effects of ramped wall temperature on unsteady two … · 2015. 2. 3. · equipment, formation and...

* Corresponding Author. Email address: [email protected]

Effects of Ramped Wall Temperature on Unsteady Two-Dimensional

Flow Past a Vertical Plate with Thermal Radiation and Chemical

Reaction

V. Rajesh

1, A. J. Chamkha

2*

(1) Department of Engineering Mathematics, GITAM University Hyderabad Campus, Rudraram, Patancheru

Mandal, Medak Dist.-502 329, A.P. India.

(2) Manufacturing Engineering Department, The Public Authority for Applied Education and Training, Shuweikh

70654, Kuwait.

Copyright 2014 © V. Rajesh and A. J. Chamkha. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

The interaction of free convection with thermal radiation of a viscous incompressible unsteady flow past a

vertical plate with ramped wall temperature and mass diffusion is presented here, taking into account the

homogeneous chemical reaction of first order. The fluid is gray, absorbing-emitting but non-scattering

medium and the Rosseland approximation is used to describe the radiative flux in the energy equation. The

dimensionless governing equations are solved using an implicit finite-difference method of the Crank–

Nicolson type, which is stable and convergent. The velocity profiles are compared with the available

theoretical solution and are found to be in good agreement. Numerical results for the velocity, the

temperature, the concentration, the local and average skin friction, the Nusselt number and Sherwood

number are shown graphically. This work has wide application in chemical and power engineering and

also in the study of vertical air flow into the atmosphere. The present results can be applied to an important

class of flows in which the driving force for the flow is provided by combination of the thermal and

chemical species diffusion effects.

Keywords: Heat and mass transfer, unsteady flow, thermal radiation, first-order chemical reaction, ramped wall

temperature, implicit finite-difference method.

1 Introduction

Theoretical investigation of natural convection flow past a vertical flat plate continues to receive

attention in the literature due to its industrial and technological applications. Unsteady free convection

flow past a vertical plate is investigated by a number of researchers considering different sets of thermal

Communications in Numerical Analysis 2014 (2014) 1-17

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Volume 2014, Year 2014 Article ID cna-00218, 17 Pages

doi:10.5899/2014/cna-00218

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conditions at the bounding plate. Mention may be made of the research studies of Georgantopoulos [1],

Raptis and Singh [2, 3], Sacheti et al. [4], Chandran et al. [5, 6], Ganesan and Palani [7] and Makinde and

Olanrewaju [8]. The present trend in the field of chemical reaction analysis is to give a mathematical

model for the system to predict the reactor performance. A large amount of research work has been

reported in this field. In particular, the study of heat and mass transfer with chemical reaction is of

considerable importance in chemical and hydrometallurgical industries. Chemical reaction can be codified

as either heterogeneous or homogeneous processes. This depends on whether they occur at an interface or

as a single phase volume reaction. A few representative fields of interest in which combined heat and mass

transfer with chemical reaction effect plays an important role, are design of chemical processing

equipment, formation and dispersion of fog, distribution of temperature and moisture over agricultural

fields and groves of fruit trees, damage of crops due to freezing, food processing and cooling towers. For

example, formation of smog is a first order homogeneous chemical reaction. Consider the emission of

NO2 from automobiles and other smoke-stacks. This NO2 reacts chemically in the atmosphere with

unburned hydrocarbons (aided by sunlight) and produces peroxyacetylnitrate, which forms an envelope of

what is termed as photochemical smog. All nature and most chemical fibers possess a shrinking potential,

i.e., as soon as they with water and/or warmth come into contact, they change their form and run in. The

order of the chemical reaction depends on several factors. The simplest chemical reaction is the first order

reaction in which the rate of reaction is directly proportional to the species concentration. Chambre and

Young [9] have analyzed a first order chemical reaction in the neighborhood of a horizontal plate. Dass et

al. [10] have studied the effect of homogeneous first order chemical reaction on the flow past an

impulsively started infinite vertical plate with uniform heat flux and mass transfer. Again, mass transfer

effects on moving isothermal vertical plate in the presence of chemical reaction studied by Dass et al. [11].

The dimensionless governing equations were solved by the usual Laplace Transform technique.

Muthucumaraswamy and Ganesan [12] studied the problem of on impulsive motion of a vertical plate with

heat flux and diffusion of chemically reactive species. Rajesh and Varma [13] presented chemical reaction

effects on free convection flow past an exponentially accelerated vertical plate. Rajesh [14] studied the

Effects of mass transfer on flow past an impulsively started infinite vertical Plate with Newtonian heating

and chemical reaction.

In the context of space technology and in processes involving high temperatures, the effects of radiation

are of vital importance. Recent developments in hypersonic flights, missile re-entry, rocket combustion

chambers, power plants for inter planetary flight and gas cooled nuclear reactors, have focused attention on

thermal radiation as a mode of energy transfer, and emphasize the need for improved understanding of

radiative transfer in these processes. The interaction of radiation with laminar free convection heat transfer

from a vertical plate was investigated by Cess [15] for an absorbing, emitting fluid in the optically thick

region, using the singular perturbation technique. Arpaci [16] considered a similar problem in both the

optically thin and optically thick regions and used the approximate integral technique and first-order

profiles to solve the energy equation. Cheng and Ozisik [17] considered a related problem for an

absorbing, emitting and isotropically scattering fluid, and treated the radiation part of the problem exactly

with the normal-mode expansion technique. Raptis [18] has analyzed the thermal radiation and free

convection flow through a porous medium by using perturbation technique. Hossain and Takhar [19]

studied the radiation effects on mixed convection along a vertical plate with uniform surface temperature

using Keller Box finite difference method. In all these papers the flow is considered to be steady. Mansour

[20] studied the radiative and free convections effects on the oscillatory flow past a vertical plate. Raptis

and Perdikis [21] studied the effects of thermal radiation and free convective flow past moving plate. Das

et al. [22] have analyzed the radiation effects on flow past an impulsively started infinite isothermal

vertical plate. The governing equations were solved by the Laplace transform technique. Chamkha et al.

[23] have studied the radiation effects on free convection flow past a semi-infinite vertical plate with mass




transfer. Rajesh et al. [24] studied numerically the effects of radiation and chemical reaction on unsteady

MHD free convection flow of a dissipative fluid past an infinite vertical Plate with Newtonian heating.

In all these investigations, the analytical or numerical solution is obtained assuming the velocity and

temperature conditions at the plate as continuous and well defined. However, several practical problems

may require non uniform or arbitrary wall conditions. Keeping in view this fact, several researchers [25-

28] considered free convection from a vertical plate with step discontinuities in the surface temperature.

Chandran et al. [29] investigated unsteady natural convection flow of a viscous incompressible fluid near a

vertical plate with ramped wall temperature. Rajesh [30] discussed radiation effects on MHD free

convection flow near a vertical plate with ramped wall temperature. Also Rajesh [31] presented Chemical

reaction and radiation Effects on the transient MHD free convection flow of dissipative fluid past an

infinite vertical porous plate with ramped wall temperature. Seth et al. [32] studied MHD natural

convection flow with radiative heat transfer past an impulsively moving plate with ramped wall

temperature. Recently Das [33] studied Magneto hydrodynamics free convection flow of a radiating and

chemically reacting fluid past an impulsively moving plate with ramped wall temperature.

The aim of this work is to study the effects of ramped wall temperature on unsteady two-dimensional free

convection heat and mass transfer flow of a viscous, radiating, chemically reacting, incompressible fluid

past a vertical plate. The order of chemical reaction in this work is taken as first-order reaction. It is hoped

that the results obtained will not only provide useful information for applications, but also serve as a

complement to the previous studies.

2 Mathematical Analysis

Consider a two-dimensional, transient, laminar, natural convection flow of an incompressible viscous

radiating fluid past a vertical plate with ramped wall temperature and mass diffusion. The fluid considered

here is a gray, absorbing-emitting radiation but non-scattering medium. The flow is assumed to be in x -

direction which is taken along the plate in the vertically upward direction, and y -axis is taken normal to

the plate. Initially, for time 0t , both the fluid and the plate are assumed to be at the same temperature

T and concentration C

. At time 0t , the temperature of the plate is raised or lowered to

0

w

tT T T

t

when 0t t , and thereafter, for 0t t , is maintained at the constant temperature wT and

the concentration level at the plate is raised to wC . It is assumed that the effect of viscous dissipation is

negligible in energy equation and there exists a homogeneous first order chemical reaction between the

fluid and species concentration. It is also assumed that the concentration C of the diffusing species in the

binary mixture is very less in comparison to the other chemical species, which are present. This leads to

the assumption that the Soret and Dufour effects are negligible. Then under the above assumptions, the

governing boundary layer equations of mass, momentum, energy and species concentration for free

convective flows with Boussinesq’s approximation are as follows:

0u v

x y

(2.1)

2

2

u u u uu v g T T g C C

t x y y

(2.2)

2

2

1 r

p

T T T T qu v

t x y y C y

(2.3)




2

2 l

C C C Cu v D K C C

t x y y

(2.4)

The initial and boundary conditions

0u , 0v , T T , C C

for 0t

0u , 0v , wC C at 0y for 0t

0

w

tT T T T

t

at 0y for 00 t t

wT T at 0y for 0t t

0u , T T , C C

, at 0x for 0t

0u , T T , C C

, as y for 0t (2.5)

We now assume Rosseland approximation (Brewster [34]), which leads to the radiative heat flux rq is

given by

44

3

sr

e

Tq

k y

(2.6)

where s is the Stefan–Boltzmann constant and ek is the mean absorption coefficient, respectively. It

should be noted that by using the Rosseland approximation we limit our analysis to optically thick fluids.

If temperature differences within the flow are sufficiently small, then equation (2.6) can be linearized by

expanding 4T into the Taylor series aboutT

, which after neglecting higher order terms takes the form:

3 44 4 3T T T T (2.7)

In view equations (2.6) and (2.7), equation (2.3) reduces to

32 2

2 2

16

3

s

e p

TT T T T Tu v

t x y y k C y

(2.8)

The local as well as average values of the skin friction, Nusselt number and the Sherwood number are as

follows:

0y

u

y

(2.9)

00

1

y

L

L

udx

L y

(2.10)




0y

x

w

Tx

yNu

T T

(2.11)

0

0

y

L

L

w

T

yNu dx

T T

(2.12)

0y

x

w

Cx

ySh

C C

(2.13)

0

0

y

L

L

w

C

ySh dx

C C

(2.14)

On introducing the following non-dimensional quantities:

0

yy

t

,

0

xx

t

,

0

tt

t

, 0tu u

, 0tv v

,

w

T TT

T T

,

p

r

CP

, 0r lK K t

34

e

s

kF

T

,w

C CC

C C

, cS

D

,

3

20w

r

g T T tG

,

3

* 20w

m

g C C tG

, m

r

GN

G (2.15)

Equations (2.1), (2.2), (2.8) and (2.4) are reduced to the following non-dimensional form

0u v

x y

(2.16)

2

2

u u u uu v T NC

t x y y

(2.17)

2

2

1 41

3r

T T T Tu v

t x y P F y

(2.18)

2

2

1r

c

C C C Cu v K C

t x y S y

(2.19)

The corresponding initial and boundary conditions are




0u , 0v , 0T , 0C for 0t

0u , 0v , 1C at 0y for 0t

T t at 0y for 0 1t

1T at 0y for 1t

0u , 0T , 0C at 0x for 0t

0u , 0 , 0C as y for 0t (2.20)

According to the above non-dimensionalization process, the characteristic time 0t can be defined as

1

3

0 2

3w

t

g T T

(2.21)

Using the non-dimensional quantities specified in (2.15), the local as well as the average values of the skin

friction, Nusselt number and the Sherwood number are as follows:

00 y

x

w

u

yg T T t

(2.22)

0

1

0 y

udx

y

(2.23)

0y

x

TNu x

y

(2.24)

0

1

0 y

TNu dx

y

(2.25)

0y

x

CSh x

y

(2.26)

0

1

0 y

CSh dx

y

(2.27)

3 Numerical Technique

In order to solve these unsteady, non-linear coupled equations (2.16) to (2.19) under the conditions

(2.20), an implicit finite difference scheme of the Crank–Nicolson type has been employed. The finite

difference equations corresponding to Eqs. (2.16) – (2.19) are as follows:




1 1 1 1 1 1

, 1, , 1, , 1 1, 1 , 1 1, 1 , , 1 , , 10

4 2

n n n n n n n n n n n n

i j i j i j i j i j i j i j i j i j i j i j i ju u u u u u u u v v v v

x y

(3.28)

1 1 1 1 1

, , , 1, , 1, , 1 , 1 , 1 , 1

, ,

1 1 1

, 1 , , 1 , 11 1

, , , ,

2 4

21

2 2

n n n n n n n n n n

i j i j i j i j i j i j i j i j i j i jn n

i j i j

n n n n

i j i j i j i jn n n n

i j i j i j i j

u u u u u u u u u uu v

t x y

u u u uNT T C C

, , 1

2

2

2

n n

i j i ju u

y

(3.29)

1 1 1 1 1

, , , 1, , 1, , 1 , 1 , 1 , 1

, ,

1 1 1

, 1 , , 1 , 1 , , 1

2

2 4

2 21 41

P 3 2

n n n n n n n n n n


i j i j

n n n n n n

i j i j i j i j i j i j

r

T T T T T T T T T Tu v

t x y

T T T T T T

F y

(3.30)

1 1 1 1 1

, , , 1, , 1, , 1 , 1 , 1 , 1

, ,

1 1 1

, 1 , , 1 , 1 , , 1

,2

2 4

2 21

22

n n n n n n n n n n


i j i j

n n n n n n

i j i j i j i j i j i j nri j

C C C C C C C C C Cu v

t x y

C C C C C C KC

Sc y

1

,

n

i jC

(3.31)

The region of integration is considered as a rectangle with sides maxx (=1) and maxy (=14), where maxy

corresponds to y which lies very well outside the momentum, energy and concentration boundary

layers. The maximum of y was chosen as 14 after some preliminary investigations, so that the last two of

the boundary conditions (2.20) are satisfied. Here, the subscript i - designates the grid point along the x-

direction, j - along the y-direction and the superscript n along the t-direction. An appropriate mesh size

considered for the calculation is 0.05x , 0.05y , and time step 0.02t . During any one-time

step, coefficients ,

n

i ju and ,

n

i jv appearing in the difference equations are treated as constants. The values of

u, v, T and C are known at all grid points at t = 0 from the initial conditions. The computations of u, v, T

and C at time level (n+1) using the known values at previous time level (n) are calculated as follows: The

finite difference equation (3.31) at every internal nodal point on a particular i -level constitute a tri-

diagonal system of equations. Such a system of equations is solved by Thomas algorithm as described in

Carnahan et al. [35]. Thus, the values of C are found at every nodal point on a particular i at (n+1) th time

level. Similarly the values of T are calculated from Eq. (3.30). Using the values of C and T at (n+1) th time

level in Eq. (3.29), the values of U at (n+1) th time level are found in a similar manner. Thus the values of

C, T and U are known on a particular i -level. The values of v are calculated explicitly using Eq. (3.28) at

every nodal point on a particular i -level at (n+1) th time level. This process is repeated for various i -




levels. Thus the values of C, T, u and v are known at all grid points in the rectangular region at (n+1) th

time level. The derivatives involved in Eqs. (2.22) – (2.27) are evaluated using five-point approximation

formula and integrals are evaluated using Newton– Cotes closed integration formula. In order to check the

accuracy of numerical results, the present study (in the absence of thermal radiation and first order

chemical reaction) is compared to the available theoretical solution of Narahari and Anwar Beg [36] in

Fig. 1 and they are found to be in good agreement.

Figure 1: Comparison of velocity profiles

4 Results and Discussion

In order to get a physical insight into the problem, a representative set of numerical results is shown

graphically in Figs. 2–16, to illustrate the influence of physical quantities such as radiation parameter F,

Buoyancy ratio parameter N, Chemical reaction parameter Kr, Schmidt number Sc, time t on velocity,

temperature and concentration in the boundary layer region. All the computations are carried out for Pr =

0.71 (i.e., for air), Pr = 7 (i.e., for water), and the corresponding values of Sc are chosen as Sc = 0.6, 2, 7,

500. The velocity profiles for various values of Buoyancy ratio parameter N, radiation parameter F,

Chemical reaction parameter Kr, Schmidt number Sc and time t are presented in Figs. 2, 3 and 4 for both

cases of ramped temperature and isothermal plates. It is noticed from Figs. 2, 3 and 4 that for both cases of

ramped temperature and isothermal plates, fluid velocity attains distinctive maximum value in the vicinity

of the plate surface and then decreases properly on increasing boundary layer coordinate y to approach the

free stream value. Also fluid velocity is slower in the case of ramped temperature plate than that in case of

isothermal plate. This is expected since in the case of ramped wall temperature the heating of the fluid

takes place more gradually than in the isothermal plate case. This feature is important in, for example,

achieving better flow control in nuclear engineering applications, since ramping of the enclosing channel

walls can help to decrease velocities.

It is observed from Fig. 2 that an increase in N leads to an increase in fluid velocity in the boundary layer

region. This implies that Buoyancy ratio parameter tends to accelerate fluid flow for both ramped

temperature and isothermal plates. It is also observed from Fig. 2 that fluid velocity decreases on

increasing radiation parameter F in the boundary layer region which implies that radiation parameter has a

decelerating influence on the fluid flow for both ramped temperature and isothermal plates. It is revealed

from Fig. 3 that fluid velocity u decreases on increasing Chemical reaction parameter in the boundary layer

region. This implies that Chemical reaction parameter decelerates fluid velocity for both ramped




temperature and isothermal plates. It is also observed from Fig. 3 that the velocity decreases with

increasing Schmidt number in the boundary layer region. An increasing Schmidt number implies that

viscous forces dominate over the diffusion effects. Schmidt number in free convection flow regimes, in

fact, represents the relative effectiveness of momentum and mass transport by diffusion in the velocity

(momentum) and concentration (species) boundary layers. Therefore an increase in Sc will counter-act

momentum diffusion since viscosity effects will increase and molecular diffusivity will be reduced. The

flow will therefore be decelerated with a rise in Sc. It is noticed from Fig. 4 that fluid velocity u increases

on increasing time t in the boundary layer region, which implies that there is an enhancement in fluid

velocity as time progresses for both ramped temperature and isothermal plates.

The temperature profiles for different values of radiation parameter F and time t are presented in Figs.5

and 6 for both cases of ramped temperature and isothermal plates. It is noticed from Figs. 5 and 6 that the

fluid temperature is maximum at the surface of the plate for both ramped temperature and isothermal plates

and it decreases on increasing boundary layer coordinate y to approach free stream value. Also fluid

temperature T is lower in the case of ramped temperature plate than that in the case of isothermal plate. It

is observed from Fig. 5 that fluid temperature T decreases on increasing radiation parameter F in the

boundary layer region which implies that radiation tends to reduce fluid temperature for both ramped

temperature and isothermal plates. It is noticed from Fig. 6 that fluid temperature T increases on increasing

time t in the boundary layer region which implies that there is an enhancement in fluid temperature as time

progresses for both ramped temperature and isothermal plates.

The Concentration profiles for different values of Chemical reaction parameter Kr, Schmidt number Sc

and time t are presented in Figs.7 and 8 for both cases of ramped temperature and isothermal plates. It is

noticed from Figs. 7 and 8 that the fluid concentration is maximum at the surface of the plate for both

ramped temperature and isothermal plates and it decreases on increasing boundary layer coordinate y to

approach free stream value. Also fluid concentration C in the case of ramped temperature plate is same as

in the case of isothermal plate. It is observed from Fig. 7 that fluid concentration C decreases on increasing

chemical reaction parameter Kr or Schmidt number Sc in the boundary layer region which implies that

chemical reaction or Schmidt number tends to reduce fluid concentration for both ramped temperature and

isothermal plates. It is noticed from Fig. 8 that fluid concentration C increases on increasing time t in the

boundary layer region which implies that there is an enhancement in fluid concentration as time progresses

for both ramped temperature and isothermal plates.

The local skin friction profiles for different N, F, Kr and Sc against axial coordinate x are plotted in Figs. 9

and 10 for both cases of ramped temperature and isothermal plates. It is noticed from Figs. 9 and 10 that

local skin friction is lower in the case of ramped temperature plate than that in the case of isothermal plate.

The local skin friction increases with the increasing values of N and decreasing values of F, Kr and Sc for

both ramped temperature and isothermal plates. The local Nusselt number for different F against axial

coordinate x is presented in Fig. 11 for ramped temperature plate. The local Nusselt number increases with

the increasing values of F. The local Sherwood number for different Kr against axial coordinate x is

presented in Fig. 12 for ramped temperature plate. The local Sherwood number increases with the

increasing values of Kr.

The average values of skin friction for different N, F, Kr and Sc against time t are plotted in Figs. 13 and

14 for both ramped temperature and isothermal plates. It is noticed from Figs. 13 and 14 that average skin

friction is lower in the case of ramped temperature plate than that in the case of isothermal plate, since

ramping decelerates the flow and lowers skin friction. The average skin friction increases with the

increasing values of N and decreasing values of F, Kr and Sc for both ramped temperature and isothermal

plates. The average Nusselt number for different F against time t is presented in Fig. 15 for ramped

temperature plate. It is observed from Fig. 15 that the average Nusselt number increases with the

increasing values of F. It is also observed that the Nusselt number increases for a range 0 1t and




decreases for 1t for all F. The average Sherwood number for different Kr against time t is presented in

Fig. 16 for ramped temperature plate. It is observed from Fig. 16 that the average Sherwood number

increases with the increasing values of Kr.

Figure 2: Effects of the N and F on the Velocity Profiles Figure 3: Effects of the Kr and Sc on the Velocity Profiles

Figure 4: Effect of the t on the Velocity Profiles Figure 5: Effects of the F on the Temperature Profiles

Figure 6: Effect of the t on the Temperature Profiles Figure 7: Effects of the Kr and Sc on the Concentration Profiles




Figure 8: Effect of the t on the Concentration Profiles Figure 9: Effects of the N and F on the Local Skin friction

Figure 10: Effects of the Kr and Sc on the Local Skin friction Figure 11: Effects of the F on the Local Nusselt number

Figure 12: Effects of the Kr on the Local Sherwood number Figure 13: Effects of the N and F on the Average Skin friction




Figure 14: Effects of the Kr and Sc on the Average Skin friction Figure 15: Effects of the F on the Average Nusselt number

5 Conclusions

A detailed numerical study has been carried out for the ramped wall temperature effects on two-

dimensional, transient, laminar, natural convection flow of an incompressible viscous fluid past a vertical

plate with thermal radiation and chemical reaction of first order. The fluid is gray, absorbing-emitting but

non scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the

energy equation. A family of governing partial differential equations is solved by an implicit finite

difference scheme of Crank–Nicolson type, which is stable and convergent. The results are obtained for

different values of buoyancy ratio N, radiation parameter F, chemical reaction parameter Kr, Schmidt

number and time t. Conclusions of this study are as follows.

1. Fluid velocity is slower in the case of ramped temperature plate than that in case of isothermal plate.

2. Fluid velocity increases with the increasing N or t and decreases with the increasing values of F, Kr and

Sc for both ramped temperature and isothermal plates.

3. Fluid temperature is lower in the case of ramped temperature plate than that in the case of isothermal

plate.

4. Fluid temperature decreases with the increasing F and increases with the increasing values of t for both

ramped temperature and isothermal plates.

5. Fluid concentration decreases with the increasing Kr or Sc and increases with the increasing values of t

for both ramped temperature and isothermal plates.

6. The local and average skin friction increases with the increasing N and decreases with the increasing

values of F, Kr and Sc for both ramped temperature and isothermal plates.

7. The local and average Nusselt number increases with the increasing values of F.

8. The local and average Sherwood number increases with the increasing values of Kr.

Acknowledgements

The authors are thankful to the reviewers for the positive comments and the valuable suggestions, which

led to definite improvement in the paper.




Appendix

Nomenclature

C concentration

C dimensionless concentration

pC specific heat at constant pressure

D mass diffusion coefficient

F radiation parameter

g acceleration due to gravity

Gm solutal Grashof number

Gr thermal Grashof number

rK chemical reaction parameter

N buoyancy ratio parameter

xNu local Nusselt number

LNu average Nusselt number

xNu dimensionless local Nusselt number

Nu dimensionless average Nusselt number

Pr Prandtl number

rq radiative heat flux

Sc Schmidt number

xSh local Sherwood number

LSh average Sherwood number

xSh dimensionless local Sherwood number

Sh dimensionless average Sherwood number

T temperature

T dimensionless temperature

t time

t dimensionless time

0t characteristic time

,u v velocity components in ,x y -directions, respectively

x spatial coordinate along the plate

x dimensionless spatial coordinate along the plate

y spatial coordinate normal to the plate

y dimensionless spatial coordinate normal to the plate

Greek symbols

thermal diffusivity

volumetric coefficient of thermal expansion

* volumetric coefficient of expansion with concentration

dynamic viscosity

kinematic viscosity




thermal conductivity of the fluid

density

local skin friction

L average skin friction

x dimensionless local skin friction

dimensionless average skin friction

Subscripts

w conditions on the wall

free stream conditions

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