Data Analysis and Heat Transfer in Nanoliquid Thin Film Flow over an Unsteady Stretching Sheet

Transcription

1 Data Analysis and Heat Transfer in Nanoliquid Thin Film Flow over an Unsteady Stretching Sheet Prashant G Metri Division of Applied Mathematics, Mälardalen University, Västerås, Sweden prashant.g.metri@mdh.se ISPMAM-2017 April 26, 2017 Prashant G Metri (MDH) April 26, / 22

2 Overview 1 Introduction 2 Mathematical formulation 3 Results and Discussions 4 Conclusions Prashant G Metri (MDH) April 26, / 22

3 Introduction 1 Boundary layer theory 2 Heat transfer 3 Nanoliquid 4 Viscous dissipation 5 Magnetohydrodynamics Prashant G Metri (MDH) April 26, / 22

4 Magnetohydrodynamics(MHD) Applications of MHD 1 The generation of electrical power with help of an electrically conducting fluid through magnetic field. 2 MHD is used in biological system in describing the rheological behaviour of blood. 3 The concept of MHD is applied in Geo-Physics to study the flow pattern in the core of earth. 4 Magnetic fields play a key role in star formation. Prashant G Metri (MDH) April 26, / 22

5 Velocity and temperature components U(x, t) = bx 1 αt, (1) The surface temperature T s of the stretching sheet is assumed to vary with the distance x from the slit as [ ] bx 2 T s (x, t) = T 0 T ref (1 αt) 3 2, (2) 2ν The applied magnetic field is assumed to be of variable kind and is chosen in its special form as B(x, t) = B 0 (1 αt) 1 2. (3) Prashant G Metri (MDH) April 26, / 22

6 Mathematical formulation Figure: Schematic representation of a nanoliquid film on an elastic sheet Prashant G Metri (MDH) April 26, / 22

7 Governing equations u x + v y = 0 (4) u t + u u x + v u y = µ nf ρ nf 2 u y 2 σb2 0 ρ nf u (5) T t + u T x + v T y = K nf 2 T (ρcp) nf y 2 + µ nf (ρc p ) nf ( ) u 2 (6) y Prashant G Metri (MDH) April 26, / 22

8 Expressions for nanoliquids ρ nf = (1 φ)ρ f + φρ s (7) µ f µ nf = (1 φ) 2.5 (8) [ ] Ks + 2K f 2φ(K f K s ) K nf = K f (9) K s + 2K f + φ(k f K s ) (ρc p ) nf = (1 φ)(ρc p ) nf + (ρc p ) s (10) The associated boundary conditions for Eqs. (4)-(6) are u = U, v = 0, T = T w at y = 0 (11) u y = T = 0 at y = h (12) y v = h at y = h (13) t Prashant G Metri (MDH) April 26, / 22

9 Similarity variables ( ) νf b 1/2 ψ(x, y, t) = xf (η) 1 αt (14) [ ] bx 2 T (x, y, t) = T 0 T ref (1 αt) 3/2 θ(η) 2ν f (15) [ ] b 1/2 η = y ν f (1 αt) (16) The velocity components u and v in terms of the Stream function ψ(x, y, t) are given by u = ψ ( ) bx y = f (η) (17) 1 αt v = ψ ( ) x = νf b 1/2 f (η) (18) 1 αt Prashant G Metri (MDH) April 26, / 22

10 Film thickness and reduced ODE b β = (1 αt) 1/2 h (19) ν f which gives dh dt = αβ νf 2 b (1 αt) 1/2 (20) Substituting similarity variable (14)-(16) into Eqs.(4)-(6) ( η f + φ 1 [ff f 2 S 2 f + f ) + 1 ] Mf = 0 (21) φ 2 θ + Pr ( Knf K f ) φ 3 [f θ 2f θ S2 ] (3θ + ηθ) + 1φ4 Ec(f ) 2 = 0 (22) corresponding boundary conditions are f (0) = 0, f (0) = θ(0) = 1, (23) f (β) = θ (β) = 0, f (β) = Sβ 2 (24) Prashant G Metri (MDH) April 26, / 22

11 Nanoliquid volume fraction the constants φ 1, φ 2, φ 3 and φ 4 that depends on the volume fraction are respectively given by ( )] φ 1 = (1 φ) [(1 2.5 ρs φ) + φ (25) ( ρs φ 2 = 1 φ + φ ( ) (ρcp ) s φ 3 = 1 φ + φ (ρc p ) f [ φ 4 = (1 φ) φ + φ (ρc ] p) s (ρc p ) f ρ f ) ρ f (26) (27) (28) Prashant G Metri (MDH) April 26, / 22

12 Thermo-physical properties of liquid and nanoparticle ρ(kg/m 3 ) C p (J/kgK) k(w /mk) Pure water Aluminium oxide(al 2 O 3 ) Silver(Ag) Titanium oxide(tio 2 ) Table: Thermo-physical properties of liquid and nanoparticle Prashant G Metri (MDH) April 26, / 22

13 Numerical solution df 0 dη = f 1, (29) df 1 dη = f 2, (30) [ df ( 2 dη = φ 1 S f 1 + η ) 2 f 2 + (f 1 ) 2 f 0 f ] Mf 1, φ 2 (31) dθ 0 dη = θ 1, (32) ( ) dθ 1 dη = φ Kf 3Pr K nf [ S 2 (3θ 0 + ηθ 1 ) + 2f 1 θ 0 θ 1 f 0 1 ] Ecf2 2, (33) φ 4 Prashant G Metri (MDH) April 26, / 22

14 Corresponding boundary conditions take the form, f 1 (0) = 1, f 0 (0) = 0, θ 0 (0) = 1, (34) f 2 (β) = 0, θ 1 (β) = 0, (35) f 0 (β) = Sβ 2. (36) Prashant G Metri (MDH) April 26, / 22

15 Results and discussion Figure: Variation of film thickness β with S Prashant G Metri (MDH) April 26, / 22

16 (a) S = 0.8 (b) S = 1.2 Figure: Effect of φ on temperature profile Prashant G Metri (MDH) April 26, / 22

17 (a) S = 0.8 (b) S = 1.2 Figure: Effects of magnetic field M on temperature profile Prashant G Metri (MDH) April 26, / 22

18 Types of nanoliquids S φ = 0.0 φ = 0.1 φ = 0.2 Al 2 O Ag TiO Table: Skin friction coefficient f (0) for various values of S and φ with Pr = and M = 2 Prashant G Metri (MDH) April 26, / 22

19 Types of nanoliquids S φ = 0.0 φ = 0.1 φ = 0.2 Al 2 O Ag TiO Table: Nussuelt number θ (0) for various values of S and φ with Pr = and M = 2 Prashant G Metri (MDH) April 26, / 22

20 Conclusions 1 The film thickness β can be affected seriously by S, β decreases dramatically with increasing S. 2 The film thinning rate decreases with the increase of the nanoparticle volume fraction. 3 Viscous dissipation enhances the thermal boundary layer thickness. 4 The dimensionless film thickness increases in magnetic field parameter M. 5 The wall temperature gradient(nusselt number) θ (0) is a decreasing function of in all the considered nanoliquids while the opposite is true for skin friction f (0). Prashant G Metri (MDH) April 26, / 22

21 References C.Y. Wang. Liquid film on an unsteady stretching surface. Quart. Appl. Math. 48, , B. Santra and B. S. Dandpat. Unsteady thin film flow over a heated stretching sheet, Int. J. H. and M. Tra. 52, , Khanafer, K., Vafai, K.: A critical synthesis of thermophysical characteristics of nanofluids. Int. J. Heat. Mass. Transfer. 54, (2011). Sakiadas, B. C.: Boundary layer behavior on continuous solid surfaces: I Boundary layer equations for two dimensional and flow. AIChE. J. 7, (1961). Prashant G Metri (MDH) April 26, / 22

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