Mass Transpiration in Nonlinear MHD Flow Due to Porous Stretching Sheet

Motivated from numerous practical applications, the present theoretical and numerical work investigates the nonlinear magnetohydrodynamic (MHD) laminar boundary layer flow of an incompressible, viscous fluid over a porous stretching sheet in the presence of suction/injection (mass transpiration). The flow characteristics are obtained by solving the underlying highly nonlinear ordinary differential equation using homotopy analysis method. The effect of parameters corresponding to suction/injection (mass transpiration), applied magnetic field, and porous stretching sheet parameters on the nonlinear flow is investigated. The asymptotic limits of the parameters regarding the flow characteristics are obtained mathematically, which compare very well with those obtained using the homotopy analysis technique. A detailed numerical study of the laminar boundary layer flow in the vicinity of the porous stretching sheet in MHD and offers a particular choice of the parametric values to be taken in order to practically model a particular type of the event among suction and injection at the sheet surface.


Motivated from numerous practical applications, the present theoretical and numerical work investigates the nonlinear magnetohydrodynamic (MHD) laminar boundary layer flow of an incompressible, viscous fluid over a porous stretching sheet in the presence of suction/injection (mass transpiration)
. The flow characteristics are obtained by solving the underlying highly nonlinear ordinary differential equation using homotopy analysis method. The effect of parameters corresponding to suction/injection (mass transpiration), applied magnetic field, and porous stretching sheet parameters on the nonlinear flow is investigated. The asymptotic limits of the parameters regarding the flow characteristics are obtained mathematically, which compare very well with those obtained using the homotopy analysis technique. A detailed numerical study of the laminar boundary layer flow in the vicinity of the porous stretching sheet in MHD and offers a particular choice of the parametric values to be taken in order to practically model a particular type of the event among suction and injection at the sheet surface.
The steady, laminar MHD boundary layer flows driven by moving boundaries are among the classical problems of theoretical fluid mechanics (see Schlichting 1 ). The usual stretching sheet problems arise in polymer extrusion processes that involve the cooling of continuous strips extruded from a dye by drawing them horizontally through a stagnant cooling fluid 2 .
The phenomena of momentum transfer in steady, laminar boundary layer flows have received much attention due to their wide applications which include drawing of plastics and elastic sheets, metal and polymer expulsion forms, paper creation, and cooling of metallic sheets etc. 2,3 .
Viscous fluid flow past a linear stretching sheet is a classical problem of laminar boundary layer flow. Blasius 4 first discovered the boundary layer flow on a flat plate using similarity transformations. Sakiadis 5,6 investigated the steady laminar boundary layer flow on a moving plate in a quiescent liquid and obtained both closed form as well as approximate solutions. Crane 7 considered the flow due to stretching of plastic sheets in the polymer industry and obtained an analytical solution of the laminar boundary layer equations. Recently, Al-Housseiny and Stone 8 have investigated the laminar boundary layer flow due to motion of stretching sheets by taking into account both the fluid motion as well as the motion of the sheet.
The control of the boundary layer flow due to stretching sheet can be enhanced by introducing magnetohydrodynamic (MHD) effects. This can be done by taking an electrically conducting fluid above the sheet and applying a magnetic field perpendicular to the plane of the sheet. In this connection, Pavlov 9 was the first to investigate the MHD flow past a stretching sheet using the Hartman formulation. He found that the applied magnetic field and permeability cause depletion of the boundary layer thickness near the sheet surface. Chakrabarti and Gupta 10 extended the classical work of Crane 7 to include the effect of a transverse magnetic field and obtained the analytical solutions to the MHD flow over a stretching sheet.
Boundary layer flows through saturated porous media and MHD has gained significant attention in the recent times because of their promising engineering applications, such as in moisture transport in thermal insulation,

Mathematical Model
We consider the steady two dimensional flow of an electrically conducting, incompressible Newtonian fluid through a porous medium over a stretching sheet issuing from a slit at the origin of the rectangular coordinates as shown in Fig. 1. The sheet is assumed to be horizontal and coincides with the plane y = 0. Two equal and opposite forces are applied to the sheet to stretch it along the x-axis. The velocity of the stretching is (U(x),0), where n 0 = such that n > −1 is the stretching parameter; U 0 and L are the characteristic scales for measuring the velocity of the stretching and distance, respectively. The porous medium is assumed to have permeability and is subjected to an external vertical magnetic field where κ 0 and H 0 denote the characteristic scales for measuring the permeability of the porous medium and magnetic field, respectively. The electrical conductivity σ of the fluid is assumed to be small so that the induced magnetic field in the fluid is weak as compared to the applied magnetic field. The sheet is permeable and subjected to suction velocity (0,V(x)) (see [ 1 ,Ch. 11,pp. 302  with suction/injection. The system is subjected to a vertical magnetic field H 0 . 1 In the present work, we have used the homotopy perturbation method which is a particular case of the more general homotopy analysis method (HAM) developed by Prof. Liao 24 . So we prefer to use the phrase HAM throughout. and the boundary layer approximation of the momentum equation where ν is the kinematic viscosity and ρ is the fluid density. The boundary conditions for the flow are given by y where U(x) and V(x) are as in (1) and (4). We use similarity transformations to convert the system (5) and (6) where f(η) denotes the dimensionless form of the stream function, we have is the dimensionless measure of suction/injection known as the mass transpiration parameter 22 . Using (9) and (10) in (5) and (6), we have the following nonlinear third order system

Q Q R R
The symbol κ is the dimensionless form of the porosity of the medium so that κ −1 is the dimensionless measure of permeability. Also, the parameter Q Q is related to the Chandrasekhar number Q by Q Q R = Q e / , where Q = σL 2 H 0 2 /(ρν). Observe that Q ≥ 0 and κ > 0. If no magnetic field is applied, Q Q = 0 so that M κ = . Increasing the permeability of the porous medium lowers the value of κ and hence of M. If the me0dium is not porous, the term containing porosity is absent which correspond to κ → 0 and Q Q M = . On the other hand, if the medium is not porous and no magnetic field is applied then M → 0. Also note that V c < 0 for suction and V c > 0 for injection. Finally, recall that n > −1.

Solution
To solve the nonlinear system (12), we use the well known HAM 25,31 which is described as follows. For p ∈ [0,1] as the homotopy embedding parameter, we consider the following boundary value problem where α ≠ 0 is the unknown scalar to be determined. For p = 0, the system (14) gives the linear system f ′′′−α 2 f ′ = 0 and for p = 1, it is the nonlinear system (12). Now assume a solution of (14) in the form and compare the like powers of p to obtain the following sequence of boundary value problems www.nature.com/scientificreports www.nature.com/scientificreports/ The system (15) and (16) can be solved recursively starting with k = 0, where at each later stage, one needs to solve a linear boundary value problem using the solutions obtained from all of the previous stages. We have obtained the solutions of first four of these problems in closed form as follows. where β = + n n   www.nature.com/scientificreports www.nature.com/scientificreports/ The expressions for the functions f k for k > 3 are lengthy; so, we omit them. The order of approximation of f using HAM is the integer N such that In the present work, we have obtained up to seventh order approximation of f. All numerical computations have been done using computer programming in MATLAB.
The HAM allows us to choose the scalar α appropriately. We take , which can be solved numerically to obtain an approximate value of α, depending upon N and the dimensionless parameters. For N = 1, (32) gives the following closed form expression To compare the first order approximation of f ′′(0) by (33) with the higher order approximations, we have obtained Fig. 2

(a) which shows the variation of
with V c for M = 0 and n = 1, 5, 10. The solid thick curves and the thin dashed curves have been obtained for N = 6 and N = 1, respectively. Clearly for each n, the formula (33) gives good approximation to f ′′(0). To quantify this approximation, we define which is the relative difference in the values of f ′′(0) = −α obtained using first order approximation with respect to the N-th order approximation. Figure 2(b) shows the variation of E R 6 with V c for the same parametric values as in Fig. 2(a). It is clear from the Fig. 2(a) that the relative difference E R 6 = 0 for n = 1, and it remains less than 5% for the other two values of n. Table 1 shows the numerical value of α obtained with the higher order approximations for V c → 0, M = 0, and n = 10. Clearly the method converges for N = 3 within the tolerance of 10 −2 . For N = 7 and V c → 0, we have f ′′(0) ≈ −1.2349677, which is close to the corresponding numerical value −1.234875 obtained by Vajravelu and Cannon 35 and Cortell 36 .
To compare the third order approximation (N = 3) of HAM to the solution with the higher order approximations, we have plotted f(η) with respect to η in Fig. 3 for n = 10 and M = 0. We have taken M = 0 since the convergence of the method is comparatively rapid for M > 0. The other parametric values are chosen to test the extreme case where the error can possibly be maximum. The three curves in each subfigure correspond to N = 3, 4, and 5, respectively as shown in the legend. Clearly, the three curves are indistinguishable for each value of V c which shows that the method converges for N = 3 and justifies that the third order analytic approximation to f using HAM are sufficient to describe the solution correctly. For rest of the numerical calculations, we have taken 4 ≤ N ≤ 7.

Asymptotic Analysis
To understand the full parametric dependence of the present boundary layer flow, we obtain approximate analytic solution f for the following extreme cases.  www.nature.com/scientificreports www.nature.com/scientificreports/ The solution of (36) is given by The function f − V c as obtained from (37) and (12) for large M is shown in Fig. 4(a) for V c = 1.5 and n = 1.5. In each case, the points marked * correspond to the asymptotic solution (37), while the solid curves correspond to the numerical solution of (12). The asymptotic solution is in a maximum relative error of 0.6% for M = 10 2 , which decreases rapidly on increasing M. Thus the two solutions are in good agreement for M ≥ 10 2 .
Case II: and h is a nonzero scalar. Assume   www.nature.com/scientificreports www.nature.com/scientificreports/ where q < 0, which for V c >> 1 requires for a balance, 1 + q = 0 or q = −1. Thus, for V c → ∞, we have Thus, in this case the asymptotic solution is given by

Results and Discussion
The numerical results have been obtained for a wide range of parameters. For most of the numerical calculations, we have taken −2 ≤ V c ≤ 2, 0 ≤ n ≤ 5, and M < ≤ 0 100. Even higher values of M are permissible since the method converges faster as M becomes larger, which can be seen from (33) through (16a)-(16d), since α = |f ′′(0)| which in turn rises with M for large M. Also, we have taken at least the fourth order approximation of f, that is N ≥ 4 for the rest of the numerical calculations to meet the convergence issues using HAM. For obtaining the streamline patterns, we have taken 7th order approximation (N = 7) to f. Skin friction coefficient at sheet wall. If τ denotes the shear stress near the stretching sheet due to the fluid flow, we have Then the coefficient C f of the frictional drag which is also known as the skin friction parameter is defined for In the literature, f ′′(0) is generally taken as a measure of the skin friction for a fixed x, which in the present situation holds only if there is no suction/injection, that is, V c ≈ 0. If this is the case, we have www.nature.com/scientificreports www.nature.com/scientificreports/ We first discuss the variation of C f with x in the following three cases. Case I: n ≥ 1. Figure 5 depicts the variation of |C f | with x/L for M = 1 and = e 2 R . Each subfigure corresponds to one value of V c among 2 0 . , 1 5 . , 1 0  . , . 0 5  , and 0 3  . . The different curves (labeled with different color and style as shown in the legend) in each subfigure correspond to different values of n ≥ 1 among 1, 1.5, 2.0, 2.5 and 3.0.
We first explain the curves for V c = −2.0 which correspond to suction. We observe from (41) that C f = 0 at x = x 0 (see the black marks • in Fig. 5

R
Clearly, C f < 0 for x < x 0 and C f > 0 for x > x 0 . The skin friction coefficient C f is negative for x in the interval (0,x 0 ], where |C f | decreases rapidly from ∞ to 0. For x > x 0 , C f is positive and starts increasing with further increase of x till a maximum is reached at about x/L = x m /L (see the points marked as in Fig. 5), where (x m /L) (n+1)/2 is a real positive root of the following polynomial equation in t Beyond x/L = x m /L, |C f | falls down with further increase in x/L and approaches the value 0 at about x/L = 1. The entire pattern of these curves shifts towards right in (x/L,C f )-plane on increasing n. Thus the magnitude of skin friction |C f | may increase or decrease on increasing n, depending upon the horizontal distance from the slit. Similar is the variation of C f with x/L for the other negative values of V c . The dependence of |C f | on suction can be easily understood from Fig. 6, which has been drawn for n = 1.5 and the fixed parametric values of M and e R same as in Fig. 5. Here in Fig. 6, each curve corresponds to one value of V c . It is clear from the first subfigure of Fig. 6 that on increasing suction, the point of maximum (x m ,|C f (x m )|) shifts downward in the (x/L,C f ) plane and for any fixed value of x/L sufficiently away from the slit, |C f | decreases on increasing suction. This is due to the comparative values of the two terms −f ′′(0) and (n−1)V c /(x/L) (n+1)/2 in shear stress τ or in C f as can be seen from (41).
We now get back to Fig. 5 to explain the case of injection (V c > 0) for which the behavior of C f is different from the case of suction. Here, C f remains positive for all x > 0. For any fixed value of n ≥ 1, the skin friction coefficient C f is a decreasing function of x/L in the considered range of x/L and C f . By correlating Figs. 5 and 6 it can be observed that the skin friction decreases on increasing injection. This observation is also in accordance with the physical expectation that the injection at the sheet wall enhances the vertical component of the fluid velocity near it, which in turn results in lowering of the frictional drag near the wall.
Case II: 0 ≤ n < 1. Observe from (41) that for (n−1) < 0, the term (n−1)V c is positive for suction while it is negative for injection. Here, the effect of change of n is more prominent in the case of injection than that in suction in view of the variation of |C f | with x/L (see the second column of Fig. 7). as n → −1 + for all values of V c and all x > 0. On comparing all the three columns of Fig. 7, it can be inferred that the unusual behavior of significant increase or decrease of |C f | with x/L is apparent in the case of (i) suction for n > 1 (ii) injection for 0 < n < 1 and (iii) both suction and injection for n < 0. These observation are useful to model the nonlinearity of the underlying boundary layer flow for the two cases of suction and injection using an appropriate value of the stretching sheet parameter n. , where . However, for V c = −1, C f remains positive for all permissible values of Re. For a fixed value of n in the range 0 ≤ n ≤ 1, |C f | decreases continuously with Re in the case of suction (V c = −1), while in the case of injection (V c = 1), C f increases with Re initially, attains rapidly a maximum and then it decreases slowly with Re. For n > 1 and V c = −1, |C f | decreases rapidly with e R to 0 and then starts increasing with Re and slowly attains a maximum value at larger Reynolds number, after which it decreases further on increasing Re. The variation for the case of injection is similar as it is for n < 1 except that on increasing n, |C f | increases for a fixed value of Re.
Role of magnetic parameter M. The effect of M on C f can be observed from (33). Clearly, |f′′(0)| is an increasing function of M. Larger M contributes towards greater resistance to the flow in x-direction. This indicates that the skin friction near the sheet wall is expected to rise for x/L away from 0 when M is sufficiently large. www.nature.com/scientificreports www.nature.com/scientificreports/ To see this effect explicitly, we have obtained Fig. 9 which shows variation of C f with x/L for = Re 2, where we have taken V c = −1.5 for suction and V c = 1.5 for injection. In each of the case for suction/injection, the subfigures have been drawn for n = 1, 1.5, 2, and 5. In each of the subfigures, the different curves correspond to M = 1, 5, 10, 20, 50, 10 2 , 10 3 , and 10 4 . Each arrow head denotes the direction of increase of the magnetic parameter M.
For the case of suction, for n = 1 and for each value of M, |C f | is a decreasing function of x/L. However, the variation of |C f | with x/L is different for n > 1, where |C f | decreases on increasing x/L. The variation of |C f | for n = 1.5, 2, and 5 is similar, but the maximum of |C f | decreases on increasing n.
For the case of injection, C f remains positive for all the considered values of n and M. In either of the cases of suction and injection, we see that |C f | increases significantly on increasing M, which shows enhancement in the skin friction caused by increasing the applied magnetic field and decreasing the permeability of the porous medium.
Streamline pattern. Figure    and for y > h(x), u = 0. So in the region y > h(x), the stream lines will be vertical. In the presence of suction at the boundary, the fluid particles in contact with the sheet start moving with the velocity of the sheet, advance towards left-upward before they are along the path normal to the sheet surface. The particles farther from the slit move through greater distance along the inclined curves. So the boundary layer thickness h(x) increases with the distance from the slit. This happens for all the considered values of n and is due to the strengthened fluid sheet velocity. These observations together show that the boundary layer flow in the presence of suction is significantly affected by changing the stretching parameter n.
On the other hand, in the presence of injection at the sheet ( Fig. 10(b)), the fluid particles in contact with the sheet start moving with the velocity of the sheet, advance a little towards right-upward and attain paths normal to the sheet. Here, the boundary layer thickness h(x) is smaller than in the case of suction, where at a fixed distance from the slit, h(x) diminishes with increase in n. For each value of n, h(x) increases with the distance from the slit.
To observe the effect of porosity and applied magnetic field on the boundary layer flow pattern, we have obtained Fig. 11 which shows the streamline pattern in the xy-plane, for different values of n, V c = −1 and e 2 = R . The various patterns (a)-(d) correspond to M = . 0 5, 1, 2, and 5 respectively. Clearly, for a fixed value of n, the boundary layer thickness decreases on incrementing M from 0.5 to 5.
It will be useful to see the explicit dependence of the velocity profiles on M for the two cases of suction and injection. This has been shown in Fig. 12, which shows the variation of the two components of velocity, u/U(x) and v/V(x) with η for n = 1.5 and Re 2 = . Here, we have taken V c = −1 for suction and V c = 1 for injection. Each curve is drawn for one value of M among 0.5, 1, 2, 5, 10, and 20, and the arrowhead in each subfigure shows the direction of increase of M. Clearly, on increasing M, the horizontal velocity profile shifts downwards in the (η,u/U(x-))-plane under suction as well as injection. On the other hand, the profile for the velocity component v/V(x) goes up on increasing M for the case of suction, while opposite to this variation occurs under injection. On correlating all the four subfigures, we see that under suction, the applied magnetic field and the permeability of the porous medium tend to enhance the fluid velocity in the direction of suction, while it hinders the fluid motion along the sheet. Under the case of injection, the applied magnetic field hinders both the vertical as well as horizontal movement of the fluid in the vicinity of the sheet wall. The value of y corresponding to least value η for which u/U(x) becomes zero, is the boundary layer thickness. So increase in M decreases the boundary layer thickness. www.nature.com/scientificreports www.nature.com/scientificreports/ conclusions The present work generalizes the classical work of Crane 7 , Pavlov 9 , Gupta and Gupta 34 , Hayat et al. 14 , and Rashidi 15 on the flow of Newtonian fluid driven by stretching sheet with external magnetic field through porous medium with suction/injection. The underlying dynamical system is described by the nonlinear boundary layer equations, which are transformed into a system of nonlinear ordinary differential equations via similarity transformations. The resulting system is solved analytically and numerically using highly efficient HAM.