HIEMENZ MAGNETIC FLOW BY DIFFERENTIAL TRANSFORMATION METHOD AND PADE APPROXIMANT

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1 ACTA TECHNICA CORVINIENSIS Bulletin of Engineering Tome IX [6], Fascicule [January arch] ISSN: K. NAYAK,. G.C. DASH HIEENZ AGNETIC FLOW BY DIFFERENTIAL TRANSFORATION ETHOD AND PADE APPROXIANT. Deartment of Physics, Radharishna Institute of Technology and Engineering, Bhubaneswar, INDIA. Deartment of athematics, I.T.E.R., Sisha O Anusandhan University, Bhubaneswar, INDIA Abstract: The agnetohydrodynamic (HD) Hiemenz boundary layer flow over a flat late embedded in a orous medium in the resence of transverse magnetic field has been studied. The governing equations are solved by differential transformation method with Pade aroximant (DT-Pade) and Runge-Kutta method along with shooting technique. The results of these two methods are comared with the results obtained by finite difference method in conjunction with quasilinearization technique reorted earlier in case of the flow without orous medium. It is found that the results of DT-Pade, Runge-Kutta and quasilinearization technique agree with each other within a certain degree of accuracy. The convergence of the method in attaining the ambient state is faster in case of Runge-Kutta method than the DT-Pade which can be imroved by emloying higher dimension Pade aroximant matrices. It is also remared that both magnetic field and orous matrix enhance the velocity field as well as sin friction. Keywords: Hiemenz magnetic flow; Porous medium; DT; Finite difference; Quasilinearization INTRODUCTION Flows in which the velocity of the incoming fluid is erendicular to a lane surface is nown as Hiemenz flow []. If in addition the fluid is electrically conducting, the flow is then called Hiemenz magnetic flow. The solution of this roblem is of interest because it is one of the few exact solutions of Navier-Stoes equation in magnetohydrodynamics. Further, the governing equations of the Hiemenz magnetic flow are nonlinear. An effective method of solution is the method of finite difference in conjunction with quasilinearization as resented in NA []. Alied athematics, Physics and roblems related to engineering exhibit nonlinear henomena. ost of nonlinear equations do not have a recise analytical solution; so numerical methods are usually alied to solve the governing equations. Some of the analytical methods are erturbation techniques [3], Adomian decomosition method (AD) (Dehghan [4-5]), homotoy analysis method (HA), DT and variational iteration method (VI). He [6] and Rashidi [7] have studied the generalized differential transformation method to solve differential equations governing flow of fluids. The HD flow finds numerous alications in industries such as HD ower generation and HD ums (Hayat et al. ([8]) etc. Further, in the field of heat transfer, the concet of flow through orous media is of great consequence in the modern technology as the orous matrix acts as a good insulator to revent energy loss. The two relevant roerties associated with the study of flow through orous media are orosity and ermeability. Porosity basically describes the fraction of total volume which is occuied by the holes. Permeability is a measure of the caacity with which fluids will flow through a orous material. Table resents the numerical values of effective orosity and ermeability of materials of common use. Table : Porosity and ermeability of tyical orous materials aterial Effective orosity Permeability Bric Coer owder Leather Limestone Sand Sand stone Silica owder Soil Wire crims coyright Faculty of Engineering Hunedoara, University POLITEHNICA Timisoara

2 ACTA TEHNICA CORVINIENSIS Fascicule [January arch] Bulletin of Engineering Tome IX [6] The objective of the resent study is to aly DT, DT-Pade aroximant and Runge-Kutta method to solve modified Navier-Stoes equation for Hiemenz magnetic flow through orous media and to comare the results obtained by the resent methods of solution with the results reorted in [] emloying finite difference method in conjunction with quasilinearization technique. HIEENZ AGNETIC FLOW The boundary layer equations for Hiemenz magnetic Darcy flows for viscous fluid following [9] are: Continuity: u v + = () x y omentum: u u u σb υ u + v = a x + υ + ( ax u) + ( ax u) () * x y y ρ K subject to the boundary conditions: u =, v = at y = (3) u = ax, at y * where uv,, υρ,, K and a are resectively the x - comonent of the velocity, the y -comonent of the velocity, the viscosity, the density, the ermeability of the medium and a constant nown as initial stretching rate (characteristic of the incoming flow) with dimension (time) -. Further, σ and B are resectively the electrical conductivity and magnetic induction. The last term reresents the additional resistance due to orosity of the orous medium. The flow geometry is shown in Figure. Figure. Flow geometry The first two rescribed boundary conditions reresent neither sli nor mass transfer on the surface where the conditions are at infinity (i.e. ambient state) means that the velocity of the fluid aroaches a linear relation with x. Introducing the variables df u = ax and v= aυ f( η), (4) where a η = y. (5) υ the equations () and (3) become 3 d f d f df df + f = (6) df f =, = atη = df = atη (7) B where = σ a ρ is the magnetic arameter and K * ak = υ is the dimensionless ermeability arameter. Case I: =, K (for large value of K ), the roblem reduces to the flow of fluids without magnetic field and orous medium. Case-II: = and K finite (for small value of K ), the roblem reresents Newtonian flow through orous medium without magnetic field. Case-III: and K finite (for small value of K ), the roblem reresents the Darcy flow of conducting fluid in the resence of magnetic field with low magnetic arameter. Solution of equation (6) with boundary conditions (7) is obtained by emloying differential transformation method with Pade aroximant (DT-Pade) and Runge-Kutta method. The results of these two methods are comared with the results obtained by finite difference method in conjunction with quasilinearization []. DIFFERENTIAL TRANSFORATION ETHOD Differential transformation method is a numerical method based on Taylor s exansion. This method determines the coefficients of series exansion of unnown function by using the initial data on the roblem. The concet of differential transformation method was first roosed by Zhou []. The DT- Pade was alied to electric circuit analysis roblems and also it was alied to several systems of differential equations for examle, initial value roblems [], difference equations [], integrodifferential equations [3], and artial differential equations [4]. Definition. The one dimensional differential transform of a function f ( η) at the oint η = η is defined as d F ( ) = { f( ) }! η d (8) η η = η where f ( η ) is the original function and F ( ) is the transformed function. 5

3 ACTA TEHNICA CORVINIENSIS Fascicule [January arch] Bulletin of Engineering Tome IX [6] Definition. The differential inverse transform of F ( ) is defined as f( η) = F ( )( η η ) (9) = The other roerties are enlisted in Table. Table : Some roerties of differential transformation method. Original function f( η= ) f ( η± ) f ( η ) f( η ) =λf ( η ) f( η ) = f ( η) f ( η ) n d f ( ) η f ( η ) = n f ( η) ( η ) = f( η) d f f ( η ) = ( η η ) m Transformed function F ( ) = F ( ) ± F( ) F ( ) = λ F ( ) F () = FrF () ( r) r= ( + n)! F ( ) = F ( + n)! F( ) = ( r+ )( r+ ) F( r) F ( r+ ) r=, = m F ( ) =δ( m) =, m PADE APPROXIANT Some techniques exist to accelerate the convergence of a given series. Among them the so-called Pade aroximant is widely alied (Baer and orris, [5]). Suose that a function f ( η ) is reresented by a ower series, i f( η) = cη () i= This exression is the fundamental oint of any analysis using Pade aroximant. The notation ci, i =,, is reserved for the given set of coefficients and f ( η ) is the associated function. [ L/ ] Pade aroximant is a rational fraction, defined as L a + aη+ aη + + alη f ( η) = b + bη+ bη + + bη, () which has a aclaurin exansion, agrees with equation () as far as ossible. It is noticed that in equation () there are L+ numerator and + denominator coefficients. So there are L+ indeendent numerator and indeendent denominator coefficients, maing L++ unnown coefficients in all. This number suggests that normally [ L/ ] ought to fit the ower series L equation () through the orders, ηη, η +. In the notation of formal ower series L i a + aη+ aη + + alη L+ + ciη = + O( η )() b + bη+ bη + + b η i= ( b + bη+ + b η )( c + cη+ ) = a + a + + a + O + + i L L η Lη ( η ) (3) Equating the coefficients of η L+, η L+,, η L+ m we get, bcl + + b cl bc L+ bc L+ =, bcl + + b cl bc L+ + bc L+ =,... (4)... bcl + b cl+ + + bc L+ + bc L+ =, If j <, we define c i = for consistency. Since b =, equation (4) becomes a set of linear equations for unnown denominator coefficients. cl + cl + cl b cl+ cl + cl + 3 cl+ b cl+ = (5) cl cl+ cl+ b cl+ From these equations, b i may be found. The numerator coefficients a, a,, a L, follow immediately from equation (3) by equating the L coefficients of, ηη,,, η + such as, a = c, a= c+ bc, a= c+ bc + bc, (6) min[ L / ] al= cl+ bc i L i. i= Thus equations (5) and (6) normally determine the Pade numerator and denominator and are called Pade equations. The [L/] Pade aroximant is constructed which agrees with the equation () through the orderη L+. SOLUTION OF THE PROBLE Analytical solution Consider the equation (4) f ( η) + f( η) f ( η) + ( f ( η) ) + + ( f ( η )) =, (7) with boundary conditions f() =, f () =, f ( ) = (8) Combination of the series obtained by DT and Pade aroximant will yield the numerical value of f () so as to reduce the resent boundary value roblem (BVP) into an initial value roblem (IVP). The diagonal Pade aroximants of degree [ / ] is emloyed to determine the aroximate solution. 53

4 ACTA TEHNICA CORVINIENSIS Fascicule [January arch] Bulletin of Engineering Tome IX [6] Let f () = A, where A is a ositive constant. Now, the differential transform method (DT) will be alied to equation (7) as follows: ( + )( + )( + 3) F ( + 3) + {( r+ )( r+ ) F( r) F( r+ ) r= ( r+ )( r+ ) Fr ( + ) F ( r+ )} ( ) + ( + ) F ( + ) = δ (9) The differential transform of boundary conditions are F() =, F() =, F() = A. () Alying the differential inverse transform, f( η) = F ( ) η = A = Aη η + η A + + A A η + η 6 8 () Case I : ( =.5, K = ) The DT exression () becomes 3 3 A 4 A 5 A 6 A A 7 f( η) = Aη η + η + η + η + + η A 5A 9 η + η + () Now our aim is to determine A using the boundary condition lim f ( η) = (3) η Alying the boundary condition (3) to [/] Pade aroximant of the derivative of the olynomial solution (), we get 4 64A + 7A 8 Aη+ η 7 6 lim A = η 3 3A+ 8A 7 + A η 7 6A 4(7 6 A ) η which gives A = Table 3: Determination of A K A Similarly, the following values of A are obtained for the various values of and K as deicted in Table 3. NUERICAL SOLUTION The governing equation is solved numerically by alying fourth order Runge-Kutta method along with shooting technique. This method has been roven to be adequate and gives accurate results for boundary layer equation. The solution is comuted for the dimensionless velocity and shown grahically. Let f( η ) = y, f '( η ) = y and f "( η ) = y3. so that y' 3 = yy 3 + y + ( y) with ya() =, ya() =, yb() =. RESULTS AND DISCUSSION The DT-Pade aroximant and Runge-Kutta method with shooting technique have been alied for solving Hiemenz magnetic flow through orous medium. The solution for the flow without orous medium has been derived as a articular case and the results are comared with the results obtained by the method of finite difference in conjunction with quasilinearization technique in Na []. Figure. Velocity rofiles (DT) Figures, 3 and 4 exhibit the results obtained by DT, DT-Pade and Runge-Kutta method associated with shooting technique. It is found that the effect of magnetic arameter is to decrease the velocity of the fluid irresective of the resence or absence of orous matrix. On careful observation it is further remared that resence of orous matrix also decreases the velocity rofiles at all the oints. Therefore, it is concluded that resence of orous matrix as well as magnetic field both decrease the velocity of the fluid at all oints of the flow domain. Further, it is seen that rate of decrease is more with 54

5 ACTA TEHNICA CORVINIENSIS Fascicule [January arch] Bulletin of Engineering Tome IX [6] the combined effect of the magnetic field and orous matrix. From figures 3 and 4 it is observed that the attainment of ambient state is faster in case of DT- Pade in comarison with DT and it is still faster in case of Runge-Kutta method. Figure 3. Velocity rofiles (DT-Pade). Figure 4. Velocity rofiles (Runge-Kutta method). Table 4: Sin friction coefficient (A) K DT- Runge- Quasilinearization Pade Kutta NA [] Now, the Table-4 resents the numerical values of sin friction comuted by different methods. It is evident that sin friction (in magnitude) increases with an increase in the values of magnetic arameter with or without orous medium. It is also seen that resence of orous matrix increases the sin friction (in magnitude) for a fixed value of magnetic arameter. Table-4 further reveals that the values of sin friction obtained by (i) Runge-Kutta and (ii) Finite difference in conjunction with quasilinearization agree u to first decimal lace. This shows the consistency of the methods alied in the resent analysis to solve the modified HD Hiemenz flow. It is suggested that the accuracy of DT-Pade method can be imroved by emloying higher degree diagonal Pade aroximants. CONCLUSION The DT-Pade and Runge-Kutta method are consistent within certain degree of accuracy to solve non-linear boundary value roblems and convergence of the method can be accelerated with higher dimension Pade aroximant matrices so as to attain the ambient state of the flow which is also assisted by the resence of magnetic field and orous medium. It is further concluded that resence of magnetic field and orous medium is found to be counterroductive in reducing the sin friction at the surface of the late. References [.] K. Hiemenz, Die Grenzschicht an einem n den gleichformigen Flussigeitsstrom eingetauchten geraden Kreiszylinder, Dingl. Polytech. J. 36 (9) 3. [.] T.Y. NA, Comutational methods in Engineering Boundary Value Problems, Academic ress, New Yor (979) -. [3.] J. H. He, Homotoy erturbation technique, Comuter ethods in Al. ech. Eng., 78 (3-4) (999) [4.]. Dehghan, F. Shaeri, Solution of an integrodifferential equation arising in oscillating magnetic fields using He's homotoy erturbation method, Progress in Electromagnetic Research, 78 (8) [5.]. Dehghan, F. Shaeri, Use of He's homotoy erturbation method for solving a artial differential equation arising in modeling of flow in orous media, Journal of Porous edia, (8) [6.] J. H. He, A new aroach to non-linear artial differential equations, Comm. Nonlinear Sci. Numer. Simulation, (4) (997) [7.].. Rashidi, H. Shahmohamadi, Analytical solution of three-dimensional Navier Stoes equations for the flow near an infinite rotating dis, Comm. Nonlinear Sci. Numer. Simulat., 4 (7) (9) [8.] T. Hayat, C. Fetecau,. Sajid, Analytic solution for HD transient rotating flow of a second grade fluid in a orous sace, Nonlinear Analysis: Real World Alications, 9 (4) (8)

6 ACTA TEHNICA CORVINIENSIS Fascicule [January arch] Bulletin of Engineering Tome IX [6] [9.] Dj. S. SjuKic, Hiemenz magnetic flow of owerlaw fluids, J. Al. ech. Trans. ASE, 4 (974) [.] J. K. Zhou, Differential transformation and its alications for electrical circuits, Huazhong University Press, Wuhan, China, 986. [.]. J. Jang, C. L. Chen, Y. C. Liu, On solving the initial value roblems using the differential transformation method, Al. ath. Comt., 5 () [.] A. Arioglu, I. Ozol, Solution of difference equations by using differential transform method, Al. ath. Comt., 74 (6) 6 8. [3.] P. Darania, A.Ebadian, A method for the numerical solution of integro-differential equations, Al. ath. Comt., 88 (7) [4.] F. Ayaz, On the two-dimensional differential transform method, Al. ath. Comt., 43 (- 3) (3) [5.] G.A. Baer, P. Graves-orris, Pade Aroximants, Cambridge U.P., 996 coyright University POLITEHNICA Timisoara, Faculty of Engineering Hunedoara, 5, Revolutiei, 338, Hunedoara, ROANIA htt://acta.fih.ut.ro 56