^{1}

^{2}

In this paper, the magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates is studied . The governing partial differential equation was transformed in to a system of dimensionless non-similar coupled ordinary differential equation. The transformed conservations equations were solved by using new algorithm. Basically , this new algorithm depends mainly on the Taylor expansion application with the coefficients of power series resulting from integrating th e order differential equation. Results obtained from new algorithm are compared with the results of numerical Range-Kutta fourth-order algorithm with help of the shooting algorithm. The comparison revealed that the resulting solutions were excellent agreement. Thermo-diffusion and diffusion-thermo effects were investigated to analyze the behavior of temperature and concentration profile. Also the influences of the first order chemical reaction and the rate of mass and heat transfer were studied. The computed analytical solution result for the velocity, temperature and concentration distribution with the effect of various important dimensionless parameters w as analyzed and discussed graphically.

The importance of thermal-diffusion and diffusion-thermo effects for various fluid flows has been studied by Eckert and Drake [

Consider the flow of an incompressible fluid due to source or sink that is located at the intersection of two rigid plane walls angled 2 α apart. Radial and symmetric nature of the flow is taken into consideration. Induced magnetic field is ignored and an applied magnetic field is considered that is applied across the flow direction. Under the aforesaid assumptions velocity field takes the form V = [ u r , 0 , 0 ] , where u r is a function of both r and θ . Soret and Dufour effects are also considered that are incorporated in energy and concentration equations respectively. The fluid is also assumed to be chemically reacting. Also, the temperature and concentration are also the function of both r and θ . While the results of angle opening α on concentration profile show that the increase in angle gives a decrease in concentration profile. The governing equations for mass, motion, energy, and mass transfer in polar coordinates under imposed assumptions become [

1 r ∂ ∂ r ( r u r ) = 0 , (1)

u r ∂ u r ∂ r = − 1 ρ ∂ p ∂ r + υ [ ∂ 2 u r ∂ r 2 + 1 r ∂ u r ∂ r + 1 r 2 ∂ 2 u r ∂ θ 2 − u r r 2 ] − σ B 0 2 ρ u r , (2)

− 1 ρ r ∂ p ∂ θ + 2 υ r 2 ∂ u r ∂ θ = 0 , (3)

ρ c p u r ∂ T ∂ r = k [ ∂ 2 T ∂ r 2 + 1 r ∂ T ∂ r + 1 r 2 ∂ 2 T ∂ θ 2 ] + μ [ 4 ( ∂ u r ∂ r ) 2 + ( ∂ u r ∂ θ ) 2 ] + D K T C s ∂ 2 C ∂ r 2 + 1 r ∂ C ∂ r + 1 r 2 ∂ 2 C ∂ θ 2 (4)

u r ∂ C ∂ r = D [ ∂ 2 C ∂ r 2 + 1 r ∂ C ∂ r + 1 r 2 ∂ 2 C ∂ θ 2 ] + D K T T m [ ∂ 2 T ∂ r 2 + 1 r ∂ T ∂ r + 1 r 2 ∂ 2 T ∂ θ 2 ] (5)

the boundary conditions are,

u r = U , ∂ u r ∂ θ = 0 , ∂ T ∂ θ = 0 , ∂ C ∂ θ = 0 , at θ = α ,

u r = 0 , T = T m , C = C w , at θ = α , (6)

where p is the fluid pressure, υ = μ ρ is kinematic viscosity, D in order are the

specific heat and coefficient of mass diffusivity. K , K T , D correspondingly are the thermal conductivity, thermal-diffusion ratio and coefficient of mass diffusivity. Further C s , T w , T m , C w , K 1 represent the concentration susceptibility, temperature at wall, mean fluid temperature, concentration at the wall and the chemical reaction constant respectively. From the continuity Equation (1), we can write

F ( θ ) = r u r ( r , θ ) , (7)

with the use of dimensionless parameters award [

f ( η ) = F ( θ ) r U , η = θ α , β ( η ) = T T w , ϕ ( η ) = C C w . (8)

Eliminating p from Equations (1) and (2) using Equations (7) and (8), we get a system of nonlinear ordinary differential equation for the normalized velocity profile f ( η ) , temperature profile β ( η ) and concentration profile ϕ (η)

f ‴ ( η ) + 2 α R e f ( η ) f ′ ( η ) + ( 4 − H a ) α 2 f ′ ( η ) = 0,

β ″ ( η ) + E c P r [ 4 α 2 f 2 ( η ) + ( f ′ ( η ) ) 2 ] + D f P r ϕ ″ ( η ) = 0,

ϕ ″ ( η ) + S c S r β ″ − S c γ α 2 ϕ ( η ) = 0, (9)

f ( 0 ) = 1 , f ′ ( 0 ) = 0 , f ( 1 ) = 0 ,

β ′ ( 0 ) = 0 , β ( 1 ) = 1 ,

ϕ ( 0 ) = 0 , ϕ ( 1 ) = 1 , (10)

where,

R e = U r α μ { divergent channel : α > 0 , U > 0 convergent channel : α < 0 , U < 0 , H a = σ B 0 2 μ , P r = μ c p K .

E c = U 2 c p T w , D f = D K T C w υ c p C s T w , S c = υ D , S r = D K T T w υ T w C w , represent Reynolds, Hartmman, Prandtl, Eckert, Dufour, Schmidt and Soret number respectively while γ = K 1 υ is the first order chemical reaction parameter. The local Nusselt and Sherwood numbers are define by

N u = q w | η = 1 K T w = − 1 α β ′ ( 1 ) , (11)

S h = M w | η = 1 D C w = − 1 α ϕ ′ ( 1 ) . (12)

This section describes how to obtain a new scheme to calculate the coefficients of the power series solution resulting from solving nonlinear ordinary differential equations to find analytical-approximate solution. These coefficients are important basis to construct the solution formula, therefore they can be computed recursively by differentiation ways. To illustrative the computation and operations for these coefficients and derivation the new scheme, we summarized the detail a new outlook in the following steps.

Step (1): Consider the non-linear differential equation as follows:

H ( f ( η ) , f ′ ( η ) , f ″ ( η ) , f ‴ ( η ) , ⋯ , f ( n − 1 ) ( η ) , f ( n ) ( η ) ) = 0, (13)

integrating Equation (13) with respect to η on [ 0, η ] yield

f ( η ) = f ( 0 ) + f ′ ( 0 ) η + f ″ ( 0 ) η 2 2 ! + ⋯ + f ( n − 1 ) ( 0 ) η n − 1 ( n − 1 ) ! + L − 1 G [ f ( η ) ] , (14)

where,

G [ f ( η ) ] = H ( f ( η ) , f ′ ( η ) , f ″ ( η ) , ⋯ , f ( n − 1 ) ( η ) ) , L − 1 = ∫ 0 η ∫ 0 η ⋯ ∫ 0 η ( d η ) n . (15)

Step (2): We take Taylor series expansion of the function G [ f ( η ) ] about η = η 0 as follows

G [ f ( η ) ] = ∑ n = 0 ∞ ( Δ η ) n n ! d n G ( f 0 ( η ) ) d η n , (16)

rewriting the Equation (16)

G [ f ( η ) ] = G [ f 0 ( η ) ] + Δ η 1 ! G ′ [ f 0 ( η ) ] + ( Δ η ) 2 2 ! G ″ [ f 0 ( η ) ] + ( Δ η ) 3 3 ! G ‴ [ f 0 ( η ) ] + ⋯ (17)

Now, we assume that Δ η = max { η , η 0 } and substituting Equation (17) in Equation (14), we obtain

f ( η ) = f 0 + f 1 + f 2 + f 3 + f 4 + ⋯ , (18)

where,

f 0 = f ( 0 ) + f ′ ( 0 ) η + f ″ ( 0 ) η 2 2 ! + ⋯ + f ( n − 1 ) ( 0 ) η ( n − 1 ) ( n − 1 ) ! , f 1 = L − 1 G [ f 0 ( η ) ] ,

f 2 = L − 1 max { η , η 0 } 1 ! G ′ [ f 0 ( η ) ] , f 3 = L − 1 ( max { η , η 0 } ) 2 2 ! G ″ [ f 0 ( η ) ] ,

f 4 = L − 1 ( max { η , η 0 } ) 3 3 ! G ‴ [ f 0 ( η ) ] ⋯ (19)

Step (3): We focus on computing the derivatives of G with respect to η which is the crucial part of the proposed method. Let start calculating

G [ f ( η ) ] , G ′ [ f ( η ) ] , G ″ [ f ( η ) ] , G ‴ [ f ( η ) ] , ⋯ .

G [ f ( η ) ] = H ( f ( η ) , f ′ ( η ) , f ″ ( η ) , f ‴ ( η ) , f ‴ ' ( η ) , ⋯ , f ( n − 1 ) ( η ) ) , (20)

G ′ [ f ( η ) ] = d G [ f ( η ) ] d η = G 1 f ⋅ f η + G 1 f ′ ⋅ ( f η ) ′ + ⋯ + G f ( n − 1 ) ⋅ ( f η ) ( n − 1 ) , (21)

G ″ [ f ( η ) ] = d 2 G [ f ( η ) ] d z 2 = G f f ⋅ ( f η ) 2 + G f f ′ ⋅ ( f η ) ′ f η + G f f ″ ⋅ f η ( f η ) ′ ′ + ⋯ + G f f ( n − 1 ) ⋅ ( f η ) ( f η ) ( n − 1 ) + G f ⋅ f η η + G f ′ f ⋅ ( f η ) ′ ⋅ f η + G f ′ f ′ ⋅ ( f η ) ′ 2 + ⋯ + G f ′ f ( n − 1 ) ⋅ ( f η ) ′ ( f η ) ( n − 1 ) + G f ′ ⋅ ( f η η ) ′ + G f ″ f ⋅ ( f η ) ′ ′ ⋅ f η

+ G f ′ ′ f ′ ⋅ ( f η ) ′ ( f η ) ′ ′ + G f ″ f ″ ⋅ ( f η ) ′ ′ 2 + G f ″ f ( n − 1 ) ⋅ ( f η ) ′ ′ ( f η ) ( n − 1 ) + G f ″ ⋅ ( f η η ) ′ ′ + ⋯ + G f ( n − 1 ) f ⋅ ( f η ) ( n − 1 ) ⋅ f η + G f ( n − 1 ) f ′ ⋅ ( f η ) ( n − 1 ) ⋅ ( f η ) ′ + ⋯ + G f ( n − 1 ) f ( n − 1 ) ⋅ ( f η ) ( n − 1 ) 2 + G f ( n − 1 ) ⋅ ( f η η ) ( n − 1 ) (22)

G ‴ [ f ( η ) ] = d 3 G [ f ( η ) ] d η 3 = G f f f ⋅ ( f η ) 3 + G f f f ′ ⋅ ( f η ) 2 ( f η ) ′ + ⋯ + G f f f ( n − 1 ) ⋅ ( f η ) 2 ⋅ ( f η ) ( n − 1 ) + G f f ⋅ 2 ( f η ) ⋅ f η η + G f f ′ f ⋅ ( f η ) ′ ( f η ) 2 + G f f ′ f ′ ⋅ ( f η ) ′ 2 ( f η ) + ⋯ + G f f ′ f ( n − 1 ) ⋅ ( f η ) ′ ( f η ) ⋅ ( f η ) ( n − 1 ) + G f f ′ ⋅ [ ( f η η ) ′ ⋅ f η + ( f η ) ′ ⋅ f η η ] + G f f ″ f ⋅ ( f η ) ′ ′ ( f η ) 2 + G f f ″ f ′ ⋅ ( f η ) ′ ′ ( f η ) ⋅ ( f η ) ′ + ⋯

+ G f f ″ f ( n − 1 ) ⋅ ( f η ) ′ ′ ( f η ) ⋅ ( f η ) ( n − 1 ) + G f f ″ ⋅ [ f η η ⋅ ( f η ) ′ ′ + f η ⋅ ( f η η ) ′ ′ ] + ⋯ + G f f ( n − 1 ) f ⋅ ( f η ) 2 ⋅ ( f η ) ( n − 1 ) + G f f ( n − 1 ) f ′ ⋅ ( f η ) ⋅ ( f η ) ′ ⋅ ( f η ) ( n − 1 ) + ⋯ + G f f ( n − 1 ) f ( n − 1 ) ⋅ ( f η ) ⋅ ( f η ) ( n − 1 ) 2 + G f f ( n − 1 ) ⋅ [ ( f η η ) ⋅ ( f η ) ( n − 1 ) + ( f η ) ( f η η ) ( n − 1 ) ]

+ G f f ⋅ f η η ⋅ ( f η ) + G f f ′ ⋅ f η η ⋅ ( f η ) ′ + ⋯ + G f f ( n − 1 ) ⋅ f η η ⋅ ( f η ) ( n − 1 ) + G η ⋅ f η η η + G f ′ f f ⋅ ( f η ) ′ ( f η ) 2 + G f ′ f ′ f ⋅ ( f η ) ′ 2 ( f η ) + ⋯ + G f ′ f f ( n − 1 ) ⋅ ( f η ) ′ ( f η ) ⋅ ( f η ) ( n − 1 ) + G f ′ f ⋅ [ ( f η η ) ′ ⋅ f η + ( f η ) ′ ⋅ f η η ] + G f ′ f ′ f ⋅ ( f η ) ′ 2 ⋅ f z + G f ′ f ′ f ′ ⋅ ( f η ) ′ 3 + ⋯ + G f ′ f ′ f ( n − 1 ) ⋅ ( f η ) ′ 2 ⋅ ( f η ) ( n − 1 ) + G f ′ f ′ ⋅ 2 ( f η ) ′ ⋅ ( f η η ) ′ + ⋯ + G f ( n − 1 ) f ( n − 1 ) f ⋅ ( f η ) ( n − 1 ) 2 ⋅ f η + G f ( n − 1 ) f ( n − 1 ) f ′ ⋅ ( f η ) ( n − 1 ) 2 ⋅ ( f η ) ′ + ⋯

+ G f ( n − 1 ) f ( n − 1 ) f ( n − 1 ) ⋅ ( f η ) ( n − 1 ) 3 + G f ( n − 1 ) f ( n − 1 ) ⋅ 2 ⋅ ( f η ) ( n − 1 ) ⋅ ( f η η ) ( n − 1 ) + G f ( n − 1 ) f ⋅ ( f η η ) ( n − 1 ) ⋅ f η + G f ( n − 1 ) f ′ ⋅ ( f η η ) ( n − 1 ) ⋅ ( f η ) ′ + ⋯ + G f ( n − 1 ) f ( n − 1 ) ⋅ ( f η η ) ( n − 1 ) ⋅ ( f η ) ( n − 1 ) + G f ( n − 1 ) ⋅ ( f η η η ) ( n − 1 ) (23)

⋮

The calculations are more complicated in the second and third derivatives because of the product rules. Consequently, the systematic structure on calculation is extremely important. Fortunately, due to the assumption that the operator G and the solution f are analytic functions, then the mixed derivatives are equivalence.

We note that the derivatives function to f unknown, so we suggest the following hypothesis

f η = f 1 = L − 1 G [ f 0 ( η ) ] , f η η = f 2 = L − 1 max { η , η 0 } 1 ! G ′ [ f 0 ( η ) ] ,

f η η η = f 3 = L − 1 ( max { η , η 0 } ) 2 2 ! G ″ [ f 0 ( η ) ] ,

f η η η η = f 4 = L − 1 ( max { η , η 0 } ) 3 3 ! G ‴ [ f 0 ( η ) ] ,

f η η η η η = f 5 = L − 1 ( max { η , η 0 } ) 4 4 ! G ‴ ' [ f 0 ( η ) ] , ⋯ (24)

Therefore Equations (20)-(23) are evaluated by

G [ f 0 ( η ) ] = H ( f 0 ( η ) , f ′ 0 ( η ) , f ″ 0 ( η ) , ⋯ , f 0 ( n − 1 ) ( η ) ) , (25)

G ′ [ f 0 ( η ) ] = G f 0 ⋅ f 1 + G f ′ 0 ⋅ ( f 1 ) ′ + G f ″ 0 ⋅ ( f 1 ) ′ ′ + ⋯ + G f 0 ( n − 1 ) ⋅ ( f 1 ) ( n − 1 ) , (26)

G ″ [ f 0 ( η ) ] = G f 0 f 0 ⋅ ( f 1 ) 2 + G f 0 f ′ 0 ⋅ ( f 1 ) ′ f 1 + G f 0 f ″ 0 ⋅ f 1 ( f 1 ) ′ ′ + ⋯ + G f 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ( f 1 ) ( n − 1 ) + G f 0 ⋅ f 2 + G f ′ 0 f 0 ⋅ ( f 1 ) ′ ⋅ f 1 + G f ′ 0 f ′ 0 ⋅ ( f 1 ) ′ 2 + ⋯ + G f ′ 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ ( f 1 ) ( n − 1 ) + G f ′ 0 ⋅ ( f 2 ) ′ + G f ″ 0 f 0 ⋅ ( f 1 ) ′ ′ ⋅ f 1 + G f ″ 0 f ′ 0 ⋅ ( f 1 ) ′ ( f 1 ) ′ ′ + G f ″ 0 f ″ 0 ⋅ ( f 1 ) ′ ′ 2

+ ⋯ + G f ″ 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ ′ ( f 1 ) ( n − 1 ) + G f ″ 0 ⋅ ( f 2 ) ′ ′ + G f ( n − 1 ) f ⋅ ( f 1 ) ( n − 1 ) ⋅ f 1 + G f 0 ( n − 1 ) f ′ 0 ⋅ ( f 1 ) ( n − 1 ) ⋅ ( f 1 ) ′ + ⋯ + G f 0 ( n − 1 ) f 0 ( n − 1 ) ⋅ ( f 1 ) ( n − 1 ) 2 + ⋯ + G f 0 ( n − 1 ) ⋅ ( f 2 ) ( n − 1 ) , (27)

G ‴ [ f 0 ( η ) ] = G f 0 f 0 f 0 ⋅ ( f 1 ) 3 + G f 0 f 0 f ′ 0 ⋅ ( f 1 ) 2 ( f 1 ) ′ + ⋯ + G f 0 f 0 f 0 ( n − 1 ) ⋅ ( f 1 ) 2 ⋅ ( f 1 ) ( n − 1 ) + G f 0 f 0 ⋅ 2 ( f 1 ) ⋅ f 2 + G f 0 f ′ 0 f 0 ⋅ ( f 1 ) ′ ( f 1 ) 2 + G f 0 f ′ 0 f ′ 0 ⋅ ( f 1 ) ′ 2 ( f 1 ) + ⋯ + G f 0 f ′ 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ ( f 1 ) ⋅ ( f 1 ) ( n − 1 ) + G f 0 f ′ 0 ⋅ [ ( f 2 ) ′ ⋅ f z + ( f 1 ) ′ ⋅ f 2 ] + G f 0 f ″ 0 f 0 ⋅ ( f 1 ) ′ ′ ( f 1 ) 2 + G f 0 f ″ 0 f ′ 0 ⋅ ( f 1 ) ′ ′ ( f 1 ) ⋅ ( f 1 ) ′ + ⋯

+ G f 0 f ″ 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ ′ ( f 1 ) ⋅ ( f 1 ) ( n − 1 ) + G f 0 f ″ 0 ⋅ [ f 2 ⋅ ( f 1 ) ′ ′ + f 1 ⋅ ( f 2 ) ′ ′ ] + ⋯ + G f 0 f 0 ( n − 1 ) f 0 ⋅ ( f 1 ) 2 ⋅ ( f 1 ) ( n − 1 ) + G f 0 f 0 ( n − 1 ) f ′ 0 ⋅ ( f 1 ) ⋅ ( f 1 ) ′ ⋅ ( f 1 ) ( n − 1 ) + ⋯ + G f 0 f 0 ( n − 1 ) f 0 ( n − 1 ) ⋅ ( f 1 ) + ⋯ + G f 0 f 0 ( n − 1 ) ⋅ f 2 ⋅ ( f 1 ) ( n − 1 ) + G f 0 ⋅ f 3 + G f ′ 0 f 0 f 0 ⋅ ( f 1 ) ′ ( f 1 ) 2 + G f ′ 0 f ′ 0 f 0 ⋅ ( f 1 ) ′ 2 ( f 1 ) + ⋯ + G f ′ 0 f 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ ( f 1 ) ⋅ ( f 1 ) ( n − 1 ) + G f ′ 0 f 0 ⋅ [ ( f 2 ) ′ ⋅ f 1 + ( f 1 ) ′ ⋅ f 1 ] + G f ′ 0 f ′ 0 f 0 ⋅ ( f 1 ) ′ 2 ⋅ f 1 + G f ′ 0 f ′ 0 f ′ 0 ⋅ ( f 1 ) ′ 3 + ⋯

+ G f ′ 0 f ′ 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ 2 ⋅ ( f 1 ) ( n − 1 ) + G f ′ 0 f ′ 0 ⋅ 2 ( f 1 ) ′ ( f 2 ) ′ + ⋯ + G f 0 ( n − 1 ) f 0 ( n − 1 ) f 0 ⋅ ( f 1 ) ′ 3 + ⋯ + G f ′ 0 f ′ 0 f 0 ( n − 1 ) ⋅ ( f 1 ) ′ 2 ⋅ ( f 1 ) ( n − 1 ) + G f ′ 0 f ′ 0 ⋅ 2 ( f 1 ) ′ ⋅ ( f 2 ) ′ + ⋯ + G f 0 ( n − 1 ) f 0 ( n − 1 ) f 0 ⋅ ( f 1 ) ( n − 1 ) 2 ⋅ f 1 + G f 0 ( n − 1 ) f 0 ( n − 1 ) f ′ 0 ⋅ ( f 1 ) ( n − 1 ) 2 ⋅ ( f 1 ) ′ + ⋯

+ G f 0 ( n − 1 ) f 0 ( n − 1 ) f 0 ( n − 1 ) ⋅ ( f 1 ) ( n − 1 ) 3 + G f 0 ( n − 1 ) f 0 ( n − 1 ) ⋅ 2 ⋅ ( f 1 ) ( n − 1 ) ⋅ ( f 2 ) ( n − 1 ) + G f 0 ( n − 1 ) f 0 ⋅ ( f 2 ) ( n − 1 ) ⋅ f 1 + G f 0 ( n − 1 ) f ′ 0 ⋅ ( f 2 ) ( n − 1 ) ⋅ ( f 1 ) ′ + ⋯ + G f 0 ( n − 1 ) f 0 ( n − 1 ) ⋅ ( f 2 ) ( n − 1 ) ⋅ ( f 1 ) ( n − 1 ) + G f 0 ( n − 1 ) ⋅ ( f 3 ) ( n − 1 ) (28)

⋮

Step (4): Substituting Equations (25)-(28) in Equation (18) we get the required analytical-approximate solution for the Equation (13).

The new algorithm described in the previous section can be used as a powerful solver to the nonlinear differential Equations (9)-(10) and to find new an analytical-approximate solution. From step (1) we have

f ( η ) = f ( 0 ) + f ′ ( 0 ) η + f ″ ( 0 ) η 2 2 ! + L 1 − 1 [ − 2 α R e f ( η ) f ′ ( η ) − ( 4 − H a ) α 2 f ′ ( η ) ] ,

β ( η ) = β ( 0 ) + β ′ ( 0 ) η + L 2 − 1 [ − E c P r [ 4 α 2 f 2 ( η ) − ( f ′ ( η ) ) 2 ] − D f P r ϕ ″ ( η ) ] ,

ϕ ( η ) = ϕ ( 0 ) + ϕ ′ ( 0 ) η + L 3 − 1 [ − S c S r β ″ + S c γ α 2 ϕ ( η ) ] , (29)

rewrite the Equation (29) as follows

f ( η ) = A 1 + A 2 η + A 3 η 2 2 ! + L − 1 G 1 [ f ( η ) ] ,

β ( η ) = B 1 + B 2 η + L − 1 G 2 [ β ( η ) ] ,

ϕ ( η ) = C 1 + C 2 η + L − 1 G 3 [ ϕ ( η ) ] , (30)

where,

A 1 = f ( 0 ) , A 2 = f ′ ( 0 ) , A 3 = f ″ ( 0 ) ,

B 1 = β ( 0 ) , B 2 = β ′ ( 0 ) , C 1 = ϕ ( 0 ) , C 2 = ϕ ′ ( 0 ) ,

G 1 [ f ] = − 2 α R e f ( η ) f ′ ( η ) − ( 4 − H a ) α 2 f ′ ( η ) ,

G 2 [ β ] = − E c P r [ 4 α 2 f 2 ( η ) − ( f ′ ( η ) ) 2 ] − D f P r ϕ ″ ( η ) ,

G 3 [ ϕ ] = − S c S r β ″ + S c γ α 2 ϕ ( η ) ,

and L 1 − 1 ( . ) = ∫ 0 η ∫ 0 η ∫ 0 η ( d η ) 3 , L 2 − 1 ( . ) = L 3 − 1 ( . ) = ∫ 0 η ∫ 0 η ( d η ) 2 . (31)

From the boundary conditions the Equation (30) becomes

f ( η ) = 1 + A 3 η 2 2 ! + L − 1 G 1 [ f ( η ) ] ,

β ( η ) = B 1 + L − 1 G 2 [ β ( η ) ] ,

ϕ ( η ) = C 1 + L − 1 G 3 [ ϕ ( η ) ] . (32)

From step (2) suppose that Δ η = max { 1 , 0 } = 1 , yield

f 0 = 1 + A 3 η 2 2 ! f 1 = L 1 − 1 G 1 [ f 0 ( η ) ] , f 2 = L 2 − 1 G ′ 1 [ f 0 ( η ) ] , ⋯ ,

β 0 = B 1 , β 1 = L 2 − 1 G 2 [ β 0 ( η ) ] , β 2 = L 2 − 1 G ′ 2 [ β 0 ( η ) ] , ⋯ ,

f 0 = C 1 , ϕ 1 = L 3 − 1 G 3 [ ϕ 0 ( η ) ] , ϕ 2 = L 3 − 1 G ′ 3 [ ϕ 0 ( η ) ] , ⋯ , (33)

and the analytical-approximate solution are

f ( η ) = f 0 + f 1 + f 2 + f 3 + ⋯ ,

β ( η ) = β 0 + β 1 + β 2 + β 3 + ⋯ ,

ϕ ( η ) = ϕ 0 + ϕ 1 + ϕ 2 + ϕ 3 + , ⋯ (34)

From step (3) yields

G 1 [ f ( η ) ] = − 2 α R e f ( η ) f ′ ( η ) − ( 4 − H a ) α 2 f ′ ( η ) ,

G 2 [ β ( η ) ] = − E c P r [ 4 α 2 f 2 ( η ) − ( f ′ ( η ) ) 2 ] − D f P r ϕ ″ ( η ) ,

G 3 [ ϕ ( η ) ] = − S c S r β ″ + S c γ α 2 ϕ ( η ) , (35)

G ′ 1 [ f ( η ) ] = d G 1 [ f ( η ) ] d η = G 1 f ⋅ f η + G 1 f ′ ⋅ ( f η ) ′ ,

G ′ 2 [ β ( η ) ] = d G 2 [ β ( η ) ] d η = G 2 f ⋅ f η + G 2 f ′ ⋅ ( f η ) ′ + G 2 ϕ ″ ⋅ ( ϕ η ) ′ ′ ,

G ′ 3 [ ϕ ( η ) ] = d G 3 [ ϕ ( η ) ] d η = G 3 β ″ ⋅ ( β η ) ′ ′ + G 3 ϕ ⋅ ( ϕ η ) , (36)

G ″ 1 [ f ( η ) ] = d 2 G 1 [ f ( η ) ] d η 2 = G f f ⋅ ( f η ) 2 + 2 ⋅ G f f ′ ⋅ f η ( f η ) ′ + G f ′ f ′ ⋅ ( f η ) ′ 2 + G f ⋅ f η η + G f ′ ⋅ ( f η η ) ′

G ″ 2 [ β ( η ) ] = d 2 G 2 [ β ( η ) ] d η 2 = G f f ⋅ ( f η ) 2 + 2 ⋅ G f f ′ ⋅ f η ( f η ) ′ + G f ′ f ′ ⋅ ( f η ) ′ 2 + G f ⋅ f η η + G f ′ ⋅ ( f η η ) ′ + G 2 ϕ ″ ϕ ″ ⋅ ( ϕ η ) ′ ′ 2 + 2 ⋅ G 2 ϕ ″ f ⋅ ( ϕ η ) ′ ′ ⋅ f η + 2 ⋅ G 2 ϕ ″ f ′ ⋅ ( ϕ η ) ′ ′ ⋅ f ′ η + G 2 ϕ ″ ⋅ ( ϕ η η ) ′ ′ ,

G ″ 3 [ ϕ ( η ) ] = d 2 G 3 [ ϕ ( η ) ] d η 2 = G 3 β ″ β ″ ⋅ ( β η ) ′ ′ 2 + G 3 ϕ ϕ ⋅ ( ϕ η ) 2 + 2 G 3 β ″ ϕ ⋅ ( β η ) ′ ′ ⋅ ϕ η + G 3 ϕ ⋅ ( ϕ η η ) + G 3 β ″ ⋅ ( β η η ) ′ ′ , (37)

G ‴ 1 [ f ( η ) ] = d 3 G 1 [ f ( η ) ] d η 3 = G 1 f f f ( f η ) 3 + 3 ⋅ G 1 f f f ′ ( f η ) 2 ⋅ ( f η ) ′ + 3 ⋅ G 1 f f ′ f ′ ⋅ ( f η ) ( f η ) ′ 2 + G 1 f ′ f ′ f ′ ⋅ ( f η ) ′ 3 + 3 ⋅ G 1 f f ⋅ f η η ⋅ f η + 3 ⋅ G 1 f f ′ ⋅ f η η ⋅ ( f η ) ′ + G 1 f ⋅ f η η η + 3 ⋅ G 1 f ′ f ⋅ ( f η η ) ′ ( f η ) + 3 ⋅ G 1 f ′ f ′ ⋅ ( f η η ) ′ ( f η ) ′ + G 1 f ′ ⋅ ( f η η η ) ′ ,

G ‴ 2 [ β ( η ) ] = d 3 G 1 [ β ( η ) ] d η 3 = G 2 f f f ( f η ) 3 + 3 ⋅ G 2 f f f ′ ( f η ) 2 ⋅ ( f η ) ′ + 3 ⋅ G 2 f f ′ f ′ ⋅ ( f η ) ( f η ) ′ 2 + G 2 f ′ f ′ f ′ ⋅ ( f η ) ′ 3 + 3 ⋅ G 2 f f ⋅ f η η ⋅ f η + 3 ⋅ G 2 f f ′ ⋅ f η η ⋅ ( f η ) ′ + G 2 f ⋅ f η η η + 3 ⋅ G 2 f ′ f ⋅ ( f η η ) ′ ( f η ) + 3 ⋅ G 2 f ′ f ′ ⋅ ( f η η ) ′ ( f η ) ′ + G 2 f ′ ⋅ ( f η η η ) ′ + G 2 ϕ ″ ϕ ″ ϕ ″ ⋅ ( ϕ η ) ′ ′ 2

+ 3 ⋅ G 2 ϕ ″ ϕ ″ f ⋅ ( ϕ η ) ′ ′ 2 ⋅ ( f η ) + 3 ⋅ G 2 ϕ ″ ϕ ″ f ′ ⋅ ( ϕ η ) ′ ′ 2 ⋅ ( f η ) ′ + 3 ⋅ G 2 ϕ ″ ϕ ″ ⋅ ( ϕ η ) ′ ′ ⋅ ( ϕ η η ) ′ ′ + 3 ⋅ G ϕ ″ f ⋅ ( ϕ η η ) ′ ′ ⋅ f η + 3 ⋅ G ϕ ″ f ′ ⋅ ( ϕ η η ) ′ ′ ⋅ ( f η ) ′ + 4 ⋅ G 2 ϕ ″ f f ′ ⋅ ( ϕ η ) ′ ′ ⋅ f η ⋅ ( f η ) ′ + 2 ⋅ G ϕ ″ f f ⋅ ( ϕ η ) ′ ′ ⋅ ( f η ) 2 + 2 ⋅ G ϕ ″ f ′ f ′ ⋅ ( ϕ η ) ′ ′ ⋅ ( f η ) ′ 2 + 2 ⋅ G 2 ϕ ″ f ⋅ ( ϕ η ) ′ ′ ⋅ f η η + 2 ⋅ G 2 ϕ ″ f ′ ⋅ ( ϕ η ) ′ ′ ⋅ ( f η η ) ′ + G 2 ϕ ″ ⋅ ( ϕ η η η ) ′ ′ ,

G ‴ 3 [ ϕ ( η ) ] = d 3 G 3 [ ϕ ( η ) ] d η 3 = G 3 β ″ β ″ β ″ ⋅ ( β η ) ′ ′ 3 + 3 ⋅ G 3 β ″ β ″ ϕ ⋅ ( β η ) ′ ′ 2 ϕ η + 3 ⋅ G 3 β ″ β ″ ⋅ ( β η ) ′ ′ ⋅ ( β η η ) ′ ′ + G 3 ϕ ϕ ϕ ⋅ ( ϕ η ) 3 + 3 ⋅ G 3 ϕ ϕ β ″ ⋅ ( ϕ η ) 2 ⋅ β ″ η + 3 ⋅ G 3 ϕ ϕ ⋅ ( ϕ η ) ϕ η η + 3 ⋅ G 3 ϕ β ⋅ ( ϕ η η ) ⋅ β η + G 3 ϕ ⋅ ( ϕ η η η ) + 3 ⋅ G 3 β ″ ϕ ⋅ ( β η η ) ′ ′ ⋅ ϕ η + G 3 β ″ ⋅ ( β η η η ) ′ ′ . (38)

⋮

We note that the derivatives of f with respect η that are given in (19), can be computing by Equations (35)-(38) as

G 1 [ f 0 ( η ) ] = − 2 α R e f 0 ( η ) f ′ 0 ( η ) − ( 4 − H a ) α 2 f ′ 0 ( η ) ,

G 2 [ β 0 ( η ) ] = − E c P r [ 4 α 2 f 0 2 ( η ) − ( f ′ 0 ( η ) ) 2 ] − D f P r ϕ ″ 0 ( η ) ,

G 3 [ ϕ 0 ( η ) ] = − S c S r β ″ 0 + S c γ α 2 ϕ 0 ( η ) , (39)

G ′ 1 [ f 0 ( η ) ] = G 1 f 0 ⋅ f 1 + G 1 f ′ 0 ⋅ ( f 1 ) ′ ,

G ′ 2 [ β 0 ( η ) ] = G 2 f 0 ⋅ f 1 + G 2 f ′ 0 ⋅ ( f 1 ) ′ + G 2 ϕ ″ 0 ⋅ ( ϕ 1 ) ′ ′ ,

G ′ 3 [ ϕ 0 ( η ) ] = G 3 β ″ 0 ⋅ ( β 1 ) ′ ′ + G 3 ϕ 0 ⋅ ( ϕ 1 ) , (40)

G ″ 1 [ f 0 ( η ) ] = G f 0 f 0 ⋅ ( f 1 ) 2 + 2 ⋅ G f 0 f ′ 0 ⋅ f 1 ( f 1 ) ′ + G f ′ 0 f ′ 0 ⋅ ( f 1 ) ′ 2 + G f 0 ⋅ f 2 + G f ′ 0 . ( f 2 ) ′

G ″ 2 [ β 0 ( η ) ] = G f 0 f 0 ⋅ ( f 1 ) 2 + 2 ⋅ G f 0 f ′ 0 ⋅ f 1 ( f 1 ) ′ + G f ′ 0 f ′ 0 ⋅ ( f 1 ) ′ 2 + G f 0 ⋅ f 2 + G f ′ 0 ⋅ ( f 2 ) ′ + G 2 ϕ ″ 0 ϕ ″ 0 ⋅ ( ϕ 1 ) ′ ′ 2 + 2 ⋅ G 2 ϕ ″ 0 f 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ f 1 + 2 ⋅ G 2 ϕ ″ 0 f ′ 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ ( f 1 ) ′ + G 2 ϕ ″ 0 ⋅ ( ϕ 2 ) ′ ′

G ″ 3 [ ϕ 0 ( η ) ] = G 3 β ″ 0 β ″ 0 ⋅ ( β 1 ) ′ ′ 2 + G 3 ϕ 0 ϕ 0 ⋅ ( ϕ 1 ) 2 + 2 G 3 β ″ 0 ϕ 0 ⋅ ( β 1 ) ′ ′ ⋅ ϕ 1 + G 3 ϕ 0 ⋅ ( ϕ 2 ) + G 3 β ″ 0 ⋅ ( β 2 ) ′ ′ (41)

G ‴ 1 [ f 0 ( η ) ] = G 1 f 0 f 0 f 0 ( f 1 ) 3 + 3 ⋅ G 1 f 0 f 0 f ′ 0 ( f 1 ) 2 . ( f 1 ) ′ + 3 ⋅ G 1 f 0 f ′ 0 f ′ 0 ⋅ ( f 1 ) ( f 1 ) ′ 2 + G 1 f ′ 0 f ′ 0 f ′ 0 ⋅ ( f 3 ) ′ 3 + 3 ⋅ G 1 f 0 f 0 ⋅ f 2 ⋅ f 1 + 3 ⋅ G 1 f 0 f ′ 0 ⋅ f 2 ⋅ ( f 1 ) ′ + G f 0 ⋅ f 3 + 3 ⋅ G 1 f ′ 0 f 0 ⋅ ( f 2 ) ′ ( f 1 ) + 3 ⋅ G 1 f ′ 0 f ′ 0 ⋅ ( f 2 ) ′ ( f 1 ) ′ + G 1 f ′ 0 ⋅ ( f 3 ) ′ ,

G ‴ 2 [ β 0 ( η ) ] = G 2 f 0 f 0 f 0 ( f 1 ) 3 + 3 ⋅ G 2 f 0 f 0 f ′ 0 ( f 1 ) 2 ⋅ ( f 1 ) ′ + 3 ⋅ G 2 f 0 f ′ 0 f ′ 0 ⋅ ( f 1 ) ( f 1 ) ′ 2 + G 2 f ′ 0 f ′ 0 f ′ 0 . ( f 1 ) ′ 3 + 3 ⋅ G 2 f 0 f 0 ⋅ f 2 ⋅ f 1 + 3 ⋅ G 2 f 0 f ′ 0 ⋅ f 2 ⋅ ( f 1 ) ′ + G 2 f ⋅ f 3 + 3 ⋅ G 2 f ′ 0 f 0 ⋅ ( f 2 ) ′ ( f 1 ) + 3 ⋅ G 2 f ′ 0 f ′ 0 ⋅ ( f 2 ) ′ ( f 1 ) ′ + G 2 f ′ 0 ⋅ ( f 3 ) ′ + G 2 ϕ ″ 0 ϕ ″ 0 ϕ ″ 0 ⋅ ( ϕ 1 ) ′ ′ 2 + 3 ⋅ G 2 ϕ ″ 0 ϕ ″ 0 f 0 ⋅ ( ϕ 1 ) ′ ′ 2 ⋅ ( f 1 ) + 3 ⋅ G 2 ϕ ″ 0 ϕ ″ 0 f ′ 0 ⋅ ( ϕ 1 ) ′ ′ 2 ⋅ ( f 1 ) ′ + 3 ⋅ G 2 ϕ ″ 0 ϕ ″ 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ ( ϕ 2 ) ′ ′

+ 3 ⋅ G ϕ ″ 0 f 0 ⋅ ( ϕ 2 ) ′ ′ ⋅ f 1 + 3 ⋅ G ϕ ″ 0 f ′ 0 ⋅ ( ϕ 2 ) ′ ′ ⋅ ( f 1 ) ′ + 4 ⋅ G 2 ϕ ″ 0 f 0 f ′ 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ f 1 ⋅ ( f 1 ) ′ + 2 ⋅ G ϕ ″ 0 f 0 f 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ ( f 1 ) 2 + 2 ⋅ G ϕ ″ 0 f ′ 0 f ′ 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ ( f 1 ) ′ 2 + 2 ⋅ G 2 ϕ ″ 0 f 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ f 2 + 2 ⋅ G 2 ϕ ″ 0 f ′ 0 ⋅ ( ϕ 1 ) ′ ′ ⋅ ( f 2 ) ′ + G 2 ϕ ″ 0 ⋅ ( ϕ 3 ) ′ ′ ,

G ‴ 3 [ ϕ 0 ( η ) ] = G 3 β ″ 0 β ″ 0 β ″ 0 ⋅ ( β 1 ) ′ ′ 3 + 3 ⋅ G 3 β ″ 0 β ″ 0 ϕ 0 ⋅ ( β 1 ) ′ ′ 2 ϕ 1 + 3 ⋅ G 3 β ″ 0 β ″ 0 ⋅ ( β 1 ) ′ ′ ⋅ ( β 2 ) ′ ′ + G 3 ϕ 0 ϕ 0 ϕ 0 ⋅ ( ϕ 1 ) 3 + 3 ⋅ G 3 ϕ 0 ϕ 0 β ″ 0 ⋅ ( ϕ 1 ) 2 ⋅ β ″ 1 + 3 ⋅ G 3 ϕ 0 ϕ 0 ⋅ ( ϕ 1 ) ϕ 2 + 3 ⋅ G 3 ϕ 0 β 0 ⋅ ( ϕ 2 ) ⋅ β 1 + G 3 ϕ 0 ⋅ ϕ 3 + 3 ⋅ G 3 β ″ 0 ϕ 0 ⋅ ( β 2 ) ′ ′ ⋅ ϕ 1 + G 3 β ″ 0 ⋅ ( β 3 ) ′ ′ . (42)

⋮

The extraction of the first derivatives of G can be represented as:

G 1 f 0 = − 2 α R e f ′ 0 ( η ) , G f 0 f 0 = 0, G f 0 f ′ 0 = − 2 α R e ,

G 1 f 0 f 0 f 0 = G 1 f 0 f ′ 0 f 0 = G 1 f 0 f ′ 0 f ′ 0 = G 1 f 0 f 0 f ′ 0 = 0 ,

G 1 f ′ 0 = − 2 α R e f 0 ( η ) − ( 4 − H a ) α 2 , G 1 f ′ 0 f 0 = − 2 α R e , G 1 f ′ 0 f ′ 0 = 0,

G 1 f ′ 0 f 0 f 0 = G 1 f ′ 0 f ′ 0 f 0 = G 1 f ′ 0 f 0 f ′ 0 = G 1 f ′ 0 f ′ 0 f ′ 0 = 0 , (43)

G 2 f 0 = − 8 E c P r α 2 f 0 ( η ) , G 2 f 0 f 0 = − 8 E c P r α 2 , G 2 f 0 f ′ 0 = 0,

G 2 f 0 f 0 f 0 = G 2 f 0 f ′ 0 f 0 = G 2 f 0 f ′ 0 f ′ 0 = G 2 f 0 f 0 f ′ 0 = 0 , G 2 ϕ ″ 0 = − D f P r ,

G 2 f ′ 0 = − 8 E c P r α 2 f ′ 0 ( η ) , G f ′ 0 f 0 = 0, G f ′ 0 f ′ 0 = − 8 E c P r α 2 ,

G f ′ 0 f 0 = G f ′ 0 f ′ 0 f 0 = G f ′ 0 f 0 f ′ 0 = G f ′ 0 f ′ 0 f ′ 0 = 0 , G 2 ϕ ″ 0 f 0 = 0 , G 2 ϕ ″ 0 ϕ ″ 0 = 0 (44)

G β ″ 0 = − S c S r , G β ″ 0 β ″ 0 = 0 , G β ″ 0 ϕ 0 = 0 , G ϕ 0 = S c γ α 2 ,

G ϕ 0 ϕ 0 = 0 , G ϕ 0 ϕ 0 ϕ 0 = G β ″ 0 ϕ 0 ϕ 0 = G ϕ 0 β ″ 0 ϕ 0 = G ϕ 0 ϕ 0 β ″ 0 = 0 , (45)

from Equation (33) by using Equations (39)-(42), gives the following,

f 0 = 1 2 A 3 η 2 + 1 ,

β 0 = B 1 ,

ϕ 0 = C 1 , (46)

f 1 = − 1 120 α R e A 3 2 η 6 − ( 1 12 α R e A 3 + 1 6 α 2 A 3 − 1 24 α 2 H a A 3 ) η 4 ,

β 1 = − 1 30 α 2 P r E c A 3 2 η 6 − 1 12 ( 4 A 3 α 2 + A 3 2 ) P r E c η 4 − 2 α 2 P r E c η 2 ,

ϕ 1 = 1 2 γ α 2 S c C 1 , (47)

f 2 = 1 10800 α 2 R e 2 A 3 3 η 10 + ( 1 280 α 3 R e A 3 2 − 1 1120 α 3 R e H a A 3 2 + 1 560 α 2 R e 2 A 3 2 ) η 8 + ( 1 180 α 2 R e 2 A 3 + 1 45 α 3 R e A 3 − 1 180 α 3 R e H a A 3 + 1 45 α 4 A 3 + 1 90 α 4 R e H a A 3 + 1 720 α 4 R e H a 2 A 3 ) η 6 ,

β 2 = 1 2700 α 2 R e 2 A 3 3 P r E c η 10 + 1 1680 α A 3 2 P r E c ( − 5 H a α 3 + 12 α 2 R e + 20 α 3 + 3 A 3 R e ) η 4 + 2 S r S c P − r E c α 2 η 2 ,

ϕ 2 = 1 30 α 2 P r E c S r S c A 3 2 η 6 + ( 1 12 P r E c S r S c ( 4 A 3 α 2 + A 3 2 ) + 1 24 S c 2 γ 2 α 4 C 1 ) η 4 , (48)

f 3 = − 1 1572480 α 3 R e 3 A 3 4 η 14 − ( 359 9979200 α 4 R e 2 A 3 3 − 359 39916800 α 4 R e 2 H a A 3 3 + 359 19958400 α 3 R e 3 A 3 3 ) η 12 + ( 29 226800 α 4 R e 2 H a A 3 2 + 29 113400 α 5 R e H a A 3 2 − 29 907200 α 5 R e H a 2 A 3 2 − 29 56700 α 4 R e 2 A 3 2 − 29 226800 α 3 R e 3 A 3 2 − 29 56700 α 5 R e A 3 2 ) η 10 − ( 1 10080 α 3 R e 3 A 3 + 1 1680 α 4 R e 2 A 3

− 1 6720 α 4 R e 2 H a A 3 + 1 840 α 5 R e A 3 − 1 1680 α 5 R e H a A 3 + 1 13440 α 5 R e H a 2 A 3 + 1 1260 α 6 A 3 − 1 1680 α 6 H a A 3 + 1 6720 α 6 H a 2 A 3 − 1 80640 α 6 H a 3 A 3 ) η 8 ,

β 3 = − 1 393120 P r E c α 4 R e 2 A 3 4 η 14 − 1 9979200 P r E c A 3 3 α 2 R e ( − 345 H a α 3 + 718 R e α 2 + 1380 α 3 + 259 A 3 R e ) η 12 + 1 453600 P r E c A 3 2 α 2 ( − 49 H a 2 α 4 + 214 H a R e α 3 + 392 H a α 4 + 120 A 3 H a R e α − 232 R e 2 α 2 − 856 R e α 3 − 784 α 4 − 240 A 3 R e 2 − 480 A 3 R e α ) η 10 − 1 20160 P r E c A 3 α 3 ( 2 H a 2 α 4

+ 13 A 3 H a 2 α 2 − 8 H a R e α 3 − 16 H a α 4 − 52 A 3 H a R e α − 104 A 3 H a α 2 + 8 R E 2 α 2 + 32 R e α 3 + 32 α 4 + 52 A 3 R E 2 + 208 A 3 R e α + 208 A 3 α 2 ) η 8 − 1 60 P r 2 D f S r S c E c α 2 A 3 2 η 6 − 1 24 P r D f ( S r S c P r E c A 3 ( 4 α 2 + A 3 ) + 1 2 S 2 γ 2 α 4 C 1 ) η 4 − P r 2 D f S r S c E c α 2 η 2

ϕ 3 = − 1 5400 S r S c P r E c α 3 R e A 3 3 η 10 + 1 16 ( − 1 210 S r S c P r E c α A 3 2 ( − 5 H a α 3 + 12 R e α 2 + 20 α 3 + 3 A 3 R e ) + a 1 210 S c 2 γ α 4 S r P r E c A 3 2 ) η 8 + 1 12 ( 1 15 S r S c P r E c α A 3 ( H a α 3 + A 3 H a α − 2 R e α 2 − 4 α 3 − 2 A 3 α ) + 1 5 S c γ α 2 ( 1 12 S r S c P r A 3 ( 4 α 2 + A 3 ) + 1 24 S c 2 γ 2 α 4 C 1 ) ) η 6 + 1 12 S c 2 γ S − r P r E c α 4 η 4 + 1 4 S r S c P r D f γ α 2 R e C 1 η 2 (49)

⋮

From step (4) substitution Equations (46)-(42) in Equation (34), the analytical-approximate solution can be resulted as follows:

f ( η ) = 1 + 1 2 A 3 η 2 − ( 1 12 α R e A 3 + 1 6 α 2 A 3 − 1 24 α 2 H a A 3 ) η 4 + ( 1 180 α 2 R e 2 A 3 − 1 120 α R e A 3 2 + 1 45 α 3 R e A 3 − 1 180 α 3 R e H a A 3 + 1 45 α 4 A 3 + 1 90 α 4 R e H a A 3 + 1 720 α 4 + R e H a 2 A 3 ) η 6 + ( 1 280 α 3 R e A 3 2 − 1 1120 α 3 R e H a A 3 2 + 1 560 α 2 R e 2 A 3 2 − ( 1 10080 α 3 R e 3 A 3 + 1 1680 α 4 R e 2 A 3 − 1 6720 α 4 R e 2 H a A 3

+ 1 840 α 5 R e A 3 − 1 1680 α 5 R e H a A 3 + 1 13440 α 5 R e H a 2 A 3 + 1 1260 α 6 A 3 − 1 1680 α 6 H a A 3 + 1 6720 α 6 H a 2 A 3 − 1 80640 α 6 H a 3 A 3 ) η 8 + 1 10800 α 2 R e 2 A 3 3 η 10 + ⋯

β ( η ) = B 1 + ( − 2 α 2 P r E c + 2 S r S c P r E c α 2 ) η 2 + ( − 1 12 ( 4 A 3 α 2 + A 3 2 ) P r E c + 1 1680 α A 3 2 P r E c ( − 5 H a α 3 + 12 α 2 R e + 20 α 3 + 3 A 3 R e ) ) η 4 − 1 30 α 2 P r E c A 3 2 η 6 + 1 2700 α 2 R e 2 A 3 3 P r E c η 10 + ⋯

ϕ ( η ) = C 1 + 1 2 γ α 2 S c C 1 η 2 + ( 1 12 P r E c S r S c ( 4 A 3 α 2 + A 3 2 ) + 1 24 l S c 2 γ 2 α 4 C 1 ) η 4 + 1 30 α 2 P r E c S r S c A 3 2 η 6 + ⋯ (50)

Here, the analysis of convergence for the analytical-approximate solution (50) that was resulted from the application of new power series algorithm for solving the problem has been extensively studied.

Definition (1): Suppose that H is Banach space, R is the real numbers and G [ F , H , P ] = ( G 1 [ F ] , G 2 [ H ] , G 2 [ P ] ) is a nonlinear operators defined by G [ F , H , P ] : H 3 → R 3 . Then the sequence of the solutions generated from a new algorithm can be written as

F n + 1 = G 1 [ F n ] , F n = ∑ k = 0 n f k , n = 0 , 1 , 2 , 3 , ⋯ H n + 1 = G 2 [ H n ] , H n = ∑ k = 0 n β k , n = 0 , 1 , 2 , 3 , ⋯ P n + 1 = G 3 [ P n ] , P n = ∑ k = 0 n ϕ k , n = 0 , 1 , 2 , 3 , ⋯ (51)

Definition (2): 1 Suppose that G [ F , H , P ] satisfies Lipschitz condition such that for 0 ≤ γ 1 , γ 2 , γ 3 < 1 , γ 1 , γ 2 , γ 3 ∈ R , we have

‖ G 1 [ F n ] − G 1 [ F n − 1 ] ‖ ≤ γ 1 ‖ F n − F n − 1 ‖ , ‖ G 2 [ H n ] − G 2 [ H n − 1 ] ‖ ≤ γ 2 ‖ H n − H n − 1 ‖ , ‖ G 3 [ P n ] − G 3 [ P n − 1 ] ‖ ≤ γ 3 ‖ P n − P n − 1 ‖ , (52)

Now, we assume that G [ F n , H n , P n ] = G ( n ) for simplify with γ = γ 1 + γ 2 + γ 3 , 0 ≤ γ < 1 yield,

‖ G ( n ) − G ( n − 1 ) ‖ ≤ γ ‖ ( F n , H n , P n ) − ( F n − 1 , H n − 1 , P n − 1 ) ‖ . (53)

The sufficient condition for convergent of the series analytical-approximate solutions F n , H n , P n is given in the following theorems.

Theorem (1): 2 The series of the analytical-approximate solution { S n = ( F n , H n , P n ) } 0 ∞ generated from new algorithm converge if the following condition is satisfied:

‖ S n − S m ‖ → 0 , as m → ∞ , for 0 ≤ γ < 1 , (54)

Proof. From the above definition, the next equation can be written as

‖ S n − S m ‖ = ‖ ( F n , H n , P n ) − ( F m , H m , P m ) ‖ = ‖ ( ∑ k = 0 n f k , ∑ k = 0 n β k , ∑ k = 0 n ϕ k ) − ( ∑ k = 0 m f k , ∑ k = 0 m β k , ∑ k = 0 m ϕ k ) ‖ = ‖ ( f 0 + L − 1 ∑ k = 0 n Δ k k ! d ( k ) G 1 [ f 0 ( η ) ] d z ( η ) , β 0 + L − 1 ∑ k = 0 n Δ k k ! d ( k ) G 2 [ β 0 ( η ) ] d η ( k ) , ϕ 0 + L − 1 ∑ k = 0 n Δ k k ! d ( k ) G 3 [ ϕ 0 ( η ) ] d η ( k ) − ( f 0 + L − 1 ∑ k = 0 m Δ k k ! d ( k ) G 1 [ f 0 ( η ) ] k ! d η ( k ) , β 0 + L − 1 ∑ k = 0 m Δ k k ! d ( k ) G 2 [ β 0 ( η ) ] k ! d η ( k ) , ϕ 0 + L − 1 ∑ k = 0 m Δ k k ! d ( k ) G 3 [ ϕ 0 ( η ) ] k ! d η ( k ) ) ‖

= ‖ L − 1 G [ ∑ k = 0 n − 1 f k , ∑ k = 0 n − 1 β k , ∑ k = 0 n − 1 ϕ ] k − L − 1 G [ ∑ k = 0 m − 1 f k , ∑ k = 0 m − 1 β k , ∑ k = 0 m − 1 ϕ k ] ‖ ≤ | L − 1 | ‖ G [ ∑ k = 0 n − 1 f k , ∑ k = 0 n − 1 β k , ∑ k = 0 n − 1 ϕ k ] − G [ ∑ k = 0 m − 1 f k , ∑ k = 0 m − 1 β k , ∑ k = 0 m − 1 ϕ k ] ‖ ≤ | L − 1 | ‖ G [ F n − 1 , H n − 1 , P n − 1 ] − G [ F m − 1 , H m − 1 , P m − 1 ] ‖ ≤ γ ‖ ( F n − 1 , H n − 1 , P n − 1 ) − ( F m − 1 , H m − 1 , P m − 1 ) ‖ = γ ‖ S n − 1 − S m − 1 ‖ , (55)

since G[F,H,P] satisfies Lipschitz condition. Let n = m + 1 , then

‖ F m + 1 − F m ‖ ≤ γ 1 ‖ F m − F m − 1 ‖ , ‖ H m + 1 − H m ‖ ≤ γ 2 ‖ H m − H m − 1 ‖ , ‖ P m + 1 − P m ‖ ≤ γ 3 ‖ P m − P m − 1 ‖ , (56)

hence,

‖ F m − F m − 1 ‖ ≤ γ 1 ‖ F m − 1 − F m − 2 ‖ ≤ ⋯ ≤ γ 1 m − 1 ‖ F 1 − F 0 ‖ , ‖ H m − H m − 1 ‖ ≤ γ 2 ‖ H m − 1 − H m − 2 ‖ ≤ ⋯ ≤ γ 2 m − 1 ‖ H 1 − H 0 ‖ , ‖ P m − P m − 1 ‖ ≤ γ 3 ‖ P m − 1 − P m − 2 ‖ ≤ ⋯ ≤ γ 3 m − 1 ≤ ‖ P 1 − P 0 ‖ , (57)

from Equation (57) we get

‖ F 2 − F 1 ‖ ≤ γ 1 ‖ F 1 − F 0 ‖ , ‖ H 2 − H 1 ‖ ≤ γ 2 ‖ H 1 − H 0 ‖ , ‖ P 2 − P 1 ‖ ≤ γ 3 ‖ P 1 − P 0 ‖ , ‖ F 3 − F 2 ‖ ≤ γ 1 2 ‖ F 1 − F 0 ‖ , ‖ H 3 − H 2 ‖ ≤ γ 2 2 ‖ H 1 − H 0 ‖ , ‖ F 3 − F 2 ‖ ≤ γ 3 2 ‖ F 1 − F 0 ‖ , ‖ F 4 − F 3 ‖ ≤ γ 1 3 ‖ F 1 − F 0 ‖ , ‖ H 4 − H 3 ‖ ≤ γ 2 3 ‖ H 1 − H 0 ‖ , ‖ P 4 − P 3 ‖ ≤ γ 3 3 ‖ P 1 − P 0 ‖ ⋮ ‖ F m − F m − 1 ‖ ≤ γ 1 m − 1 ‖ F 1 − F 0 ‖ , ‖ H m − H m − 1 ‖ ≤ γ 2 m − 1 ≤ ‖ H 1 − H 0 ‖ , ‖ P m − P m − 1 ‖ ≤ γ 3 m − 1 ‖ P 1 − P 0 ‖ (58)

By using triangle inequality, we find that as m → ∞ , we have ‖ S n − S m ‖ → 0 , then S n is a Cauchy sequence in Banach space H^{3}.

Theorem (2): 3 Let G = ( G 1 , G 2 , G 3 ) be a nonlinear operator satisfies Lipschitz condition from H^{3} to H^{3}. If the series analytical-approximate solution { S n } converges, then it is converged to the solution of the problem (9)-(10).

Proof.

‖ G [ S 2 ] − G [ S 1 ] ‖ = ‖ ( G 1 [ F 2 ] , G 2 [ H 2 ] , G 3 [ P 2 ] ) − ( G 1 [ F 1 ] , G 2 [ H 1 ] , G 3 [ P 1 ] ) ‖ = ‖ G 1 [ F 2 ] − G 1 [ F 1 ] , G 2 [ H 2 ] − G 2 [ H 1 ] , G 3 [ P 2 ] − G 3 [ P 1 ] ‖ = ‖ G 1 [ F 2 ] − G 1 [ F 1 ] ‖ + ‖ G 2 [ H 2 ] − G 2 [ H 1 ] ‖ + ‖ G 3 [ P 2 ] − G 3 [ P 1 ] ‖

≤ γ 1 ‖ F 2 − F 1 ‖ + γ 2 ‖ H 2 − H 1 ‖ + γ 3 ‖ P 2 − P 1 ‖ ≤ ( γ 1 + γ 2 + γ 3 ) ‖ ( F 2 , H 2 , P 2 ) − ( F 1 , H 1 , P 1 ) ‖ = γ ‖ S 2 − S 1 ‖

Therefore, from the Banach fixed-point theorem, there is a unique solution of the problem (9)-(10). We will prove that { S n } 0 ∞ converges to S.

G [ S ] = G [ ∑ k = 0 ∞ S k ] = lim n → ∞ G [ ∑ k = 0 n S k ] = lim n → ∞ G [ S n ] = lim n → ∞ S n + 1 = S .

In practice, the theorems (1) and (2) suggest to compute the value of γ 1 , γ 2 , γ 3 , as described in the following definition.

Definition (1): 4 for k = 1 , 2 , 3 , ⋯

γ 1 k = ( ‖ F k + 1 − F k ‖ ‖ F 1 − F 0 ‖ = ‖ f k + 1 ‖ ‖ f 1 ‖ , ‖ f 1 ‖ ≠ 0, 0, ‖ f 1 ‖ = 0,

γ 2 k = ( ‖ H k + 1 − H k ‖ ‖ H 1 − H 0 ‖ = ‖ β k + 1 ‖ ‖ β 1 ‖ , ‖ β 1 ‖ ≠ 0 , 0 , ‖ β 1 ‖ = 0 ,

γ 3 k = ( ‖ P k + 1 − P k ‖ ‖ P 1 − P 0 ‖ = ‖ ϕ k + 1 ‖ ‖ ϕ 1 ‖ , ‖ ϕ 1 ‖ ≠ 0 , 0 , ‖ ϕ 1 ‖ = 0. (59)

Now, the definition (1) can be applied on the magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates to find convergence, then to obtain for examples as below.

If we choose R e = 30 , α = 3 ∘ , H a = 500 , S r = S c = 0.1 , P r = 0.1 , D f = 0.01 , E c = 0.01 , γ = 0.04 then obtain:

‖ F 2 − F 1 ‖ 2 ≤ γ 1 ‖ F 1 − F 0 ‖ 2 ⇒ γ 1 = 0.08954317 < 1 , ‖ F 3 − F 2 ‖ 2 ≤ γ 1 2 ‖ F 1 − F 0 ‖ 2 ⇒ γ 1 2 = 0.005099704 < 1 , ‖ F 4 − F 3 ‖ 2 ≤ γ 1 3 ‖ F 1 − F 0 ‖ 2 ⇒ γ 1 3 = 0.0001956583630 < 1 , ⋮

‖ H 2 − H 1 ‖ 2 ≤ γ 2 ‖ H 1 − H 0 ‖ 2 ⇒ γ 2 = 0.2496673982 < 1 , ‖ H 3 − H 2 ‖ 2 ≤ γ 2 2 ‖ H 1 − H 0 ‖ 2 ⇒ γ 2 2 = 0.03173895000 < 1 , ‖ H 4 − H 3 ‖ 2 ≤ γ 2 3 ‖ H 1 − H 0 ‖ 2 ⇒ γ 2 3 = 0.002169482685 < 1 , ⋮

‖ P 2 − P 1 ‖ 2 ≤ γ 3 ‖ P 1 − P 0 ‖ 2 ⇒ γ 3 = 0.73232384 < 1 , ‖ P 3 − P 2 ‖ 2 ≤ γ 3 2 ‖ P 1 − P 0 ‖ 2 ⇒ γ 3 2 = 0.09141845239 < 1 , ‖ P 4 − P 3 ‖ 2 ≤ γ 3 3 ‖ P 1 − P 0 ‖ 2 ⇒ γ 3 3 = 0.003873822310 < 1 , ⋮

‖ F 2 − F 1 ‖ + ∞ ≤ γ ‖ F 1 − F 0 ‖ + ∞ ⇒ γ = 0.07399291 < 1 , ‖ F 3 − F 2 ‖ + ∞ ≤ γ 2 ‖ F 1 − F 0 ‖ + ∞ ⇒ γ 2 = 0.004719598 < 1 , ‖ F 4 − F 3 ‖ + ∞ ≤ γ 3 ‖ F 1 − F 0 ‖ + ∞ ⇒ γ 3 = 0.0001557525428 < 1 , ⋮

‖ H 2 − H 1 ‖ + ∞ ≤ γ ‖ H 1 − H 0 ‖ + ∞ ⇒ γ = 0.2384340096 < 1 , ‖ H 3 − H 2 ‖ + ∞ ≤ γ 2 ‖ H 1 − H 0 ‖ + ∞ ⇒ γ 2 = 0.02467876 < 1 , ‖ H 4 − H 3 ‖ + ∞ ≤ γ 3 ‖ H 1 − H 0 ‖ + ∞ ⇒ γ 3 = 0.001715583134 < 1 , ⋮

‖ P 2 − P 1 ‖ + ∞ ≤ γ ‖ P 1 − P 0 ‖ + ∞ ⇒ γ = 0.732255116 < 1 , ‖ P 3 − P 2 ‖ + ∞ ≤ γ 2 ‖ P 1 − P 0 ‖ + ∞ ⇒ γ 2 = 0.087297019 < 1 , ‖ P 4 − P 3 ‖ + ∞ ≤ γ 3 ‖ P 1 − P 0 ‖ + ∞ ⇒ γ 3 = 0.003011824761 < 1 , ⋮

Also, if we get α = − 2 ∘ , R e = 10 , H a = 110 , P r = D f = 0.2 , γ = 0.4 , E c = 0.01 .

‖ F 2 − F 1 ‖ 2 ≤ γ 1 ‖ F 1 − F 0 ‖ 2 ⇒ γ 1 = 0.03060256592 < 1 , ‖ F 3 − F 2 ‖ 2 ≤ γ 1 2 ‖ F 1 − F 0 ‖ 2 ⇒ γ 1 2 = 0.1113608591 < 1 , ‖ F 4 − F 3 ‖ 2 ≤ γ 1 3 ‖ F 1 − F 0 ‖ 2 ⇒ γ 1 3 = 0.4872382121 < 1 , ⋮

‖ H 2 − H 1 ‖ 2 ≤ γ 2 ‖ H 1 − H 0 ‖ 2 ⇒ γ 2 = 0.0004683459276 < 1 , ‖ H 3 − H 2 ‖ 2 ≤ γ 2 2 ‖ H 1 − H 0 ‖ 2 ⇒ γ 2 2 = 0.005594660398 < 1 , ‖ H 4 − H 3 ‖ 2 ≤ γ 2 3 ‖ H 1 − H 0 ‖ 2 ⇒ γ 2 3 = 0.02712979918 < 1 , ⋮

‖ P 2 − P 1 ‖ 2 ≤ γ 3 ‖ P 1 − P 0 ‖ 2 ⇒ γ 3 = 0.000005903459683 < 1 , ‖ P 3 − P 2 ‖ 2 ≤ γ 3 2 ‖ P 1 − P 0 ‖ 2 ⇒ γ 3 2 = 0.0001292820508 < 1 , ‖ P 4 − P 3 ‖ 2 ≤ γ 3 3 ‖ P 1 − P 0 ‖ 2 ⇒ γ 3 3 = 0.0004542885489 < 1 , ⋮

‖ F 2 − F 1 ‖ + ∞ ≤ γ 1 ‖ F 1 − F 0 ‖ + ∞ ⇒ γ 1 = 0.02756015475 < 1 , ‖ F 3 − F 2 ‖ + ∞ ≤ γ 1 2 ‖ F 1 − F 0 ‖ + ∞ ⇒ γ 1 2 = 0.1104543115 < 1 , ‖ F 4 − F 3 ‖ + ∞ ≤ γ 1 3 ‖ F 1 − F 0 ‖ + ∞ ⇒ γ 1 3 = 0.4872217326 < 1 , ⋮

‖ H 2 − H 1 ‖ + ∞ ≤ γ 2 ‖ H 1 − H 0 ‖ + ∞ ⇒ γ 2 = 0.00041798914 < 1 , ‖ H 3 − H 2 ‖ + ∞ ≤ γ 2 2 ‖ H 1 − H 0 ‖ + ∞ ⇒ γ 2 2 = 0.005302953669 < 1 , ‖ H 4 − H 3 ‖ + ∞ ≤ γ 2 3 ‖ H 1 − H 0 ‖ + ∞ ⇒ γ 2 3 = 0.02690804181 < 1 , ⋮

‖ P 2 − P 1 ‖ + ∞ ≤ γ 3 ‖ P 1 − P 0 ‖ + ∞ ⇒ γ 3 = 0.00000534538 < 1 , ‖ P 3 − P 2 ‖ + ∞ ≤ γ 3 2 ‖ P 1 − P 0 ‖ + ∞ ⇒ γ 3 2 = 0.00010743626 < 1 , ‖ P 4 − P 3 ‖ + ∞ ≤ γ 3 3 ‖ P 1 − P 0 ‖ + ∞ ⇒ γ 3 3 = 0.00043058757 < 1 , ⋮

Then ∑ k = 0 ∞ f k ( η ) , ∑ k = 0 ∞ β k ( η ) and ∑ k = 0 ∞ ϕ k ( η ) converge to the solutions f ( η ) , β ( η ) and ϕ ( η ) respectively when 0 ≤ γ 1 k , γ 2 k , γ 3 k < 1 , k = 1 , 2 , ⋯ .

This section is dedicated to study the influence of various non dimensional physical parameters on velocity field f ( η ) , temperature field β ( η ) and concentration field ϕ ( η ) . Also the influence of different parameters on rate of heat transfer and rate of mass transfer are under observation for diverging and converging channels. In

Approximation | A 3 | B 1 | C 1 |
---|---|---|---|

1 term | −2.1999729 | 1.00040723 | 0.99999452 |

2 term | −2.1998869 | 1.00034130 | 0.99999045 |

3 term | −2.1993584 | 1.00034466 | 0.99999078 |

4 term | −2.1993648 | 1.00034465 | 0.99999077 |

5 term | −2.1993648 | 1.00034465 | 0.99999077 |

6 term | −2.1993648 | 1.00034465 | 0.99999077 |

7 term | −2.1993648 | 1.00034465 | 0.99999077 |

8 term | −2.1993648 | 1.00034465 | 0.99999077 |

Approximation | A 3 | B 1 | C 1 |
---|---|---|---|

1 term | −1.89052474 | 1.00059930 | 0.99995131 |

2 term | −1.88870911 | 1.00065724 | 0.99992738 |

3 term | −1.88872478 | 1.00065993 | 0.99992621 |

4 term | −1.88872513 | 1.00065996 | 0.99992618 |

5 term | −1.88872513 | 1.00065996 | 0.99992618 |

6 term | −1.88872513 | 1.00065996 | 0.99992618 |

7 term | −1.88872513 | 1.00065996 | 0.99992618 |

η | f ( η ) | ( R − K 4 ) | β ( η ) | ( R − K 4 ) | ϕ ( η ) | η |
---|---|---|---|---|---|---|

0.0 | 1.0000000 | 1.0000000 | 1.00034465 | 1.00034465 | 0.99999077 | 0.99999077 |

0.1 | 0.9890195 | 0.9890195 | 1.0003445 | 1.0003445 | 0.9999908 | 0.9999908 |

0.2 | 0.9562698 | 0.9562698 | 1.0003437 | 1.0003437 | 0.9999910 | 0.9999910 |

0.3 | 0.9022985 | 0.9022985 | 1.0003409 | 1.0003409 | 0.9999913 | 0.9999913 |
---|---|---|---|---|---|---|

0.4 | 0.8279407 | 0.8279407 | 1.0003338 | 1.0003338 | 0.9999917 | 0.9999917 |

0.5 | 0.7341800 | 0.7341800 | 1.0003195 | 1.0003195 | 0.9999924 | 0.9999924 |

0.6 | 0.6220550 | 0.6220550 | 1.0002170 | 1.0002170 | 0.9999932 | 0.9999932 |

0.7 | 0.4923839 | 0.4923839 | 1.0001958 | 1.0001958 | 0.9999944 | 0.9999944 |

0.8 | 0.3456799 | 0.3456799 | 1.0001958 | 1.0001958 | 0.9999958 | 0.9999958 |

0.9 | 0.1818466 | 0.1818466 | 1.0011290 | 1.0011290 | 0.9999977 | 0.9999977 |

1.0 | 0.0000000 | 0.0000000 | 1.0000000 | 1.0000000 | 1.0000000 | 1.0000000 |

η | f ( η ) | ( R − K 4 ) | β ( η ) | ( R − K 4 ) | ϕ ( η ) | ( R − K 4 ) |
---|---|---|---|---|---|---|

0.0 | 1.0000000 | 1.0000000 | 1.0003447 | 1.0003447 | 0.9999908 | 0.9999908 |

0.1 | 0.9905498 | 0.9905498 | 1.0006599 | 1.0006599 | 0.9999262 | 0.9999262 |

0.2 | 0.9621219 | 0.9621219 | 1.0006587 | 1.0006587 | 0.9999282 | 0.9999282 |

0.3 | 0.9144866 | 0.9144866 | 1.0006544 | 1.0006544 | 0.9999308 | 0.9999308 |

0.4 | 0.8472720 | 0.8472720 | 1.0006434 | 1.0006434 | 0.9999346 | 0.9999346 |

0.5 | 0.7599801 | 0.7599801 | 1.0006202 | 1.0006202 | 0.9999399 | 0.9999399 |

0.6 | 0.6520119 | 0.6520119 | 1.0005776 | 1.0005776 | 0.9999469 | 0.9999469 |

0.7 | 0.52269974 | 0.52269974 | 1.0005067 | 1.0005067 | 0.9999560 | 0.9999560 |

0.8 | 0.3713515 | 0.3713515 | 1.0003962 | 1.0003962 | 0.9999675 | 0.9999675 |

0.9 | 0.1973059 | 0.1973059 | 1.0002326 | 1.0002326 | 0.9999820 | 0.9999820 |

1.0 | 0.0000000 | 0.0000000 | 1.0000000 | 1.0000000 | 1.0000000 | 1.0000000 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

R e | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0 | 0.015725 | −0.025891 | 0.015725 | −0.025890 |

10 | 0.015501 | −0.023525 | 0.016295 | −0.028446 |

20 | 0.015403 | −0.021341 | 0.017951 | −0.031205 |

30 | 0.015357 | −0.019331 | 0.023679 | −0.034198 |

40 | 0.015334 | −0.017490 | 0.047894 | −0.037470 |

50 | 0.015300 | −0.015814 | 0.166257 | −0.041098 |

α | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |
---|---|---|---|---|---|

0˚ | 0.670000 | −0.133889 | 0˚ | 0.670000 | −0.133889 |

−2˚ | 0.698657 | −0.120838 | 2˚ | 0.645605 | −0.146431 |

−4˚ | 0.728034 | −0.107761 | 4˚ | 0.628332 | −0.158178 |

−6˚ | 0.754221 | −0.095311 | 6˚ | 0.620146 | −0.168901 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

H a | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0 | 0.733702 | −0.106982 | 0.601853 | −0.171361 |

200 | 0.747908 | −0.096138 | 0.642952 | −0.156091 |

400 | 0.754835 | −0.086380 | 0.678122 | −0.141432 |

600 | 0.755444 | −0.077668 | 0.706215 | −0.127721 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

S c | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0.5 | 0.744752 | −0.101423 | 0.630359 | −0.163681 |

1.0 | 0.749722 | −0.203428 | 0.642586 | −0.329127 |

1.5 | 0.754690 | −0.306011 | 0.654809 | −0.496338 |

2.0 | 0.759656 | −0.409171 | 0.667028 | −0.665313 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

S r | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0.5 | 0.744695 | −0.050600 | 0.630303 | −0.081579 |

1.0 | 0.749573 | −0.101190 | 0.642440 | −0.163447 |

1.5 | 0.754450 | −0.151922 | 0.654576 | −0.245755 |

2.0 | 0.759327 | −0.202797 | 0.666713 | −0.328503 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

P r | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0.5 | 0.741769 | −0.020314 | 0.623021 | −0.032670 |

1.0 | 1.487440 | −0.040499 | 1.255751 | −0.065259 |

1.5 | 2.237014 | −0.060706 | 1.898192 | −0.097917 |

2.0 | 2.990488 | −0.080937 | 2.550341 | −0.130647 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

E c | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0.5 | 0.370903 | −0.010234 | 1.059363 | −0.016412 |

1.0 | 0.741769 | −0.020314 | 2.118688 | −0.032670 |

1.5 | 1.112635 | −0.030395 | 3.178012 | −0.048929 |

2.0 | 1.483501 | −0.040476 | 4.237338 | −0.065188 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

D f | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0.5 | 0.741769 | −0.020314 | 0.623021 | −0.032670 |

1.0 | 0.743759 | −0.020339 | 0.627914 | −0.032706 |

1.5 | 0.745747 | −0.020338 | 0.632806 | −0.032742 |

2.0 | 0.747736 | −0.020350 | 0.637691 | −0.032778 |

α = − 5 ∘ | α = 5 ∘ | |||
---|---|---|---|---|

γ | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) | α N u = β ′ ( 1 ) | α S h = ϕ ′ ( 1 ) |

0.2 | 0.741807 | −0.020467 | 0.623059 | −0.032821 |

0.4 | 0.741883 | −0.020771 | 0.623134 | −0.033125 |

0.6 | 0.741959 | −0.021075 | 0.623201 | −0.033429 |

0.8 | 0.742035 | −0.213797 | 0.623285 | −0.033732 |

In addition to this section highlights the major outcomes of the analytical study presented by new algorithm. The analysis of the variations in temperature and concentration profiles for different parameters is prepared. For that purpose, Figures are plotted for varying several parameters. Moreover we have been divided this section into two subsections follow as.

・ Channel divergent ( α > 0 ).

In Figures 2-9 are plotted to show the behavior curves of velocity, temperature and concentration profiles under the impact of different physical parameters. An increasing the opening angle α gives variations in velocity, temperature and concentration profiles as displayed in

・ Channel convergent ( α < 0 ).

For the converging channel, the variations in velocity, temperature and concentration profile due to the varying parameters are depicted in Figures 10-17, the behavior of velocity and temperature for changing angle opening α and Reynolds number R e is quite opposite to the behavior of f ( η ) , β ( η ) and ϕ ( η ) in diverging channel as seen in

Prandtl and Dufour numbers on the temperature profile is similar for the effect in diverging channel. Also these Figures demonstrated the concentration profile possess same effect when there are changing in Hartmann number, Schimdt number, Soret number and chemical reaction parameter. Physical explanations can be provided the temperature profile show that the temperature at the central region increases with increasing angle opening. This can be attributed to that for fixed Reynold number, increasing angle opening leads to increase the cross-sectional flow area. This in turn leads to decrease the flow velocity and this mean the flow will be decelerated. Therefore, the heat dissipation will be reduced which leads to increase the temperature of the fluid. The inertia force of the fluid increases with increasing Reynold number which leads to enhance the parabolic behavior (increasing central temperature) for diverging channel with the opposite view in converging channel. Hartmman number increase in this case Lorentz force is also increasing for diverging and converging channels. This force imports extra drag to the flow. Therefore the temperature profile becomes more flat which means decreasing the temperature within the central region. The thin boundary layers that are near to the wall lead to that the temperature gradient at the highest level. Furthermore to the existence of the thick boundary layer in central region lead to that the temperature gradient at low level. Eckert number increases with increased temperature and thus produces an increase in Kinetic energy. The change of the temperature profile with Prandtl number, and the increase of temperature with Prandtl number result from increasing of the momentum diffusivity. The Dufour number shows to increase shows less effect on temperature, increasing Dufour leads to increase the thermal energy of the fluid thus the temperature increase. The rate of most chemical reactions increases with a decrease the concentration of reactants. As for temperature, it increases if a reaction is heat-emitting and decreases when the reaction absorbs heat.

In this paper, the unsteady and two-dimensional magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates is studied analytically using a new algorithm. The solution obtained by new algorithm is an infinite power series for appropriate initial approximation. The construction of this algorithm possessed good convergent series and the convergence of the results is explicitly shown. Graphical results and tables are presented to investigate the influence of physical parameters on velocity, temperature and concentration. Analysis of the converge confirms that the new algorithm is an efficient technique as compared to Range-Kutta algorithm with help of Shooting algorithm. The new algorithm that is widely applied to solve ordinary differential equations lead to the solutions resulting from this algorithm is compatible with numerical solution. Effects of different parameters on temperature and concentration profiles are analyzed and presented graphically. The conclusions can be drawn from the analysis presented:

・ The behavior of temperature and concentration profiles are the same results α , R e , E c , P r and D f for diverging channel.

・ Hartmann number H a can be used to reduce the temperature of the flow fluid. Also, concentration of the fluid can also be controlled by employing a strong magnetic field.

・ For converging channel, the variations in temperature are opposite for diverging channel with an increase in channel opening α and R e .

・ For diverging channel, Nusselt number drops with a rise in angle opening and increases with a rise in Reynolds number and behaves oppositely for convergent channel.

・ Increase in heat transfer rate is observed for increasing P r , E c , S r , S c , D f and γ in both channels.

・ Increase in Reynolds number and Angle opening gives a drop to mass transfer rate for diverging channel and a rise for the converging channel.

・ The rate of mass transfer decreased for both channels with an increase in Schmidt, Soret, Prandtl, Eckert, Dufour numbers and chemical reaction parameter.

・ Results obtained by new algorithm are in excellent agreement with numerical solution obtained.

The authors declare no conflicts of interest regarding the publication of this paper.

Al-Saif, A.-S.J.A. and Jasim, A.M. (2019) New Analytical Study of the Effects Thermo-Diffusion, Diffusion-Thermo and Chemical Reaction of Viscous Fluid on Magneto Hydrodynamics Flow in Divergent and Convergent Channels. Applied Mathematics, 10, 268-300. https://doi.org/10.4236/am.2019.104020