9. Nano fluid
Nanofluid contains very small-sized particles in
Nanometer-sized. In a base fluid, these types of fluids
are made of colloidal suspensions of the nanoparticles.

10. Hybrid Nanofluid
It can be defined as the mixing of nanoparticles with
different physical and chemical bonds forms a nanofluid
called a hybrid nanofluid

11. Tri-hybrid Nanofluid
The tri-hybrid nanofluid is devolved by three different
nanomaterials with different physical properties and
chemical bonds.

12. Consider a flow on stretching sheet through
incompressible tri hybrid nanofluid TiO2-SiO2-
Al2O3-H2O which is formed by combining TiO2,
Al2O3 and SiO2 with water by velocity q with
having components (u, v) along the directions
(x, y).
The physical configuration is defined by using
Cartesian coordinates system and the sheet is
supposed to be stretching with speed Ux =ax and
temperature is of sheet is organized by T∞ and
the temperature of tri hybrid nanofluid is
defined by T described in fig.

15.  As boundary layer flow and heat transfer of fluid find many applications in industrial usage, the
researcher was motivated to find more sources to enhance the heat transfer over the stretching sheet,
etc. Different researchers discussed the heat transfer of fluid theoretically and numerically over the
drawing of a polymer sheet, cooling a metallic plate of a bath, and the stretching sheet, etc. Industries
caused global warming, and air pollution and became a global issue. They worked to find the sources
of heat transfer that can transfer more heat as well as less environmental pollution. Therefore, the
enhancement in heat transfer properties and heat conducting abilities of tri-hybrid nano fluid motivated
the researchers to consider tri-hybrid nano fluid in industrial applications. In this view, Sakiadis [1]
studied the boundary layer flow on a continuous solid surface which is different from boundary layer
flow on a surface having a finite length. He discussed the behavior of boundary layer flow on the
continuous surface and also derived the differential and integral equations of boundary layer flow for
the such surface.

16.  Equations solved on moving continuous surface and cylindrical moving surface for both laminar and
turbulent flow on boundary layer and it was experimentally supported by Tsou et al. [2]. They also
discussed flow on moving surfaces analytically. They suggested its ability to drive equations in
solution. Rollins et al. [3] discussed the heat transfer properties of a second-order fluid flowing past a
stretching sheet in two cases e.g (i) the sheet with prescribed surface temperature (PST-case) and (ii)
the sheet with prescribed surface heat flux (PHF-case). The solution and heat transfer properties were
computed by using Kummir‘s function. They further concluded that no solution exists for smaller
values of the Prandtl number ()while the opposite trend for larger values of for both PST and PHF
cases. Sheridan et al. [4] discussed the MHD flow on an unsteady stretching sheet numerically by
transforming the PDEs into ODEs and solved the obtained equation by using Keller Box Method. They
further discussed the effects of magnetic parameters on the velocity of flow and concluded that
enhancement in the values of parameters decreases the velocity of flow.

17.  Puneethet al. [5] investigated the induced magnetic field that results from the three-dimensional bio
convective flow of a Casson nanofluid containing gyrotactic microorganisms along a vertical stretching
sheet. These microorganisms’ motion leads to bio convection and they serve as bioactive mixers that
aid in stabilizing nanoparticle suspension. Abel et al. [6] discussed the heat flow and temperature on
the non-isothermal stretching sheet with variable viscosity. They transformed the proposed model into
ODEs and solved using the 4th order RKF-method. They analyzed the effects of fluid viscosity and
other various parameters like Visco- fluid, etc for two cases of PST and PHF. Keblinskia et al. [7]
discussed the heat flow over the stretching sheet for various heat fluxes subjected to injection and
suction. The formed equations were transformed into ODEs and solved by RKF-Method with various
parameters and . They also discussed the effects of these parameters. They resulted that heat flow on
the surface is increased because of injection and decreased because of suction.

18.  [8] discussed the factors that affect the rate of flow of the hybrid nanofluid. They concluded that the
performance of the nanofluid system increased by different factors. Manjunatha et al. [9] explored how
we can increase the flow of heat through the hybrid nanofluid deviation of the general properties of the
general fluid in the presence of Lorentz force on the flow. They used the RKF method to solve the
equations numerically. They further analyzed the consequence of the temperature-dependent
viscosity onward with free convection and magnetic parameter on an increase in heat flow in a
boundary layer field with the assistance of a hybrid nanofluid. Manjuntha et al. [10] analyzed the
complication magnetohydrodynamic (MHD) flow and heat flow of the viscous, incompressible, and
electronically dominant dusty flow through the unsteady stretching sheet numerically. The derived
equations are transformed in ODEs that are solved by the RKF method. The results were presented
for certain parameters including the Nusselt number (), skin coefficient of friction, and different flow
parameters included for both VWT and VHF.

19.  Tayebi et al. [11] studied the consequence of a magnetic field on the decay formation and natural
convection inside the cage through the hybrid nanofluid containing conducting wavy solid block. They
further investigated the effects of fluid-solid in the research. Ghalambaz et al. [12] studied the hybrid
nanofluid (/water) in conjugate natural convection. The physical model is described in terms of PDEs
and further converted into a dimensionless form solved by using the finite element method (FEM). The
results depictedthe heat flow being enhanced by adding hybrid nanoparticles in the convection regime
(low Rayleigh number).
 Khanet al. [13] discussed the heat flow in nanofluid over the stretching sheet numerically. This was the
1st research paper in which working on stretching sheets in nanofluid was discussed. They purposed
similarity equation solutions that depend on the variation of and Sherwood numbers on the values of
constant numbers like , Lewis number (), Brownian number (), and thermophoresis number (). They
discussed the values function in graphical forms. They concluded that is decreasing function on the
different values of constants and the Sherwood number is increasing on the greater values of .
Various researchers work to enhance the heat flow rates.

20.  Manjunatha et al. [14] purposed a thermotical model for the enhancement of heat flow using tri-hybrid
nanofluid by using a combination of nanoparticles and with water and formed a hybrid nanofluid
which decomposed the harmful particles' environmental purity and other various appliance used for
cooling. They presented the thermal conductivity and specific heat capacitance by mathematical
equations that are transformed in ODEs and further the mathematical equations are solved by using
the mathematical RKF method. They represented the results using graphs that clearly observed tri
hybrid nanofluid can transfer more heat than hybrid nanofluid.

21. 1. Sakiadis, B. C. (1961). Boundary‐layer behavior on continuous solid surfaces: I. Boundary‐layer equations for two‐dimensional and
axisymmetric flow. AIChE Journal, 7(1), 26-28.
2. Tsou, F. K., Sparrow, E. M., & Goldstein, R. J. (1967). Flow and heat transfer in the boundary layer on a continuous moving
surface. International Journal of Heat and Mass Transfer, 10(2), 219-235.
3. Rollins, D., & Vajravelu, K. (1991). Heat transfer in a second-order fluid over a continuous stretching surface. Acta
mechanica, 89(1), 167-178.
4. Sharidan, S., Mahmood, M., & Pop, I. (2006). Similarity solutions for the unsteady boundary layer flow and heat transfer due to a
stretching sheet. Applied Mechanics and Engineering, 11(3), 647.
5. Puneeth, V., Manjunatha, S., Gireesha, B. J., & Gorla, R. S. R. (2021). Magneto convective flow of Casson nanofluid due to Stefan
blowing in the presence of bio-active mixers. Proceedings of the Institution of Mechanical Engineers, Part N: Journal of
Nanomaterials, Nanoengineering and Nanosystems, 235(3-4), 83-95.
6. Abel, M. S., Khan, S. K., & Prasad, K. V. (2002). Study of visco-elastic fluid flow and heat transfer over a stretching sheet with
variable viscosity. International journal of non-linear mechanics, 37(1), 81-88.
7. Elbashbeshy, E. M. (1998). Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied
Physics, 31(16), 1951.
8. Sakiadis, B. C. (1961). Boundary‐layer behavior on continuous solid surfaces: I. Boundary‐layer equations for two‐dimensional and
axisymmetric flow. AIChE Journal, 7(1), 26-28.
9. Tsou, F. K., Sparrow, E. M., & Goldstein, R. J. (1967). Flow and heat transfer in the boundary layer on a continuous moving
surface. International Journal of Heat and Mass Transfer, 10(2), 219-235.
10. Rollins, D., & Vajravelu, K. (1991). Heat transfer in a second-order fluid over a continuous stretching surface. Acta
mechanica, 89(1), 167-178.
11. Sharidan, S., Mahmood, M., & Pop, I. (2006). Similarity solutions for the unsteady boundary layer flow and heat transfer due to a
stretching sheet. Applied Mechanics and Engineering, 11(3), 647.
12. Puneeth, V., Manjunatha, S., Gireesha, B. J., & Gorla, R. S. R. (2021). Magneto convective flow of Casson nanofluid due to Stefan
blowing in the presence of bio-active mixers. Proceedings of the Institution of Mechanical Engineers, Part N: Journal of
Nanomaterials, Nanoengineering and Nanosystems, 235(3-4), 83-95.
13. Abel, M. S., Khan, S. K., & Prasad, K. V. (2002). Study of visco-elastic fluid flow and heat transfer over a stretching sheet with
variable viscosity. International journal of non-linear mechanics, 37(1), 81-88.
14. Elbashbeshy, E. M. (1998). Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied
Physics, 31(16), 1951.