U. S. Rajput and Gaurav Kumar

Transcription

1 MATEMATIKA, 2018, Volume 34, Number 2, c Penerbi UTM Press. All righs reserved Effec of Hall Curren on Unseady Magneo Hydrodynamic Flow Pas an Exponenially Acceleraed Inclined Plae wih Variable Temperaure and Mass Diffusion U. S. Rajpu and Gaurav Kumar Deparmen of Mahemaics and Asronomy Universiy of Lucknow, Lucknow U.P, India Corresponding auhor: rajpugauravlko@gmail.com Aricle hisory Received: 22 Sepember 2016 Received in revised form: 29 July 2018 Acceped: 2 Augus 2018 Published on line: 1 December 2018 Absrac The presen sudy is carried ou o examine he effec of Hall curren on unseady flow of a viscous, incompressible and elecrically conducing fluid pas an exponenially acceleraed inclined plae wih variable wall emperaure and mass diffusion in he presence of ransversely applied uniform magneic field. The plae emperaure and he concenraion level near he plae increase linearly wih ime. The governing equaions involved in he presen analysis are solved by he Laplace-ransform echnique. The velociy profile is discussed wih he help of graphs drawn for differen parameers like hermal Grashof number, mass Grashof number, Prandl number, Hall curren parameer, acceleraion parameer, he magneic field parameer and Schmid number, and he numerical values of skin-fricion have been abulaed. I is observed ha he flow paern is affeced significanly wih plae acceleraion, Hall curren. The imporance of he problem can be seen in cooling of elecronic componens of a nuclear reacor, bed hermal sorage and hea sink in he urbine blades. Keywords Magneo hydrodynamic flow; inclined plae; variable emperaure; mass diffusion; Hall curren. Mahemaics Subjec Classificaion 76W05, 76D05. 1 Inroducion The MHD flow problems play imporan roles in differen area of science and echnology, like biological science, peroleum engineering, chemical engineering, mechanical engineering, biomechanics, irrigaion engineering and aerospace echnology. The influence of magneic field on viscous, incompressible and elecrically conducing fluid is of grea imporance in many applicaions such as magneic maerial processing, glass manufacuring conrol processes and purificaion of crude oil. The response of laminar skin fricion and hea ransfer o flucuaions in he sream velociy was sudied by Lighhill [1. Rajpu and Kumar [11 have analyzed MHD 34:2 (2018) eissn

2 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) flow pas an impulsively sared verical plae wih variable emperaure and mass diffusion. Free convecion flow pas an exponenially acceleraed verical plae was considered by Singh and Kumar[4. Mass ransfer effecs on he flow pas an exponenially acceleraed verical plae wih consan hea flux was sudied by Basan e al. [5. Muhucumaraswamy e al. [9 have considered hea ransfer effec on flow pas an exponenially acceleraed verical plae wih variable emperaure. Muhucumaraswamy e al. [10 have sudied mass ransfer effecs on exponenially acceleraed isohermal verical plae. Hall curren effec on MHD flow is also significan in many cases. Some such problems already sudied are menion here. Kaagiri [2 has invesigaed he effec of Hall curren on he magneo hydrodynamic boundary layer flow pas a semi-infinie fas plae. Pop [5 has considered he effec of Hall currens on hydromagneic flow near an acceleraed plae. Hall effec on magneohydrodynamic boundary layer flow over a coninuous moving fla plae was invesigaed by Pop and Waanabe [6. Aia [7 has analyzed he effec of variable properies on he unseady Harmann flow wih hea ransfer considering he Hall effec. Furher, Aia and Sayed [8 have sudied he Hall effec on unseady MHD couee flow wih hea ransfer of a bingham fluid wih sucion and injecion. Thamizhsudar and Pandurangan[12 have considered combined effecs of radiaion and Hall curren on MHD flow pas an exponenially acceleraed verical plae in he presence of roaion. Srinivas and Naikoi [13 has analyzed Hall effec on unseady MHD free convecion flow over a sreching shee wih variable viscosiy and viscous dissipaion. We are considering he unseady MHD flow pas an exponenially acceleraed inclined plae wih variable emperaure and mass diffusion in he presence of Hall curren. The resuls are shown wih he help of graphs and able. 2 Mahemaical Analysis The geomerical model of he problem is shown in Figure-1 Figure 1: Physical Model Consider an unseady flow of a viscous, incompressible, elecrically conducing fluid pas an impulsively sared non-conducing inclined fla plae. The x axis is aken along he verical

3 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) plane and z axis is normal o i. Thus he z axis lies in he horizonal plane. The plae is inclined a angle αfrom verical. A ransverse magneic field B 0 of uniform srengh is applied on he flow. Iniially i has been considered ha he plae as well as he fluid is a he same emperaure T. The species concenraion in he fluid is aken as C. A ime >0, he plae sars exponenially acceleraing in is own plane wih velociy u=u 0 e b, and emperaure of he plae is raised o T w. The concenraion C near he plae is raised linearly wih respec o ime. The flow model is as follows: u = u υ 2 z + gβ cosα(t T ) + gβ cos α(c C 2 ) σb2 0 (u + mv), (1) ρ(1 + m 2 ) v = v υ 2 z + σb2 0(mu v), (2) 2 ρ(1 + m 2 ) C = C D 2 z, (3) 2 T ρc p = T k 2 z. (4) 2 The iniial and boundary condiions are 0 : u = 0, v = 0, T = T, C = C, for all z, > 0 : u = u 0 e b, v = 0, T = T + (T w T ) u2 0 υ, C = C + (C w C ) u2 0 υ, z = 0, (5) u 0, v 0, T T, C C as z. The following non-dimensional quaniies are inroduced o ransform equaions (1), (2), (3) and (4) ino dimensionless form: z = zu 0 υ, ū = u, v = v, θ = (T T ) u 0 u 0 (T w T ), S c = υ D, µ = ρυ, P r = µc p k, G r = gβυ(t w T ) u 3, 0 M = σb2 0 υ ρu 2, G m = gβ υ(c w C ) 0 u 3, C = (C C ) 0 (C w C ), b = bυ (6) u 2, = u2 0 0 υ, m = ω eτ e. Thus he model becomes ū = 2 ū z + G M(ū + m v) 2 r cos αθ + G m cos α C (1 + m 2 ), (7) v = 2 ū M(mū v) + z 2 (1 + m 2 ), (8) C = 1 2 C S c z, 2 (9) θ = 1 2 θ P r z 2. (10) The corresponding boundary condiions (5) become: 0 : ū = 0, v = 0, θ = 0, C = 0, for all z, > 0 : ū = e b, v = 0, θ =, C =, a, z = 0, ū 0, v 0, θ 0, C 0, as z. (11)

4 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) Dropping bars in he above equaions, we ge u = 2 u z + G M(u + mv) rcosαθ + G 2 m CosαC (1 + m 2 ), (12) v = 2 v M(mu v) + z2 (1 + m 2 ), (13) C = 1 2 C S c z, 2 (14) θ = 1 2 θ P r z2. (15) The boundary condiions are 0 : u = 0, v = 0, θ = 0, C = 0, for all z, > 0 : u = e b, v = 0, θ =, C = a z = 0, u 0, v 0, θ 0, C 0, as z. Wriing he equaions (12) and (13) in combined form (using q = u + iv) (16) q = 2 q z + G 2 r cos αθ + G m cos αc qa, (17) C = 1 2 C S c z, 2 (18) θ = 1 2 θ P r z2. (19) The boundary condiions become: 0 : q = 0, θ = 0, C = 0, for all z, > 0 : q = e b, θ =, C =, a z = 0, q 0, θ 0, C 0, as z. The dimensionless governing equaions (17) o (19), subjec o he boundary condiions (20), are solved by he usual Laplace - ransform echnique. The soluion obained is as follows: { C = (1 + z2 S [ } c 2 )erfc Sc 2 z S c e z2 4 Sc, { π θ = (1 + z2 P [ } r 2 )erfc Pr 2 z P r e z2 4 Pr, π q= 1 2 eb a+bz A 15 + cosα [ πgr { 4a 2 A 9 z+ ae az A 2 z e } az A 1 P r + 2A 13 A 3 P r π 2A 13 A 3 G r P r { aa 10 z+ 1 Pr A 13 πa4 + 2 πa 11 Pr 2a πa 11 Pr + πg m { A 9 z ae az A 2 z 2e az A 1 S c + 2A 14 A 5 S c } + S c G m { aa 16 z + (20) 1 2 } πp r A 11 A 13 πa8 Pr πa 12 1 } + 2A 12 πsc. πsc A 14 A 7 Sc A 14 πsc A 6

5 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) The expressions for he symbols involved in he above soluions are given in he appendix. 3 Skin Fricion The dimensionless skin fricion a he plae is ( ) dq = τ x + iτ y. dz The numerical values of τ x and τ y, for differen parameers are given in Table-1. 4 Resul and Discussions z=0 The velociy profile for differen parameers like, hermal Grashof number (Gr), mass Grashof number (Gm), magneic field parameer (M), Hall parameer (m), acceleraion parameer (b), Prandl number (Pr) and ime () is shown in figures 1.1 o 2.8. I is observed from figures 1.1 and 2.1 ha he primary and secondary velociies of fluid decrease when he angle of inclinaion of pla (α) is increased. I is observed from figure 1.2 and 2.2, when he mass Grashof number Gm is increased hen he velociies are increased. From figures 1.3 and 2.3 i is deduced ha when hermal Grashof number Gr is increased hen he velociies ge increased. If Hall curren parameer m is increased hen he primary velociy increased and secondary velociy decreased (figures 1.4 and 2.4). I is observed from figures 1.5 and 2.5 ha he effec of increasing values of he parameer M resuls in decreasing u and increasing v. I is deduced ha when acceleraion parameer is increased hen he velociies are increased (figures 1.6 and 2.6). Furher, i is observed ha velociies decrease when Prandl number is increased (figures 1.7 and 2.7). When he Schmid number increased hen he velociies ge decreased (figures 1.8 and 2.8). Furher, from figures 1.9 and 2.9 i is observed ha velociies increase wih ime. Skin fricion is given in Table 1. The value of τ x increases wih he increase in hermal Grashof number, mass Grashof Number, Hall curren parameer and ime, and i decreases wih he angle of inclinaion of plae, he magneic field parameer, acceleraion parameer, Prandl number and Schmid number. Similar effec is observed wih τ x, excep magneic field parameer, acceleraion parameer and Hall parameer, in which case τ y increases wih magneic field parameer and acceleraion parameer, and decreases wih Hall parameer. 5 Conclusion The conclusions of he sudy are as follows: Primary velociy increases wih he increase in hermal Grashof number, mass Grashof number, Hall curren parameer, acceleraion parameer and ime. Primary velociy decreases wih angle of inclinaion of plae, he magneic field, Prandl number and Schmid number. Secondary velociy increases wih he increase in hermal Grashof number, mass Grashof number, he magneic field, acceleraion parameer and ime.

6 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) Secondary velociy decreases wih he angle of inclinaion of plae, Hall curren parameer, Prandl number and Schmid number. τ x increases wih he increase in Gr, Gm, m and, and i decreases wih α, M, b, Pr and Sc. τ y increases wih he increase in Gr, Gm, b, M and, and i decreases α, m, Pr and Sc. Figure 1.1: Velociy u for Differen Values of α Figure 1.2: Velociy u for Differen Values of Gm Figure 1.3: Velociy u for Differen Values of Gr Figure 1.4: Velociy u for Differen Values of m Figure 1.5: Velociy u for Differen Values of M Figure 1.6: Velociy u for Differen Values of b

7 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) Figure 1.7: Velociy u for Differen Values of Pr Figure 1.8: Velociy u for Differen Values of Sc Figure 1.9 Velociy u for Differen Values of Figure 2.1: Velociy u for Differen Values of α. Figure 2.2: Velociy u for Differen Values of Gm Figure 2.3: Velociy u for Differen Values of Gr

8 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) Figure 2.4: Velociy v for Differen Values of m Figure 2.5: Velociy v for Differen Values of M. Figure 2.6: Velociy v for Differen Values of b Figure 2.7: Velociy v for Differen Values of Pr Figure 2.8: Velociy v for Differen Values of Sc Figure 2.9: Velociy v for Differen Values of

9 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) Table 1: Skin Fricion for Differen Parameers (α in degree) α M m Pr Sc Gm Gr b τ x τ y References [1 Lighhill, M. J. The response of laminar skin fricion and hea ransfer o flucuaions in he sream velociy. Proc. R. Soc. A , [2 Kaagiri M, The effec of Hall curren on he magneo hydrodynamic boundary layer flow pas a semi-infinie fas plae. Journal of he Physical Sociey of Japan (4): [3 Pop, I. The effec of Hall currens on hydromagneic flow near an acceleraed plae. J. Mah. Phys. Sci : [4 Singh, A. K. and Naveen Kumar. Free convecion flow pas an exponenially acceleraed verical plae. Asrophysics and Space Science : [5 Basan, K. J., Prasad, R and Rai, S. Mass ransfer effecs on he flow pas an exponenially acceleraed verical plae wih consan hea flux. Asrophysics and Space Science : [6 Pop, I. and Waanabe, T. Hall effecs on magneohydrodynamic boundary layer flow over a coninuous moving fla plae. Aca Mech : [7 Aia, H. A. The effec of variable properies on he unseady Harmann flow wih hea ransfer considering he Hall effec. Appl. Mah. Model (7):

10 U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) [8 Aia Hazem Ali, Ahmed Mohamed Eissa Sayed. The Hall effec on unseady MHD Couee flow wih hea ransfer of a Bingham fluid wih sucion and injecion. Applied Mahemaical Modelling : [9 Muhucumaraswamy, R., K.E. Sahappan, Naarajan, R. Hea ransfer effecs on flow pas an exponenially acceleraed verical plae wih variable emperaure. Theore. Appl. Mech (4): [10 Muhucumaraswamy, R., Sahappan, K. E. and Naarajan, R. Mass ransfer effecs on exponenially acceleraed isohermal verical plae. In. J. of Appl. Mah. and Mech (6): [11 Rajpu, U. S. and Kumar Surendra. MHD Flow pas an impulsively sared verical plae wih variable emperaure and mass diffusion. Applied Mahemaical Sciences (3): [12 Thamizhsudar, M. and Pandurangan, J. Combined effecs of radiaion and Hall curren on MHD flow pas an exponenially acceleraed verical plae in he presence of roaion. Inernaional Journal of Innovaive Research in Compuer and Communicaion Engineering : Issue 12. [13 Maripala Srinivas and Naikoi Kishan. Hall effecs on unseady MHD free convecion flow over a sreching shee wih variable viscosiy and viscous dissipaion. World Applied Sciences Journal (6): Appendix A 1 = 1 A 16 e 2 az (1 A 17 ), A 2 = 1 + A 16 e 2 az (1 A 17 ), A 8 = A 4, A 3 = 1 + A 20 A 18 (1 A 21 ), A 4 = 1 + A 23 + A 18 (1 A 24 ), A 5 = 1 + A 25 A 19 (1 A 26 ), A 6 = 1 A 27 A 19 (1 + A 28 ), A 7 = A 6, az A 9 = 2e A 1 (1 a), A 10 = ( [ 2e z2 Pr ) z Pr 4 + πza11 Pr, A 11 = 1 + erf z 2 [ q q z Sc A 12 = 1 + erf 2, A 13 = e a 1+Pr z apr 1+Pr, A14 = e a 1+Sc z asc 1+Sc, [ [ 2 a z 2 a + z A 15 = 1 + A 29 + e 2 a+bz A 30, A 16 = erf 2, A 17 = erf 2, q q A 18 = e 2z apr 1+Pr, A19 = e 2z asc 1+Sc, A20 = erf z 2 ap r 1+P r, 2 A 21 = erf z + 2 ap r [ 1+P r 2 a 1+P, A 22 = erf r z P r, 2 2 A 23 = erf A 25 = erf [ 2 a 1+P r z P r 2 z 2 2 as c 1+S c, A 24 = erf, A 26 = erf [ 2 a 1+P r + z P r 2 z as c 1+S c,,,

11 A 27 = erf A 29 = erf U. S. Rajpu and Gaurav Kumar / MATEMATIKA 34:2 (2018) [ 2 a 1+S c 2 [ S c 2 a 1+S, A 28 = erf c + 2 S c, 2 2 [ [ 2 a + b z 2 a + b + z 2 Lis of Symbols u v g m T C b D T w C w B 0 K G r G m P r S c M C p ū v b C α β β ν ρ µ σ ω e τ e θ, A 30 = erf 2, a = The primary velociy of he fluid The secondary velociy of he fluid The acceleraion due o graviy Time Hall curren parameer Temperaure of he fluid Species concenraion in he fluid Acceleraion parameer Mass diffusion coefficien Temperaure of he plae a z= 0 Species concenraion a he plae z= 0 Magneic field Thermal conduciviy of he fluid Thermal Grashof number Mass Grashof number Prandl number Schmid number The magneic parameer Specific hea a consan pressure Dimensionless Primary velociy Dimensionless secondary velociy Dimensionless acceleraion parameer Dimensionless ime Dimensionless concenraion Angle of inclinaion of plae Volumeric coefficien of hermal expansion Volumeric coefficien of concenraion expansion Kinemaic viscosiy Densiy Coefficien of viscosiy Elecrical conduciviy Cycloron frequency of elecrons Elecron collision ime Dimensionless emperaure M(1 im) 1 + m 2.