Magnetohydrodynamic Natural Convection Flow with Newtonian Heating and Mass Diffusion over an Infinite Plate that Applies Shear Stress to a Viscous Fluid

Magnetohydrodynamic Natural Convection Flow with Newtonian Heating and Mass Diffusion over an Infinite Plate that Applies Shear Stress to a Viscous Fluid

Dumitru Vieru^a, Corina Fetecau^a, Constantin Fetecau^b,c, and Niat Nigar^d

aDepartment of Theoretical Mechanics, Technical University of Iasi, Iasi 700050, Romania

bDepartment of Mathematics, Technical University of Iasi, Iasi 700050, Romania

cAcademy of Romanian Scientists, Bucuresti 050094, Romania

dAbdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan Reprint requests to Con. F.; E-mail:c_fetecau@yahoo.com

Z. Naturforsch.69a, 714 – 724 (2014) / DOI: 10.5560/ZNA.2014-0068 Received September 2, 2014 / published online November 20, 2014

Unsteady magnetohydrodynamic natural convection flow with Newtonian heating and constant mass diffusion over an infinite vertical plate that applies an arbitrary time-dependent shear stress to a viscous optically thick fluid is studied in the presence of a heat source. Radiative effects are taken into consideration and exact solutions for the dimensionless velocity and temperature are established under Boussinesq approximation. The solutions that have been obtained, uncommon in the literature, satisfy all imposed initial and boundary conditions and can generate exact solutions for any motion problem with technical relevance of this type. For illustration, a special case is considered and the influence of pertinent parameters on the fluid motion is graphically underlined.

Key words:Natural Convection Flow; Newtonian Heating; Mass Diffusion; Boundary Shear Stress.

1. Introduction

Theoretical study of magnetohydrodynamic (MHD) natural convection flows over an infinite plate con- tinues to receive extensive attention due to their in- dustrial and technological applications. Such flows have important applications in polymer industry and metallurgy where hydromagnetic techniques are being used. In astrophysics and geophysics, MHD is applied to the study of stellar and solar structures, interstel- lar matter, radio propagation through the ionosphere and so on. Soundalgekar et al. [1,2] and Raptis and Singh [3] seem to be the first authors who took into consideration the influence of magnetic field in their works.

The radiative heat transfer also has an important role in manufacturing industries and the interaction of nat- ural convection with the thermal radiation was studied by Mansour [4] in an oscillatory flow over a vertical plate. Effects of thermal radiation on the flow near an infinite vertical oscillating isothermal plate in the pres- ence of a transversely applied magnetic field have been

studied by Chandrakala and Bhaskar [5]. Many other studies on MHD natural convection flow under dif- ferent physical situations have been developed. Some of the most recent exact solutions for such flows are those of Ghosh and Beg [6], Toki [7], Rajesh [8], Seth et al. [9], Samiulhaq et al. [10], Fetecau et al. [11,12], and Narahari and Debnath [13].

However, in all these papers the flow is driven by a prescribed surface temperature or by a sur- face heat flux. In the following we assume that the flow is set up by a Newtonian heating from the plate, i.e. the heat transfer from the surface is proportional to the local surface temperature. The Newtonian heating, with extensive applications in many important engineering devices, seems to be initiated by Merkin [14]. The effects of Newto- nian heating on the natural convection flow over an infinite plate have been investigated by Lesnic et al. [15], and Pop et al. [16]. Meantime, many interesting studies regarding the natural convection with Newtonian heating appeared (see for instance Chaudary and Jain [17], Salleh et al. [18], Narahari and

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Ishak [19], Narahari and Nayan [20], and Hussanan et al. [21]).

MHD natural convection flows combined with heat and mass transfer also have important applications in engineering and chemical and biological sciences (see for instance Jaluria [22] and Kays et al. [23]). The mass transfer appears in the theory of stellar struc- tures, and observable effects are detectable on the so- lar surface. Early enough Soundalgekar et al. [1] pro- vided an exact analysis of combined effects of heat and mass transfer on the MHD flow of a viscous fluid. An analytical study on mass transfer effects on the flow over an accelerated vertical plate was performed by Jha et al. [24]. Similar studies continuously appeared in the literature. However, the most recent and inter- esting results seem to be those of Muthucumaraswamy and Vijayalakshmi [25], Hayat et al. [26], Rajput and Kumar [27,28], and Kishore et al. [29]. Newtonian heating and mass transfer effects on the free con- vection flow past an uniformly accelerated or oscil- lating plate in the presence of thermal radiation are also studied by Narahari et al. [30], respectively, Hus- sanan et al. [31]. It is worth pointing out that in all these works boundary conditions for velocity are used although in some physical situations the force with which the plate is moved is prescribed. Furthermore, in the Newtonian mechanics force is the cause and kine- matics is the effect (see Rajagopal [32] for a detailed discussion on the problem) and the ‘no slip’ bound- ary condition may not be applicable to flows of some polymeric fluids. In such cases a boundary condition on the shear stress can be used (see for instance Tal- houk [33]).

Based on the above mentioned remarks, the pur- pose of the present work is to establish exact solu- tions for the unsteady MHD natural convection flow in- duced by an infinite plate that applies an arbitrary time- dependent shear stress to a viscous fluid with New- tonian heating. The viscous dissipation is neglected but the analysis of heat and mass transfer is devel- oped in the presence of radiative effects and a heat source. The general solutions that have been obtained satisfy all imposed initial and boundary conditions and they can generate a large class of exact solutions cor- responding to different problems with technical rel- evance. Consequently, the problem of natural con- vection flow of a viscous fluid over an infinite plate that applies a shear stress to the fluid with Newto- nian heating and constant mass diffusion is complete

solved. Finally, the case when the plate applies a con- stant shear stress to the fluid is considered and the effects of pertinent parameters on the dimensionless velocity and temperature fields are graphically under- lined.

2. Mathematical Formulation

Let us consider the unsteady natural convection flow of an incompressible, radiating, and electrically con- ducting viscous optically thick fluid over an infinite vertical plate with Newtonian heating and constant mass diffusion. They-axis is normal to the plate and the x-axis is taken along the plate in the vertically upward direction. Initially, at time t =0, the whole system is at rest at the temperature T∞ with a con- centration levelC_∞ at all points. After time t =0⁺ the plate applies an arbitrary time-dependent shear stress τ0f(t) to the fluid in the presence of a heat source [13] and the concentration level at the plate is Cw. Here τ0 is a constant shear stress and the function f(·) is piecewise continuous and f(0) =0.

The heat transfer from the plate surface is assumed to be proportional to the local surface temperature T.

A magnetic field of uniform strength Bis applied perpendicular to the plate and the magnetic Reynolds number is assumed to be small enough so that the in- duced magnetic field can be neglected [34]. In addi- tion, we also make the following assumptions: i) the radiative effects in the flow direction are negligible in comparison to those in they-direction; ii) the iner- tia terms, viscous dissipation heat, and Soret and Du- four effects are negligible; iii) the optically thick radi- ation limit is considered, and there is no applied elec- tric field. As the plate is infinite, all physical quantities are functions ofyandt only. In these conditions un- der the usual Boussinesq approximation, i.e. the den- sity changes with temperature, which gives rise to the buoyancy force, this unsteady flow is governed by the following partial differential equations [35]:

∂u(y,t)

∂t =ν∂²u(y,t)

∂y² +gα[T(y,t)−T_∞] +gβ[C(y,t)−C_∞]−σB²

ρ u(y,t); y,t>0, (1)

ρc_p∂T(y,t)

∂t =k∂²T(y,t)

∂y² −∂q_r(y,t)

∂y

−Q[T(y,t)−T_∞]; y,t>0,

(2)

∂C(y,t)

∂t =D∂²C(y,t)

∂y² ; y,t>0, (3) where u and T are velocity and temperature of the fluid,Cis the species concentration in the fluid,ν is the kinematic viscosity,g is the gravitational acceler- ation, α is the volumetric coefficient of thermal ex- pansion, β is the volumetric coefficient of mass ex- pansion, σ is electrical conductivity, ρ is the den- sity of the fluid, c_p is the specific heat at constant pressure, k is the thermal conductivity, q_r is the ra- diative heat flux, Q is the dimensional heat absorp- tion/generation coefficient, andDis the species diffu- sion coefficient.

Of course, the radiative heat flux can be simplified as

q_r=−4σ_s 3k_R

∂T⁴

∂y , (4)

if the Rosseland approximation [36] is used. Here σs

is the Stefan–Bolzmann constant, and k_R is the mean absorption coefficient. If further the temperature dif- ferences within the flow are sufficiently small, (2) can be reduced to [20]

ρc_p∂T(y,t)

∂t =k

1+16σ_sT_∞³ 3k k_R

∂²T(y,t)

∂y²

−Q[T(y,t)−T_∞]; y,t>0.

(5)

The corresponding initial and boundary conditions are

u(y,0) =0, T(y,0) =T∞,

C(y,0) =C∞; y≥0, (6)

∂u(y,t)

∂y _y=0

=τ₀ µ f(t),

∂T(y,t)

∂y _y=0

=−h kT(0,t), C(0,t) =C_w; t>0,

(7)

u(y,t)→0, T(y,t)→T_∞,

C(y,t)→C_∞ as y→∞, (8)

wherehis the heat transfer coefficient for Newtonian heating.

By introducing the next non-dimensional variables and functions

y^∗=h

ky, t^∗=ν h

k 2

t, u^∗= k

νhGru, T^∗=T−T∞

T_∞ , C^∗= C−C_∞

Cw−C_∞, Q^∗= Qk (1+Nr)h², f^∗(t^∗) = τ0

ν µGr k

h 2

f k²t^∗

νh²

,

(9)

and dropping out the star notation, we attain to the next non-dimensional initial boundary-value problem:

∂u(y,t)

∂t =∂²u(y,t)

∂y² −Mu(y,t) +T(y,t) +NC(y,t); y,t>0,

(10)

Pr_eff∂T(y,t)

∂t =∂²T(y,t)

∂y² −QT(y,t);

Sc∂C(y,t)

∂t =∂²C(y,t)

∂y² ; y,t>0,

(11)

u(y,0) =0, T(y,0) =0 ;

C(y,0) =0 ; y≥0, (12)

∂u(y,t)

∂y y=0

=f(t),

∂T(y,t)

∂y _y=0

=−[1+T(0,t)], C(0,t) =1 ; t>0,

(13)

u(y,t)→0, T(y,t)→0,

C(y,t)→0 as y→∞. (14)

Into (10) and (11), we have denoted by M=σB²

µ k

h 2

, N=Gm

Gr , Gr=gαT∞

ν² k

h 3

,

Gm=gβ(C_w−C_∞) ν²

k h

3

, Pr_eff= Pr

1+Nr, Pr=µc_p k , Nr=16σ_sT_∞³

3k k_R , Sc=ν D,

(15)

the magnetic parameter, the buoyancy ratio parameter, the thermal Grashof number, the mass Grashof num- ber, the effective Prandtl number [37], the Prandtl num- ber, the radiation parameter, and the Schmidt number,

respectively. Pr_effand Sc are transport parameters rep- resenting the thermal and mass diffusivity as compared to the momentum diffusivity while N represents the relative contribution of the mass transport rate on nat- ural convection flow [38].

3. Solution of the Problem

The two partial differential equations (11) are not coupled to the momentum equation (10) and the species concentration in the fluid, namely

C(y,t) =erfc y√ Sc 2√

t

!

, (16)

where erfc(·)is the complementary error function, has been previously determined by different authors (see [38, Eq. (13a)], for instance). In the following, by us- ing the Laplace transform technique, we determine the fluid temperature and then its velocity.

3.1. Calculation of the Temperature Field

Applying the Laplace transform with respect to the temporal variablet to (11)₁and using the correspond- ing initial and boundary conditions, we find that

Pr_eff(q+Q)T¯(y,q) =∂²T¯(y,q)

∂y² ; y>0, (17) where q is the transform parameter, and the Laplace transform ¯T(y,q)ofT(y,t)has to satisfy the conditions

∂T¯(y,q)

∂y _y=0

=−

T¯(0,q) +1 q

and T¯(y,q)→0 as y→∞.

(18)

The solution of the differential equation (17) subjected to the conditions (18) is given by

T¯(y,q) = 1 q[p

Pr_eff(q+Q)−1]

·exph

−yp

Pr_eff(q+Q)i .

(19)

Now, writing ¯T(y,q)in a suitable form, applying the inverse Laplace transform, and using (A1)–(A3) from Appendix(see also [39] and [40, Eq. (B.2)]) as well as

the convolution theorem, we obtain for the temperature fieldT(y,t)the simple expression

T(y,t) = 1 QPr_eff−1

1+√ QPr_eff

2 e^−y

√QPr_eff

·erfc y√

Pr_eff 2√

t −p Qt

+1−√ QPr_eff 2

·e^y

√QPrefferfc y√

Pr_eff 2√

t +p Qt

−e^{−y+Prefft −Qt}erfc y√

Pr_eff 2√

t −

√t

√Pr_eff

; Q6= 1

Pr_eff.

(20)

As expected, in absence of the heat source whenQ=0, (20) reduces to the known form [38, Eq. (14)]. Further- more, the expression (20) for temperature is not valid forQ=1/Pr_eff. In order to determine the solution in this case, we takeQ=1/Pr_effinto (19) and follow the same way as before. The corresponding temperature field is (see (A1) – (A6) from Appendixor Hetnarski [39, p. 252]).

T(y,t) = √

Pr_eff+1 2Pr_eff t−y

2+

√Pr_eff 4

·e^−yerfc y√

Pr_eff 2√

t −

√t

√Pr_eff

+ 1−√

Pr_eff 2Pr_eff t−

√Pr_eff 4

·e^yerfc y√

Pr_eff 2√

t +

√t

√Pr_eff

+

√t

√

πPr_effexp

−y²Pr_eff 4t − t

Pr_eff

.

(21)

Direct computations show that T(y,t), given by (20) or (21), satisfies all imposed initial and boundary con- ditions. The common expression of the two terms of (13)₂, obtained by two different manners for the gen- eral case (20), is given by

∂T(y,t)

∂y _y=0

= 1

QPr_eff−1

·

exp

1−QPr_eff Pr_eff t

erfc

−

√t

√Pr_eff

−p

QPr_efferfp Qt

−QPr_effi

=−[1+T(0,t)] as QPr_eff6=1,

(22)

where erf(·)is the error function of Gauss.

3.2. Calculation of the Velocity Field

Applying the Laplace transform to (10) and using the initial condition (12)₁, we find that

(q+M)u(y,¯ q) =∂²u(y,q)¯

∂y² +T¯(y,q) +NC(y¯ ,q); y>0,

(23)

where ¯u(y,q) and ¯C(y,q)are the Laplace transforms of u(y,t) andC(y,t), respectively. According to the boundary conditions (13)₁and (14)₁, ¯u(y,q)has to sat-

isfy the conditions

∂u(y,q)¯

∂y _y=0

=f¯(q) and

¯

u(y,q)→0 as y→∞,

(24) where ¯f(q) is the Laplace transform of the function f(t). The expression of ¯C(y,q)is immediately obtained applying the Laplace transform to (11)₂, using the ini- tial condition (12)₃, and taking into account the bound- ary conditions (13)₃and (14)₃.

In order to determine the expression of ¯u(y,q), namely

u(y,¯ q) =− f¯(q)

√q+M e^−y

√q+M+N

√q+Me^−y

√qSc−√ qSc e^−y

√q+M

q[(1−Sc)q+M]√ q+M +

√q+Me^−y

√

Pr_eff(q+Q)−p

Pr_eff(q+Q)e^−y

√q+M

q[(1−Pr_eff)q+M−QPr_eff][p

Pr_eff(q+Q)−1]√

q+M; Pr_eff6=1, Sc6=1,

(25)

we introduce (19) and the expression of ¯C(y,q)into (23) and use the corresponding boundary conditions.

Finally, applying the inverse Laplace transform to (25) and using the convolution theorem together with (A1) – (A3), (A7) and (A8), we obtain after lengthy but straightforward computations

u(y,t) =u_m(y,t) +u_T(y,t) +u_C(y,t);

Pr_eff6=1, Sc6=1, M6=0, (26)

where

u_m(y,t) =− 1

√ π

Z _t 0

f(t−s)

√s

·exp

−y² 4s−Ms

ds,

(27)

u_T(y,t) = 1 (Pr_eff−1)√

π

· Z _t

0

√1 s

exp

−y² 4s−Ms

− 1

√Pr_effexp

−y²Pr_eff 4s −Qs

·g

t−s; QPr_eff−M

Pr_eff−1 ,Q,− 1

√Pr_eff

ds, (28)

u_C(y,t) = N M

rdSc π

· Z t

0

√1 sexp

−y²

4s−Ms+d(t−s)

·erfp

d(t−s) ds + N

2M (

2erfc y√ Sc 2√

t

!

−e^dt

"

e^y

√

dScerfc y√ Sc 2√

t +√ dt

!

+ e^−y

√

dScerfc y√ Sc 2√

t −√ dt

!#) ,

(29)

what could be called the mechanical, thermal, and mass components of the dimensionless velocity field u(y,t);d=M/(Sc−1)and

g(t;a,b,c) = b a(b−c²) + b−a

a(c²−b+a)e^−at− c√ b a(b−c²)erfc

√ bt

− c√ b−a

a(c²−b+a)e^−aterfcp

(b−a)t

+ c²

(c²−b)(c²−b+a)e^(c2−b)terfc c√ t

. (30)

Direct computations clearly show thatu(y,t)given by (26) satisfies all imposed initial and boundary condi- tions. However, in order to prove the boundary condi- tion (13)₁, we must use the identity

y 2√ π

Z t 0

h(t−s) s√

s exp

−y² 4s−As

ds=

√2 π

Z ∞

y/(2√ t)

h

t− y² 4s²

exp

−s²−Ay² 4s²

ds, (31)

where h(·) is an arbitrary function, and A is con- stant.

Case Pr_eff=1

In this case by making Pr_eff=1 into (25) and fol- lowing the same way as before, we observe that the components u_m(y,t) and u_C(y,t) remain unchanged while

u_T(y,t) = 1 (Q−1)(M−Q)

· 1+√

Q 2 e^−y

√ Qerfc

y 2√

t−p Qt

+1−√ Q 2 e^y

√Qerfc y

2√ t+p

Qt

−e^{−y−(Q−1)t}erfc y

2√ t−√

t

+ 1

2(M−Q)√ M

e^y

√ Merfc

y 2√

t+√ Mt

−e^−y

√ Merfc

y 2√

t−√ Mt

+ 1

√ π

Z _t 0

"

e^−Q(t−s)

√t−s +e^t−serfc −√ t−s

#

·

e^y

√ Merfc

y 2√

s+√ Ms

−e^−y

√ Merfc

y 2√

s−√ Ms

ds

if Q6=M and Q6=1.

(32)

Case Sc=1

Simple computations show that one obtains the solu- tion corresponding to this case by substitutingu_C(y,t) from (26) with

uC(y,t) = N M

erfc

y 2√

t

−1 π

Z t 0

1

ps(t−s)exp

−y² 4s−Ms

ds

)

;

M6=0. (33)

CaseM=0 (Absence of Magnetic Field)

In this case the first two components of the velocity are given by (27) and (28) withM=0, while

u_C(y,t) = N Sc−1

·h√

ScF₁(y,t)−F₁(y√ Sc,t)i

,

(34)

F₁(y,t) =

t+y² 2

erfc

y 2√

t

−y√

√t π exp

−y² 4t

.

(35)

In particular, if the heat source is absent (namelyQ= 0), the componentuT(y,t)can be reduced to the sim- pler form

u_T(y,t) = Pr_eff Pr_eff−1

h F₂(yp

Pr_eff,t)

−p

Pr_effF₂(y,t)−F₃(yp Pr_eff,t) +p

Pr_effF3(y,t)i +

√Pr_eff Pr_eff−1

·h F₄(yp

Pr_eff,t)−p

Pr_effF₄(y,t)i

+ 1

Pr_eff−1 h

F₁(yp

Pr_eff,t)−p

Pr_effF₁(y,t)i , (36)

where the functionsF₂,F₃,F₄are defined as F2(y,t) =erfc

y 2√

t

, F₃(y,t) =exp

t

Pr_eff− y

√Pr_eff

·erfc y

2√ t−

√t

√Pr_eff

, F₄(y,t) =2√

√ t π exp

−y² 4t

−yerfc y

2√ t

. (37)

The solution corresponding to this last case, as it was to be expected, is in accordance with the result ob- tained by Narahari and Dutta [38, Eq. (15a)]. More precisely, the velocity componentsu_T(y,t)andu_C(y,t) given by (34) and (36) are identically as form with those from [38, Eq. (15a)] and the additional factors

√

Sc and √

Pr_eff appearing in front of the functions

F1(y,t), F2(y,t), F3(y,t), and F4(y,t) into (34) and (36) are due to the shear stress boundary condition (13)₁.

4. Some Numerical Results and Discussion

In order to get some physical insight of our results, we consider the special case when the plate applies a constant shear to the fluid and determine the influence of the essential parameters Pr_eff,M,N, andQon the velocity and temperature. In this case the thermal and mass components of the dimensionless velocityu(y,t) remain unchanged, while the function f(·)from (27) has to be the Heaviside unit step functionH(·). Conse- quently, the mechanical componentu_m(y,t)of the ve- locity becomes

u_m(y,t) =− 1

√ π

Z t 0

√1 s

·exp

−y² 4s−Ms

ds,

(38)

or equivalently (see (A3) fromAppendix) u_m(y,t) = 1

2√ M

e^y

√ Merfc

y 2√

t+√ Mt

−e^−yMerfc y

2√ t−√

Mt

; M6=0.

(39)

Fig. 1. Temperature profiles for different values of t and Preff.

Since the solutions corresponding to this problem are well defined, the distributions of velocity and tempera- ture fields can be easy determined for any set of values of physical parameters.

The dimensionless temperature profiles againstyat two values of the timetare graphically represented for three different values of Pr_eff in Figure1 in the pres- ence of heat absorption (Q>0). It is observed that the temperature of the fluid is increasing with time, de- creases with respect to Pr_eff, and smoothly decreases from a maximum value at the boundary to a minimum value for large values of y. Furthermore, an increase of the effective Prandtl number implies a decreasing of the thermal boundary layer thickness. Consequently, greater values of Pr_effare equivalent to decreasing the thermal conductivity and therefore heat can easier dif- fuse away from the heated surface at smaller values of the effective Prandtl number. The influence ofQon the temperature is shown in Figure2for the case of heat absorption (cooling of the plate). As expected, in the presence of heat absorption within the boundary layer the temperature of the fluid decreases. Of course, an opposite effect appears in the case of heat generation whenQ<0 but the corresponding graphs are not in- cluded here.

The velocity profiles for different values oft, Pr_eff, M, Sc,N, andQare illustrated in Figures3–8. A series of calculations was performed for different situations with typical values. It clearly results from Figure3that

Fig. 2. Temperature profiles for different values of t and Q>0 (heat absorption).

Fig. 3. Velocity profiles for different values oft.

the fluid velocity against yincreases with increasing time. Near the surface of the plate the fluid velocity in- creases, becomes maximum and then decreases to an asymptotic value for large value ofy. The influence of Pr_eff andM on the fluid motion is underlined in Fig- ures4 and5. The dimensionless velocity of the fluid is a decreasing function with respect to both numbers.

A stronger decrease appears with regard to the effective Prandtl number but in both cases, as well as in Figure1, there are velocity over-shoots close to the plate. Then the velocity profiles smoothly descend to their lowest

Fig. 5. Velocity profiles for different values ofM.

Fig. 4. Velocity profiles for different values of Pr_eff.

values for increasingy. In all cases the values of ve- locity at any distanceyare always higher or lower for distinct values oft, Pr_efforM.

The effects of the buoyancy ratio parameterN and the Schmidt number Sc on the motion, as it results from Figures6 and7, are opposite. The fluid velocity de- creases with respect to Sc and increases for increasing N. An increase of Sc, as it is mentioned in [38], means increase in the hydrodynamic boundary layer thickness for a fixed species diffusivity and this will cause the decrease in velocity. Physical significance of the effect

Fig. 6. Velocity profiles for different values of Sc.

Fig. 7. Velocity profiles for different values ofN.

of N on the velocity distribution is clearly explained by Narahari and Dutta [38]. The buoyancy force due to the species diffusion adds to the thermal buoyancy force andN=0 corresponds to the case when the natu- ral convection arises from the thermal buoyancy force only. The influence of Qon velocity profiles is pre- sented in Figure8. It is observed that the fluid velocity decreases in the case of heat absorption. Moreover, all velocity profiles assume a parabolic shape in the vicin- ity of the plate and then smoothly decrease to zero for ygoing to infinity.

5. Conclusions

The main purpose of this work is to provide exact solutions for the unsteady natural convection flow of an incompressible viscous fluid along an infinite verti- cal plate with Newtonian heating, mass diffusion, heat source, and shear stress boundary conditions. More ex- actly, contrary to what is usually assumed, after time t =0 the plate applies an arbitrary time-dependent shear stress to the fluid. For completion, the magnetic and radiation effects are taken into consideration and exact solutions for the dimensionless temperature and velocity fields are obtained in the absence of internal dissipation. In the absence of a heat source, the tem- perature distribution reduces to that obtained by Nara- hari and Dutta [38, Eq. (14)] and according to Mag- yari and Pantokratoras [37] the heat transfer character-

Fig. 8. Velocity profiles for different values ofQ>0 (heat absorption).

istics can be brought to light by means of the effective Prandtl number Pr_eff only. The velocityu(y,t)is pre- sented as a sum of its mechanical, thermal, and mass components. Moreover, bearing in mind an important remark from [12, Sect. 2], it is worth pointing out that the fluid velocity corresponding to the same motion of linearly viscous fluids through a porous medium does not depend on magnetic and permeability parameters independently, but only by a combination of them that can be called the effective permeabilityK_eff.

Finally, in order to obtain some physical insight of present results, the case when the plate applies a con- stant shear stress to the fluid is considered and the ef- fects of pertinent parameters on the dimensionless tem- perature and velocity are graphically underlined. The main results are: i) Exact solutions for the dimension- less temperature and velocity are established. Due to its generality, (26) can generate a large class of exact solutions corresponding to any motion of this type with technical relevance; ii) The velocity of the fluid is pre- sented as a sum of its mechanical, thermal, and mass components. Consequently, the influence of each of them on the fluid motion is delimited; iii) The thermal boundary layer thickness, as well as the fluid temper- ature, increases in time and decreases with respect to the effective Prandtl number Pr_eff; iv) The fluid veloc- ity is a decreasing function with respect to parameters Pr_eff,M, and Sc and increases for increasingN. It also decreases in the case of heat absorption.

Appendix

L⁻¹ (e^−a

√q+b

q )

=1 2

e^−a

√ berfc

a 2√

t−√ bt

+e^a

√ berfc

a 2√

t+√ bt

,

(A1)

L⁻¹ e^−a

√q

√q+b

= 1

√πtexp

−a² 4t

−be^ab+b2terfc a

2√ t+b√

t

,

(A2)

Z _t 0

√1 se^u2s−x

2

s ds=

√ π 2iu

e^−2iuxerfc x

√t−iu√ t

−e^2iuxerfc x

√t+iu√ t

,

(A3)

L⁻¹ (e^−a

√q+b

q² )

= 1

2

t− a 2√

b

e^−a

√ berfc

a 2√

t−√ bt

+

t+ a 2√

b

e^a

√ berfc

a 2√

t+√ bt

,

(A4)

L⁻¹n√ qe^−a

√qo

= a²−2t 4t²√

πtexp

−a² 4t

; Re(a²)>0,

(A5)

L⁻¹ (√

q+be^−a

√q+b

q²

)

=

−a 4

e^−a

√ berfc

a 2√

t−√ bt

+e^a

√ berfc

a 2√

t+√ bt

(A6) +1+2bt

4√ b

e^−a

√ berfc

a 2√

t−√ bt

−e^a

√ berfc

a 2√

t+√ bt

+

√t

√ πexp

−a² 4t −bt

, L⁻¹

(√

q+be^−a

√q+b

q−c )

=

√b+c 2

·e^ct

e^−a

√ b+cerfc

a 2√

t−p (b+c)t

−e^a

√b+cerfc a

2√ t+p

(b+c)t

+ 1

√πtexp

−a² 4t −bt

,

(A7)

Z t 0

1 s√

se^u2s−x

2

s ds=

√ π 2x

·

e^−2iuxerfc x

√t−iu√ t

+e^2iuxerfc x

√t+iu√ t

.

(A8)

Acknowledgement

The author Niat Nigar is highly thankful to the Abdus Salam School of Mathematical Science, GC University, Lahore and Higher Education Commission of Pakistan for generous supporting and facilitating her research work.

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