An mhd viscous liquid stagnation point flow and heat transfer with thermal radiation and transpiration

Accepted Manuscript An mhd viscous liquid stagnation point flow and heat transfer with thermal radiation and transpiration U.S. Mahabaleshwar, K.R. Nagaraju, P.N. Vinay kumar, M.N. Nadagoud, R. Bennacer, Dumitru Baleanu PII: DOI: Article Number: Reference:

S2451-9049(19)30194-5 https://doi.org/10.1016/j.tsep.2019.100379 100379 TSEP 100379

To appear in:

Thermal Science and Engineering Progress

Received Date: Revised Date: Accepted Date:

12 May 2019 9 June 2019 7 July 2019

Please cite this article as: U.S. Mahabaleshwar, K.R. Nagaraju, P.N. Vinay kumar, M.N. Nadagoud, R. Bennacer, D. Baleanu, An mhd viscous liquid stagnation point flow and heat transfer with thermal radiation and transpiration, Thermal Science and Engineering Progress (2019), doi: https://doi.org/10.1016/j.tsep.2019.100379

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AN MHD VISCOUS LIQUID STAGNATION POINT FLOW AND HEAT TRANSFER WITH THERMAL RADIATION AND TRANSPIRATION U. S. MAHABALESHWAR1, K. R. NAGARAJU2 , P. N. VINAY KUMAR3 , M.N. NADAGOUD4 , R. BENNACER5 & DUMITRU BALEANU6 1

Department of Mathematics, Shivagangothri, Davangere University, Davangere 577 002, INDIA, e-mail: [email protected] 2

Department of Mathematics, Government Engineering College, Hassan -573 201, INDIA, e-mail:[email protected] 3Department

of Mathematics, SHDD Government First Grade College, Paduvalahippe, Hassan-573 211, INDIA, e-mail: [email protected] 4

Department of Mechanical and Materials Engineering, Wright State University, Dayton,

Ohio 45324, USA, e-mail:[email protected] 5LMT/ENS-Cachan/CNRS/Universite

Paris, Saclay, 61, av.du President Wilson, 94235, Cachan, FRANCE, e-mail: [email protected] 6Department

of Mathematics, Cankaya University, Faculty of Arts & Science, Balgat 0630, Ankara, Turkey & Institute of Space Science, Magurele-Bucharest, Romania E-mail: [email protected] ABSTRACT The impact of kinematic parameters magnetohydrodynamic (MHD) and thermal radiation on the unsteady flow of a Newtonian liquid through stagnation point due to a linear sheet with mass transpiration is considered. The characteristics of heat and MHD impinging on the sheet are analyzed theoretically. The flow of an electrically conducting liquid through stagnation point has gained considerable interest due to its industrial relevance. In the chemical engineering applications involving cooling of the liquid namely glass blowing, food processing, metal thinning, polymer extrusion, silicon chip manufacturing and applications of similar kind. In all these chemical engineering applications, the interplay between the regulating kinematic parameters and the nature of the fluid is of at most priority. The flow problem is modelled into nonlinear unsteady Navier-Stokes’ partial differential equations. The similarity solution for the velocity distribution is obtained. Depending on the type of boundary heating, the analytical solutions for temperature distribution is derived by means of a power series (Gauss hypergeometric). Temperature distribution for two types of boundary heating processes viz., prescribed time-dependent constant surface temperature (PTDCST) and prescribed time-dependent wall heat flux (PTDWHF) is discussed. There found to exist branching of solutions for both velocity and temperature distribution for certain range of controlling parameters. In fact there exists dual solution for both cases of stretching/shrinking sheet and these are analyzed to see the impacts of various physical parameters on the solution domain. The impact of various regulating parameters on the velocity as well as temperature is analyzed by means of numerous plots. Keywords: mass transpiration; thermal radiation; stagnation point; viscous liquid; heat transfer; kinematic parameters

Nomenclature a

constant, stretching rate (s-1)

ai 's

arbitrary constant

A1 & A2

constants

b

constant(s-1)

B0

magnetic field (W m-2 )

BBL

backward boundary layer flow

Cp

constant pressure (W kg-1K-1)

f

similarity variable

f  0 

wall shear stress

F  

  f      another similarity variable

FBL

forward boundary layer flow

M

  1  bt     B0  Hartmann number also called Chandrasekhar number   a   

NR

 16 *T3     radiation number  3 r  C p b 

p

pressure (Nm-2)

Pr

       Prandtl  

qr

wall radiative heat flux (Wm-2)

qw

wall heat flux (Wm-2)

Q

   B02   Chandrasekhar number 1   a 1  bt    

T

temperature (K)

t

time (s)

number

T4

Taylor’s series expansions

Tref

  Tw  T  reference

T

temperature away from the sheet (K)

Tw

temperature at wall (K)

Ue

external free stream velocity at wall (ms-1)

U

moving velocity at wall (ms-1)

Uw

external stretching/shrinking velocity at wall (ms-1)

u

axial velocity (ms-1)

v

transverse velocity (ms-1)

Vc

 1  bt   V  x, t  a 

Vc  0

suction

Vc  0

injection

Vc  0

impermeable fixed wall

Vw  x, t 

wall transpiration velocity (ms-1)

x

axial axis (m)

y

transverse axis (m)

temperature (K)

  constant mass transpiration 

Greek symbols α

     thermal diffusivity (m2 s-1)  C P  

r

mean absorption coefficient (m-2 )



 b    unsteady parameter  a

 0

accelerating flow

 0

decelerating flow

 0

steady state



     velocity vector    u, v, t  .stream function (m2 s-1)

η

Blasius similarity variable



constant

1 1

  F [a, b, c, z ] Kummers’ function

θ

 T  T    for the PTDCWT case  Tw  T 



dimensionless wall temperature gradient



thermal conductivity (m2s-1)



 Uw     (constant) wall moving parameter U  

 0

aiding flows for stretching sheet

0

opposing flows for shrinking sheet

 0

Falkner-Skan flow case

µ

viscosity Kgm 1s 1



  2 1    kinematic viscosity  m s   



new independent variable



stretching/shrinking parameter

ρ

density Kgm 3



conductivity m 1

σ*

constant of Stefan-Boltzman (5.67x10-8 Wm-2K-4)

















τw

shearing stress at wall m 2 s 1

  

 T  Tw   (PTDWHF)   T T  w 

ψ

    x, y, t   physical stream function  m s  2 1

Subscripts/superscripts c

constant

ref

reference

e

external free stream velocity

O

origin

w

wall



far away from the pole

*

dimensionless quantities

f

derivative with respect to 

1. INTRODUCTION In fluid dynamics, the unsteady MHD liquid flow through Homann stagnation point over a stretching/shrinking sheet with radiation is considered as one of the classical problems as these flow problems appears many branches of flow fields of engineering and technological applications. The laminar flows past a stretching/shrinking sheet along the plane with linear velocity component proportion to the stagnation point have attracted many researchers and scientists due to its significant relevance in the industrial processes involving cooling/heating of liquids like polymer extrusion, metal thinning, hot rolling etc (See Fischer

1976; Altan et al. 1979). In all these process, the desired outcome depends on the choice of regulating parameters in order to control the heat transfer in the fluid. Sakiadis (1961a, b, c) initiated the laminar flow of an ambient liquid due to a continuous solid sheet. He investigated the flow problem due to inextensible surfaces, but in practical situation there are flow problems with extensible surfaces. In the classical works of Crane (1970), the closed form solution of the laminar flows due to an accelerating sheet with linear velocity acting on it is derived. In 1973 McCormack and Crane, thoroughly examined the boundary layer flows past extensible surfaces and provided the analytical solutions. In 1977, Gupta and Gupta extended the Classical works of Crane and discussed the velocity and energy characteristics for constant surface temperature. Subsequently many authors have coined many physical models to see the effect of various aspects of stretching/ shrinking sheet (see Goldstein 1965, Cheng and Minkowycz 1977, Grubka and Bobba 1985, Chen and Char 1988, Wang 1990, Anderrson 1992, Cortel 2005,Miklavcic and Wang 2006, Wang 2008, Fang et al. 2010). In the technological processes involving cooling of the liquid, stagnation point flow of the liquid past a continuously moving surface is widely studies due to their practical relevance. In 1911, Hiemenz is first to study the stagnation point flow for two dimensional case. Later, Falkner-Skan (1931), obtained the analytical solution for steady external inviscid liquid flow ue  x   ax . In fact, when n = 0 and n =1 these results transforms into n

Blasius (1908) and Hiemenz (1911) respectively. Homann (1936) investigated the stagnation point flow for axisymmetric three dimensional and provided the similarity solution. The steady/unsteady velocity as well as MHD liquid flows past a stretching/shrinking sheet has wide applications in space technology, cooling of nuclear

reactors, heat exchangers, petroleum production and metallurgical processes (Brewster 1972) .

There are two ways to conduct MHD formulation problem namely, Chandrasekhar

formulation or Hartman formulation. Pavlov (1973, 74) discussed the MHD flows due to a stretching sheet using Hartman formulation. Chakrabarthi and Gupta (1979), extended the classical works of Crane and obtained the analytical solution by inclusion of transverse MHD. Chaim (1997, 1998), discussed the stagnation point MHD fluid flow over a nonisothermal stretching sheet with heat transfer and derived the analytical solution. Raptis (1998, 99) investigated the viscous flows with radiation by applying Rosseland approximation. In fact the thermal radiation is an important controlling parameter in obtaining the desired outcome in the manufacturing industries especially in polymer extrusion processes. Pop et. al.(2004) investigated the effect of radiation through the stagnation point flow due to stretching sheet and demonstrated that the in comparison to velocity and temperature profiles, the laminar boundary layer thickness increases with the radiation. Many authors have discussed the physical flow models involving thermal radiation as one of the regulating parameters (see Brewster 1992, Raptil 1999 and Makinde & Animasun 2016). In 2005, Siddheshwar and Mahabaleshwar discussed the analytical solutions of MHD liquid (Walter’s liquid B) flow past a stretching sheet is analyzed using Chandrasekhar formulation (see Chandrasekhar 1961). Later, many authors have discussed about the effects MHD flows past a stretching/shrinking sheet with various controlling parameter (see Andersson 1995; Liao 2003; Siddheshwar and Mahabaleshwar 2005; Cortell 2005; Mahabaleshwar et al. 2016). In view of the above discussion, the impacts of unsteady, radiative MHD viscous Homann stagnation point liquid flow past a stretching/shrinking sheet is investigated theoretically. As far as the authors’ knowledge, so far there are no results provided by

inclusion of MHD and radiation. An attempt is carried theoretically to see that this model fits into the realistic problems of engineering processes. 2. Mathematical Model: An unsteady laminar stagnation point MHD viscous liquid flow with radiation due to a stretching/shrinking sheet is considered. The two dimensional coordinates (x, y) is considered as shown in Figure 1 by applying a magnetic field of uniform strength normal to the sheet. The physical model of the flow problem is depicted in Figure 1. Two forces of equal strength but opposite in direction are applied along the surface.

Fig. 1: Schematic diagram of the physical flow problem. The unsteady free stream velocity far away from the sheet is considered as follows:

U   ax 1  bt  , 1

(1)

here, a and b are constants with, a>0 for the case of stretching sheet, a<0 for shrinking sheet case. Assume that U w and Vw are the moving velocity at wall and the normal transpiration velocity is considered. The moving external free-stream velocity at wall is given by,

U w  U  ,

(2)

here,  denotes the constant to be determined. Based on the choice of  , the following cases arises: if   0 , aiding the flow (stretching sheet case), if   0 , opposes the flow (shrinking

sheet case) and if   0 , converts into classical Falkner-Skan flows. The temperature distribution is governed by the specific heat flux at wall is given by, qw  qw  x   

T or y

prescribing constant sheet temperature Tw  Tw  x  for Tw  T heated sheet as well as

Tw  T for cooled sheet. The basic equations for continuity, momentum and temperature are (Wang 2008; Grubka and Bobba 1985; Magyari 2009, 2010; Su et al. 2012, Fang et al. . 2012): u v   0, x y

(3)

2 ue u u u ue  2u  B0  x, t  u, u v   ue  2   t x y t x y

(4)

1 qr T T T  2T , u v  2  t x y y  CP y

(5)

with associated governing boundary conditions:

u  x, y, t   U w ( x, t )  uw ( x, t ), T  Tw, v  x, y, t   Vw  x, t  , at

y0

u  x , y , t   ue ( x , t )

y

as

(6a, b)

here, B0 denotes the applied magnetic field is due to the Hartmann formulation, u and v denotes the axial as well as transverse velocities respectively and one can refer nomenclature for rest of the parameters. Based on the choice of stretching/shrinking velocity factor a, the boundary layer flow is classified into two types namely forward (FBL) and backward boundary layer flows (BBL) (Goldstein 1965). The rise in the flow due to sheet issuing from the slot and moving towards  is known as FBL, where as in the case of BBL, the continuous stretching coming from  and entering into the slot. These types of boundary layers have distinct physical phenomena which have significant applications in the industrial and technological processes (see Fisher 1976, Altan and Gegel 1979). Also, it follows that

u w  x , t   ue ( x , t ) 

ax since the Eq.(3) – (5) with the governing boundary conditions 1  bt 

admits the similarity solutions. The wall transpiration velocity Vc  Vw  x, t 

1  bt , here Vc a

represents the mass transpiration parameter with the choice, Vc  0 , Vc  0 and Vc  0 for suction, injection and impermeable sheet respectively. Using the Rosseland (1931) approximation, the radiative heat flux, qr is modelled as (Brewster 1972, Siddheshwar & Mahabaleshwar 2005, Mastroberardino & Mahabaleshwar 2013),

 

4 4 *  T , qr   3 r  y

(7)

all physical quantities are referred in the nomenclature. The Taylor’s series expansions for T 4 is given by,

T 4  T4  4 T3 T  T   6T2 T  T      . 2

(8)

The expression for T 4 upon neglecting the higher order terms is given by, T 4   3T4  4 T3T .

(9)

and thus the resulting expression is,

 qr 16  * T3  2T  y 3  C p r  y 2 Using Eq. (10) in (5), transforms into:

(10)

16 *T3 T T T  u v    t x y  3 r  CP

  2T  2  y

(11)

In order to facilitate the similarity solutions for equation (3)- (5), the physical steam function and dimensionless variables,

  x, y, t   f   x

a a T  Tw 1  bt  ,  y ,      ,  1  bt  1  bt Tref

with usual basic definitions u 

(12 a, b, c)

  and v   is introduced and Tref is the reference x y

temperature. In terms of the similarity variables Eq. (12), the velocities are derived as,

u

a ax f   . f   and v   1  bt 1  bt

(13)

Using these similarity variables Eq.(12)-(13) in Eqs. (3) – (6) and Eq. (11), the following coupled equations are derived:

   f  f f  f 2    f  f  1  Q f  Q   2  0, 2  

1  N R   Pr  f 



 2

 

    0 ,

(14)

(15)

with imposed governing conditions:

f    Vc ,

f     ,     1

f    ,     0 The dimensionless numbers  ,   

at

 0

as

 

.

(16a, b)

 H 02  Uw 16 *T3 , Pr  , N R  and Q  in 1  U 3 r  C p b  a 1  bt 

the Eqs. (14) - (16) are respectively stretching/shrinking sheet parameter, Prandtl number,

wall moving parameter, radiation parameter and the Chandrasekhar number. The present analysis is carried with the assumption that  is unity. Since,   0 for the stretching sheet case and also, the heat transfer in the stagnation point flow is examined when the ratio of free flow velocity to the stretching velocity is unity. Based on the choice of the unsteady parameter,  

b the flow is classified into accelerating flow and decelerating flow when a

  0 and   0 respectively,   0 for steady state. The results in the present study are , restricted only to the case   0 . When, Q    N R  0 , the results are converted to the classical stagnation point problems due to Hermann (1936) and when Q  N R  0 , the results are converted into Fang et al (2012). The present results are converted into Su et al (2012 if the temperature is absent. Hardy (1939) , proved the non-existence of solution for

  2 , which can be extended the case of any positive  . Thus, the specific values  are considered in the analysis. The analytical expression for momentum equation (14) is derived for the decelerating liquid flow with   2 and   1 . The resulting nonlinear differential equation is as follows:

   f  f f  1  f 2  2  f  f  1   Q f  Q   0 .   2  Hiemenz Lorentz force

(17)

unsteady effect

The 1st factor f  f f  1  f 2  0 , in the equation (17), converts into results of Hiemenz(1911) with the imposed conditions f (0)  f  0   0, f     1 . The 2nd and 3rd factor in the equation (17), represents the unsteady effect and the opposition to the flow due to Lorentz force respectively. Another similarity variable F    f     is introduced in the system to assist the analysis in order to solve the momentum equation and the resulting governing equation is given by,

F  F F  F 2   4  Q  F  0 .

(18)

The imposed governing boundary conditions are also transforms into:

F    Vc ,

F    1  

F    0

at

 0

as

 

.

(19a, b)

The imposed condition in Eq.19(b) advocates choosing F  exp    , where  is constant obtained from Eq. (19). The exact analytical solution for equation (18) with associated boundary condition (19) together with the above choice of F , the required solution is assumed to be of the form,

 1    (1  exp ( ))  F    VC   ,   0,   

(20)

for the Eq.(18) and the imposed boundary condition (19). The physically feasible solution is possible if the constant  must be positive definite and satisfies the quadratic equation (see Abramowitz and Stegun 1972, Birkhoff and MacLane 1996),

 2  Vc     Q  3  0

(21)

.

The solutions of Eq.(21) are given by, 1

2  2 Vc  Vc           Q  3   . 2  2  

(22)

2

V  The Eq.(21) has real solutions provided  c      Q  3  0 . The existence of two zeroes 2 of Eq.(21) advocates the possibility of dual solution of the Newtonian liquid flow due to a sheet for different choices of Vc and  . The critical value  c is given by

Vc2   c     Q  3 . For   Q  3 , there exists only one real positive solution for the  4  Eq.(21). For   Q  3 , there exists two positive solutions say 1 and  2 . For   Q  3 ,

1  Vc and  2  0 are the solutions to Eq. (21). The solution branches into upper and lower for the shrinking boundary when    c and 1   2 . Negative solution for the equation (21) is not discussed as it has no significant physical relevance for the flow problem. 3. Stokes stream function The two-dimensional stream function is the incompressible flow. The stream function

    x, y, t  was developed by Stoke’s in 1842. The physical stream function in 2dimensional form in the incompressible flow is considered and the velocity profiles along axial and transverse axis in terms of   x, y, t  are given by, u 

  and v   y x

respectively. Further, the value of stream function must be constant for the stream lines that are tangent to the flow velocity vector. The vector expression for stream function is given by,     where,     x, y, t  , for the flow velocity vector    u , v, t  .

Further,

for fixed physical parameters, the expression for   x, y, t  reduces to,



x  1  2t

  x, y , t   

  y       1  2t

  y      1     1 exp V         c       .       1  2t   

(23)

The temperature distribution of unsteady forced convective liquid flow due to a stretching/ shrinking sheet is discussed in the following section. 4.

Heat transfer and solution The temperature distribution for two types of boundary heating namely prescribed time-

dependent constant wall temperature (PTDCWT) and prescribed time-dependent wall heat

flux (PTDWHF) for the viscous liquid flow due to a stretching/shrinking sheet is investigated. 4.1 Prescribed time-dependent constant wall temperature (PTDCWT) The governing conditions for the thermal boundary layer in this case is, T  Tw ,

at y  0   T  T , as y   

,

(23a, b)

for physical quantities one can refer nomenclature. The following is the expression for temperature function    in non-dimensional form,

T    T  TW  T    .

(24)

The temperature equation (23) with associated boundary condition (24) is transformed into the homogeneous ODE with variable coefficients   2 is substituted along with Eq.(15) , the specific value of  is considered due to the non-existence of solution for positive values of  and the transformed equations are as follows : The governing thermal boundary layer equation is given by,

1  N R   Pr  f     0 ,

(25)

subjected to the transformed boundary condition given by,

  0   1 and      0 .

(26)

In order to support the analysis, new independent variable  defined by,

 1    Pr     exp    , 2    1  N R   

(27)

is introduced in the equation (25) and (26) along with equation (20) yields,



 d d 2  Pr  4   1 2    1  0,  2  1  N     1  N   d  d R R  

(28)

with corresponding boundary conditions converting into,

 1    Pr   1 and   0   0 . 2    1  N R   

 

(29)

Using the Kummer’s function, the solution of Eq.(28) is obtained and is given by, 

     ai k i , here ai 's

represents the arbitrary constant and k, constant to be discussed

i 0

(see Abramowitz and Stegun 1972). Upon substitution of    in Eq.(28) and equating the coefficients of each of the terms of the power series are equated to zero to get the arbitrary constants ai 's . The resultant solution is given by,



  2  4Q    ,   , 2   1  N R   

    A1  A2   Pr  

(30)

 Pr  4  Q    1 , 0 1  N R    2    where, A1  ,  Pr  4  Q    Pr  4  Q  1    Pr    1  2  , 0     1  2  , 1 N  1 N   1  N R   2           R R    A2 

1 are constants and  Pr  4  Q    Pr  4  Q  1    Pr    1  2  , 0     1  2  , 1 1 N  N   1  N R   2          R R     

  a, x    e  t t a 1dt represents the upper incomplete gamma function(see Abramowitz and x

Stegun 1972 page no. 259). The Blasius similarity solution of temperature equation (PTDCWT case) is given by,



   2  4  Q  1    Pr , exp     .  2  1  N   2 1 N      R R  

    A1  A2   Pr  

(31)

Upon substitution of A1 and A2 in Eq.(31), the resultant expression for    is,

 Pr  4  Q    Pr  4  Q  1    Pr   1  , 0   1  , exp        1  N R    2    1  N R    2  1  N R   2  .      Pr  4  Q    Pr  4  Q  1    Pr    1  2  , 0     1  2  , 1  N  1  N   1  N R   2         R R   

(32)

The following is the temperature gradient  at wall in the non-dimensional form obtained from Eq.(32),  

Pr

 4 Q   1 

   1 N R    2   1    Pr  Pr   2 exp      1  N R   1  N R   2    .    0    Pr  4  Q    Pr  4  Q  1    Pr    1  2  , 0     1  2  ,      1  N R   2   1  N R    1  N R  

(33)

The above Eqs.(32) and (33) are evaluated graphically for the physical parameters Pr, Q, NR, λ and Vc through various plots. In the next section, the temperature distribution for PTDWHF case is discussed. 4.2 PTDWHF The governing equation for PTDWHF boundary heating case is same as that of the PTDCWT boundary heating case except the imposed boundary condition for PTDWHF case differs from that of PTDCWT case and a different notation is used for PTDWHF case than the one used for PTDCWT case. The imposed thermal boundary conditions for the PTDWHF case is given by,



T qw  x   at y =0 and T  T as y   . y 1  bt

Substitution of    , temperature variable is given by, T  T 

(34a, b) qw     with thermal  a

boundary conditions Eq.(34) in Eq.(15) yields the governing thermal boundary layer equation for PTDWHF case as follows:

1  N R    Pr  f      0

(35)

and the associated boundary condition takes the form,

  0   1 and      0 .

(36)

Since, the governing equation (35) is identical to the equation (25) except for the imposed boundary condition. Thus, the similarity solution for    is obtained by using the identical

variable as in Eq.(27) and using the equations (35)-(36), one can obtain the following equations:

   

 Pr  4  Q     1 Pr     Pr  4  Q   1 , 0 1 , exp                 2    2  1  N R   2  1  N R       1  N R    1    Pr   2   1  N R   

 

 Pr  4 Q    1 2       1 N R  

.

(37)

 1    Pr  exp   2    1  N R   

The temperature distribution for PTDWHF case is evaluated graphically by using the Eq.(37) for physical parameters Pr, Q, NR, λ and Vc through different plots. 5.

Outcomes of the investigation: The theoretical investigation to analyze the impacts of radiative MHD unsteady

viscous Homann stagnation point liquid flow over a sheet with mass transpiration is carried out. The analytical solutions for velocity and temperature are obtained by modeling the physical flow problem into the unsteady Navier-stokes’ equation.

The governing equations

are set of nonlinear PDE’s which are then mapped into nonlinear ODE’s via similarity transformation with variable coefficient. The temperature distribution is examined for two types of boundary heating process viz. PTDCWT and PTDWHF. The exact solutions for thermal boundary value problems are obtained in terms of incomplete gamma function. The effect of on both velocity and heat transfer is shown graphically in Figs2-14. The flow characteristics for velocity, temperature and dimensionless wall skin friction drag are presented for various controlling parameters namely, Pr, Q, NR, λ and Vc. Figs.2a-2c and Figs.3a-3c depicts the impact of mass transpiration parameter and the Chandrasekhar number over the liquid flow driven by stretching/shrinking of the sheet. These figures depict the solution domain as a plot of  in terms of  by keeping of one of Q or Vc and varying the other. As discussed earlier in the section 2. In the case of shrinking boundary, the induced mass suction Vc  0  parameter the solution branches into upper and

lower for   Q  3 , where as for the induced mass injection Vc  0  parameter and in the impermeable wall Vc  0  case, no branching of solution takes place and a unique solution exists for   Q  3 . The two branch solutions upper as well as lower are denoted by solid and dotted line respectively. In the present results, the branching of the solution is studied with MHD and the presence of MHD impacts significantly to the dual solution space. In Fig.2a-2c, the increase in Vc forces  to increase with decreasing  . Also, the same effect is found to be in Fig.3a-3c, as the increasing value of Q with the decrease in  is due to the resistance offered to the liquid flow and also on the solution domain due to the presence of Lorentz force. But, this trend reverses in the lower branch solution case. Depending on the choice of combined value of Q and  , the region of existence of dual solution varies.

Fig.2a. The region of solution of  versus  , the wall moving parameter for various choice of mass transpiration Vc  in the absence of Chandrasekhar number  Q  .

Fig.2b. The region of solution of  versus  for Q  2 and different choice of Vc .

Fig.2c. The region of solution of  versus  for Q  5 and different choice of Vc .

Fig.3a. The region of solution of  versus  for the impermeable sheet.

Fig.3b. The region of solution of  versus  for mass suction.

Fig.3c. The region of solution of  versus  for mass injection. Figs. 4a-4d demonstrates the axial velocity boundary  f  impacted by mass transpiration Vc  on liquid flow past a stretching/shrinking sheet for different choice of regulating physical parameters. Through these plots, it is evident that the increase in Vc leads to the flattening of velocity boundary where as for the decreasing values of Vc leads to the thicker velocity boundary as the fluid blows away from the wall. Figs.4a-4b shows the branching of the solution, the increasing values of Vc leading to the flattening velocity boundary where as the opposite trend can be seen for the lower branch. In the impermeable case, the choice of maximum value for Vc leads to flattening of velocity in the upper solution branch where as in the lower branch solution, the thicker velocity boundary as in Fig.4b. Figs. 5a-5b demonstrates the impact of Chandrasekhar number  Q  on f . Through these plots, it can be concluded that the effect of Q produces similar effect as that of Vc on both the solution branches. But, in comparison to Vc , the increased magnetic field produces increased flattening/thickening of the velocity boundary for different branch of solutions.

This is because of the presence of MHD, which produces the drag force namely Lorentz force which offers significant opposition to the liquid flow.

Fig.4a. The impact of Vc on f with  when   Q  3 .

Fig.4b. The impact of Vc with MHD on f in the absence of  when   Q  3 .

Fig.4c. The impact of Vc with MHD on f in the presence of  when   Q  3 .

Fig.4d. The impact of Vc with MHD on f in the presence of  when   Q  3 .

Fig.5a. The impact of Q on f with  .

Fig.5b. The impact of Q on f without  .

The effect of wall shear stress coefficient f nn  0  on the axial velocity boundary is demonstrated on the in the Figs.6a – 6c. The wall shear stress coefficient is derived from the

Eq. (20) as, f 0   (1   ) . These plots give a good comparison of the closed form solutions with that of the numerical solution. Here only the closed form solution is considered for discussion and also in future analysis. The velocity distributions are complex in nature for the non-exponential case in comparison with the exponential case. As it is a well known fact that most of the velocity functions are not monotonic in nature and in fact the, velocity overshoots in the boundary. The combination of Vc , Q and  greatly affects the flow boundary as well as the shear stress in the fluid.

Fig.6a: The evaluation of axial velocity distribution in upper and lower branch for wall shear stress coefficient f  0  in the absence of Q and  .

Fig.6b: The evaluation of axial velocity distribution in upper and lower branch for f  0  in the presence of Q and for small value of  .

Fig.6c: The evaluation of axial velocity distribution in upper and lower branch for f  0  in the presence of Q and large value of  .

(a)

(b)

(c) (d) Fig.7. Comparison of streamline patterns both branch of solutions (upper branch Figs. (a)-(b) and lower branch Figs. (c)-(d)) at t = 0 in the absence/presence of MHD with Vc  5 and   1 . The stream line patterns of the branching solutions for both upper and lower are demonstrated in Figs. 7a-7b and Figs.7c- 7d as well as Figs.8a-8b and Figs. 8c-8d in the

absence/presence of MHD respectively with other physical quantities kept constant for different times. These plots clearly demonstrate that there is a flow separation in both the solution branch due to the fact that there is a velocity arising over the stretching sheet which acts against the free stream. In addition to this, the increased strength of magnetic field results in strong drag known as Lorentz force which offers significant opposition for the free flow and resulting in the flattening of the velocity boundary. Further, the transverse velocity decays from the wall to a more flattening close to the surface boundary. In the upper solution branch the reversal flow velocity occurs where as in the thicker layer for the case of lower branch. Using Eq.(23), the velocity components in each of the coordinates is as follows, 2    y Vc  Vc   x           u  x, y , t      Q 1 1 exp 3         2  2   1  2t      1  2t   v  x, y , t  

   y  1 1    Vc  2 1  2t  1  2t  Vc  Vc         Q  3     2  2   

(a)

       

   Vc 2  V  c         Q 1 exp 3        2 2         

(b)

   y  . 1  2t     

(c)

(d)

Fig.8. Comparison of streamline patterns both branch of solutions (upper branch Figs. (a)-(b) and lower branch Figs. (c)-(d)) at t = 1 in the absence/presence of MHD with Vc  5 and   1 Figs.9a-9d and Figs.10a-10d depicts the effect of Q and Vc respectively for both the solution branches. It can be seen in both the cases that, in the upper branch solution case, there is no impact of wall stretching on the solution domain as the velocity is approaching towards zero. But, liquid flows along the axis and then approaches towards zero positively for lower branch solution. The perturbation induced by the stretching in the backward boundary layer flows (Goldstein 1965) is lost in the liquid due to the physical difference in both the boundary layers. The transverse velocities for both branches seizes to zero for different velocity time and further for both upper and lower branch of solutions, it can be positive at other locations due to this backward boundary layer flows.

(a)

(b)

(c)

(d)

Fig.9. Axial and transverse velocity components of the both branch solution (upper and lower) for different choices of Q with Vc  5 and   1 .

(a)

(b)

(c)

(d)

Fig.10. Axial and transverse velocity components of the both branch solution (upper and lower) for different choices of Vc with Q  1 and   1 .

Figs. 11-14 depicts the effect of physical parameters Pr, Vc , NR and Q on temperature profiles. From these plots both type of boundary heating processes PTDCWT and PDTWHF may be analyzed. Figure 11demonstrates the impact of Pr on heat transfer. These plots also, demonstrates, that thinner thermal boundary in both cases of boundary heating PTDCWT and PDTWHF for larger values of Pr. Indeed, it is developing a slow rate of thermal diffusion. These plots together with the earlier plots, yields the conclusion that, in comparison to the thickness of viscous boundary layer, the thermal boundary layer thickness is much thinner. In the liquids with lesser Pr value, the heat diffuses to the vertical surfaces much faster than in the fluids with higher Pr value. The similar effects as that of Pr in both the boundary heating processes can be seen for Vc and Q in stretching and shrinking sheet case respectively. In contrast to the effect of Pr, the effect Vc for shrinking sheet, Q for stretching sheet and NR for both stretching as well as shrinking sheet can be seen. In comparison to PTDCWT case, the thermal boundary is quantitatively larger than that of in PDTWHF case. Hence, it can be concluded that PTDCWT type of boundary heating is much advantageous than that of the PDTWHF type of boundary heating. Further, these physical characteristics are in line with the case of regions away from the sheet.

Fig.11: Impact of Pr on thermal boundary ( for PTDCWT and PDTWHF (upper solution branch)) with other physical parameters kept constant.

(a)

(b)

(c)

(d)

Fig.12: Impact of Vc on thermal boundary (for PTDCWT and PDTWHF) with other physical parameters kept constant.

(a)

(b)

(c)

(d)

Fig. 13: Impact of NR on thermal boundary (PTDCWT and PDTWHF) with other physical parameters kept constant.

Fig.14: Impact of Q on thermal boundary (PTDCWT and PDTWHF (upper solution branch)) with other physical parameters kept constant. 6. Concluding remarks: The unsteady, radiative MHD Homann stagnation point flow of a viscous liquid due to a stretching/shrinking sheet is considered in the present study. The physical flow problem is modelled into governing equations which obey the boundary conditions. Using similarity transformations, these are mapped into nonlinear ODE’s. The closed form exact analytical solutions for both velocity and temperature are derived based on the choice of regulating parameters. The closed form solutions for temperature distribution for PTDCWT and PDTWHF type of boundary heating processes and the solutions so obtained are much suitable for the realistic problems. Further, to provide the physical insights and to see the

impacts of controlling parameters on viscous boundary as well as thermal boundary many plots are drawn. From these plots, it can be concluded that applied magnetic field should be of minimum strength in order to regulate the cooling of the liquid. In addition to this, it should be ensured that the occurrence of radiation is minimized to obtain the desired outcome. Further, it can be concluded that the PTDCWT type of boundary heating is much suited for cooling of the fluid than PDTWHF type of boundary heating.

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The physical flow problem is modelled into a nonlinear partial differential equations

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Finds applications in the chemical industries dealing with fluids and elastic materials.

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The analytical solutions of momentum and thermal boundary layer are obtained.

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The temperature profiles are obtained for both type of heating processes.

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The effects of physical parameters on velocity and temperature are analyzed.