Adv. Studies Theor. Phys., Vol. 5, 2011, no. 7, 337-342 An Exact Solution of MHD Boundary Layer Flow over a Moving Vertical Cylinder Alvaro H. Salas Universidad de Caldas, Manizales, Colombia Universidad Nacional de Colombia asalash2002@yahoo.com FIZMAKO Research Group Abstract The steady flow of an incompressible electrically conducting fluid over a semi infinite moving vertical cylinder in the presence of an uniform transverse magnetic field is analyzed. The partial differential equations governing the flow are reduced to an ordinary nonlinear differential equation by using the self-similarity transformation. By using elementary methods, a closed form solution to the problem is given. Mathematics Subject Classification: 76W05 Keywords: MHD, boundary layer flow, uniform transverse magnetic field 1 Introduction Boundary layer flow behavior on a cylinder moving in a Newtonian fluid was initially studied by Sakiadis [2], and obtained a numerical solution using a similarity transformation. Later, this problem has received the attention of certain researchers (see [3, 4, 5]). the problem of MHD flow over infinite surfaces has become more important due to the possibility of applications in areas like nuclear fusion, chemical en- gineering, medicine, and high-speed, noiseless printing. Problem of MHD flow in the vicinity of infinite plate has been studied intensively by a number of investigators (see, e.g., [6, 7, 8, 9,10] and the references therein). But only very few authors studied the flow past semi-infinite vertical cylinder (see, e.g., [11, 12, 13, 14, 15] and the references therein). Most of previous investigations were concerned with numerical studies and there are only few papers in the literature that deal with a theoretical analysis
338 A. H. Salas of problem of MHD flow along a vertical cylinder, however, an important number of theoretical investigations are concerned with flow past vertical and flat plates without magnetic field. The subject of the present paper is to give an analytic solution to the problem of boundary layer in a laminar flow of a viscous incompressible and electrically conducting fluid past a permeable moving vertical semi-infinite cylinder under the action of a uniform magnetic field in the case of a linear external velocity [1]. The governing boundary layer equations with initial and boundary conditions are reduced to an ordinary nonlinear differential equation which is solved in an elementary way by using a special ansatz. 1.1 Mathematical formulation We consider a steady laminar and incompressible viscous MHD flow past a moving per- meable semi-infinite vertical cylinder of radius R. The applied transverse magnetic field B is assumed to be uniform. All fluid properties are assumed to be constant and the magnetic Reynolds number is assumed to be small so that the magnetic field can be neglected. No electric field is assumed to exist. Axial coordinate χ is measured along the axis of the cylinder. The radial coordinate r is measured normal to the axis of cylinder. We denote by u e (x) =u x the external velocity with u > 0. Under these assumptions and with the boundary layer approximation, the governing equations describing the problem are (ru) x + (rv) r =0, u u x + v u x = γ u (r r r r )+u du e e dx + σb2 ρ (u e u), (1) with initial and boundary conditions u(r, x) =u ω x, v(r, x) = v ω,u(,x)=u e (x), (2) we denote by u and v the velocity components in the χ and r directions, respectively. γ is the kinematic viscosity, ρ is the fluid density, and σ is the electric conductivity of the fluid. On the other hand, v ω is the suction/injection parameter, with v ω > 0 corresponding to the wall suction, v ω < 0 corresponding to the wall blowing, and the case v ω = 0 characterizing the impermeable wall. In our following analysis we assume that v ω > 0 and u ω > 0. The stream function ψ is defined by ru = ψ/ r and rv = ψ/ x. Substituting these expressions in (1)-(2), the continuity equation is automatically
MHD flow over a vertical cylinder 339 satisfied and we obtain the boundary value problem 1 ψ r 2 r 2 ψ x r + 1 ψ ψ r 3 x = γ r 3 ψ r γ r 2 2 ψ r 2 + γ r ψ r (R, x) = u ωx, r 1 ψ 2 ψ r 2 x r 2 3 ψ r + u 3 e ψ x (R, x) =Rv ω, du e dx + σb2 ρ (u e 1 r ψ ), (3) r lim ( 1 ψ r r r (r, x)) = u x. (4) We look for self-similar solutions in the form vu R ψ(r, x) = xf(t), (5) 2 where f is the dimensionless stream function and t = u R/2v((r 2 R 2 )/R) is the similarity variable. In terms of this variable, the governing equation and boundary conditions (3) are transformed into (Kt +2R)f (t)+kf (t)+f(t)f (t) f (t) 2 M(f (t) 1) + 1 = 0, (6) f(0) = a, f (0) = b, f ( ) =1, (7) with 2v K =2 u R,a= v ωr vu R,b= u ω 2u,M= σb2 ρu. (8) Note that M>0 is the magnetic parameter and a>0 plays the role of suction parameter. 2 Main result Our main objective is to give an exact solution to the problem described in previuos section. To this end, we will suppose a solution to Eq. (6) in the form f(t) =t + p + q exp( μt), (9) where p, q and μ are constants and μ>0. Observe that f (t) =1 qμ exp( μt),
340 A. H. Salas so condition f ( ) = 1 is satisfied, since f ( ) = lim (1 qμ exp( μt)) = 1. t + On the other hand, conditions hold if f(0) = a, f (0) = b p = (2u u ω ) vu 2 Rμu v ω 2μ vu 3 and q = 2u u ω 2μu. (10) Thus, we seek solution to (6) in the form f(t) =t (2u u ω ) vu 2 Rμu v ω 2μu vu + 2u u ω 2μu exp( μt). (11) Substituting (11) into (6), we obtain a polynomial equation in the variables t and ζ = exp( μt). Equating its coefficients to zero we obtain a polynomial system in the variables B and μ. Solving it gives ( B = ± 1 ρ uω u (u R 2 8v) 2u 3 v ωr 2v ) u 3 ω 2 vσ u R, μ = u ω 8v. (12) Since u, u ω, u, v, ρ, σ and R are positive constants, the expression for B in (12) is defined for 0 <v< R 2 u 2 2(4u + u ω ) and 0 <v ω < R2 u 2 2v(4u + u ω ). 2u 3 R Therefore, an exact solution to Eq. (6) subject to conditions (7) is given by f(t) =t + Rv ω vu 3 2 2vu vuω + ( ) 2v 3 u 3 ω 2v v (2u u ω ) u R + exp Ru ω u 3 Rvu 3 8v t. Finally a self-similar solution to the boundary-value problem (3)-(4) is ( u vr u R r 2 R 2 ψ(r, x) = + Rv ω vu 3 2 2vu vuω + 2v 3 u 3 ω 2 2v R v + Ru ω u 3 2 ( v (2u Rvu 3 u ω ) exp u ) ) 4v (r2 R 2 ) x,
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342 A. H. Salas [14] Magnetic field effect on a moving vertical cylinder with constant heat flux, Heat and Mass Transfer 39 (2003), no. 5-6, 381-386. [15] M. A. Hossain and M. Ahmed, MHD forced and free con vection boundary layer flow near the leading edge, International Journal of Heat and Mass Transfer 33 (1990), no. 3, 571-575. Received: January, 2011