BOUNDARY LAYER ANALYSIS ALONG A STRETCHING WEDGE SURFACE WITH MAGNETIC FIELD IN A NANOFLUID

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1 Proceedings o the International Conerence on Mechanical Engineering and Reneable Energy 7 (ICMERE7) 8 December, 7, Chittagong, Bangladesh ICMERE7-PI- BOUNDARY LAYER ANALYSIS ALONG A STRETCHING WEDGE SURFACE WITH MAGNETIC FIELD IN A NANOFLUID M. Ali *, M. A. Alim and R. Nasrin Chittagong University o Engineering & Technology, Assistant Proessor, Department o Mathematics, Bangladesh Bangladesh University o Engineering and Technology, Proessor, Department o Mathematics, Bangladesh ali.mehidi93@gmail.com *, maalim@math.buet.ac.bd, raity@gmail.com Abstract- The present problem perorms the boundary layer lo, heat and mass transer in a nanoluid along a stretching edge ith the eect o magnetic ield. The equation is ormulated in terms o momentum, energy and nanoparticle concentration along ith boundary conditions. The mathematical ormulation is based on Falkner Skan boundary layer equation and Buongiorno model. The governing partial dierential equations are transormed into ordinary dierential equations by applying local similarity transormations and solve them numerically. The results o velocity, temperature and nanoparticle concentration proiles are displayed in graphically and also the local skin riction coeicient, rate o heat and mass transer are in tabular orm. Keyords: MHD, Hall current, Viscous Dissipation. INTRODUCTION MHD laminar boundary layer lo over an inclined stretching sheet has noticeable applications in glass bloing, continuous casting, paper production, hot rolling, ire draing, draing o plastic ilms, metal and polymer extrusion, metal spinning and spinning o ibbers. During its manuacturing process a stretched sheet interacts ith the ambient luid thermally and mechanically. Both the kinematics o stretching and the simultaneous heating or cooling during such processes has a decisive inluence on the quality o the inal products. In the extrusion o a polymer sheet rom a die, the sheet is some time stretched. By draing such a sheet in a viscous luid, the rate o cooling can be controlled and the inal product o the desired characteristics can be achieved. In vie o its signiicant application various authors has been done a lot o orks related to this ield such as, Venkatesulu and Rao [] has considered the eect o Hall Currents and Thermo-diusion on convective heat and mass ttranser viscous lo through a porous medium past a vertical porous plate, Sudha Mathe et al.[] studied the Hall eects on heat and mass transer through a porous medium in a rotating channel ith radiation, Kumar and Singh [3] have studied Mathematical modeling o Soret and Hall eects on oscillatory MHD ree convective lo o radiating luid in a rotating vertical porous channel illed ith porous medium, Chauhan and Rastogi [4] analyzed the eect o Hall current on MHD slip lo and heat transer through a Porous medium over an accelerated plate in a rotating system and Nazmul Islam & Alam [5] studied Duour and Soret eects on steady MHD ree convection and mass transer luid lo through a porous medium in a rotating system, Raptiset et al. [6] have studied the viscous lo over a non-linearly stretching sheet in the presence o a chemical reaction and magnetic ield. Tan et al. [7] studied various aspects o this problem, such as the heat, mass and momentum transer in viscous los ith or ithout suction or bloing. Abel and Mahesh [8] presented an analytical and numerical solution or heat transer in a steady laminar lo o an incompressible viscoelastic luid over a stretching sheet ith poer-la surace temperature, including the eects o variable thermal conductivity and non-uniorm heat source and radiation. So the present paper is ocused on steady MHD ree convection, heat and mass transer nanoluid lo o an incompressible electrically conducting luid along a stretching edge shape surace.. GOVERNING EQUATIONS OF THE PROBLEM ICMERE7

2 AND SIMILARITY ANALYSIS Let us consider steady to dimensional MHD laminar boundary layer lo o an incompressible, electrically conducting, viscous Netonian luid past a stretching edge surace hich is electrically non conducting semi ininite sheet ith heat and mass transer. The stretching sheet is permeable to allo or possible bloing or suction, and is continuously stretching in the direction o x axis. The lo is along the edge surace hich is measured the x-axis and y-axis is perpendicular to it. To equal and opposite orces are applied along the x-axis so that the all is stretched ith a velocity u u x ax and keeping the origin ixed. The m surace temperature T and nanoparticle concentration C are maintained at non-uniorm temperature hich is greater than the ree stream temperature T and nanoparticle concentration C. The uniorm transverse magnetic ield B o is imposed parallel to the y-axis and the induced magnetic ield due to the motion o the electrically conducting luid is negligible since or small magnetic Reynolds number. It is also assumed that the external electric ield is zero and the electric ield due to polarization o charges is negligible. The total angle o the edge is u x, the. The velocity o the edge surace is Ux, the temperature o the edge ree stream velocity is is T and nanoparticle concentration C are respectively deined as ollos m m m m u x ax, U x bx, T T bx, C C bx Where a and b are positive constant and the exponent m (pressure gradient parameter) is a unction o the edge angle parameter β here the total apex angle o the edge is βπ such that β m m or β. β m Thereore, the governing partial dierential equations o continuity, momentum, energy and nanoparticle concentration are as ollos presence o Hall current are: Equation o continuity: Nanoparticle concentration equation: C C C D T u v D x y T T B y y The above equations are subject to the olloing boundary conditions: u u x, v, T T,C C at y and u U x,t T,C C as y here u and v are the velocity components along x and y directions, is the kinematic viscosity o the base luid, ρ is the density o the base luid, σ is the electrical conductivity, B is the magnetic ield intensity, g is the acceleration due to gravity, is the thermal diusivity o the base luid, DB is the Bronian diusion coeicient, DT is the thermophoresis diusion coeicient. Here τ is the ratio o the eective heat capacity o the nanoparticle material and the heat capacity o the ordinary luid, T is the luid temperature and C is the nanoparticle concentration respectively. To convert the governing equations into a set o ordinary dierential equations, e introduce the olloing similarity transormations: U m xνu T T η y, ψ η,θη, x m T T C C ψ ψ φη,u and v C C y x By applying the above similarity transormations, the partial dierential Eq. () Eq.(4) are transormed into non-dimensional, nonlinear and coupled ordinary dierential equations as ollos: * - + β - M + K + m - (5) (4) u v x y Momentum equation: u u U u σb u v U ν U u x y x y ρ Energy equation: T T T T C DT T u v α τ D B + x y y y y T y () () (3) (6) θ Pr Nb θ φ + Nt θ θ - β θ Nt φ θ Le Pr φ-β φ (7) Nb The transormed boundary conditions: ' = = λ,θ = =, = at η =, ' and,θ = as η Where ICMERE7

3 m σb x a B M,λ, Nb ρ a b ν τdtt-t m Nt =, β, Pr, T ν m α Le =, K D B * νx Ka m τd C -C =, are the magnetic parameter, velocity ratio parameter, Bronian motion parameter, thermophoresis parameter, pressure gradient parameter, Prandtl number,leis number and porosity parameter respectively. The important physical quantities o this problem are skin riction coeicient C, the local Nusselt number Nu and the local Sherood number Sh hich are proportional to rate o velocity, rate o temperature and rate o nanoparticle concentration respectively. 3. METHODOLOGY The governing undamental equations o momentum, thermal and concentration in Netonian luids are essentially nonlinear coupled ordinary or partial dierential equations. Generally, the analytical solution o these nonlinear dierential equations is almost diicult, so a numerical approach must be made. Hoever no single numerical method is applicable to every nonlinear dierential equation. The various types o methods that are available to solve these nonlinear dierential equations are inite dierence method, shooting methods, quasi-linearization, local similarity and non-similarity methods, inite element methods etc. Among these, the shooting method is an eicient and popular numerical scheme or the ordinary dierential equations. This method has several desirable eatures that make it appropriate or the solution o all parabolic dierential equations. Hence, the system o reduced nonlinear ordinary dierential equations together ith the boundary conditions have been solved numerically using ourth-order Runge-Kutta scheme ith a shooting technique. Thus adopting this type o numerical technique described above, a computer program ill be setup or the solution o the basic nonlinear dierential equations o our problem here the integration technique ill be adopted as the ourth order Runge-Kutta method along ith shooting iterations technique. First o all, higher order non linear dierential equations are converted into simultaneous linear dierential equations o irst order and they are urther transormed into initial value problem applying the shooting technique. Once the problem is reduced to initial value problem, then it is solved using Runge -Kutta ourth order technique. The eects o the lo parameters on the velocity, temperature and species concentration are computed, discussed and have been graphically represented in igures and also the values o skin riction, rate o temperature and rate o concentration shon in Table or various values o dierent parameters. In this regard, deining ne variables by the equations y, y, y, y y,y,y The higher order dierential equations (5), (6), and (7) may be transormed to seven equivalent irst order dierential equations and boundary conditions respectively are given belo: y = y, y = y 3 * - y = - M+K + m - y - y y - β - y 3 3 y 4 = y 5, y 5 = -Pr Nby5y 7 + Nty 5 +yy5 βyy 4 Nt y 6 = y 7, y 7 = - y5 - LePryy7 - βyy6 Nb The transormed boundary conditions: y =,y = λ,y = α,y =,y = α, y =,y =α at η = y, y, y as η 4 6 Where the unknons α,α and α 3 are determined such that y, y, y as η. The 4 6 essence o this method is that irst the boundary value problem is converted to an initial value problem and then use a shooting numerical technique to guess the values o α,α and α3 until the boundary conditions y, y, y as η are satisied. 4 6 The resulting dierential equations are then easily integrated using ourth order classical Runge-Kutta method. 4. RESULTS AND DISCUSSION Numerical solution are obtained using the above numerical scheme or the distribution o velocity, temperature and nanoparticle concentration proiles across the boundary layer or dierent values o the parameters. The velocity proiles or various dimensionless parameters have been shon in Fig. Fig. 4. From these igures it is observed that the velocity proiles increases or increasing values o magnetic parameter, pressure gradient parameter and porosity parameter as a result the boundary layer thickness decreases but the reverse result arises in case o stretching ratio parameter. Figure 5 Figure 7 shos the temperature proiles or various entering parameters. From these igures it is observed that the heat transer rate increases or increasing values o Prandtl number and Bronian motion as a result the thermal boundary layer thickness decreases but reverse trend arises or thermophoresis parameter. The nanoparticle concentration have been shon in Fig.8 Fig.. The concentration decreases or increasing values o thermophoresis parameter, stretching ratio and Leis number but increases or Bronian motion parameter. Also, the numerical values o skin riction coeicient, rate o heat transer and rate o mass transer has been shon in Table. 5. CONCLUSIONS From the above analysis the main observation is the velocity proile is exist up to.35 but in Falker Skan problem it as.98. The pressure gradient parameter, thermophoresis parameter and stretching ratio parameter is the key actor to enhance heat and mass transer rate. ICMERE7

4 = - = -.3 = - =. = M =.5 M =. M = Fig.: Velocity proile or various values o β Fig.4: Velocity proile or various values o M K*= K*=.7 K*=.5 Pr =.5 Pr =. Pr = Fig.: Velocity proile or various values o K* Fig.5: Temperature proile or various values o Pr =.3 = =.5 Nb = Nb = Nb = Fig.3: Velocity proile or various values o λ Fig.6: Temperature proile or various values o Nb ICMERE7

5 Nt = - Nt = - Nt = - Nb =.3 Nb =.5 Nb = Fig.7: Temperature proile or various values o Nt Fig. : Nanoparticle concentration proile or Nb Nt = -. Nt = - Nt = - =.3 = = Fig.8: Nanoparticle concentration proile or Nt Le =. Le = 3. Le = Fig. 9: Nanoparticle concentration proile or Le Fig. : Nanoparticle concentration proile or λ 7. REFERENCES [] D.Venkatesulu and U. R. Rao, Eect o Hall Currents and Thermo-diusion on convective Heat and Mass Transer lo o a Viscous, Chemically reacting rotating luid through a porous medium past a Vertical porous plate, International Journal o Scientiic & Engineering Research, vol. 4, no. 4, 3. [] S. Mathe, P. R. Nath and R.D. Prasad, Hall Eects on Heat and Mass Transer Through a Porous Medium in a Rotating Channel ith Radiation, Advances in Applied Science Research, vol. 3, no. 5, pp ,. [3] R. Kumar and K. D. Singh, Mathematical modeling o Soret and Hall eects on oscillatory MHD ree convective lo o radiating luid in a rotating vertical porous channel illed ith porous medium, International Journal o Applied Mathematics and Mechanics, vol. 8, no. 6, pp ,. [4] D. S. Chauhan and P. Rastogi, Hall Eects on MHD Slip Flo and Heat Transer Through a Porous Medium over an Accelerated Plate in a Rotating System, International Journal o Nonlinear Science, vol.4, no., pp. 8-36,. ICMERE7

6 [5] N. Islam and M. M. Alam, Duour and Soret eects on steady MHD ree convection and mass transer luid lo through a porous medium in a rotating system, Journal o Naval Architecture and Marine Engineering, 7. [6] A. Raptis and C. Perdikis, Viscous lo over a non-linearly stretching sheet in the presence o a chemical reaction and magnetic ield, International Journal o Nonlinear Mechanics, vol.4, pp , 6. [7] Y. Tan, X. You, Xu Hang and S. J. Liao, A ne branch o the temperature distribution o boundary layer los over an impermeable stretching plate, Heat Mass Transer, vol. 44, pp. 5-54, 8. [8] M. S. Abel and N. Mahesha, Heat transer in MHD viscoelastic luid lo over a stretching sheet ith variable thermal conductivity, non-uniorm heat source and radiation, Applied Mathematics and Modelling, vol.3, pp , NOMENCLATURE Symbol Meaning Unit MHD Megnetohydrodynamic (K) α Thermal diusivity o mm s - the base luid К Thermal conductivity m - K - Electrical conductivity sm - D Bronian diusion - B coeicient D Thermophoresis - T diusion coeicient kinematics viscosity o m s - the base luid, density o base luid, kg m -3 B Magnetic ield intensity, Am - Ψ stream unction u Velocity component ms - along x -axis v Velocity component ms - along y -axis a Stream velocity constant b Free stream velocity constant τ ratio o the eective heat capacity λ stretching ratio C nanoparticle volume kg m -3 raction C plate volume raction kg m -3 C ree stream nanoparticle volume raction T luid temperature k - T plate temperature k - T ree stream temperature ICMERE7