ACTA TECHNICA CORVINIENSIS - Bulletin of Engineering

ACTA TECHNICA CORVINIENSIS – BULLETIN of ENGINEERINGSingh [6]. Basant kumar Jha and RavindraPrasad [7] analyzed mass transfer effects on theflow past an accelerated infinite vertical platewith heat sources. Again Basant kumar Jha [8]discussed MHD free convection and masstransform flow through a porous medium.Recently Muthucumaraswamy et al. [9] studiedunsteady flow past an accelerated infinitevertical plate with variable temperature anduniform mass diffusion.The hydro magnetic free convection flow pastan accelerated vertical porous plate withvariable temperature through a porous mediumhas many technical applications. Hence it isproposed to study MHD free convection flowpast an accelerated vertical porous plate withvariable temperature through a porous medium.The dimensionless governing equations aresolved using the Laplace transform technique.MATHEMATICAL ANALYSISAn unsteady flow of an electrically conductingviscous incompressible fluid past an infinitevertical porous plate with variable temperaturethrough porous medium has been considered. Amagnetic field of uniform strength is assumed tobe applied transversely to the porous plate. Themagnetic Reynolds number of the flow is takento be small enough so that the inducedmagnetic field can be neglected. The flow isassumed to be in x′ - direction which is takenalong the vertical plate in the up ward direction.The y′ -axis is taken to be normal to the plate.Initially the plate and the fluid are at the sametemperatureT ′ . At time t ′ ∞>0, the plate isaccelerated with a velocity u′= u0t′in its ownplane and the plate temperature is raisedlinearly with time t. It is assumed that the effectof viscous dissipation is negligible. Then by usualBoussinesq’s approximation, the governingequations for the unsteady flow are2 2∂u′ ∂u′ ∂ u′σBu′0νu′+ v′ = gβ( T′ − T′∞)+ ν − −2∂t′ ∂y′ ∂y′ρ K′(1)2⎛∂T′ ∂T′⎞ ∂ T′ρCp⎜ + v′⎟=κ2⎝ ∂t′ ∂y′⎠ ∂y′(2)With the initial and boundary conditionst′ ≤ 0 , u′ = 0 , T ′= T ∞′ for all y′(3)t′ > 0 , u′ u0t′T′ = T′ ∞+ T′ w− T′∞At′ at y′ = 0u′ = 0 , T′ → T ∞′ as y′ →∞.= , ( )12⎛u 3where A 0⎞= ⎜ ⎟⎝ ν ⎠Equation (1) is valid when the magnetic lines offorce are fixed relative to the fluid.On introducing the following non-dimensionalquantities:u′⎛u u = , t t 0⎞= ′ ⎜ ⎟⎝ ν ⎠( νu ) 1 3w0∞12 3T′ −T′μC∞pθ = , PT ′ − T ′r= , MκG( ′ − ′ )gβT Tw ∞r= ,u0⎛u30 ⎞, y = y′⎜ ν2 ⎟⎝ ⎠ ,1σ B= , (4)ρ( νu) 1 3012 30ν23u02⎛u30 ⎞= ′ ⎜ 2 ⎟−v′γ = , K K⎝ νin equations (1) to (3), leads to∂u 2− γ∂u = Gurθ+ ∂ −Mu−u2∂t ∂y ∂yK2⎛∂θ ∂θ ⎞ ∂ θP r ⎜ − γ ⎟=(6)2⎝ ∂t ∂y ⎠ ∂yWith the initial and boundary conditions :t ≤ 0: u = 0 , θ = 0 for all yt > 0: u = t , θ = t at y = 0(7)u = 0 , θ → 0 as y →∞All the physical variables are defined in thenomenclature. The solution of equations (5) and(6), subject to the boundary conditions (7) by thelaplace transform technique when the prandtlnumber Pr= 1, is given by⎡ γ⎤ ⎡ γ⎤Gy d2 2y drt−y t d− +⎛ ⎞ ⎡ ⎢ ⎥ ⎛ ⎢ 2⎥⎣ ⎦+ ⎞ ⎛⎣ ⎦y−2t d⎞⎤u= ⎜1− ⎟ ⎢e erfc + e erfc⎥⎝ M′⎠2 ⎜ 2 t ⎟ ⎜ 2 t ⎟⎢⎣⎝ ⎠ ⎝ ⎠⎥⎦⎡ γ⎤ ⎡ γ⎤y G− y d+ 2 2y dry t d−⎡ ⎤⎡⎢ ⎥ ⎛ ⎢ 2⎥⎣ ⎦− ⎞ ⎛⎣ ⎦y+2t d⎞⎤− 1 e erfc e erfc4 d⎢− ⎢−⎥⎣ M′⎥⎦⎜ 2 t ⎟ ⎜ 2 t ⎟⎢⎣⎝ ⎠ ⎝ ⎠⎥⎦Gt ⎡r ⎛ y+ tγ⎞ −γy ⎛ y−tγ⎞⎤+ erfc + e erfc2M ′⎢ ⎜ ⎟ ⎜ ⎟⎥⎣ ⎝ 2 t ⎠ ⎝ 2 t ⎠⎦(8)Gy ⎡r −γy ⎛ y− tγ⎞ ⎛ y+tγ⎞⎤− e erfc −erfc2M ′⎢γ⎜ ⎟ ⎜ ⎟⎥⎣ ⎝ 2 t ⎠ ⎝ 2 t ⎠⎦⎠(5)922010/Fascicule 2/April­June /Tome III