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- 1. Welcome Date : 21.10.2016 Time : 2.30 pm 18 December 2017 Jayachandra Babu M (Colloquium) 1
- 2. Research Supervisor Dr. N. Sandeep, Assistant Professor, Department of Mathematics, VIT, Vellore. By Jayachandra Babu M, Research Scholar, 14PHD0216 (EPT). NUMERICAL STUDY OF SOME MHD FLOWS OVER A STRETCHING SURFACE 18 December 2017 Jayachandra Babu M (Colloquium) 2
- 3. Chapter 1 : Introduction and background Chapter 2 : Effect of nonlinear thermal radiation on non-aligned bio-convective stagnation point flow of a nanofluid over a stretching sheet Chapter 3 : MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion Chapter 4 : Magnetohydrodynamic dissipative flow across a slendering stretching sheet with temperature dependent viscosity Chapter 5 : Three-dimensional MHD slip flow of nanofluids over a slendering stretching sheet with thermophoresis and Brownian motion Chapter 6 : 3D MHD flow of ferrofluid over a slendering stretching sheet with thermophoresis and Brownian motion Content 18 December 2017 Jayachandra Babu M (Colloquium) 3
- 4. Chapter - 1 Introduction and Background 18 December 2017 Jayachandra Babu M (Colloquium) 4
- 5. Motivation to study the MHD Flow: Magnetohydrodynamics (MHD) is the study of the magnetic properties of electrically conducting fluids. The electrically conducting non-Newtonian fluids can be opted as a cooling liquid as their flow can be regulated by external magnetic field, which regulates the heat transfer to some extent. The interaction between the conducting fluid and the magnetic field radically modifies the flow, with consequent effects on such important flow properties as pressure drop and heat transfer Examples: plasmas, liquid metals, and salt water or electrolytes. Applications: Useful in Cooling of Nuclear Reactors (Nuclear fission reactors are often cooled by liquid sodium. Liquid sodium is often pumped around using electromagnetic forces), wireless mouse, magnetic reasoning scanning, aeronautical plasma flows, chemical engineering and electronics, magnetic material processing, geophysics and control of the cooling rate etc. 18 December 2017 Jayachandra Babu M (Colloquium) 5
- 6. 18 December 2017 Jayachandra Babu M (Colloquium) 6 Classification Of Fluid Flows Viscous flows and inviscid flows: significance of viscosity Compressible and incompressible flows : change in density with the change in pressure Laminar and Turbulent flows : The highly ordered/disordered fluid motion Steady and unsteady flows: change in flow conditions (velocity, cross section ..) with time Natural and forced flows : fluid motion is due to natural means or forced
- 7. One, Two, and Three-Dimensional Flows • A flow field is best characterized by its velocity distribution. • A flow is said to be one-, two-, or three-dimensional if the flow velocity varies in one, two, or three dimensions, respectively. • However, the variation of velocity in certain directions can be small relative to the variation in other directions and can be ignored. The development of the velocity profile in a circular pipe. V = V(r, z) and thus the flow is two-dimensional in the entrance region, and becomes one-dimensional downstream when the velocity profile fully develops and remains unchanged in the flow direction, V = V(r). 18 December 2017 Jayachandra Babu M (Colloquium) 7
- 8. 18 December 2017 Jayachandra Babu M (Colloquium) 8
- 9. Newtonian Fluid Newtonian fluid is a fluid in which the velocity gradient is directly proportional to the shear stress. Though many fluids are Newtonian fluids over a wide range of temperatures and pressures, there are some fluids which are having the departure from the simple Newtonian relationships. They are called non- Newtonian fluids. 18 December 2017 Jayachandra Babu M (Colloquium) 9 𝜏𝑦𝑧 = −𝜇 𝑑𝑢 𝑑𝑦
- 10. Non-Newtonian Fluids 18 December 2017 Jayachandra Babu M (Colloquium) 10
- 11. Dilatant fluid: The fluid in which the viscosity increases as the velocity gradient increases. Ex: pastes, suspensions like corn starch in water Pseudoplastic fluid: The fluid in which the viscosity decreases as the velocity gradient increases. Ex: Nail polish, blood, paint Bingham fluid: A viscous fluid that possesses a yield strength which must be exceeded before the fluid will flow. Ex: Lava 18 December 2017 Jayachandra Babu M (Colloquium) 11
- 12. • Nanomaterials can be metals, ceramics, polymeric materials or composite materials whose size is in the range of 1-100 nanometers (nm). • Nanofluids are suspensions of metallic or nonmetallic nano powders in base liquid and can be employed to increase heat transfer rate in various applications 18 December 2017 Jayachandra Babu M (Colloquium) 12 Nanomaterials and fluids
- 13. 18 December 2017 Jayachandra Babu M (Colloquium) 13
- 14. Heat Transfer Heat is a form of energy transfer from a high temperature location to a low temperature location. The three main methods of heat transfer - conduction, convection and radiation. The rate of heat transfer is of great importance because of the frequent need to either increase or decrease the rate at which heat flows between two locations. Ex: electricity generation. Household electricity is most frequently manufactured by using fossil fuels or nuclear fuels. The method involves generating heat in a reactor. The heat is transferred to water and the water carries the heat to a steam turbine (or other type of electrical generator) where the electricity is produced. The challenge is to efficiently transfer the heat to the water and to the steam turbine with as little loss as possible. Attention must be given to increasing heat transfer rates in the reactor and in the turbine and decreasing heat transfer rates in the pipes between the reactor and the turbine. 18 December 2017 Jayachandra Babu M (Colloquium) 14
- 15. Mass Transfer Mass transfer is the net movement of mass from one location, usually meaning stream, phase, fraction or component, to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, filtration and distillation. Some common examples of mass transfer processes are the evaporation of water from a pond to the atmosphere, the purification of blood in the kidneys and liver, and the distillation of alcohol. In industrial processes, mass transfer operations include separation of chemical components in distillation columns, absorbers such as scrubbers or stripping. Mass transfer is often coupled to additional transport processes, for instance in industrial cooling towers. These towers couple heat transfer to mass transfer by allowing hot water to flow in contact with hotter air and evaporate as it absorbs heat from the air. 18 December 2017 Jayachandra Babu M (Colloquium) 15
- 16. Soret and Dufour effects: The energy flux caused by concentration gradient is known as Dufour or diffusion-thermo effect, whereas the mass diffusion due to temperature gradient is called Soret or thermo-diffusion effect. They have their own moment in the fields such as hydrology and geosciences. The Soret effect is employed in the detachment of isotopes and in mix among the gases with light molecular weight (Hydrogen or Helium) and medium molecular weight (Hydrogen or Air). Thermophoresis: It is a mass transfer mechanism of movement of small particles due to thermal gradient. It cause small particles to lodge on the bleak surfaces. Brownian motion: Random motion of particles suspended in a fluid resulting from their collision with the fast-moving atoms or molecules in the gas or liquid. Ex: These have numerous applications in various fields like nuclear reactor safety, aerosol technology, environmental and atmosphere pollution. 18 December 2017 Jayachandra Babu M (Colloquium) 16 Other physical effects used
- 17. Numerical Methods When analytical solution of the mathematically defined problem is possible but it is time-consuming and the error of approximation we obtain with numerical solution is acceptable, then we can go for numerical methods. Analytical methods have limitations in case of nonlinear problem. In such cases numerical methods works very well. 18 December 2017 Jayachandra Babu M (Colloquium) 17
- 18. Shooting Technique 18 December 2017 Jayachandra Babu M (Colloquium) 18 Convert the BVP Into initial value problem Guess a value for the auxiliary conditions at one point of time. Solve the initial value problem using Euler, Runge-Kutta, Newton’s method… Check if the boundary conditions are satisfied, otherwise modify the guess and resolve the problem. Use interpolation in updating the guess. It is an iterative procedure and can be efficient in solving the BVP. This method is effective when the interval is [t0, tf] (initial and boundary conditions)
- 19. Runge-Kutta-Fehlberg Method 18 December 2017 Jayachandra Babu M (Colloquium) 19 The trickiest part of using Runge-Kutta methods to approximate the solution of a differential equation is choosing the right step size. Too large a step size and the error is too large and the approximation is inaccurate. Too small a step size and the process will take too long and possibly have too much round off error to be accurate. Furthermore, the appropriate step size may change during the course of a single problem. What we need is an algorithm which includes a method for choosing the appropriate step size at each step. The Runge-Kutta Fehlberg methods do just this, which is why they have largely replaced the Runge-Kutta methods in practice.
- 20. Uniqueness of the thesis: Present work • In Ch.2, we considered non-aligned stagnation flow. This has applications in medical device filtration technologies and microbial fuel cell technology areas • In Ch. 3 & 4, we considered the 2D flow over a slendering sheet. This has applications such as polymer extrusion, metallurgical processes and machine design. • In Ch. 5 & 6, we considered the 3D flow over a slendering sheet. This has applications in several fields including aerosol technology. Existed works • Existed works focused on aligned stagnation flow • Existed works focused on 2D flow over a uniform thickness sheet • Existed works focused on 3D flow over a uniform thickness sheet 18 December 2017 Jayachandra Babu M (Colloquium) 20
- 21. Chapter - 2 Effect of nonlinear thermal radiation on non-aligned bio-convective stagnation point flow of a nanofluid over a stretching sheet 18 December 2017 Jayachandra Babu M (Colloquium) 21 This paper was published in Alexandria Engineering Journal (ELSEVIER)
- 22. Aim of the study The current study covers the relative study of MHD bioconvective stagnation point flow of a nanofluid comprising gyrotactic microorganisms across a stretching sheet in the presence of nonlinear radiation parameter. Bioconvection is the process of spontaneous pattern construction in suspension of microorganisms which are a little denser compare to water and swim upward. Bioconvection is useful in bio-reactors, fuel cell technology and bio-diesel fuels. Now a days, to generate or raise convection in nanofluids, inquiries are produced on the deployment of microorganisms. As nanoparticles are not having motility, their movement will be drive by thermophoresis and Brownian motion. Thus the movement of microorganisms is free of the movement of nanoparticles. This combination attracted researchers because of its use in medical filtration device technologies and microbial fuel cell technology. 18 December 2017 Jayachandra Babu M (Colloquium) 22
- 23. Physical model of the problem Fig. 2.1 18 December 2017 Jayachandra Babu M (Colloquium) 23
- 24. Governing Equations: 2 2 * 3 2 16 3 * T f B f p D T T T T T C T u v T D x y y C k y y y y T y 2 2 2 2 T B D C C C T u v D x y T y y 2 2 c n w dW n n n C u v D n x y y C C y y (2.1) (2.2) (2.3) (2.4) (2.5) 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 = 0, 𝑢 𝜕𝑢 𝜕𝑥 + 𝑣 𝜕𝑢 𝜕𝑦 = 𝑈∞ 𝜕𝑈∞ 𝜕𝑥 + 1 𝜌𝑓 𝜕 𝜕𝑦 𝜇𝑓 𝜕𝑢 𝜕𝑦 − 𝜎𝐵0 2 𝑢 − 𝑈∞ 𝜌𝑓 , , 18 December 2017 Jayachandra Babu M (Colloquium) 24
- 25. By using the following similarity transformations (2.7) , we have converted the governing equations (2.2) – (2.5) as a set of nonlinear ordinary differential equations: '( ) , , 1 1 , , f f f w w w u c x c f G v c f C C n n T T C C n n (2.7) The corresponding boundary conditions are ( ) , 0, , , n at 0 , , , n as w f w w T u U x cx v k h T T C C n y y u U ax by T T C C n y (2.6) 18 December 2017 Jayachandra Babu M (Colloquium) 25
- 26. The following equations are the resultant transformed equations: 2 2 1 1 ''' '' ' '' ' ' 0 f f e ff f M f 2 2 ''' '' ' '' ' ' ' 0 G G e fG f G M G S '' Pr ' '' 0 Nt Le f Nb '' ' ' '' '' Pr ' 0 Pe Lb f (2.8) (2.9) (2.10) (2.11) (2.12) 3 2 2 1 1 1 '' 3 1 1 1 Pr ' Pr ' ' Pr ' 0 w w w Ra Ra Nt Nb f And the corresponding boundary conditions (2.7) changed as 1 2 0, 0, ' 1, ' 0, ' 1 , 0, 0 at 0 ' , '' , 0, 0, 0 as f G f g Bi f G (2.13) 18 December 2017 Jayachandra Babu M (Colloquium) 26
- 27. The Magnetic field parameter, Viscosity variation parameter, Radiation parameter, Prandtl number, Thermophoresis parameter, Brownian motion, Lewis number, Pecklet number, Bioconvection constant and Bioconvection Lewis number are given as follows: 2 3 0 1 16 * , , , ,Pr , , 3 * , , , , f T w f f f f B w c B n w n D B T a M m T T S Ra Nt c kk c D C C dW n Nb Le Pe Lb D D n n D (2.14) For engineering design interest, the abbreviated Nusselt number, Sherwood number and the density number of the motile microorganisms (after non-dimensionalization) are given by 1 2 1 2 1 2 ' 0 , ' 0 , ' 0 x x x x x x Nu Pe Sh Pe Nn Pe (2.15) 18 December 2017 Jayachandra Babu M (Colloquium) 27
- 28. Effect of Magnetic field parameter on velocity, temperature and concentration profile Fig. 2.2 Fig. 2.3 Fig. 2.4 Results and Discussion 18 December 2017 Jayachandra Babu M (Colloquium) 28
- 29. Fig. 2.5 Effect of thermophoresis on temperature profile Fig. 2.6 Effect of thermophoresis on concentration profile 18 December 2017 Jayachandra Babu M (Colloquium) 29
- 30. Fig. 2.8 Effect of bioconvection Lewis number on gyrotactic microorganism profile Fig. 2.7 Effect of Pecklet number on gyrotactic microorganism profile 18 December 2017 Jayachandra Babu M (Colloquium) 30
- 31. 18 December 2017 Jayachandra Babu M (Colloquium) 31 Fig. 2.9 Effect of radiation parameter on temperature profile Fig. 2.10 Effect of Brownian motion parameter on concentration profile
- 32. Table 2.1 Values of reduced Nusselt number, reduced Sherwood number and the mass transfer rate of the motile microorganisms for diverse parameters in oblique flow 𝑀 R𝑎 𝑁𝑡 𝑃𝑒 𝐿𝑏 𝐵𝑖 −𝜃′(0 −𝜙′(0 −𝜒′ 0 1 0.090411 1.654470 1.888114 3 0.089468 1.548341 1.763810 6 0.088414 1.441098 1.640789 1 0.090410 1.654493 1.888125 1.5 0.088766 1.655390 1.887038 2 0.087159 1.655816 1.885704 1 0.089672 1.616068 1.868808 2 0.087665 1.556681 1.839729 3 0.084420 1.528600 1.826674 0.3 0.090421 1.654328 1.576237 0.5 0.090421 1.654328 1.888383 0.7 0.090421 1.654328 2.207772 0.5 0.090421 1.654328 1.888383 0.6 0.090421 1.654328 2.005665 0.7 0.090421 1.654328 2.115546 0.1 0.090411 1.654470 1.888114 0.15 0.127607 1.632295 1.872169 0.2 0.158914 1.613592 1.857638 18 December 2017 Jayachandra Babu M (Colloquium) 32
- 33. Table 2.2 Values of reduced Nusselt number, reduced Sherwood number and the mass transfer rate of the motile microorganisms for diverse parameters in free stream flow 𝑀 R𝑎 𝑁𝑡 𝑃𝑒 𝐿𝑏 𝐵𝑖 −𝜃′(0 −𝜙′(0 −𝜒′ 0 1 0.090029 1.619865 1.844518 3 0.088532 1.478636 1.677737 6 0.086624 1.328780 1.505927 1 0.090025 1.619880 1.844427 1.5 0.088175 1.619641 1.842178 2 0.086314 1.618945 1.839627 1 0.089231 1.578762 1.823705 2 0.087010 1.512457 1.790153 3 0.083315 1.474527 1.770132 0.3 0.090059 1.619764 1.536353 0.5 0.090059 1.619764 1.845458 0.7 0.090059 1.619764 2.161277 0.5 0.090059 1.619764 1.845458 0.6 0.090059 1.619764 1.963004 0.7 0.090059 1.619764 2.073377 0.1 0.090029 1.619865 1.844518 0.15 0.126614 1.594756 1.825741 0.2 0.156903 1.572792 1.807985 0.090059 1.619764 1.845458 0.090059 1.619764 1.961908 0.090059 1.619764 2.078358 18 December 2017 Jayachandra Babu M (Colloquium) 33
- 34. Table 2.3 Comparison of the present results for −𝜃′(0 and −𝜙′(0 when Pr = 𝐿𝑒 = 10, , 𝐵𝑖 = 0.1, 𝛽 = 𝑅𝑎 = 𝜆1 = 𝑃𝑒 = 𝑀 = 𝐿𝑏 = 0 Nt −𝜃′(0 Nb=0.1 (O.D. Makinde et al. 2011) −𝜃′(0 Nb=0.1 (Khan et al. 2016) Present results −𝜙′(0 Nb=0.1 (O.D. Makinde et al. 2011) −𝜙′(0 Nb=0.1 (Khan et al. 2016) Present results 0.1 0.0929 0.09291 0.092912 2.27741 2.27741 2.277412 0.3 0.0925 0.09252 0.092521 2.2228 2.22281 2.222811 0.5 0.0921 0.09212 0.092120 2.1783 2.17834 2.178341 18 December 2017 Jayachandra Babu M (Colloquium) 34
- 35. Conclusions Rising values of the Magnetic field, thermophoresis and Biot number increases the temperature field. The density of the motile microorganisms is a decreasing function of Peclet number and Bioconvection Lewis number. Brownian motion and Lewis number both suppress the concentration profile. Increasing the external magnetic field lessen the heat and mass transfer rate. 18 December 2017 Jayachandra Babu M (Colloquium) 35
- 36. Chapter - 3 MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion 18 December 2017 Jayachandra Babu M (Colloquium) 36 This paper was published in Alexandria Engineering Journal (ELSEVIER)
- 37. Aim of the study The boundary layer flow across a slendering stretching sheet has gotten awesome consideration due to its inexhaustible pragmatic applications in nuclear reactor technology, acoustical components, chemical and manufacturing procedures, for example, polymer extrusion, and machine design. In this paper, we focused on the Soret and Dufour effects on the Williamson fluid flow over a stretching sheet with variable thickness in the presence of velocity slip parameter. The governing nonlinear partial differential equations of this problem transformed as the nonlinear ordinary differential equations by using suitable transformations and then solved numerically using Runge-Kutta fourth order method based shooting technique. 18 December 2017 Jayachandra Babu M (Colloquium) 37
- 38. Physical model of the problem Fig. 3.1 18 December 2017 Jayachandra Babu M (Colloquium) 38 1 2 m y A x b
- 39. Governing Equations: 0 u v x y 2 2 2 2 m T p s p D k T T k T C u v x y C C C y y 2 2 2 2 m T m m D k C C C T u v D x y T y y (3.1) (3.2) (3.3) (3.4) And the corresponding boundary conditions are * 1 * * 2 3 ( , ) ( ) , , 0, , , , 0, , w w w u u x y U x h v x y y T C T x y T x h C x y C x h y y and u T T C C at y (3.5) 18 December 2017 Jayachandra Babu M (Colloquium) 39 2 2 2 2 2 ( ) 2 u u u B x u u u u v x y y y y
- 40. (3.6) (3.7) (3.8) By using the following similarity transformations (3.6) , we have converted the governing equations (3.2) – (3.4) as a set of nonlinear ordinary differential equations: 0 ( ) '( ) m u U x b f , 1 0 1 1 ( ) '( ) ( ) 2 1 m m m v U x b f f m , 1 0 1 ( ) , 2 m m x b y U ( ) w T T T x T , ( ) w C C C x C Then the resultant transformed equations are 2 2 (1 '') ''' ' '' ' 0 1 m f f f ff M f m 1 '' Pr( ' ' '') 0 1 m f f Du m 1 '' ( ' ' '') 0 1 m Sc f f Sr m (3.9) And the corresponding boundary conditions (3.5) changed as 1 1 2 3 1 (0) 1 ''(0) , '(0) 1 ''(0) , 1 (0) 1 '(0) , (0) 1 '(0) , '( ) 0, ( ) 0, ( ) 0 m f h f f h f m h h f (3.10) 18 December 2017 Jayachandra Babu M (Colloquium) 40
- 41. 3 1 3 0 2 0 0 ( ) ( 1) , 2 ( ) ,Pr , , ( 1) ( ) ( ) , ( ) m p m T w s p w m T w m m w x b m U C B D k C C M Du U m k C C T T D k T T Sc Sr D T C C (3.11) Williamson fluid parameter, Magnetic field parameter, Prandtl number, Dufour number, Schmidt number and Soret number are given as follows: For engineering design interest, the skin friction coefficient, the local Nusselt number and the local Sherwood number (after non-dimensionalization) are given by: 0.5 0.5 0.5 1 1 2 Re ''(0). Re '(0) 2 2 1 Sh Re '(0) 2 f x x x x x m m C f Nu m (3.12) 18 December 2017 Jayachandra Babu M (Colloquium) 41
- 42. Results and Discussion 18 December 2017 Jayachandra Babu M (Colloquium) 42 Fig. 3.2 Soret effect on concentration profile Fig. 3.3 Soret effect on temperature profile
- 43. 18 December 2017 Jayachandra Babu M (Colloquium) 43 Fig. 3.4 Dufour effect on temperature profile Fig. 3.5 Dufour effect on concentration profile
- 44. Fig. 3.6 Velocity slip parameter effect on velocity profile 18 December 2017 Jayachandra Babu M (Colloquium) 44 Fig. 3.7 Temperature jump parameter effect on temperature profile
- 45. 18 December 2017 Jayachandra Babu M (Colloquium) 45 Table 3.1: Values of the skin friction coefficient, local Nusselt number and local Sherwood number for different parameters in both cases 𝑀 𝑆𝑟 𝐷𝑢 ℎ1 ℎ2 ℎ3 𝐶𝑓𝑥 −𝑁𝑢𝑥 −𝑆ℎ𝑥 𝛬 = 0 𝛬 = 0.2 𝛬 = 0 𝛬 = 0.2 𝛬 = 0 𝛬 = 0.2 1 -2.180932 -2.232527 0.491866 0.487824 0.621523 0.612501 1.5 -2.459006 -2.522645 0.455531 0.450817 0.542038 0.529592 2 -2.747013 -2.822965 0.421898 0.416492 0.452003 0.435928 0.1 -2.747014 -2.822971 0.392954 0.388011 0.828698 0.812607 0.2 -2.747014 -2.822971 0.395267 0.390203 0.732428 0.716213 0.3 -2.747014 -2.822971 0.397443 0.392227 0.634924 0.618492 0.1 -2.747014 -2.822971 0.614913 0.502342 0.576201 0.562753 0.2 -2.747014 -2.822970 0.593745 0.484776 0.507034 0.491797 0.3 -2.747014 -2.822971 0.572666 0.467279 0.436474 0.419239 0.1 -3.933728 -4.167604 0.450826 0.443378 0.579201 0.560291 0.3 -2.747014 -2.822971 0.401337 0.395672 0.436474 0.419239 0.5 -2.127096 -2.162213 0.369257 0.364538 0.335768 0.321009 0.1 -2.747014 -2.822971 0.423148 0.416551 0.412101 0.395545 0.3 -2.747014 -2.822971 0.401337 0.395672 0.436474 0.419239 0.5 -2.747014 -2.822971 0.381665 0.376787 0.458457 0.440670 0.1 -2.747014 -2.822971 0.415999 0.409827 0.510434 0.487900 0.6 -2.747014 -2.822971 0.385889 0.380608 0.358546 0.346166 1.2 -2.747014 -2.822971 0.367188 0.362161 0.264204 0.256686 M Sr Du 1 h2 h3 h x Cf x Nu x Sh 0 0.2 0 0.2 0 0.2
- 46. 18 December 2017 Jayachandra Babu M (Colloquium) 46 Table 3.2 Comparison of the values of when M=Du=Sc=Sr=0, m=0.5 𝑓′′(0 𝜆 ℎ1 Khader and Megahed [2003] Present study 0.2 0 -0.924828 -0.924829 0.25 0.2 -0.733395 -0.733396 0.5 0.2 -0.759570 -0.759570
- 47. Conclusions Soret and Dufour numbers increases the concentration and shows the opposite behavior on temperature profile. The effect of velocity slip parameter is quite opposite on velocity and temperature. It depreciated the velocity field and enhanced the temperature field. Velocity slip parameter enhances the skin friction coefficient, Sherwood number and depreciates the Nusselt number. Dufour number reduce both the heat and mass transfer rates. 18 December 2017 Jayachandra Babu M (Colloquium) 47
- 48. Chapter - 4 Magnetohydrodynamic dissipative flow across the slendering stretching sheet with temperature dependent viscosity 18 December 2017 Jayachandra Babu M (Colloquium) 48 This paper was published in Results in Physics (ELSEVIER)
- 49. Aim of the study • In this study, we analyzed the two-dimensional MHD flow across a slendering stretching sheet in the presence of variable viscosity and viscous dissipation. The sheet is thought to be convectively warmed. Convective boundary conditions through heat and mass are employed. • Similarity transformations used to convert the governing nonlinear partial differential equations as a group of nonlinear ordinary differential equations. Runge-Kutta based shooting technique is utilized to solve the converted equations. 18 December 2017 Jayachandra Babu M (Colloquium) 49
- 50. Physical model of the problem Fig. 4.1 18 December 2017 Jayachandra Babu M (Colloquium) 50
- 51. Governing Equations: 0 u v x y (4.1) (4.2) (4.3) (4.4) 2 ( ) u u u u v B x u x y y y 2 2 2 p p T T k T u u v T x y C y c y 2 0 2 ( ) m C C C u v D k C C x y y And the corresponding boundary conditions are 1 2 ( , ) ( ), , 0, , at 0 0, , w w B w u x y U x v x y T C k h T T D h C C y y y and u T T C C at y (4.5) 18 December 2017 Jayachandra Babu M (Colloquium) 51 ) ( 1 1 * 1 w T a b T T
- 52. (4.6) By using the following similarity transformations (4.6) , we have converted the governing equations (4.2) – (4.4) as a set of nonlinear ordinary differential equations: 1 1 1 ( ) , ( ) , 2 1 1 ( ) ( ) , 2 1 ( ) , ( ) n n n w w n x c df y b u b x c d n df n v b x c f d n T T T x T C C C x C The resultant transformed equations are 3 2 2 2 1 3 2 2 2 2 2 1 0 1 1 d f n d f d f df d d f a A f M A d n d d n d d d 2 2 2 2 2 1 Pr 0 1 d d n df d f f Ec d d n d d 2 2 1 2 0 1 1 d d n df Sc f Kr d d n d n (4.7) (4.8) (4.9) 18 December 2017 Jayachandra Babu M (Colloquium) 52
- 53. And the corresponding boundary conditions (4.5) changed as 0 1 2 0 0 1 (0) , 1, 1 1 (0) , 1 (0) , 0, ( ) 0, ( ) 0 n df f n d d d d d df d (4.10) Variable viscosity parameter, Magnetic field parameter, Prandtl number, Eckert number, Schmidt number and Chemical reaction parameter are given by: 2 0 1 5 1 2 2 0 1 , ,Pr , , , p w n n p m C B A b T T M b k b x c k Ec Sc Kr C D b x c (4.11) 18 December 2017 Jayachandra Babu M (Colloquium) 53
- 54. The essential physical measures of concern, the skin friction coefficient, the local Nusselt number and the Sherwood numbers are indicated as below: 0.5 0.5 2 1/2 1/2 2 0 0 0.5 1/2 0 1 1 2 Re , Re 2 2 1 Sh Re 2 f x x x x x n d f n d C Nu d d n d d (4.12) Numerical Procedure The set of transformed equations (4.7) - (4.9) with the boundary conditions (4.10) are solved by using Runge- Kutta based shooting technique. To analyze the three basic profiles (velocity, temperature, and concentration) through graphs for the impacts of different parameters, for example, viscosity variation parameter, we utilize the following: 18 December 2017 Jayachandra Babu M (Colloquium) 54 1 1 2 2, 1, 0.65, 0.5, 0.3, 1, 0.5, 0.3, 0.2, 0.2 M a n A Ec Sc Kr
- 55. Results and Discussion Fig. 4.2 Effect of Eckert number on temperature profile Fig. 4.3 Effect of chemical reaction parameter on concentration profile 18 December 2017 Jayachandra Babu M (Colloquium) 55
- 56. Effect of variable viscosity parameter on temperature and concentration profiles Fig.4.4 Fig.4.5 18 December 2017 Jayachandra Babu M (Colloquium) 56
- 57. Effect of wall thickness parameter on velocity, temperature and concentration profiles Fig.4.6 Fig.4.7 Fig.4.8 18 December 2017 Jayachandra Babu M (Colloquium) 57
- 58. Fig. 4.9 Effect of heat transfer Biot number on temperature profile Fig. 4.10 Effect of mass transfer Biot number on concentration profile 18 December 2017 Jayachandra Babu M (Colloquium) 58
- 59. 18 December 2017 Jayachandra Babu M (Colloquium) 59 Table 4.1: The effect of different parameters on the skin friction coefficient, local Nusselt and the Sherwood numbers in terms of values 𝑀 𝐸𝑐 𝜆 𝑆𝑐 𝛽1 𝛽2 𝐾𝑟 𝐴 𝑓′′ 0 −𝜃′(0 −𝜙′(0 1 -1.308178 0.115245 0.168546 2 -1.662102 0.100166 0.167888 3 -1.965126 0.086884 0.167393 0.2 -1.648289 0.107210 0.167870 0.3 -1.679902 0.089271 0.167814 0.4 -1.715608 0.070135 0.167753 1 -1.711883 0.107082 0.169993 2 -1.786025 0.115823 0.172771 3 -1.863899 0.123301 0.175137 2 -1.649401 0.108530 0.177203 3 -1.649401 0.108530 0.181474 4 -1.649401 0.108530 0.184057 0.2 -1.681797 0.088135 0.167811 0.3 -1.707280 0.112911 0.167772 0.4 -1.727150 0.131196 0.167741 0.2 -1.670091 0.095216 0.167840 0.3 -1.670091 0.095216 0.233025 0.4 -1.670091 0.095216 0.289180 0.2 -1.670091 0.095216 0.160787 0.3 -1.670091 0.095216 0.163714 0.4 -1.670091 0.095216 0.165992 0.5 -1.889172 -0.007535 0.167478 0.6 -1.900625 -0.008296 0.167467 0.7 -1.913637 -0.009229 0.167454
- 60. 18 December 2017 Jayachandra Babu M (Colloquium) 60 Table 4.2: Validation of the numerical technique by comparing with the others for −𝜃′(0 M RKS Bvp4c Bvp5c 1 0.115245 0.1152451342 0.1152451342 2 0.100166 0.1001668710 0.1001668711 3 0.086884 0.0868845644 0.0868845643 4 0.006452 0.0064522314 0.0064522314
- 61. Conclusions Magnetic field parameter, Eckert number, heat transfer Biot number and variable viscosity parameter are useful to enhance the temperature field of the flow over a slender sheet. Wall thickness parameter improves both heat and mass transfer rates. Mass transfer Biot number and variable viscosity parameter increase the concentration field. Rising values of the wall thickness parameter and Schmidt number lessen the concentration. Increasing values of the Eckert number lessen the heat transfer rate. 18 December 2017 Jayachandra Babu M (Colloquium) 61
- 62. Chapter - 5 Three-dimensional MHD slip flow of nanofluids over a slendering stretching sheet with thermophoresis and Brownian motion 18 December 2017 Jayachandra Babu M (Colloquium) 62 This paper was published in Advanced Powder Technology (ELSEVIER)
- 63. Aim of the study We considered water based Cu and CuO nanofluids. With the assistance of similarity transformations, we changed the derived governed equations as ordinary differential equations. We exhibit and explain the graphs for various parameters of interest. We discussed the skin friction coefficient, reduced Nusselt and reduced Sherwood numbers for the influence of the pertinent parameters with the assistance of tables separately for two nanofluids. (Cu-water and CuO-water). 18 December 2017 Jayachandra Babu M (Colloquium) 63
- 64. Physical model of the problem Fig. 5.1 18 December 2017 Jayachandra Babu M (Colloquium) 64
- 65. Governing Equations: 0 u v w x y z 2 2 2 B p nf B nf D T T T T C T T c u v w k D x y z z z T z z 2 2 2 2 T B D C C C C T u v w D x y z T z z (5.1) (5.2) (5.3) (5.4) (5.5) And the corresponding boundary conditions are (5.6) * * 1 1 * * 2 3 ( , ) ( ) , , ( ) , , , , 0, v=0, , w w w w u v u x y U x h v x y V x h z z T C T x y T x h C x y C x h z z and u T T C C at y 18 December 2017 Jayachandra Babu M (Colloquium) 65 2 2 2 ( ) nf nf u u u u u v w B x u x y z z 2 2 2 ( ) nf nf v v v v u v w B x v x y z z
- 66. Nanofluid parameters are given by: ( ) 1 , 1 , (1 ) (1 2.5 ) ( ) 3( 1) 3( 1) 1 , 1 2 ( 2) ( 1) ( ) where , , , , ( ) nf nf p nf f f p f nf nf f f p s s s s f p f f f c r e c k k k k c k r e k c k (5.7) By using the following similarity transformations (5.8) , we have converted the governing equations (5.2) – (5.5) as a set of nonlinear ordinary differential equations: 0.5 1 *0.5 ( ) , ( ) 2 1 1 ( ) ( ) 1 2 2 n n n f g u a x y c v a x y c a n n f g w x y c f g n 0.5 1 *0.5 ( 1) ( ) 2 n n a z x y c , ( ) w T T T x T , ( ) w C C C x C (5.8) 18 December 2017 Jayachandra Babu M (Colloquium) 66
- 67. The resultant transformed equations are 2 3 2 3 2 3 1 1 1 1 1 1 0 1 2.5 2 2 2 1 n f f f g n f f r n n f g M 2 3 2 3 2 3 1 1 1 1 1 1 0 1 2.5 2 2 2 1 n g g f g n g g r n n f g M 2 2 2 2 1 1 Pr 1 ( ) 0 1 2 2 nf f k n f g n Nb Nt d f g k n 2 2 2 2 1 0 1 Nt f g n Le f g Nb n (5.9) (5.10) (5.11) (5.12) And the changed boundary conditions are (5.13) 1 1 1 1 2 3 1 (0) 1 ''(0) , '(0) 1 ''(0) , 1 1 (0) 1 ''(0) , '(0) 1 ''(0) , 1 (0) 1 '(0) , (0) 1 '(0) , '( ) 0, '( ) 0, ( ) 0, ( ) 0 n f h f f h f n n g h g g h g n h h f g 18 December 2017 Jayachandra Babu M (Colloquium) 67
- 68. 2 0 0 0 ,Pr , , , f p B T f B B C D C D T M Nb Nt Le a k k T k D (5.14) Magnetic field parameter, Prandtl number, Brownian motion, Thermophoresis and Lewis number are given by : For engineering concern, the skin-friction coefficient, the local Nusselt number and the Sherwood numbers (after non-dimensionalization) are given by: 0.5 0.5 1/2 1/2 0.5 1/2 1 1 2 Re ''(0). Re '(0) 2 2 1 Sh Re '( ) 2 0 f x x x x x n n C f Nu n (5.15) Numerical Procedure Equations (5.9) - (5.12) subject to the conditions (5.13) are figured out with the assistance of shooting process numerically. For numerical solutions, we assigned the values for non-dimensional parameters as 18 December 2017 Jayachandra Babu M (Colloquium) 68 1 2 3 6.2, 0.65, 0 0.1, 0.1, 2, .3, 0.1, 0.3, 0.3, 0.3, 1. Pr n h h h Le Nb Nt M
- 69. Results and Discussion 18 December 2017 Jayachandra Babu M (Colloquium) 69 Fig. 5.2 Effect of thermophoresis on temperature profile Fig. 5.3 Effect of thermophoresis on concentration profile
- 70. Fig. 5.4 Effect of Brownian motion on temperature profile Fig. 5.5 Effect of Brownian motion on concentration profile 18 December 2017 Jayachandra Babu M (Colloquium) 70
- 71. Effect of volume fraction parameter on velocity, temperature and concentration profiles Fig. 5.6 Fig. 5.7 Fig. 5.8 18 December 2017 Jayachandra Babu M (Colloquium) 71
- 72. Effect of velocity slip parameter on velocity profiles Fig. 5.9 Fig. 5.10 18 December 2017 Jayachandra Babu M (Colloquium) 72
- 73. Fig. 5.12 Effect of temperature jump parameter on temperature profile Fig. 5.13 Effect of concentration jump parameter on concentration profile 18 December 2017 Jayachandra Babu M (Colloquium) 73
- 74. 18 December 2017 Jayachandra Babu M (Colloquium) 74 Table 5.1: Skin friction coefficient, reduced Nusselt number and reduced Sherwood number values of various parameters for the mixture of water and Cu 𝑁𝑏 Nt 𝜑 Le ℎ1 ℎ2 ℎ3 𝑓′′ 0 −𝜃′(0 −𝜙′(0 0.1 -1.210278 1.053276 1.416400 0.2 -1.210278 0.952262 1.568673 0.3 -1.210278 0.856031 1.618353 0.1 -1.210278 1.053276 1.416400 0.2 -1.210278 0.965144 1.232196 0.3 -1.210278 0.884537 1.106288 0.01 -1.801297 1.409106 1.618712 0.04 -1.853671 1.371369 1.626106 0.1 -1.912716 1.298678 1.643243 5 -1.208579 1.065172 0.911661 7 -1.208579 1.057251 1.160033 10 -1.208579 1.053273 1.415687 0.3 -1.208491 1.053303 1.415697 0.4 -1.062894 1.003678 1.367307 0.5 -0.950393 0.960988 1.324892 0.3 -1.210278 1.053276 1.416400 0.5 -1.210278 0.881066 1.455965 0.7 -1.210278 0.754801 1.485609 0.3 -1.210278 1.053276 1.416400 0.6 -1.210278 1.082283 0.933785 0.9 -1.210278 1.096642 0.696436
- 75. Table 5.2: Skin friction coefficient, reduced Nusselt number and reduced Sherwood number values of various parameters for the mixture of water and CuO 18 December 2017 Jayachandra Babu M (Colloquium) 75 𝑁𝑏 Nt 𝜑 Le ℎ1 ℎ2 ℎ3 𝑓′′ 0 −𝜃′(0 −𝜙′(0 0.1 -1.183941 1.065687 1.427469 0.2 -1.183941 0.964142 1.579332 0.3 -1.183941 0.867279 1.628888 0.1 -1.183941 1.065687 1.427469 0.2 -1.183941 0.976766 1.244759 0.3 -1.183941 0.895344 1.120616 0.01 -1.786772 1.410073 1.619656 0.04 -1.800547 1.375104 1.629519 0.1 -1.800750 1.307401 1.650247 5 -1.181989 1.077793 0.925419 7 -1.181989 1.069970 1.172807 10 -1.181989 1.066072 1.427291 0.3 -1.181884 1.066125 1.427333 0.4 -1.043149 1.016615 1.379317 0.5 -0.935197 0.973837 1.337055 0.3 -1.183941 1.065687 1.427469 0.5 -1.183941 0.889670 1.467211 0.7 -1.183941 0.761051 1.496913 0.3 -1.183941 1.065687 1.427469 0.6 -1.183941 1.094992 0.939183 0.9 -1.183941 1.109451 0.699769
- 76. Table 5.1 Thermo-physical attributes of water and nanoparticles 18 December 2017 Jayachandra Babu M (Colloquium) 76 𝜌 𝑘 𝑔 𝑚 2 𝐶𝑝 𝐽 𝑘𝑔 𝐾 𝑘 𝑊 𝑚 𝐾 Pure water 997.1 4179 0.613 Copper 8933 385 401 Copper oxide 6320 531.8 76.5
- 77. 18 December 2017 Jayachandra Babu M (Colloquium) 77 Table 5.3 : Comparison of the present results for reduced Nusselt number when 1 2 3 0, 0, 6.2, 1, 0, 0.1, 0 0, N h h h b Nt Pr n Le Reduced Nusselt number −𝜃′(0 Rashidi et al. [2014] Present study M Cu-water CuO-waterr Cu-water CuO-water 1 0.92890862 0.94800696 0.9289085 0.948006 2 0.88544087 0.90160918 0.8854404 0.901609 3 0.84801483 0.86211196 0.8480148 0.862111
- 78. Conclusions Heat and mass transfer performance of Cu-water nanofluid is high when compared with CuO-water nanofluid. Rising values of Nb and Le lessen the concentration field. Increasing values of ϕ, h1 depreciate velocity field. Nt and Nb both reduce the local Nusselt number but shows opposite behavior on Sherwood number. Rising values of h1 enhances the skin friction coefficient but lessen the heat and mass transfer rate. 18 December 2017 Jayachandra Babu M (Colloquium) 78
- 79. Chapter - 6 3D MHD flow of ferrofluid over a slendering stretching sheet with thermophoresis and Brownian motion 18 December 2017 Jayachandra Babu M (Colloquium) 79 This paper was published in Journal of Molecular Liquids (ELSEVIER)
- 80. Aim of the study The purpose of this paper is the theoretical analysis of the steady, three- dimensional, MHD and slip flow of a nanofluid across a slendering stretching sheet with thermophoresis and Brownian motion. Our considerations are water as base fluid, graphene and magnetite as nanoparticles and we did the simultaneous study of them on various profiles. R- K fourth order based shooting method is enforced to resolve the altered governing non-linear equations. 18 December 2017 Jayachandra Babu M (Colloquium) 80
- 81. Physical model of the problem Fig. 6.1 18 December 2017 Jayachandra Babu M (Colloquium) 81
- 82. Governing Equations: 0 u v w x y z 2 2 2 ( ) nf nf nf u u u u u v w B x u x y z z 2 2 2 ( ) nf nf nf v v v v u v w B x v x y z z 2 2 2 B p nf B nf D T T T T C T T c u v w k D x y z z z T z z 2 2 2 2 T B D C C C C T u v w D x y z T z z (6.1) (6.2) (6.3) (6.4) (6.5) And the corresponding boundary conditions are * * 1 1 * * 2 3 ( , ) ( ) , , ( ) , , , , 0, v=0, , w w w w u v u x y u x j v x y v x j z z T C T x y T x j C x y C x j z z and u T T C C at y (6.6) 18 December 2017 Jayachandra Babu M (Colloquium) 82
- 83. Nanofluid parameters are given by: ( ) 1 , 1 , (1 ) (1 2.5 ) ( ) 3( 1) 3( 1) 1 , 1 2 ( 2) ( 1) ( ) where , , , , ( ) nf nf p nf f f p f nf nf f f p s s s s f p f f f c r e c k k k k c k r e k c k (6.7) By using the following similarity transformations (5.8) , we have converted the governing equations (5.2) – (5.5) as a set of nonlinear ordinary differential equations: 0.5 1 *0.5 ( ) , ( ) 2 1 1 ( ) ( ) 1 2 2 n n n f g u a x y c v a x y c a n n f g w x y c f g n 0.5 1 *0.5 ( 1) ( ) 2 n n a z x y c , ( ) w T T T x T , ( ) w C C C x C (6.8) 18 December 2017 Jayachandra Babu M (Colloquium) 83
- 84. The resultant transformed equations are 2 3 2 3 2 3 1 1 1 1 1 1 0 1 2.5 2 2 2 1 n f f f g n f f r n n f g M 2 3 2 3 2 3 1 1 1 1 1 1 0 1 2.5 2 2 2 1 n g g f g n g g r n n f g M 2 2 2 2 1 1 Pr 1 ( ) 0 1 2 2 nf f k n f g n Nb Nt d f g k n 2 2 2 2 1 0 1 Nt f g n Le f g Nb n (6.9) (6.10) (6.11) (6.12) And the changed boundary conditions are 2 2 2 1 1 1 2 2 2 0 0 0 2 1 2 3 2 0 0 0 1 1 (0) 1 , '(0) 1 , (0) 1 , 1 1 '(0) 1 , (0) 1 , (0) 1 , 0, 0, n f f n g f j f j g j n n g g j j j f g 0, 0 as (6.13) 18 December 2017 Jayachandra Babu M (Colloquium) 84
- 85. 2 0 0 0 ,Pr , , , f p B T f B B C D C D T M Nb Nt Le a k k T k D (6.14) Magnetic field parameter, Prandtl number, Brownian motion, Thermophoresis and Lewis number are given by : For engineering concern, the skin-friction coefficient, the local Nusselt number and the Sherwood numbers (after non-dimensionalization) are given by: 0.5 0.5 1/2 1/2 0.5 1/2 1 1 2 Re ''(0). Re '(0) 2 2 1 Sh Re '( ) 2 0 f x x x x x n n C f Nu n (6.15) Numerical Procedure Equations (6.9) to (6.12) subject to the conditions (6.13) are solved with the assistance of Runge-Kutta based shooting technique numerically. For numerical solutions, we assigned the values for non-dimensional parameters as 18 December 2017 Jayachandra Babu M (Colloquium) 85 1 2 3 6.2, 0.65, 0.1, 0.1, 2, 0.3, 0.3, 0.3 0.3, 0 1 , .1, Nb Nt M j j j Pr n Le
- 86. Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Effect of nanoparticle volume fraction on velocity, temperature and concentration profiles 18 December 2017 Jayachandra Babu M (Colloquium) 86
- 87. Fig. 6.6 Fig. 6.7 18 December 2017 Jayachandra Babu M (Colloquium) 87 Effect of thermophoresis parameter on temperature and concentration profiles
- 88. Fig. 6.8 Fig. 6.9 18 December 2017 Jayachandra Babu M (Colloquium) 88 Effect of Brownian motion parameter on temperature and concentration profiles
- 89. 18 December 2017 Jayachandra Babu M (Colloquium) 89 Table 6.1: Effect of various parameters on friction factor, Nusselt number and Sherwood number for the mixture of water and magnetite 𝜑 Nt Nb 𝑗1 𝑗2 𝑗3 Le 𝑓′′ 0 −𝜃′(0 −𝜙′(0 0.01 -0.875967 1.281648 0.322377 0.05 -0.938080 1.225730 0.320365 0.1 -0.991571 1.162464 0.322377 0.1 -0.991571 1.162464 0.322377 0.2 -0.991571 1.090804 -0.156461 0.3 -0.991571 1.022743 -0.545024 0.1 -0.991571 1.162464 0.322377 0.2 -0.991571 1.071778 0.645183 0.3 -0.991571 0.982329 0.752139 0.3 -0.991571 1.162464 0.322377 0.6 -0.732947 1.067327 0.270555 0.9 -0.586166 0.998284 0.235931 0.3 -0.991571 1.162464 0.322377 0.6 -0.991571 0.872439 0.463977 0.9 -0.991571 0.695397 0.551459 0.3 -0.991571 1.162464 0.322377 0.6 -0.991571 1.169280 0.249784 0.9 -0.991571 1.173591 0.203874 1 -0.991419 1.203844 -0.141120 1.5 -0.991419 1.179337 0.124039 2 -0.991419 1.162620 0.321750
- 90. 18 December 2017 Jayachandra Babu M (Colloquium) 90 Table 6.2: Effect of various parameters on friction factor, Nusselt number and Sherwood number for the mixture of water and graphene 𝜑 Nt Nb 𝑗1 𝑗2 𝑗3 Le 𝑓′′ 0 −𝜃′(0 −𝜙′(0 0.01 -0.868158 1.283956 0.331300 0.05 -0.906959 1.235448 0.336819 0.1 -0.942829 1.178522 0.348779 0.1 -0.942829 1.178522 0.348779 0.2 -0.942829 1.108635 -0.126864 0.3 -0.942829 1.041974 -0.513716 0.1 -0.942829 1.178522 0.348779 0.2 -0.942829 1.090898 0.667649 0.3 -0.942829 1.004273 0.773376 0.3 -0.942829 1.178522 0.348779 0.6 -0.707089 1.085372 0.295228 0.9 -0.570068 1.016777 0.295228 0.3 -0.942829 1.178522 0.348779 0.6 -0.942829 0.880978 0.490583 0.9 -0.942829 0.700614 0.577627 0.3 -0.942829 1.178522 0.348779 0.6 -0.942829 1.185611 0.269088 0.9 -0.942829 1.190062 0.219039 1 -0.942607 1.217297 -0.112125 1.5 -0.942607 1.194245 0.152575 2 -0.942607 1.178656 0.348644
- 91. Table 6.3 Thermo-physical attributes of water and nano particles 18 December 2017 Jayachandra Babu M (Colloquium) 91 Pure water Graphene Magnetite 𝜌 𝑘 𝑔 𝑚2 997 2250 5180 𝐶𝑝 𝐽 𝑘𝑔 𝐾 4076 2100 670 𝑘 𝑊 𝑚 𝐾 0.605 2500 9.7 𝜎 𝑆 𝑚 0.005 1x107 1x105
- 92. Conclusions The temperature increases with the increase in the thermophoresis and Brownian motion parameters. Nanofluid volume fraction parameter improves both temperature and concentration but showed opposite behavior on velocity profiles. All profiles (velocity, temperature, concentration) decreased with the enhancement in the corresponding slip parameter values. Nanoparticle volume fraction, thermophoresis parameter, Brownian motion parameter and velocity slip parameters reduce the heat transfer rate. 18 December 2017 Jayachandra Babu M (Colloquium) 92
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- 95. S. Nadeem, R. Mehmood and N.S. Akbar, Optimized analytical solution for oblique flow of a Casson-nano fluid with convective boundary conditions, International Journal of Thermal Sciences 78 (2014) 90-100. M. Rahman, A.V. Rosca and I. Pop, Boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective boundary condition using Buongiorno’s model, International Journal of Numerical Methods for Heat & Fluid Flow 25 (2) 2015 299-319. G. Ibanez, Entropy generation in MHD porous channel with hydrodynamic slip and convective boundary conditions, International Journal of Heat and Mass Transfer 80 (2015) 274-280. R. Kandasamy, C. Jeyabalan and K.K. Sivagnana Prabhu, Nanoparticle volume fraction with heat and mass transfer on MHD mixed convection flow in a nanofluid in the presence of thermo-diffusion under convective boundary condition, Appl Nanosci (2015) doi: 10.1007/s13204-015-0435-5. M. Waqas, M. Farooq, M.I. Khan, A. Alsaedi, T. Hayat and T. Yasmeen, Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, International Journal of Heat and Mass Transfer 102 (2016) 766-772. K. Naganthran and R. Nazar, Stagnation – point flow of a nanofluid past a stretching/shrinking sheet with heat generation/absorption and convective boundary condition, 1784 (2016) 050004 doi: 10.1063/1.4966823. T. Fang, J. Zhang and Y. Zhong, Boundary layer flow over a stretching sheet with variable thickness, Applied Mathematics and Computation 218 (2012) 7241-7252. M.M. Khader and A.M. Megahed, Boundary layer flow due to a stretching sheet with a variable thickness and slip velocity, Journal of Applied Mechanics and Technical Physics 56 (2) (2015) 241-247. S.P. Anjali Devi and M. Prakash, Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet, Journal of Nigerian Mathematical Society 34 (2015) 318-330. S.P. Anjali Devi and M. Prakash, Slip flow effects over hydromagnetic forced convective flow over a slendering stretching sheet, Journal of Applied Fluid Mechanics 9 (2) (2016) 683-692. M. Jayachandra Babu and N. Sandeep, Three-dimensional MHD slip flow of nanofluids over a slendering stretching sheet with thermophoresis and Brownian motion effects, Advanced Powder Technology 27(5) (2016) 2039-2050. M. Jayachandra Babu and N. Sandeep, MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion effects, Alexandria Engineering Journal 55 (2016) 2193-2201. 18 December 2017 Jayachandra Babu M (Colloquium) 95
- 96. 18 December 2017 Jayachandra Babu M (Colloquium) 96 C.S.K. Raju, N. Sandeep, M. Jayachandra Babu, V. Sugunamma, Dual solutions for three-dimensional MHD flow of a nanofluid over a nonlinearly permeable stretching sheet, Alexandria Engineering Journal 55 (2016) 151-162. M. Turkyilmazoglu, Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids, Chemical Engineering Science 84 (2012) 182- 187. M.M. Rahman, M.A. Al-Lawatia, I.A. Eltayeb, N. Al-Salti, Hydromagnetic slip flow of water based nanofluids past a wedge with convective surface in the presence of heat generation or absorption, International Journal of Thermal Sciences 57 (2012) 172-182. A. Malvandi and D.D. Ganji, Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, International Journal of Thermal Sciences 84 (2014) 196-206. I.Y. Seini, D.O. Makinde, Boundary layer flow near stagnation-points on a vertical surface with slip in the presence of transverse magnetic field, International Journal of Numerical Methods for Heat & Fluid Flow 24 (3) (2014) 643-653. A. Malvandi, F. Hedayati, D.D. Ganji, Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet, Powder Technology 253 (2014) 377-384. A.K. Abdul Hakeem, N. Vishnu Ganesh, B. Ganga, Magnetic field effect on second order slip flow of nanofluid over a strtching/shrinking sheet with thermal radiation effect, Journal of Magnetism and Magnetic Materials 381 (2015) 243-257. K. Das, S. Jana, P.K. Kunde, Thermophoretic MHD slip flow over a permeable surface with variable fluid properties, Alexandria Engineering Journal 54 (2015) 35-44. C. Sulochana, N. Sandeep, Dual solutions for radiative MHD forced convective flow of a nanofluid over a slendering stretching sheet in porous medium, J. Naval Architecture and Marine Eng., 12 (2015) 115-124. Z. Abbas, M. Naveed, M. Sajid, Hydromagnetic slip flow of nanofluid over a curved stretching surface with heat generation and thermal radiation, Journal of Molecular Liquids 215 (2016) 756-762. S.P. Anjali Devi, M. Prakash, Slip flow effects over hydromagnetic forced convective flow over a slendering stretching sheet, Journal of Applied Fluid Mechanics 9(2) (2016) 683-692. R. Kandasamy, T. Hayat, S. Obaidat, Group theory transformation for Soret an Dufour effects on free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface in the presence of heat source/sink, Nuclear Engineering and Design, 241, 2011, 2155-2161. N. Sandeep, C. Sulochana, C.S.K. Raju, M. Jayachandra Babu, V. Sugunamma, Unsteady boundary layer flow of thermophoretic MHD nanofluid past a stretching sheet with space and time dependent internal heat source/sink, Applications and Applied Mathematics, 10 (1), 2015, 312-327.
- 97. Future Plan Linear and nonlinear Stability analysis Statistical mechanics Compressible flow modeling 18 December 2017 Jayachandra Babu M (Colloquium) 97
- 98. 18 December 2017 Jayachandra Babu M (Colloquium) 98 My publications 1. Heat and Mass transfer in MHD Eyring-Powell nanofluid flow due to cone in porous medium, International Journal of Engineering Research in Africa, 19 (2016) 57-74. 2. Effect of variable heat source/sink on chemically reacting 3D slip flow caused by a slendering stretching sheet, International Journal of Engineering Research in Africa, , 25(2016) 58-69. 3. Three-dimensional MHD slip flow of nanofluids over a slendering stretching sheet with thermophoresis and Brownian motion effects , Advanced Powder Technology, 27.5 (2016) 2039-2050. 4. UCM flow across a melting surface in the presence of double stratification and cross-diffusion effects, Journal of Molecular Liquids, 232 (2017) 27-35. 5. 3D MHD slip flow of a nanofluid over a slendering stretching sheet with thermophoresis and Brownian motion effects, Journal of Molecular Liquids, 222 (2016) 1003-1009. 6. Magnetohydrodynamic dissipative flow across the slendering stretching sheet with temperature dependent variable viscosity, Results in Physics, 2017 May 25. 7. Free convective MHD Cattaneo-Christov flow over three different geometries with thermophoresis and Brownian motion, Alexandria Engineering Journal, 2017 Feb 3. 8. Effect of nonlinear thermal radiation on non-aligned bio-convective stagnation point flow of a magnetic- nanofluid over a stretching sheet, Alexandria Engineering Journal, 55.3 (2016) 1931-1939. 9. MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion Effects, Alexandria Engineering Journal, 55.3 (2016) 2193-2201. 10. Nonlinear Thermal Radiation and Induced Magneticfield Effects on Stagnation-Point Flow of Ferrofluids, Journal of Advanced Physics, 5 (2015) 1-7.
- 99. 18 December 2017 Jayachandra Babu M (Colloquium) 99