### 4.1 Solution of momentum equation

The momentum boundary layer equation is partially decoupled from the energy and species equations. Integrating Equation (11) with *η* once over to the interval [0,\eta ], we obtain

{f}_{\eta \eta}+f\phantom{\rule{0.2em}{0ex}}{f}_{\eta}=-s+{\int}_{0}^{\eta}[3{f}_{\eta}^{2}+M{f}_{\eta}]\phantom{\rule{0.2em}{0ex}}d\eta ,

(26)

where s=-{f}_{\eta \eta}(0). Letting \eta \to \mathrm{\infty}, we obtain

s={\int}_{0}^{\mathrm{\infty}}[3{f}_{\eta}^{2}+M{f}_{\eta}]\phantom{\rule{0.2em}{0ex}}d\eta .

(27)

Integrating Equation (26) once again, we obtain

{f}_{\eta}+\frac{1}{2}{f}_{\eta}^{2}=1-s\eta +{\int}_{0}^{\eta}\left[{\int}_{0}^{{\eta}_{1}}[3{f}_{{\eta}_{2}}^{2}+M{f}_{{\eta}_{2}}]\phantom{\rule{0.2em}{0ex}}d{\eta}_{2}\right]\phantom{\rule{0.2em}{0ex}}d{\eta}_{1}.

(28)

The solution procedure of Equation (28) can be reduced to the sequential solutions of the Riccati type equation of the form

{f}_{\eta}^{(n)}+\frac{1}{2}{{f}^{(n)}}^{2}=\mathit{RHS}\left[{f}_{\eta}^{(n-1)}\right].

(29)

This iteration algorithm has to be solved by substituting suitable zero-order approximations {f}_{\eta}^{(0)}(\eta ) for {f}_{\eta}(\eta ) into the right-hand side of Equation (28). We assume a zero-order approximation as

{f}_{\eta}^{(0)}(\eta )={e}^{-{s}_{0}\eta},

(30)

which satisfies the condition at infinity. Integrating Equation (30) with respect to *η* and using the condition {f}_{\eta}^{(0)}(0)=0, we get

{f}_{\eta}^{(0)}(\eta )=\frac{1-{e}^{-{s}_{0}\eta}}{{s}_{0}}.

(31)

Using the above solution in Equation (27), the approximate value of *s* can be obtained as

{s}_{0}=\sqrt{\frac{3}{2}+M},\phantom{\rule{2em}{0ex}}{f}_{\eta \eta}^{(0)}(0)=-{s}_{0}.

(32)

Now substituting all the derivatives of zero-order approximation {f}_{\eta}^{(0)}(\eta ) into the right-hand side of Equation (28), we obtain the equation for first-order iteration {f}_{\eta}^{(1)} as follows:

{f}_{\eta}^{(1)}+\frac{1}{2}{{f}^{(1)}}^{2}=1+\frac{3}{4{s}_{0}^{2}}[{e}^{-2{s}_{0}\eta}-1]+\frac{M}{{s}_{0}^{2}}[{e}^{-{s}_{0}\eta}-1].

(33)

Further, we assume that the first-order iterate of *f* satisfies the boundary conditions on *f* as given in (14). The above non-linear Riccati type equation can be solved in terms of a confluent hypergeometric Whittaker function as discussed by Khan [19]. However, we restrict ourselves to the zero-order solution, and similarly, to heat and mass transport equations.

### 4.2 Solution of heat transfer equation

Using the zero-order approximations of *f* and {f}_{\eta} and further introducing a new variable

\xi =-\frac{\mathit{Pr}}{{s}_{0}^{2}}{e}^{-{s}_{0}\eta},

(34)

Equation (12) and the thermal boundary conditions (15) take the form

where {\mathit{Pr}}^{\ast}=\mathit{Pr}/{s}_{0}^{2} is the modified Prandtl number. The solution of Equation (35) is assumed in the form of

\theta (\xi )={\theta}_{c}(\xi )+{\theta}_{p}(\xi ),

where {\theta}_{c}(\xi ) is the complementary solution and {\theta}_{p}(\xi ) is the particular solution. The complementary solution of Equation (35) is obtained in terms of confluent hypergeometric function in the following form:

{\theta}_{c}(\xi )={C}_{0}{\xi}^{\alpha}M[\alpha -4,\alpha +1,\frac{-\xi}{1+\frac{4}{3}K}],

(37)

where

M[a,b,z]=\sum _{r=0}^{\mathrm{\infty}}\frac{a(a+1)\cdots (a+r-1)}{b(b+1)\cdots (b+r-1)}\frac{z}{r!}

is Kummer’s function (see Abramowitz and Stegun [39]) and

\alpha =\frac{{\mathit{Pr}}^{\ast}}{1+\frac{4}{3}K}.

The particular solution is obtained as

{\theta}_{p}(\xi )={a}_{0}{\xi}^{2}+{a}_{1}{\xi}^{3}+{a}_{2}{\xi}^{4},

(38)

where

Now, the complete solution can be written as

\theta (\xi )={\theta}_{c}(\xi )+{\theta}_{p}(\xi ).

(39)

Making use of the boundary conditions (36) and rewriting the solution in terms of the variable *η*, we get

\begin{array}{rcl}\theta (\eta )& =& {C}_{1}\frac{{e}^{-{s}_{0}\eta \alpha}M[\alpha -4,\alpha +1,-\alpha {e}^{-{s}_{0}\eta}]}{M[\alpha -4,\alpha +1,-\alpha ]}+{a}_{0}{{\mathit{Pr}}^{\ast}}^{2}{e}^{-2{s}_{0}\eta}\\ -{a}_{1}{{\mathit{Pr}}^{\ast}}^{3}{e}^{-3{s}_{0}\eta}+{a}_{2}{{\mathit{Pr}}^{\ast}}^{4}{e}^{-4{s}_{0}\eta},\end{array}

(40)

where

{C}_{1}=1-{a}_{0}{{\mathit{Pr}}^{\ast}}^{2}+{a}_{1}{{\mathit{Pr}}^{\ast}}^{3}-{a}_{2}{{\mathit{Pr}}^{\ast}}^{4}.

### 4.3 Solution of mass transfer equation

Using the zero-order approximation of *f* and {f}_{\eta} and further introducing a new variable

\zeta =-\frac{\mathit{Sc}}{{s}_{0}^{2}}{e}^{-{s}_{0}\eta},

(41)

Equation (13) and the thermal boundary conditions in (16) take the form

where {\mathit{Sc}}^{\ast}=\mathit{Sc}/{s}_{0}^{2} is the modified Schmidt number. Following the solution procedure discussed in the case of the energy equation, the solution of Equation (42) is obtained in terms of confluent hypergeometric function as

\varphi (\eta )=\frac{{e}^{-{s}_{0}{\mathit{Sc}}^{\ast}\eta}M[{\mathit{Sc}}^{\ast}-4,{\mathit{Sc}}^{\ast}+1,-{\mathit{Sc}}^{\ast}{e}^{-{s}_{0}\eta}]}{M[{\mathit{Sc}}^{\ast}-4,{\mathit{Sc}}^{\ast}+1,-{\mathit{Sc}}^{\ast}]}.

(44)