Transient MHD Fluid Flow Past a Moving Vertical Surface in a

Velocity Slip Flow Regime

IGHOROJE W.A. OKUYADE1,*, TAMUNOIMI M. ABBEY2

1Department of Mathematics/Statistics,

Federal Polytechnic of Oil and Gas,

Bonny Island,

NIGERIA

2Theoretical and Applied Mathematics Group, Department of Physics,

University of Port Harcourt,

Port Harcourt,

NIGERIA

*Corresponding Author

Abstract: - The problem of unsteady MHD fluid flow past a moving vertical surface in a slip flow regime is

presented. The model is built on the assumption that the flow is naturally convective with oscillating time-

dependent and exponentially decaying suction and permeability, double-diffusion, viscous dissipation, and

temperature gradient-dependent heat source, and non-zero tangential velocity at the wall; the fluid is viscous,

incompressible, Newtonian, chemically reactive, and magnetically susceptible; the surface is porous, and

electrically conductive, and thermally radiative. The governing partial differential equations are highly coupled

and non-linear. For easy tractability, the equations are reduced to one-dimensional using the one-dimensional

unsteady flow theory. The resulting equations are non-dimensionalized and solved using the time-dependent

perturbation series solutions, and the Modified Homotopy Perturbation Method (MHPM). The solutions of the

concentration, temperature, velocity, rates of mass and heat diffusion, and wall shear stress are obtained,

computed, and presented graphically and quantitatively, and analyzed. The results among others, show that the

increase in the: Schmidt number increases the fluid concentration, velocity, the rate of heat transfer to the fluid,

and the stress on the wall, but decreases the rate of mass transfer to the fluid; Magnetic field parameter

decreases the fluid velocity and stress on the wall; Slip parameter increases the flow velocity, but decreases the

stress on the wall; Permeability parameter increases the flow velocity and the stress on the wall. These results

are benchmarked with the reports in existing literature and they agree.

Key-Words: - Chemically reacting, MHD, Slip flow, Thermally radiating, Thermo-diffusion, Temperature

gradient-dependent heat source, Viscous dissipation.

MSC: 76/R

Received: February 18, 2023. Revised: December 9, 2023. Accepted: February 8, 2024. Published: March 6, 2024.

1 Introduction

Flow problems with the magnetic field, chemical

reaction, and heat source/sink effects cut across

nature, science, and engineering. They have

applications in the chemical and petroleum

industries, cooling of nuclear reactors, catalytic

reactors, and the likes.

In fluid dynamics, the no-slip condition applies

to viscous fluids, and therein it is assumed that at

solid boundaries the fluid velocity is equal to zero.

The exertion is based on the fact that fluid particles

on the surface do not move along with the flow

when the force of adhesion is stronger than the

cohesion. At the fluid-solid interface, the force of

attraction between the fluid particles and the solid

particles (adhesive force) is stronger than that

between the fluid particles (cohesive force). The

force imbalance brings the fluid velocity to zero.

The no-slip condition is a universal

assertion\assumption or a mere ideology and does

not apply to inviscid flows, where the effect of the

boundary layer is neglected. However, for

engineering applications, the concept of no-slip

condition does not always hold. For example, at

very low pressure (at high altitude) some of the fluid

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particles near the solid surface move along with the

surface, thus bringing to bear the slip velocity

condition. Furthermore, at very high altitudes fluid

particles adjacent to the surfaces (of aircraft and

rockets, say) no longer take the velocity of the solid

surface but possess finite tangential velocities

known as the slip velocities (non-zero) with which

they slip along the surface. These slip velocities are

linearly proportional to the shear stress on the plate,

while the surface boundary with slip velocity

condition is the velocity-slip regime. The slip factor

is a function of mass flow rate, fluid entry condition,

working fluid viscosity, boundary layer growth,

flow separation, etc. A common approximation for

fluid slip is

yu

wall

uu  /



, where

1

/)

1

2( mm



the slip length,

1

m

is Maxwell’s

reflection coefficient [1]. Importantly, the

Maxwell’s reflection coefficient/transmission

coefficient is a parameter that describes how the

wave is reflected by impedance discontinuity in the

transmission medium. It is the ratio of the amplitude

of the reflected wave/ transmitted wave to the

incident wave. The slip idealization was first

conceived and presented in [2], as a flow model

wherein the velocity normal to the boundary is set to

zero, while the velocity parallel to the boundary is

left free. Building on this, [3], formulated a slip

model, which has been used extensively by

researchers to date. Upon Maxwell’s model, a lot of

research has been carried out. For example, [4],

studied the slip flow at the entrance region of a

parallel plate channel; [5] considered the flow in

rectangular and annular ducts; [6] studied the MHD

steady flow in a channel with slip at the permeable

boundaries.

Specifically, concerning the flow past moving

vertical plates, [7] examined the MHD visco-elastic

flow with velocity slip when the plate is oscillating;

[8] investigated the transient flow under variable

suction, periodic temperature, and slip conditions;

[9] examined the effect of periodic heat and mass

transfer on the unsteady natural convective flow in

the slip-flow regime when the suction velocity

oscillates with time; [10] examined the flow under a

magnetic field influence when the plate is

oscillating in a slip velocity regime. Furthermore,

[11] studied the MHD flow under radiation and

temperature gradient-dependent heat source in a slip

flow regime; [12] looked into the slip boundary

layer of non-Newtonian fluid with convective

thermal boundary condition; [13] considered the

MHD convective heat and mass transfer in a

boundary layer slip flow over with thermal radiation

and chemical reaction; [14] looked at the transient

MHD flow of a third-grade fluid when the plate is

insulated, and in the presence of thermo-diffusion,

time-dependent suction, heat source, mass transfer

and slip effects. [1], studied the unsteady MHD

natural convective flow over a porous vertical plate

in the presence of radiation and temperature

gradient-dependent heat source, exponentially

decaying suction and permeability in a slip flow

regime using time-dependent perturbation method

and numerical analysis, and observed that the

velocity increases with the increase in the slip

parameter and Grashof number, but decreases with

the increase in the magnetic field, heat source,

radiation, and chemical reaction rate parameters; the

temperature decreases with the increase in the

radiation and heat source parameters; the

concentration decreases with the increase in the

Schmidt number and chemical reaction rate

parameter. [15], studied the MHD natural

convective chemically reactive flow in the presence

of thermo-diffusion, fluctuating wall temperature

and concentration, thermal radiation, and free

stream and slip velocities; [16], considered the

MHD boundary layer flow with slip near a

stagnation point; [17], considered the flow in the

presence of heat generation/absorption, slip

velocity, and temperature jump. [18], investigated

thermal diffusion and chemical reaction effects on

an unsteady flow in the presence of temperature-

dependent heat source and velocity slip condition

using the method of exponentially increasing small

perturbation law, and saw that the velocity is

enhanced by the increase in the slip and

permeability parameters, and Grashof numbers; the

temperature increases with the increase in the

Prandtl number, but decreases with increase in the

heat source parameter; the concentration increases

with the increase in the Soret number, but decreases

with the increase in the permeability parameter and

Schmidt number. [19], examined the effects of

variable viscosity and periodic boundary conditions

on a free convective flow in a slip regime; [20],

examined the flow of a micro-polar fluid over a

radiating surface in the presence of variable

viscosity in a slip regime; [21], investigated the flow

under velocity slip and time-periodic boundary

effects; [22], examined numerically the higher-order

chemical reaction effects on MHD Nano-fluid flow

with velocity slip boundary condition. More so,

[23], considered the boundary layer flow in the

presence of cross-diffusion effect in a velocity slip

regime; [24], studied numerically the flow of a

Newtonian fluid in the presence of buoyancy, order

two thermal slip and entropy generation; [25],

looked into the transient slip flow with ramped plate

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temperature and concentration, thermal radiation

and buoyancy effects. Similarly, [26], considered

the transient MHD natural convective flow in a slip

regime with periodic movement, Hall currents, and

rotation effects; [27], gave a Reynolds analogy for

the flow at different regimes.

Other reports on the convective flow over a

vertical plate with slip velocity conditions are found

in [28], [29], [30], [31], [32], [33], [34], [35], [36],

[37].

More so, the interaction of electric and magnetic

fields in a flow system results in many factors that

influence the flow. By application, when a wire

carrying alternating current is applied to a non-zero

resistive plate/conductor a voltage difference is

created between the ends of the conductor in the

electric field. The electric field accelerates the

charge carriers (electrons, ions, and holes) on the

plate in the direction of the electric field to give

kinetic energy. At collision with each other on the

plate, the charged particles are

scattered/randomized. The scattering motions of the

charged particles cause the temperature of the

plate/conductor to rise. This thermal effect is called

the Joule/Ohmic heating. By this, electric energy is

converted into thermal energy. Also, as the fluid

flows past the plate a dissipating force that works

mechanically to heat the fluid is produced. It is

noteworthy that Joule heating is limited by

viscosity, electric conductivity, and fouling deposits

on the conductor. Furthermore, the varying

alternating currents lead to the heating of the plate

non-uniformly. Similarly, the heating of the plate to

a high-temperature regime leads to the emission of

thermal radiant rays. The effects of Joules/Ohmic

heating, magnetic field, and viscous dissipation in

the problem of convective heat and mass transfer on

the flow past vertical plates have been investigated.

For example, [38], examined the viscous and Joule

heating effects on the MHD free convection flow

with variable plate temperature; [39], studied the

MHD natural convective flow of a radioactive fluid

past an inclined plate in the presence of chemical

reaction, temperature-dependent heat source, and

Joule heating using the method of regular

perturbation, and noticed that an increase in the

magnetic field parameter decreases the velocity,

whereas an increase in the permeability parameter

increases it; the increase in the Prandtl and Schmidt

numbers, respectively, condense the thermal and

concentration boundary layers.

In highly interactive systems, where magnetic

flux, convection, and chemical reaction are

significant heat and mass transfer occur

simultaneously. The simultaneous effect on the

system called double-diffusion induces buoyancy.

The differential in temperature produces Dufour

(thermo-diffusion), while the differential in

concentration produces Soret (diffusion-thermo).

The double-diffusion phenomena were developed

from the kinetic theory of gases, [40], [41]. They are

smaller than the Fourier and Ficks effects, [42]. For

their relevance, many reports bearing double-

diffusion effects exist in the literature. Specifically,

on the flow over vertical plates, [43], considered

natural convective and mass transfer effects on a

two-dimensional case using the similarity technique

and Runge-Kutta sixth-order approach; [44],

examined the effects of thermal radiation, Hall

currents, Dufour and Soret numbers on the MHD

mixed convective flow; [45] investigated the free

convective flow with double-diffusive convection

using the successive linearization method; [46]

studied a mixed convective heat and mass transfer

flow along a wavy surface in a Darcy porous

medium in the presence of cross-diffusion effects

using similarity transformation and numerical

scheme for aiding flow, opposing flow, and for both

aiding and opposing flows.

The problem of natural convective fluid flow

over a vertical plate with chemical reaction,

radiation, and temperature-dependent heat source in

a slip flow regime using the time-dependent and

exponentially decaying perturbation series solution

approach was examined by [1]. In their work, the

effects of thermo-diffusion and viscous dissipation

were neglected. As an extension of [1], this present

work investigates the flow problem in the presence

of the aforementioned parameters using the time-

dependent and exponentially decaying perturbation

series solutions and the Modified Homotopy

Perturbation Method.

This paper is presented in the following format:

Section 2 gives the problem formulation; Section 3

holds the problem Solution; Section 4 holds the

conclusion.

2 Problem Formulation

x

U, C , T

  

y

g

T

C

uB

Fig. 1: The model of a vertically accelerating plate

in a fluid

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The transient MHD natural convective flow past

a moving vertical surface in a slip flow regime is

investigated. The schematic of the flow is shown in

Figure 1. The model formulation is based on the

assumptions that: the fluid is Newtonian, thermally

radiating, chemically reacting and electrically

conducting; the physical properties of the fluid such

as the specific heat at constant pressure, thermal

conductivity, and density remain constant

throughout the fluid; the fluid is mixed with a

chemical species at a higher concentration to initiate

a chemical reaction; the plate is porous, its

permeability and suction at the wall are oscillating,

time-dependent and exponentially decaying; the

plate is connected to a wire carrying an alternating

current, which produces a voltage between the ends,

and which in turn energizes the ions, electrons and

holes on the plate to generate a Joule/Ohmic heating

that produces a mechanical force/viscous

dissipation; a magnetic field force of uniform

strength and negligible induction effect is applied

transversely to the plate; there is a convective

temperature gradient between the bottom and upper

surfaces of the plate with a heat source at the bottom

and sink at the top; the plate is heated to a high

temperature regime such that thermal rays are

emitted into the fluid; the flow is naturally

convective. In this model, the x-axis is taken to be in

the vertical direction of the plate and the y-axis is

normal to it. Therefore, if

),( vu

are the velocity

components in the spatial (

tyx ,,

) coordinates;

w

T

and

w

C

are the temperature and concentration at the

wall;

v

is the velocity along the y-axis and the

suction at the wall;



T

and



C

are the fluid

equilibrium temperature and concentration,

T

and

C

are the fluid temperature and concentration.

Then, using the unsteady one-dimensional flow

theory and Boussinesq’s approximations the

governing equations of continuity, momentum,

energy, and mass diffusion are:

o

v

y

v



0

'

(1)

 















TT

t

g

y

u

y

u

v

t

u'

2

'

2

'



 

'

2

'u

m

o

B

e

CC

c

g



























(2)

'

1

2

'

2

'

'y

r

q

p

C

y

T

p

C

k

y

T

v

t

T

















 

2

'

2

'

y

C

s

C

p

Ct

Dk

y

u

p

C

TT

y

p

C

Q



































(3)

 















CC

r

k

y

C

D

y

C

v

t

C'

2

'

2

'

(4)

with the boundary condition:

,0)0,'(',

'

)0,'(' 



tv

y

u

tu



,)0,'(' w

TtT 

w

CtC )0,'('

at

0'y

(5)

,),'(',0),'(',0),'(' 

 TtTtvtu



 CtC ),'('

at

' y

(6)



is the kinematic viscosity;



is the density;

g

is

the acceleration due to gravity;

t



is the coefficient

of volumetric expansion due to temperature;

c



is

the coefficient of volumetric expansion due to

concentration;

k

is the thermal conductivity of the

fluid;

m



is the magnetic field permittivity;

'Q

is

the heat source/sink;

e



is the electrical

conductivity of the fluid;

2

o

B

is the magnetic field

flux;

p

C

is the specific heat capacity at constant

pressure;

s

C

is the concentration susceptibility;

t

k

is

the thermal diffusivity ratio;

D

is the coefficient of

mass transfer/ mass diffusion coefficient;



is the

permittivity of the porous plate;

r

k

is the chemical

reaction term of the species.

Assuming the suction at the wall and

permeability of the medium are oscillating, time-

dependent, and exponentially decaying, then:

)

''

1( tm

e

o

vv 





(7)

)

''

1( tm

e

o





(8)

where

'm

is a positive constant,

o

v

is the steady

suction at the wall. Suction is a criterion for

determining certain flow situations. For example,

for

0v

suction (the fluid moves towards the

plate),

0v

injection (the fluid moves from the

plate), and

0v

the plate is impermeable. Similarly,

o



is the steady permeability of the wall.

Permeability is Darcian for

,1

o



and non-Darcian

for

1

o



. While porosity is a measure of the voids

in a material, permeability is a measure of the ease

of flow of a fluid through a porous solid. In other

words, porosity determines the number, sizes, and

inter-connectedness of the voids in a solid material,

while permeability determines the ease of fluid flow

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through the voids. Both porosity and permeability

are related to the number, size, and connections of

openings in solid materials, hence many a time they

are used interchangeably.

More so, radiation is seen as a heat transfer

from a high-temperature regime. In effect, it is

comparable to convective heat transfer. It is

described in terms of optical depths: depths at which

photons travel/penetrate fluids. Optical depths can

be thin or thick. A fluid is optically thin/transparent

when its density is relatively low, and the depth of

penetration/distance it allows long photon travel in

it is far less than unity (

1



). Examples of

optically thin environments include the non-

participating media in which energy is emitted from

the fluid but is not absorbed, as in gray gas. Also, a

fluid is optically thick/non-transparent when its

density is high enough to allow short photon travel

in it. The optically thick fluid emits and absorbs

radiation at the boundaries. The analysis of radiation

is based on the optic limits: thin or thick.

Importantly, the radiative heat flux is approximated

by the Roseland diffusion approximations. Now, on

the assumption that the fluid here is optically thin,

we adopt the Roseland approximation:

)

44

(

1

4

1

3' 





 TT

y

r

q

r

q



such that

)

44

(

2

'

2

1

4

'









TT

y

r

q



(9)

Taking the temperature difference within the

flow to be sufficiently small such that

,)(







TT

and



is a non-constant small temperature correction

factor, then

4

T

can be expressed as a linear function

of the temperature in the Taylor series about



T

:

4

3

4

4





TTTT

with the higher-order terms neglected.

Substituting this into equation (9) gives:

2'

'

2

3

1

16

'yT

T

y

r

q











(10)

and by equations (7), (8) and (10), equations (2) -

(4) become:

 

















TT

t

g

y

u

y

u

tm

e

o

v

t

u'

2

'

2

'

)

''

1(

'



 

'

)

''

1(

2

'u

tm

e

o

m

o

B

e

CC

c

g

































(11)

 

2

'

2

'

1

2

'

2

'

)

''

1(

'

y

C

s

C

p

Ct

Dk

y

u

p

C

TT

y

p

C

Q

y

r

q

p

C

y

T

p

C

k

y

T

tm

e

o

v

t

T























































(12)

 

















CC

r

k

y

C

D

y

C

tm

e

o

v

t

C'

2

'

2

'

)

''

1(

'



(13)

Making the problem independent of particular

units of measurement and geometry, and generating

the necessary parameters that control the flow, we

introduce the following non-dimensionalized

quantities:

,

4

'

2



t

o

v

t

,

'

,

'

o

v

u

y

o

v

y



,

2

'4

o

v

m











 T

w

TTT'

,

2

)(

,

'

o

v

T

w

T

t

g

Gr

C

w

CCC

















,

2

,

2

)(

o

v

m

o

B

e

M

o

v

C

w

C

c

g

Gc





















o

,

k



Pr

,

D

Sc





,

D

r

k







,

o

v

p

Ck Q

N



'



,

)(

2





T

w

T

p

Co

v

Ec

k

T

Ra 2

3

1

16









,

 







T

w

T

s

kC

C

w

C

t

Dk

Dr

(14)

where



and



are the non-dimensionalized

temperature and concentration, respectively;

N

is the

heat generation/absorption parameter;

Gr

is the

Grashof number due to temperature difference;

Gc

is the Grashof number due to concentration

difference;

M

is the magnetic field force;



is the

porosity parameter;

Pr

is the Prandtl number;

Dr

is the Dufour number;

Sc

is the Schmidt number;



is the chemical reaction rate) into equations (11) –

(13), (5) and (6), we have:















GcGr

y

u

y

u

mt

e

t

u

2

'

2

)1(

4

1



u

mt

e

o

M

















 )1(

1



(15)

y

N

y

mt

e

t





















 Pr

2

Pr)1(

4

1



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2

'

2

Pr y

Dr

y

u

Ec 





















(16)























Sc

y

Sc

mt

e

t2

2

)1(

4

1

(17)

where

Ra1



and the boundary conditions

1,1, 



y

u



at

0y

(18)

0,0,0 u

at

y

(19)

Other factors influencing the flow are the

Nusselt number (or thermal conductivity of the

fluid), Sherwood number (or species conductivity of

the fluid), and wall shear stress (the force the fluid

exerts on the wall), and these are prescribed non-

dimensionally as:

0

)(' 

 y

yNu

(20)

0

)(' 

 y

ySh

(21)

0

)(' 

y

yuCf



(22)

3 Problem Solution

3.1 Method of Solution

Equations (15) - (17) are reduced to ordinary

differential equations and solved using time-

dependent perturbation series solutions of the form,

[11]:

)

2

()(

1

)(),(



O

mt

eyuy

o

utyu 





(23)

)

2

()(

1

)(),(



O

mt

eyy

o

ty 





(24)

)

2

()(

1

)(),(



O

mt

eyy

o

ty 





(25)

Substituting these into equations (15) - (19)

appropriately, collecting and equating the cefficient

of the powers

,



we have:

for the zeroth order

o

Gc

o

Gr

o

uM

dy

o

du

dy

o

ud  1

2

(26)

2

Pr

2

y

o

Dr

dy

o

du

Ec

dy

o

d

dy

o

d



























(27)

0

2

2





o

Sc

dy

o

d

Sc

dy o

d



(28)

with the boundary conditions

1,1, 



oo

y

o

u

o

u



at

0y

(29)

0,0,0  ooo

u

at

y

(30)

and for the first order

1112

1

2

1

2 GcGr

dy

o

du

uM

dy

du

dy

ud

(31)

dy

o

d

m

dy

d

dy

d







Pr

1

4

Pr

1

21

2



2

'

1

2

1

Pr2y

Dr

dy

du

dy

o

du

Ec 



 













(32)

dy

o

d

Sc

dy

d

Sc

dy

d









1

2

1

2



(33)

with the boundary conditions

0

1

,0

1

,

1

1



y

u



at

0y

(34)

0,0,0 111 u

at

y

(35)

where

o

MM



1

1

,

4

1

2m

o

MM 



,

)1Pr(,1,

4NRa

m

Sc  















Equations (26), (27), (31), and (32) are still

highly coupled. A second perturbation becomes

necessary. We resort to using the Modified

Homotopy Perturbation Method (MHPM) of

solutions.

The analysis associated with the Homotopy

Perturbation Method is as follows:

Consider the nonlinear equation

     

rfvNvL 

,

r

(36)

with the boundary condition

















r

y

u

uB ,0,

(37)

where L is a linear operator, N is a nonlinear

operator, B is a boundary operator,



is the boundary

of the domain



,

 

rf

is a known analytic function.

For a Homotopy Perturbation technique, He (a

Chinese) constructed a homotopy:

 

Rprv  ]1,0[,

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which satisfies

 

     

0][][1,  rfvNvLp

o

uLvLppvH

(38)

where

]1,0[p

is an impending parameter,

o

u

is an

initial approximation that satisfies the boundary

conditions. Clearly, from equation (38), we have:

 

0][0,  o

uLvLvH

,

       

0][1,  rfvNvLvH

Importantly, the process of changing p from

zero to unity

 

prv ,

is like that of changing from

 

r

o

u

to

 

ru

, and this is called a deformation in

Topology; the

 

0][  o

uLvL

and

     

0][  rfvNvL

are called homotopic.

Here, the basic assumption is that the solution of

equation (38) can be expressed as a power series in

p:

...

2

1 vppv

o

vv

and the approximate solution of equation (36) is

obtained as, [47]; [48]:

...

21

lim  vv

o

vvu

1p

The difference between HPM and MHPM is

seen in their use of boundary conditions. In HPM,

order zero takes the boundary conditions at t<0,

order one takes the boundary conditions at t>0 for

y=0, and order two takes that at t>0 for y=∞, while

in MHPM all the orders use the boundary conditions

at t>0: y=0 and y=∞ but with little modifications, as

we shall see below.

Based on the given analysis, writing equations (26) -

(35) in MHPM form, we have

For the Zeroth Order

]

1

'''['')1( o

Gc

o

Gr

o

uM

o

u

o

up

o

up 

(39)

]''

2

)''(Pr'''['')1( o

Dr

o

uEc

oo

p

o

p



(40)

]'''['')1( o

Sc

o

Sc

o

p

o

p



(41)

such that

]

1

'['' o

Gc

o

Gr

o

uM

o

up

o

u

(42)

]''

2

)''(Pr'['' o

Dr

o

uEc

o

p

o



(43)

]'['' o

Sc

o

Scp

o



(44)

Expanding the dependent variables in terms of p, we

have:

for the zeroth order:

..

02

2

0100  uppuu

o

u

(45)

..

02

2

0100  pp

o

(46)

..

02

2

0100  pp

o

(47)

and for the first order

..

12

2

11101 uppuuu

(48)

..

12

2

11101 pp

(49)

..

12

2

11101 pp

(50)

Substituting equations (45)-(47) into equations (42)-

(44) and (29) and (30), gives:







































)

02

2

0100

(

)

02

2

0100

(

)

02

2

01

00

(

1

)'

02

2

'

01

'

00

(

''

02

2

''

01

''

00

p

pGc

p

pGr

up

puuM

up

puu

puppuu

(51)















































)''

02

2

''

01

''

00

(

]

2

)''

02

(

2

)''

01

(

2

)''

00

[(Pr

)'

02

2

'

01

'

00

(

''

02

2

''

01

''

00

p

Dr

up

uEc

p

ppp





(52)

































)

02

2

01

00

(

)'

02

2

'

01

'

00

(

''

02

2

''

01

''

00

p

Sc

p

Sc

ppp



(53)

Collecting the coefficients of the of the powers of p

in each case, we have:

0''

00 u

(54)

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0''

00 



(55)

0''

00 

(56)

0000001

'

00

''

01  GcGruMuu

(57)

''

00

2

)''

00

(Pr'

00

''

01  DruEc



(58)

00

'

00

''

01 



ScSc

(59)

0101011

'

01

''

02  GcGruMuu

(60)

''

01

2

)''

01

(Pr'

01

''

02  DruEc



(61)

01

'

01

''

02 



ScSc

(62)

with the boundary conditions

1

00

,1

00

,

00

00 



y

u



at

0y

(63)

0

00

,0

00

,0

00 u

at

y

(64)

0

01

,0

01

,

01

01 



y

u



at

0y

(65)

0

01

,0

01

,0

01 u

at

y

(66)

0

02

,0

02

,

02

02 



y

u



at

0y

(67)

0

02

,0

02

,0

02 u

at

y

(68)

The First Order

Similarly, by expressing equations (31)-(35) in

MHPM form, we substitute equations (48)-(50) into

them, and collecting and equating the coefficients of

the powers of p in the resulting equations to zero,

we obtain:

0''

10 u

(69)

0''

10 



(70)

0''

10 

(71)

1010

'

00102

'

10

''

11  GcGruuMuu

(72)

'

00

Pr

10

4

Pr

'

10

''

11  m



''

10

2

)'

10

'.

00

(Pr2 DruuEc

(73)

0010

'

10

''

11 



ScSc

(74)

1111011121112 ''''  GcGruuMuu

(75)

'

01

Pr

11

4

Pr

'

11

''

12  m



''

10

)]'

10

'.

01

()'

11

'.

00

[(Pr2 DruuuuEc

(76)

0111

'

11

''

12 



ScSc

(77)

with the boundary conditions,

0

10

,0

10

,

10

10 



y

u



at

0y

(78)

0

10

,0

10

,0

10 u

at

y

(79)

0

11

,0

11

,

11

11 



y

u



at

0y

(80)

0

11

,0

11

,0

11 u

at

y

(81)

0

12

,0

102

,

12

12 



y

u



at

0y

(82)

0

12

,0

12

,0

12 u

at

y

(83)

Equations (54)-(68) and (69)-(83) are solved

using the Mathematica 11.2 computational software,

and their solutions are found in the Appendices.

3.2 Results and Discussion

The solutions of the concentration, temperature,

velocity, Nusselt number, Sherwood number, and

wall shear stress are computed and presented

quantitatively and graphically. The effects of the

rate of chemical reaction, Schmidt number, Grashof

number, slip, magnetic field, and permeability

parameters are considered. For physically realistic

constant values of

,7.0Pr,01.0,1.0  DrEc

5.0,1,1.0,01.0,3,3,3  mtpNGcRa



and varied values of

0

,,,,,



MGrSc

, we obtained

the figures and table below.

Fig. 2: Concentration-Chemical Reaction Rate

Profiles

Fig. 3: Velocity-Chemical Reaction Rate Profiles

0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.2

0.4

0.6

0.8

1.0

y

y against y for 0.1,0.5,1.0,1.5,2.0

0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.1

0.2

0.3

0.4

0.5

u y

u y against y for 0.1,0.5,1.0,1.5,2.0

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Table 1. Some Parameters-Sherwood Number,

Nusselt Number, and Wall Shear Stress Relations

Parameters

)0('

)0('

)0('u



0.1

0.294755

0.327439

0.256117

0.5

0.267821

0.327442

0.260611

1.0

0.233099

0.327447

0.266285

1.5

0.197205

0.327451

0.271934

Sc

0.1

0.323176

0.327744

0.250048

0.5

0.280969

0.327446

0.258193

1.0

0.224703

0.327449

0.268373

1.5

0.164533

0.327451

0.278554

M

0.1

0.293304

0.5

0.260064

1.0

0.220136

1.5

0.179636



0.1

0.509107

0.5

0.382889

1.0

0.260636

1.5

0.165565

o



0.1

0.152264

0.5

0.260631

1.0

0.341636

1.5

0.368636

The effects of the chemical reaction rate on the

flow are seen in Figure 2, Figure 3 and Table 1.

They show that the increase in the rate of chemical

reaction increases the fluid concentration, velocity,

the rate of heat transfer to the fluid, the force

exerted on the wall, and the rate of mass transfer to

the fluid. A chemical reaction may lead an increase

in the interaction of fluid particles, and the

production of new species. More so, a chemical

reaction may be exothermic or endothermic, and

therein heat is generated or absorbed.

Phenomenally, this ought to increase the velocity,

thus accounting for what is seen in Figure 2, Figure

3 and Table 1.

Fig. 4: Concentration-Schmidt Number Profiles

Fig. 5: Velocity-Schmidt number Profiles

The effects of Schmidt number on the flow are

shown in Figure 4, Figure 5 and Table 1. They

depict that the increase in the Schmidt number

increases the fluid concentration, velocity, the rate

of heat transfer to the fluid, and stress on the wall,

but decreases the rate of mass transfer to the fluid.

Schmidt number is the ratio of momentum diffusion

to mass diffusion. When the mass diffusion

increases the concentration increases. Similarly,

when the momentum diffusion dominates the

system, the velocity increases. More so, with the

increase in the concentration, the velocity, as a

function of concentration increases.

Fig. 6: Velocity-Grashof Number Profiles

Fig. 7: Velocity-Slip Parameter Profiles

Sc 0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.2

0.4

0.6

0.8

1.0

y

y against y for

S

c0.1,0.5,1.0,1.5,2.0

Sc 0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.1

0.2

0.3

0.4

0.5

u y

u y against y for

S

c0.1,0.5,1.0,1.5,2.0

Gr 0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.1

0.2

0.3

0.4

u y

u y against y for

G

r0.1,0.5,1.0,1.5,2.0

0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.2

0.4

0.6

u y

u y against y for 0.1,0.5,1.0,1.5,2.0

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Volume 19, 2024

Furthermore, the effect of the Grashof number

on the flow is shown in Figure 6. It depicts that the

increase in the Grashof number increases the

velocity of the flow. When the temperature at the

plate/environmental temperature is higher than that

of the fluid at equilibrium heat is transferred from

the plate to the fluid. Now, a differential exists

between the temperature at the plate and that of the

fluid at equilibrium. In the presence of volumetric

expansion due to heat exchange and gravity,

convection currents are generated. The convection

currents induce a buoyancy force which reduces the

fluid viscosity, thus enhancing the fluid velocity; as

seen in Figure 6. This result aligns with [1] and [18].

Additionally, the effects of the slip velocity

parameter on the flow are shown in Figure 7 and

Table 1. They show that the increase in the slip

parameter increases the flow velocity, but decreases

the stress on the wall. The slip length is a function

of Maxwell’s reflection/transmission coefficient,

which describes the way wave

reflection/transmission affects the flow velocity. A

higher reflection/transmission increases the slip

length, and vice versa. Therefore, an increase in the

wave reflection/transmission increases the fluid

velocity. This result is in agreement with [1] and

[18].

Fig. 8: Velocity-Magnetic Field Profiles

Fig. 9: Velocity-Permeability Profiles

Similarly, the effects of the magnetic field on the

flow are shown in Figure 8 and Table 1. They show

that the increase in the magnetic field parameter

decreases the fluid velocity and the stress on the

wall. The particles of a chemically reactive fluid

exist as charges or ions and generate electric

currents in the presence of an applied magnetic

force. Again, the interaction of the electric currents

with the magnetic field force produces a mechanical

force called the Lorentz force (

o

BjF 

, where j

is the electric current density, and Bo is the magnetic

field flux). The Lorentz force has the potency of

freezing up the flow velocity. More so, the decrease

in the velocity must decrease the stress on the wall.

The results are in consonant with [1].

Also, the effects of permeability/porosity are

shown in Figure 9 and Table 1. They show that the

increase in the permeability factor of the porous

media increases the flow velocity and the stress on

the wall. In addition to the hydraulic conductivity

effect of the porous plate, the permeability factor

determines the ease of flow of a fluid passing

through a porous medium. The higher the

permeability parameter the easier the fluid flows

through the medium. Therefore, the velocity will

increase with the increase in the permeability

parameter. This result aligns with [18].

4 Conclusion

Transient MHD fluid flow past a moving vertical

surface in a slip flow regime is investigated. The

analysis of results shows that the increase in the:

● rate of chemical reaction increases the fluid

concentration, velocity, rate of heat transfer

to the fluid, and stress on the wall, but

decreases the rate of mass transfer to the

fluid.

● Schmidt number increases the fluid

concentration, velocity, rate of heat transfer

to the fluid, stress on the wall, and the rate

of mass transfer to the fluid.

● Grashof number increases the fluid

velocity.

● magnetic field parameter decreases the

fluid velocity and stress on the wall.

● slip parameter increases the flow velocity,

but decreases the stress on the wall

● porosity parameter increases the flow

velocity and stress on the wall.

These results are benchmarked with some reports in

existing literature, and they are in consonance.

M0.1,0.5,1.0,1.5,2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.1

0.2

0.3

0.4

0.5

u y

u y against y for 0.1,0.5,1.0,1.5,2.0

o

0.3, 0.5, 1.0, 1.5, 2.0

0.5

1.0

1.5

2.0

2.5

3.0

y

0.1

0.2

0.3

0.4

0.5

0.6

u y

u y against y for o0.3,0.5,1.0,1.5,2.0

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APPENDICES

Φ00(y) = (A-y)/A

Φ01(y) = (3 A y Sc-3 y2 Sc+2 A2 y δ Sc-3 A y2 δ

Sc+y3 δ Sc)/(6 A)

Φ02(y) = (30 A2 y -90 A y2+60 y3+30 A3 y δ

-60 A2 y2 δ +8 A4 y δ2-20 A2 y3 δ2+15 A y4

δ2-3 y5 δ2)/(360 A)

Φ10(y) = 0

Φ11(y) = (2 A2 y δ Sc-3 A y2 δ Sc+y3 δ Sc)/(6 A)

Φ12(y) = 1/(360 A λ) (-15 A3 y γ ψ+30 A y3 γ ψ-15

y4 γ ψ+30 A2 y γ Sc-90 A y2 γ Sc+60 y3 γ Sc+15 A3 y

δ λ -30 A y3 δ λ +15 y4 δ λ +8 A4 y δ2 λ -20

A2 y3 δ2 λ +15 A y4 δ2 λ -3 y5 δ2 λ );

Θ00(y) = (A-y)/A

Θ01(y) = -(((-A y+y2) γ)/(2 A λ))

Θ02(y) = 1/(12 A2 λ2) (A3 y γ2-3 A2 y2 γ2+2 A y3

γ2+3 A3 y λ Εc Pr-6 A2 y2 λ Εc Pr+4 A y3 λ Εc

Pr-y4 λ Εc Pr+6 A3 y λ ΕcGc Gr Pr-12 A2 y2 λ ΕcGc

Gr Pr+8 A y3 λ ΕcGc Gr Pr-2 y4 λ ΕcGc Gr Pr+3 A3 y λ

Εc Pr-6 A2 y2 λ Εc Pr+4 A y3 λ Εc Pr-y4 λ Εc

Pr-6 A2 y λ Dr Sc+6 A y2 λ Dr Sc-4 A3 y δ λ Dr Sc+6

A2 y2 δ λ Dr Sc-2 A y3 δ λ DrSc)

Θ10(y) = 0

Θ11(y) = -(((-A y+y2) γ)/(2 A λ))

Θ12(y) = 1/(96 A λ2) (8 A2 y γ2-24 A y2 γ2+16 y3 γ2-

A3 m y +2 A m y3-m y4-32 A2 y δ λ Dr Sc+48

A y2 δ λ Dr Sc-16 y3 δ λ Dr Sc+8 A2 y λ Pr Sc-24 A y2

λ Pr Sc+16 y3 λ Pr Sc+4 A3 y δ λ Pr Sc-16 A2 y2 δ λ Pr

Sc+16 A y3 δ λ Pr Sc-4 y4 δ λ PrSc)

u00(y) = 0

u01(y) = (2 A3 y Gc-3 A2 y2Gc+A y3 Gc+2 A3 α Gc-3

A y2 α Gc+y3 α Gc+2 A3 y Gr-3 A2 y2Gr+A y3 Gr+2

A3 α Gr-3 A y2 α Gr+y3 α Gr)/(6 A (A+α));

u02(y) = 1/(360 A (A+α) λ) (15 A4 y λ Gc-60 A3 y2 λ

Gc+60 A2 y3 λ Gc-15 A y4 λ Gc-45 A3 y α λ Gc+60 A

y3 α λ Gc-15 y4 α λ Gc+15 A4 y γ Gr-30 A2 y3 γ

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Gr+15 A y4 γ Gr+15 A3 y α γ Gr-30 A y3 α γ Gr+15

y4 α γ Gr+15 A4 y λ Gr-60 A3 y2 λ Gr+60 A2 y3 λ Gr-

15 A y4 λ Gr-45 A3 y α λ Gr+60 A y3 α λ Gr-15 y4 α

λ Gr-8 A5 y λ Gc M1+20 A3 y3 λ Gc M1-15 A2 y4 λ Gc

M1+3 A y5 λ Gc M1-48 A4 y α λ Gc M1+60 A3 y2 α λ

Gc M1-15 A y4 α λ Gc M1+3 y5 α λ Gc M1-8 A5 y λ

Gr M1+20 A3 y3 λ Gr M1-15 A2 y4 λ Gr M1+3 A y5 λ

Gr M1-48 A4 y α λ Gr M1+60 A3 y2 α λ Gr M1-15 A

y4 α λ Gr M1+3 y5 α λ Gr M1+15 A4 y λ Gc Sc-30 A2

y3 λ Gc Sc+15 A y4 λ Gc Sc+15 A3 y α λ Gc Sc-30 A

y3 α λ Gc Sc+15 y4α λ Gc Sc+8 A5 y δ λ Gc Sc-20 A3

y3 δ λ Gc Sc+15 A2 y4 δ λ Gc Sc-3 A y5 δ λ Gc Sc+8

A4 y α δ λ Gc Sc-20 A2 y3 α δ λ Gc Sc+15 A y4 α δ λ

Gc Sc-3 y5 α δ λ GcSc)

u10(y) = 0

u11(y) = 0

u12(y) = 1/(360 A (A+α)2) (15 A5 y Gc-60 A4 y2

Gc+60 A3 y3 Gc-15 A2 y4 Gc+15 A5 α Gc-45 A4 y α

Gc-60 A3 y2 α Gc+120 A2 y3 α Gc-30 A y4 α Gc-45

A4 α2 Gc+60 A y3 α2 Gc-15 y4 α2 Gc+15 A5 y Gr60

A4 y2 Gr +60 A3 y3 Gr-15 A2 y4 Gr+15 A5 α Gr -45

A4 y α Gr -60 A3 y2 α Gr +120 A2 y3 α Gr-30 A y4 α

Gr-45 A4 α2 Gr +60 A y3 α2 Gr-15 y4 α2 Gr+8 A6 y δ

Gc Sc-20 A4 y3 δ Gc Sc+15 A3 y4 δ Gc Sc-3 A2 y5 δ Gc

Sc+8 A6 α δ Gc Sc+8 A5 y α δ Gc Sc-40 A3 y3 α δ Gc

Sc+30 A2 y4 α δ Gc Sc-6 A y5 α δ Gc Sc+8 A5 α2 δ Gc

Sc-20 A2 y3 α2 δ Gc Sc+15 A y4 α2 δ Gc Sc-3 y5 α2 δ

GcSc)

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present

research at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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(Attribution 4.0 International, CC BY 4.0)

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