The set of equations (5)-(7) subject to the initial and boundary conditions (9a)-(9e) were solved analytically using Laplace transforms. The exact solutions for species concentration C(\eta ,t), fluid temperature T(\eta ,t) and fluid velocity u(\eta ,t) are, respectively,

C(\eta ,t)=erfc\left(\frac{\eta}{2}\sqrt{\frac{a}{t}}\right),

(10)

T(\eta ,t)={P}_{1}(\eta ,t)-H(t-1){P}_{1}(\eta ,t-1),

(11)

u(\eta ,t)=P(\eta ,t)+{\alpha}_{1}[{P}_{2}(\eta ,t)-H(t-1){P}_{2}(\eta ,t-1)]+{\alpha}_{2}{P}_{3}(\eta ,t),

(12)

where

\begin{array}{c}P(\eta ,t)={\mathcal{L}}^{-1}[\overline{f}(s){e}^{-\eta \sqrt{s+M}}],\hfill \\ {P}_{1}(\eta ,t)=(\frac{t}{2}+\frac{c\eta}{4\sqrt{d}}){e}^{\eta \sqrt{d}}erfc({\mathrm{t}}_{1})+(\frac{t}{2}-\frac{c\eta}{4\sqrt{d}}){e}^{-\eta \sqrt{d}}erfc({\mathrm{t}}_{2}),\hfill \\ {P}_{2}(\eta ,t)=(\frac{\alpha t-1}{2{\alpha}^{2}}+\frac{c\eta}{4\alpha \sqrt{d}}){e}^{\eta \sqrt{d}}erfc({\mathrm{t}}_{1})+(\frac{\alpha t-1}{2{\alpha}^{2}}-\frac{c\eta}{4\alpha \sqrt{d}}){e}^{-\eta \sqrt{d}}erfc({\mathrm{t}}_{2})\hfill \\ \phantom{{P}_{2}(\eta ,t)=}+\frac{{e}^{-\alpha t}}{2{\alpha}^{2}}[{e}^{\eta \sqrt{d-\alpha c}}erfc({\mathrm{t}}_{3})+{e}^{-\eta \sqrt{d-\alpha c}}erfc({\mathrm{t}}_{4})]\hfill \\ \phantom{{P}_{2}(\eta ,t)=}-(\frac{\alpha t-1}{2{\alpha}^{2}}+\frac{\eta}{4\alpha \sqrt{M}}){e}^{\eta \sqrt{M}}erfc({\mathrm{t}}_{5})-(\frac{\alpha t-1}{2{\alpha}^{2}}-\frac{\eta}{4\alpha \sqrt{M}}){e}^{-\eta \sqrt{M}}erfc({\mathrm{t}}_{6})\hfill \\ \phantom{{P}_{2}(\eta ,t)=}-\frac{{e}^{-\alpha t}}{2{\alpha}^{2}}[{e}^{\eta \sqrt{M-\alpha}}erfc({\mathrm{t}}_{7})+{e}^{-\eta \sqrt{M-\alpha}}erfc({\mathrm{t}}_{8})],\hfill \\ {P}_{3}(\eta ,t)=\frac{1}{\beta}erfc({\mathrm{t}}_{9})-\frac{{e}^{-\beta t}}{2\beta}[{e}^{\eta \sqrt{-a\beta}}erfc({\mathrm{t}}_{10})+{e}^{-\eta \sqrt{-a\beta}}erfc({\mathrm{t}}_{11})]\hfill \\ \phantom{{P}_{3}(\eta ,t)=}-\frac{1}{2\beta}[{e}^{\eta \sqrt{M}}erfc({\mathrm{t}}_{5})+{e}^{-\eta \sqrt{M}}erfc({\mathrm{t}}_{6})]\hfill \\ \phantom{{P}_{3}(\eta ,t)=}+\frac{{e}^{-\beta t}}{2\beta}[{e}^{\eta \sqrt{M-\beta}}erfc({\mathrm{t}}_{12})+{e}^{-\eta \sqrt{M-\beta}}erfc({\mathrm{t}}_{13})],\hfill \\ {\mathrm{t}}_{1},{\mathrm{t}}_{2}=\pm \sqrt{\frac{dt}{c}}+\frac{\eta}{2}\sqrt{\frac{c}{t}},\phantom{\rule{2em}{0ex}}{\mathrm{t}}_{3},{\mathrm{t}}_{4}=\pm \sqrt{\frac{(d-\alpha c)t}{c}}+\frac{\eta}{2}\sqrt{\frac{c}{t}},\hfill \\ {\mathrm{t}}_{5},{\mathrm{t}}_{6}=\pm \sqrt{Mt}+\frac{\eta}{2\sqrt{t}},\phantom{\rule{2em}{0ex}}{\mathrm{t}}_{7},{\mathrm{t}}_{8}=\pm \sqrt{(M-\alpha )t}+\frac{\eta}{2\sqrt{t}},\hfill \\ {\mathrm{t}}_{9}=\frac{\eta}{2}\sqrt{\frac{a}{t}},\phantom{\rule{2em}{0ex}}{\mathrm{t}}_{10},{\mathrm{t}}_{11}=\pm \sqrt{-\beta t}+\frac{\eta}{2}\sqrt{\frac{a}{t}},\phantom{\rule{2em}{0ex}}{\mathrm{t}}_{12},{\mathrm{t}}_{13}=\pm \sqrt{(M-\beta )t}+\frac{\eta}{2\sqrt{t}},\hfill \\ a=Sc,\phantom{\rule{2em}{0ex}}c=Pr,\phantom{\rule{2em}{0ex}}d=Pr\varphi ,\phantom{\rule{2em}{0ex}}{\alpha}_{1}=\frac{-Gr}{c-1},\phantom{\rule{2em}{0ex}}{\alpha}_{2}=\frac{-Gm}{a-1},\hfill \\ \alpha =\frac{d-M}{c-1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\beta =\frac{-M}{a-1}.\hfill \end{array}

Here, erfc(x), {\mathcal{L}}^{-1} and H(t-1) are respectively the complimentary error function, the inverse Laplace transform operator and the Heaviside unit step function.

Equations (10)-(12) represent the analytical solutions for the free convective heat and mass transfer flow of a viscous, incompressible, electrically conducting and heat absorbing fluid past a flat plate with ramped temperature in the presence of a uniform transverse magnetic field. In order to highlight the effects of the ramped temperature on the fluid flow, it is worthwhile to compare such a flow with the flow near a moving plate with constant temperature. The solution for species concentration is given by equation (10). However, the fluid temperature and velocity for free convection near an isothermal plate has the following form:

T(\eta ,t)=\frac{1}{2}[{e}^{\eta \sqrt{d}}erfc({\mathrm{t}}_{1})+{e}^{-\eta \sqrt{d}}erfc({\mathrm{t}}_{2})],

(13)

u(\eta ,t)=P(\eta ,t)+{\alpha}_{2}{P}_{3}(\eta ,t)+{\alpha}_{1}{P}_{4}(\eta ,t),

(14)

where

\begin{array}{rcl}{P}_{4}(\eta ,t)& =& \frac{1}{2\alpha}[{e}^{\eta \sqrt{d}}erfc({\mathrm{t}}_{1})+{e}^{-\eta \sqrt{d}}erfc({\mathrm{t}}_{2})]\\ -\frac{{e}^{-\alpha t}}{2\alpha}[{e}^{\eta \sqrt{(d-\alpha c)}}erfc({\mathrm{t}}_{3})+{e}^{-\eta \sqrt{(d-\alpha c)}}erfc({\mathrm{t}}_{4})]\\ -\frac{1}{2\alpha}[{e}^{\eta \sqrt{M}}erfc({\mathrm{t}}_{5})+{e}^{-\eta \sqrt{M}}erfc({\mathrm{t}}_{6})]\\ +\frac{{e}^{-\alpha t}}{2\alpha}[{e}^{\eta \sqrt{M-\alpha}}erfc({\mathrm{t}}_{7})+{e}^{-\eta \sqrt{M-\alpha}}erfc({\mathrm{t}}_{8})].\end{array}

(15)

The physical quantities of engineering interest are the Sherwood number *Sh*, the Nusselt number *Nu* and skin-friction *τ*. The Sherwood number measures the rate of mass transfer at the plate and is given by

Sh=-{\left(\frac{\partial C}{\partial \eta}\right)}_{\eta =0}=\sqrt{\frac{a}{\pi t}}.

(16)

The Nusselt number measures the rate of heat transfer at the plate, and for a ramped temperature plate it is

Nu=-{\left(\frac{\partial T}{\partial \eta}\right)}_{\eta =0}=-[{F}_{1}(t)-H(t-1){F}_{1}(t-1)],

(17)

where

{F}_{1}(t)=-(\sqrt{d}t+\frac{c}{2\sqrt{d}})erf\left(\sqrt{\frac{dt}{c}}\right)-\sqrt{\frac{ct}{\pi}}{e}^{-\frac{dt}{c}}.

(18)

In the case of an isothermal plate, the Nusselt number is

N{u}_{i}=(\sqrt{d})erf\left(\sqrt{\frac{dt}{c}}\right)+\sqrt{\frac{c}{\pi t}}{e}^{-\frac{dt}{c}}.

(19)

Equations (10)-(14) represent the analytical solutions for the flow induced by a general time-dependent movement of the vertical flat plate. To gain some practical understanding of the flow dynamics, some particular cases of time-dependent movements of the plate are discussed below.

### 3.1 Plate movement with uniform velocity

Assuming that the plate moves with uniform velocity f(t)=H(t), the fluid velocity for the ramped temperature plate is obtained as

u(\eta ,t)={P}_{5}(\eta ,t)+{\alpha}_{1}[{P}_{2}(\eta ,t)-H(t-1){P}_{2}(\eta ,t-1)]+{\alpha}_{2}{P}_{3}(\eta ,t),

(20)

while the isothermal plate has the velocity

u(\eta ,t)={P}_{5}(\eta ,t)+{\alpha}_{1}{P}_{4}(\eta ,t)+{\alpha}_{2}{P}_{3}(\eta ,t),

(21)

where

{P}_{5}(\eta ,t)=\frac{1}{2}[{e}^{\eta \sqrt{M}}erfc({\mathrm{t}}_{5})+{e}^{-\eta \sqrt{M}}erfc({\mathrm{t}}_{6})].

(22)

The skin friction for the ramped temperature plate is

{\tau}_{1}={\left(\frac{\partial u}{\partial \eta}\right)}_{\eta =0}={Q}_{1}(t)+{\alpha}_{1}[{F}_{2}(t)-H(t-1){F}_{2}(t-1)]+{\alpha}_{2}{F}_{3}(t),

(23)

and for the isothermal plate

{\tau}_{1i}={Q}_{1}(t)+{\alpha}_{1}{F}_{4}(t)+{\alpha}_{2}{F}_{3}(t),

(24)

where

\begin{array}{c}{Q}_{1}(t)=-\sqrt{M}erf(\sqrt{Mt})-\frac{{e}^{-Mt}}{\sqrt{\pi t}},\hfill \\ {F}_{2}(t)=-[\left(\frac{\alpha t-1}{{\alpha}^{2}}\right)\sqrt{d}+\frac{c}{2\alpha \sqrt{d}}]erf\left(\sqrt{\frac{dt}{c}}\right)-\frac{\sqrt{d-\alpha c}}{{\alpha}^{2}}erf\left(\sqrt{\left(\frac{d-\alpha c}{c}\right)t}\right){e}^{-\alpha t}\hfill \\ \phantom{{F}_{2}(t)=}+[\left(\frac{\alpha t-1}{{\alpha}^{2}}\right)\sqrt{M}+\frac{1}{2\alpha \sqrt{M}}]erf(\sqrt{Mt})+\frac{\sqrt{M-\alpha}}{{\alpha}^{2}}erf\left(\sqrt{(M-\alpha )t}\right){e}^{-\alpha t}\hfill \\ \phantom{{F}_{2}(t)=}-\frac{{e}^{-\frac{dt}{c}}}{\alpha}\sqrt{\frac{ct}{\pi}}+\frac{{e}^{-Mt}}{\alpha}\sqrt{\frac{t}{\pi}},\hfill \\ {F}_{3}(t)=\frac{1}{\beta}[\sqrt{-a\beta}erf(\sqrt{-\beta t}){e}^{-\beta t}+\sqrt{M}erf(\sqrt{Mt})-\sqrt{M-\beta}erf\left(\sqrt{(M-\beta )t}\right){e}^{-\beta t}],\hfill \\ {F}_{4}(t)=\frac{-\sqrt{d}}{\alpha}erf\left(\sqrt{\frac{dt}{c}}\right)+\frac{\sqrt{d-\alpha c}}{\alpha}erf\left(\sqrt{\left(\frac{d-\alpha c}{c}\right)t}\right){e}^{-\alpha t}+\frac{\sqrt{M}}{\alpha}erf(\sqrt{Mt})\hfill \\ \phantom{{F}_{2}(t)=}-\frac{\sqrt{M-\alpha}}{\alpha}erf\left(\sqrt{(M-\alpha )t}\right){e}^{-\alpha t}.\hfill \end{array}

The solutions (11), (13), (17), (19), (20), (21), (23) and (24), in the absence of heat absorption (\varphi =0) and mass transfer (Gm=0), are in agreement with the results obtained by Seth *et al.* [24] when {K}_{1}=0, N=0. The results are also consistent with those of Seth and Ansari [23] for a non-porous medium {K}_{1}=0 in the absence of mass transfer and with the results reported by Chandran *et al.* [22] in the absence of a magnetic field M=0, mass transfer and heat absorption.

### 3.2 Plate movement with single acceleration

Assuming that the plate moves with single acceleration f(t)=tH(t), the fluid velocity for a ramped temperature plate was obtained as

u(\eta ,t)={P}_{6}(\eta ,t)+{\alpha}_{1}[{P}_{2}(\eta ,t)-H(t-1){P}_{2}(\eta ,t-1)]+{\alpha}_{2}{P}_{3}(\eta ,t),

(25)

and for the isothermal plate

u(\eta ,t)={P}_{6}(\eta ,t)+{\alpha}_{1}{P}_{4}(\eta ,t)+{\alpha}_{2}{P}_{3}(\eta ,t),

(26)

where

{P}_{6}(\eta ,t)=(\frac{t}{2}+\frac{\eta}{4\sqrt{M}}){e}^{\eta \sqrt{M}}erfc({\mathrm{t}}_{5})+(\frac{t}{2}-\frac{\eta}{4\sqrt{M}}){e}^{-\eta \sqrt{M}}erfc({\mathrm{t}}_{6}).

(27)

The skin friction for the ramped temperature plate is

{\tau}_{2}={Q}_{2}(t)+{\alpha}_{1}[{F}_{2}(t)-H(t-1){F}_{2}(t-1)]+{\alpha}_{2}{F}_{3}(t).

(28)

The corresponding skin friction for the isothermal plate is

{\tau}_{2i}={Q}_{2}(t)+{\alpha}_{1}{F}_{4}(t)+{\alpha}_{2}{F}_{3}(t),

(29)

where

{Q}_{2}(t)=-(\frac{1}{2\sqrt{M}}+t\sqrt{M})erf(\sqrt{Mt})-\sqrt{\frac{t}{\pi}}{e}^{-Mt}.

(30)

### 3.3 Plate movement with periodic acceleration

Assuming that the plate moves with periodic acceleration f(t)=cos\omega tH(t), the fluid velocity for the ramped temperature plate is

u(\eta ,t)={P}_{7}(\eta ,t)+{\alpha}_{1}[{P}_{2}(\eta ,t)-H(t-1){P}_{2}(\eta ,t-1)]+{\alpha}_{2}{P}_{3}(\eta ,t),

(31)

and the fluid velocity for the isothermal plate is

u(\eta ,t)={P}_{7}(\eta ,t)+{\alpha}_{1}{P}_{4}(\eta ,t)+{\alpha}_{2}{P}_{3}(\eta ,t),

(32)

where

\begin{array}{rl}{P}_{7}(\eta ,t)=& \frac{1}{4}{e}^{-i\omega t}[{e}^{\eta \sqrt{M-i\omega}}erfc(\sqrt{(M-i\omega )t}+\frac{\eta}{2\sqrt{t}})\\ +{e}^{-\eta \sqrt{M-i\omega}}erfc(-\sqrt{(M-i\omega )t}+\frac{\eta}{2\sqrt{t}})]\\ +\frac{1}{4}{e}^{i\omega t}[{e}^{\eta \sqrt{M+i\omega}}erfc(\sqrt{(M+i\omega )t}+\frac{\eta}{2\sqrt{t}})\\ +{e}^{-\eta \sqrt{M+i\omega}}erfc(-\sqrt{(M+i\omega )t}+\frac{\eta}{2\sqrt{t}})].\end{array}\}

(33)

The skin friction for the ramped temperature plate is expressed as

{\tau}_{3}={Q}_{3}(t)+{\alpha}_{1}[{F}_{2}(t)-H(t-1){F}_{2}(t-1)]+{\alpha}_{2}{F}_{3}(t),

(34)

and the skin friction for the isothermal plate is

{\tau}_{3i}={Q}_{3}(t)+{\alpha}_{1}{F}_{4}(t)+{\alpha}_{2}{F}_{3}(t),

(35)

where

\begin{array}{rl}{Q}_{3}(t)=& -\frac{{e}^{-Mt}}{\sqrt{\pi t}}-\frac{1}{2}\sqrt{M-i\omega}{e}^{-i\omega t}erf\left(\sqrt{(M-i\omega )t}\right)\\ -\frac{1}{2}\sqrt{M+i\omega}{e}^{i\omega t}erf\left(\sqrt{(M+i\omega )t}\right).\end{array}

(36)