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

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, 2017 1 / 22

Overview 1 Introduction 2 Mathematical formulation 3 Results and Discussions 4 Conclusions Prashant G Metri (MDH) April 26, 2017 2 / 22

Introduction 1 Boundary layer theory 2 Heat transfer 3 Nanoliquid 4 Viscous dissipation 5 Magnetohydrodynamics Prashant G Metri (MDH) April 26, 2017 3 / 22

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, 2017 4 / 22

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, 2017 5 / 22

Mathematical formulation Figure: Schematic representation of a nanoliquid film on an elastic sheet Prashant G Metri (MDH) April 26, 2017 6 / 22

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, 2017 7 / 22

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, 2017 8 / 22

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, 2017 9 / 22

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, 2017 10 / 22

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) 2.5 1 φ + φ (ρc ] p) s (ρc p ) f ρ f ) ρ f (26) (27) (28) Prashant G Metri (MDH) April 26, 2017 11 / 22

Thermo-physical properties of liquid and nanoparticle ρ(kg/m 3 ) C p (J/kgK) k(w /mk) Pure water 997.1 4179 0.613 Aluminium oxide(al 2 O 3 ) 3790 765 40 Silver(Ag) 10500 235 429 Titanium oxide(tio 2 ) 4250 686.2 8.9538 Table: Thermo-physical properties of liquid and nanoparticle Prashant G Metri (MDH) April 26, 2017 12 / 22

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 2 + 1 ] 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, 2017 13 / 22

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, 2017 14 / 22

Results and discussion Figure: Variation of film thickness β with S Prashant G Metri (MDH) April 26, 2017 15 / 22

(a) S = 0.8 (b) S = 1.2 Figure: Effect of φ on temperature profile Prashant G Metri (MDH) April 26, 2017 16 / 22

(a) S = 0.8 (b) S = 1.2 Figure: Effects of magnetic field M on temperature profile Prashant G Metri (MDH) April 26, 2017 17 / 22

Types of nanoliquids S φ = 0.0 φ = 0.1 φ = 0.2 Al 2 O 3 0.3 6.84816095 6.88304732 6.91315758 0.7 2.93846813 2.95772743 2.97386683 1.1 1.63692196 1.64313584 1.6480675 1.5 0.81243729 0.81364587 0.81459189 1.8 0.30812787 0.30826542 0.09508691 Ag 0.3 6.84816274 6.95034032 7.02005426 0.7 2.93846813 2.9924374 3.02431103 1.1 1.6369238 1.65347101 1.66223297 1.5 0.81243568 0.81561332 0.81717883 1.8 0.30812787 0.308484 0.30864766 TiO 2 0.3 6.84817172 6.88833479 6.92259184 0.7 2.93846813 2.9606241 2.97867719 1.1 1.63692196 1.164404822 1.649456 1.5 0.81243568 0.81380491 0.81486948 1.8 0.30813015 0.30828994 0.30839695 Table: Skin friction coefficient f (0) for various values of S and φ with Pr = 6.8173 and M = 2 Prashant G Metri (MDH) April 26, 2017 18 / 22

Types of nanoliquids S φ = 0.0 φ = 0.1 φ = 0.2 Al 2 O 3 0.3 9.49448354 8.21914135 7.21097505 0.7 4.69013197 4.0857758 3.62021146 1.1 3.193028 2.79876989 2.50815207 1.5 2.22126221 1.96091387 1.78149254 1.8 1.28527789 1.11922258 1.0131573 Ag 0.3 9.49448603 6.30230848 4.10734857 0.7 4.69013197 3.17798981 2.4335424 1.1 3.1930316 2.21094727 1.59709454 1.5 2.2212578 1.57471104 1.19441778 1.8 0.34672556 0.8803575 0.66593056 TiO 2 0.3 9.49449848 8.20801037 7.159999524 0.7 4.69013197 4.08360774 3.60509909 1.1 3.193028 2.80147178 2.50985349 1.5 2.2212578 1.96989346 1.79957274 1.8 1.28528742 1.12952079 1.03448954 Table: Nussuelt number θ (0) for various values of S and φ with Pr = 6.8173 and M = 2 Prashant G Metri (MDH) April 26, 2017 19 / 22

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, 2017 20 / 22

References C.Y. Wang. Liquid film on an unsteady stretching surface. Quart. Appl. Math. 48, 601 610, 1990. B. Santra and B. S. Dandpat. Unsteady thin film flow over a heated stretching sheet, Int. J. H. and M. Tra. 52, 1965 1970, 2009. Khanafer, K., Vafai, K.: A critical synthesis of thermophysical characteristics of nanofluids. Int. J. Heat. Mass. Transfer. 54, 4410 4428 (2011). Sakiadas, B. C.: Boundary layer behavior on continuous solid surfaces: I Boundary layer equations for two dimensional and flow. AIChE. J. 7, 26 28 (1961). Prashant G Metri (MDH) April 26, 2017 21 / 22

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