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Keywords:

  • magnetohydrodynamics;
  • chemical reaction;
  • viscoelastic fluid;
  • mixed convection;
  • porous medium

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES

A numerical model is developed to analyse the combined effects of mixed convection magnetohydrodynamic heat and mass transfer in a viscoelastic fluid in a porous medium with chemical reaction, non-uniform heat source/sink and viscous-Ohmic dissipation. The governing boundary layer equations for momentum, energy and species transfer are transformed to a set of nonlinear ordinary differential equations by using similarity solutions which are then solved numerically. The effects of magnetic field on local skin-friction, local Nusselt number and local Sherwood number are studied for various physical parameters. The results show that there are significant effects of various physical parameters on velocity profiles, temperature and concentration profiles.


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES

The problem of boundary layer flow over a stretching surface has received great interest among researchers due to its wide range of applications in several engineering processes, for example materials manufactured by extrusion, glass-fibre and paper production. In industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a wind-up roller, which is located at a finite distance away. In these cases, the final product of desired characteristics depends on the rate of cooling in the process. Sakiadis[1] introduced the study of boundary layer flow over a continuous solid surface moving with constant velocity. The dynamics of the laminar boundary layer flow of a Newtonian fluid caused by a flat elastic sheet whose velocity varies linearly with distance from the fixed point of the sheet was originated from the pioneering work of Crane.[2] Soundalgekar[3] investigated the Stokes problem for a viscoelastic fluid. A considerable amount of attention was drawn by Gupta and Gupta[4] in the stretching sheet problem with constant surface temperature. Rajagopal et al.[5] have analysed the flow of a viscoelastic fluid over a stretching sheet. Pillai et al.[6] investigated the effects of work done by deformation in viscoelastic fluid in porous media with uniform heat source. Cortell[7, 8] studied the flow and heat transfer of a viscoelastic fluid over a stretching surface considering both constant sheet temperature and prescribed sheet temperature. Hayat et al.[9] also investigated the effects of work done by deformation in second grade fluid with partial slip condition, in this no account of heat source has been taken into consideration.

The study of MHD flow of viscoelastic fluids over a continuously moving surface has a wide range of application in the production of synthetic sheet, of plastic sheet, cooling of metallic sheet, aerodynamic extrusion etc. Andersson[10] examined the flow of viscoelastic fluid over a stretching sheet under the influence of uniform magnetic field. Abel et al.[11] studied convective heat and mass transfer in a viscoelastic fluid flow through a porous medium over a stretching sheet. Makinde[12] analysed hydromagnetic boundary layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux. Computational dynamics of hydromagnetic stagnation-point flow towards a stretching sheet was studied by Makinde and Charles.[13] Sibanda and Makinde[14] described MHD flow and heat transfer due to a rotating disk in a porous medium with Ohmic heating and viscous dissipation.

Cortell[15] investigated viscous fluid flow and heat transfer over a nonlinearly stretched sheet in the presence of thermal radiation and viscous dissipation in the case of prescribed wall heat flux, whereas Makinde and Ogulu[16] carried out an analysis to study the effects of variable viscosity and thermal radiation on heat and mass transfer past a vertical porous plate in the presence of a transverse magnetic field. A new dimension has been added to this investigation by Abo-Eldahab and El Aziz[17] to study the effects of space-dependent heat source/sink in addition to the temperature-dependent heat source/sink in their study of viscous flow. Study of viscoelastic fluid flow and heat transfer over a stretching sheet with variable viscosity was given by Abel et al.[18] Nandeppanavar et al.[19] analysed the heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet in the presence of thermal radiation and non-uniform heat source/sink.

In view of the above discussions, authors envisage to investigate the steady two-dimensional MHD mixed convection heat and mass transfer flow past a semi-infinite vertical porous plate embedded in a viscoelastic fluid-saturated porous medium in the presence of variable thermal conductivity, non-uniform heat source/sink parameter, chemical reaction, viscous and Ohmic dissipation since these parameters have significant contribution to convective transport process. The problem addressed here is a fundamental one that arises in many practical situations such as polymer extrusion process and combined effects of the physical parameters that will have a large impact on the heat and mass transfer characteristics. It is assumed that the stretching velocity, surface temperature and surface concentration vary linearly with the distance from the point. The nonlinearity of the basic equations and additional mathematical difficulties associated with it has led to the use of numerical method. The transformed dimensionless governing equations are solved numerically by using fifth-order Runge-Kutta-Fehlberg method (RKF45) with shooting technique.

MATHEMATICAL FORMULATIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES

Consider steady two-dimensional MHD laminar convective heat and mass transfer flow in a viscoelastic fluid-saturated porous medium over a vertical stretching sheet in the presence of a uniform transverse magnetic field, non-uniform heat source sink, viscous-Ohmic dissipation and chemical reaction. The flow is generated due to the stretching of the sheet, caused by the simultaneous application of two equal and opposite forces along the x-axis. Keeping the origin fixed, the sheet is then stretched with a speed varying linearly with the distance from the slit. A uniform transverse magnetic field B0 is imposed along the y-axis as shown in Figure 1. The x-axis is taken in the direction of the main flow along the plate and the y-axis is normal to the plate with velocity components u, v in these directions. We adopt Walter's liquid B′ model for the constitutive equation of viscoelastic fluid as given by (in Cartesian tensor notation):

  • display math(1)

where τij is the stress tensor, p is the pressure, eik is the rate-of-strain tensor, µ is the coefficient of viscosity and k0 is elastic parameter. Under the usual boundary layer approximation, along with Walter's liquid B′ approximations, the governing equations describing the conservation of mass, momentum, energy and concentration are governed by the following equations[7, 8]:

image

Figure 1. Physical Model and coordinate system.

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(i) Equation of continuity:

  • display math(2)

(ii) Equation of momentum:

  • display math(3)

(iii) Equation of energy:

  • display math(4)

(iv) Equation of momentum:

  • display math(5)

where ρ is the density of the liquid, T and C are the temperature and concentration, respectively. Cb is the form of drag coefficient which is independent of viscosity and other properties of the fluid, but is dependent on the geometry of the medium, k is the permeability of the porous medium, βt and βc are the coefficients of thermal and concentration expansions, respectively, cp is the specific heat at constant pressure, inline image is the kinematic viscosity, σ is the electrical conductivity of the fluid, B0 is the externally imposed magnetic field strength in the y-direction, D is the molecular diffusivity, κ is the thermal conductivity, inline image is the rate of internal heat generation/absorption coefficient. The other terms have their usual meanings as given in the nomenclature.

The dimensionless form of inline image can be defined as:

  • display math(6)

where inline image and inline image are parameters of space and temperature-dependent internal heat generation/absorption. It is to be noted that inline image and inline image correspond to internal heat generation while inline image and inline image correspond to internal heat absorption, Tw is the temperature of the sheet and T is the constant temperature far away from the sheet.

The appropriate physical boundary conditions for the problem under study are given by:

  • display math(7)
  • display math(8)
  • display math(9)
  • display math(10)

where A0, A1 are constants, l is the characteristic length, Tw is the wall temperature of the fluid and T is the temperature of the fluid far away from the sheet, Cw is the wall concentration of the solute and C is the concentration of the solute far away from the sheet.

Similarity Solution for the Equation of Momentum

The similarity solution of governing Equations (2) and (3) are of the form:

  • display math(11)

where f is the dimensionless stream function and η is the similarity variable. u and v satisfy Equation (2). Now, substituting (11) in Equation (3), we obtain the following fourth order nonlinear ordinary differential equation as follows:

  • display math(12)

and the appropriate boundary conditions (7) and (8) become:

  • display math(13)
  • display math(14)

where inline image is the viscoelastic parameter, inline image is inverse Darcy number, inline image is the Hartmann number, inline image is the thermal Grashof number, inline image is the solutal Grashof number. In case of viscous fluid (k1 = 0), but for a viscoelastic fluid (k1 ≠ 0). The boundary conditions (13) and (14) are not sufficient to solve the Equation (12). Thus we required one more boundary condition, as Mahapatra et al.[20] reasonably argued that in the case of boundary layer flow of viscoelastic fluid, the characteristic time scale associated with the motion is large compared with the relaxation time of the fluid. Thus terms of orders inline image and higher orders may be neglected and therefore we may seek the solution of Equation (12) in the form:

  • display math(15)

Substituting Equation (15) in Equation (12) and equating the coefficients of inline image we get:

  • display math(16)
  • display math(17)

Using Equations (13) and (14), the boundary conditions for f0 and f1 are:

  • display math(18)
  • display math(19)

Now we find the zeroth-order stream function Equation (16) as a third order equation of f0(η) for which three boundary conditions are prescribed by Equation (18). The first-order stream function Equation (17) is also third-order equation of f1(η) for which three boundary conditions are prescribed by Equation (19). Since, the order of the differential equations (16) and (17) matches well with the number of boundary conditions prescribed by Equations (18) and (19), respectively, Equations (16) and (17) would produce unique solutions.

Similarity Solution of Energy and Mass-Diffusion Equations

The thermal conductivity of fluid κ is assumed to vary linearly with temperature and it is of the form:

  • display math(20)

we consider non-isothermal temperature boundary condition as follows:

  • display math(21)

We now define dimensionless temperature variable θ and concentration φ of the form:

  • display math(22)

Using Equation (22) we have from Equations (4) and (5) as:

  • display math(23)
  • display math(24)

Using perturbation technique, we take inline image and inline image (see Ref.[20]). Thus we obtain the following nonlinear ordinary differential equations for inline image and inline image and inline image by using perturbation to Equations (23) and (24). Equating inline image and x0 order terms, we obtain:

  • display math(25)
  • display math(26)

Equating k1th and x2 order terms, we get:

  • display math(27)
  • display math(28)

Corresponding thermal boundary conditions become:

  • display math(29)

where inline image is the Prandtl number, inline image is the Eckert number, inline image is the Schmidt number, inline image is the chemical parameter.

Skin-Friction Coefficient, Nusselt Number and Sherwood Number

The most important physical quantities for the problem are skin-friction coefficient (Cf), local Nusselt number (Nux) and local Sherwood number (Shx) which are defined by the following relations:

  • display math(30)

The skin-friction on the flat plate τw, rate of heat transfer qw and rate of mass transfer mw are given by:

  • display math(31)

Substituting Equation (31) in Equation (30) thus we get:

  • display math(32)

where inline image is the local Reynolds number.

NUMERICAL SOLUTIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES

The coupled ordinary differential Equations (16), (17) and (25)-(28) are third order in f0, f1, second order in θ0(η), θ1(η) and φ0(η), φ1(η) which have been reduced to a system of 14 simultaneous equations of first-order for 14 unknowns. In order to solve this system of equations numerically we require 14 initial conditions but two initial conditions on f0 and f1, one initial condition each on θ0, θ1 and φ0, φ1 are known. However, the values of inline image θ0, θ1 and φ, φ1 are known as η[RIGHTWARDS ARROW] ∞. These six end conditions are utilised to produce six unknown initial conditions at η = 0 by using shooting technique. The most crucial factor of this scheme is to choose the appropriate finite value of η. In order to estimate the value of η, we start with an initial guess value and solve the boundary value problem consisting of Equations (16), (17) and (25)-(28) to obtain inline image. The solution process is repeated with another large value of η until two successive values of inline image differ only after desired significant digit. The last value of η is taken as the final value of η for a particular set of physical parameters for determining velocity f(η), temperature θ(η) and concentration φ(η) in the boundary layer. After knowing all the seven initial conditions, we solve this system of simultaneous equations using fifth-order Runge-Kutta-Fehlberg integration scheme with automatic grid generation scheme which ensures convergence at a faster rate. The results are provided in several tables and graphs.

RESULTS AND DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES

The present nonlinear boundary value problem cannot be solved in closed-form, so numerical method is adopted in order to describe the physics of the problem well. The resulting nonlinear ordinary differential equations are integrated by fourth-order Runge-Kutta-Fehlberg method with shooting technique. In order to assess the accuracy of the numerical results, the validity of the numerical code has been checked under some limiting cases. Comparison of skin friction coefficient f″(0) for various values of k1 are made with Bhattacharyya et al.[21] and Mahapatra et al.[20] in the absence of viscous dissipation, buoyancy force, porous parameter, magnetic parameter and Eckert number which shows a very good agreement as seen from Table 1.

Table 1. Comparison of skin friction coefficient f″(0), Ha = 0, Sc = 0.0 and various values of k1 with Bhattacharyya et al.[21] and Mahapatra et al.[20]
k1Bhattacharyya et al.[21]Mahapatra et al.[20]Present results
0.0011.00050.99641.0002
0.0051.00250.99841.0022
0.011.00501.00091.0050

Figure 2 shows the behaviour of velocity profiles for different values of viscoelastic parameter k1 = 0.001, 0.005, 0.01. It is seen from this figure that the transverse component velocity increases with increasing values of k1. Figure 3 shows the behaviour of velocity profiles for different values of Hartmann number Ha with viscoelastic parameter k1 = 0.0, 0.01. It is observed from this figure that fluid velocity is higher in viscous fluid than viscoelastic fluid. The Hartmann number represents the importance of the magnetic field on the flow field. It is observed from this figure that the effect of the magnetic field decreases the velocity profiles, due to the fact that the application of transverse magnetic field normal to the flow direction has a tendency to give rise to a resistive type force called the Lorentz force and hence results in retarding the velocity profile. Thus, the increase in the Hartmann number results in the increase in the Lorentz force due to which the velocity profile decreases. So the effect of the magnetic field is to retard the velocity of the fluid. Figure 4 depicts that the horizontal velocity profile increases with the increase in the value of the thermal Grashof number Grt and the peak is observed near the stretching sheet which exponential decreases away from the stretching surface.

image

Figure 2. Effects of k1 on the velocity profile.

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image

Figure 3. Effects of k1 with Ha on the velocity profile.

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image

Figure 4. Effects of Grt on the velocity profile.

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Figure 5 shows the velocity profiles for various values of solutal buoyancy force whose effect is taken into account by the parameter Grc (solutal Grashof number). It is noticed that the effect of increasing Grc is to increase the velocity profile. Thus what actually is observed is that the velocity overshoots (i.e. the velocity at a certain value of η which exceeds the velocity at the edge of the boundary layer) in the boundary layer region and buoyancy force acts like a favourable pressure gradient which accelerates the fluid velocity within the boundary layer. Figure 6 displays the influence of the inverse Darcy number Da−1 on the velocity profile. It is seen that the increasing the value of inverse Darcy number is to decrease the velocity profiles which is due to the presence of porous medium that causes higher retardation to the fluid velocity. Figure 7 represents the effect of ∈ and Ha on velocity profiles. It is observed from this figure that the effect of magnetic field is to decrease the velocity profile and similar effect of ∈ is seen on velocity profiles.

image

Figure 5. Effects of Grc on the velocity profile.

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image

Figure 6. Effects of Da−1 on the velocity profile.

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image

Figure 7. Effects of ϵ and Ha on the velocity profile.

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The effect of viscous dissipation on temperature profile is demonstrated in Figure 8. We observe by analysing the graph that the effect of increasing the value of Eckert number Ec is to increase the temperature distribution in the flow region, which is responsible for increasing temperature in boundary layer. Figure 9 depicts the effect of space-dependent heat source/sink parameter inline image. It is observed that the boundary layer generates the energy, which causes the temperature profile to increase with increasing values of inline image (heat source) whereas in the case inline image (absorption) reverse effects are seen. Figure 10 presents the variation of temperature profiles with increase in the temperature-dependent heat source/sink parameter inline image. It is observed that increase in the value of inline image (heat generation) results in an increase in the temperature in the thermal boundary layer whereas reverse effect is seen during heat absorption (i.e. inline image). Figure 11 illustrates the variation of temperature profile for various values of Prandtl number. From this figure it is seen that the temperature decreases with increasing the values of Prandtl number Pr in the boundary layer. From this plot it is evident that the temperature in the boundary layer falls very quickly for large value of the Prandtl number, because of the fact that thickness of the boundary layer decreases with the increase in the value of the Prandtl number. Figure 12 depicts the effect of Hartmann number Ha and ϵ on temperature profiles in the thermal boundary layer. It is observed from the plot that temperature increases with the increase in the value of Hartmann number. The reason behind the increase in the temperature in the boundary layer is due to the fact that a body force (called Lorentz force) is produced which opposes the motion in the presence of transverse magnetic field and the resistance offered to the flow is thus responsible for the increase in the temperature. Further, it is noted that the boundary layer thickness increases in the presence of transverse magnetic field, whereas the reverse effect is seen due to the increase in the value of ϵ, that is temperature profiles decrease with the increase in the value of ϵ which in turn is responsible for the decrease in the thermal boundary layer.

image

Figure 8. Effects of Ec on the temperature profile.

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image

Figure 9. Effect of inline image on the temperature profile.

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image

Figure 10. Effect of inline image on the temperature profile.

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image

Figure 11. Effects of Pr with Ha on the temperature profile.

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image

Figure 12. Effects of ϵ with Ha on the temperature profile.

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Figure 13 shows the plot of skin-friction coefficient with Schmidt number Sc for various values of Hartmann number. It is examined from this figure that there is significant effect of Hartmann number on f″(0), that is skin-friction coefficient decreases with increase in the value of Hartmann number for all values of Sc whereas no significant effect of Sc is seen on the skin-friction coefficient. The effect of Hartmann number and Schmidt number Sc is demonstrated in Figure 14. We observe by analysing the graph that the effect of increasing the value of Hartmann number Ha is to decrease the rate of heat transfer in the flow region, which is responsible for decreasing Nusselt number in the thermal boundary layer. Whereas the increase in Schmidt number Sc is to decrease the Nusselt number in the boundary layer. Figure 15 shows the graph of Sherwood number for different values of Schmidt number Sc and Ha. Increase of Schmidt number means decrease of molecular diffusivity D, which results in decrease in the Schmidt number in the solutal boundary layer. Figure 16 depicts the variation of chemical reaction parameter on concentration profile. It is seen that R1 produces a decrease in concentration profile since chemical reaction decelerates the concentration of species.

image

Figure 13. Variations of f″(0) with Sc for different values of Ha.

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image

Figure 14. Variations of Nusselt number with Sc for different values of Ha.

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image

Figure 15. Variations of Sherwood number with Sc for different values of Ha.

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image

Figure 16. Effects of R1 on the concentration profile.

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CONCLUSIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES

The problem of two-dimensional mixed convection flow due to a viscoelastic fluid-saturated porous medium in the presence of inverse Darcy number, Prandtl number, Schmidt number, non-uniform heat source/sink parameter, magnetic parameter, chemical reaction parameter, local thermal Grashof number and local solutal Grashof number effects was investigated. The resulting partial differential equations, describing the problem, are transformed into ordinary differential equations by using similarity transformations. These equations are more conveniently solved numerically using Runge-Kutta Fehlberg method with shooting technique for the computation of the flow, heat and mass transfer characteristics for various values of physical parameters. The numerical results obtained and compared with previously reported cases available in the open literature and they are found to be in very good agreement. From the present investigation, the following conclusions are drawn:

  1. Increase in the viscoelastic parameter k1 is to increase the velocity in the momentum boundary layer.
  2. Increase in the inverse Darcy number Da−1 is to decrease the velocity profile.
  3. Increasing thermal buoyancy parameter leads to increase the value of velocity profiles.
  4. Velocity profiles are strongly influenced by the magnetic field in the momentum boundary layer, which decreases with the increase in the Hartmann number.
  5. Prandtl number influences the temperature profiles in the thermal boundary layer as a result the temperature decreases in the thermal boundary layer.
  6. Chemical reaction parameter deteriorates the concentration profile.
  7. Skin-friction and local Nusselt number decrease with the increase in Hartmann number whereas local Sherwood number increases with Hartmann number.
  8. Local Nusselt number and local Sherwood number decrease with the increase in the Schmidt number.

NOMENCLATURE

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES
A0, A1

constants in Equation (10)

inline image

transverse magnetic field

B0

applied magnetic field strength

C

concentration of the solute

Cf

local skin-friction coefficient

D

molecular diffusivity

Da−1

inverse Darcy number

Ec

Eckert number

Grc

local concentration Grashof number

Grt

local temperature Grashof number

Ha

Hartmann number

k

permeability of the porous medium

k1

viscoelastic parameter

l

characteristic length

Nux

Nusselt number

Pr

Prandtl number

inline image

non-uniform heat source/sink

R1

chemical reaction parameter

Sc

Schmidt number

Shx

Sherwood number

Tw

wall temperature of the fluid

T

ambient temperature of the fluid

u, v

velocity components along x and y directions

x

flow directional coordinate along the stretching sheet

y

distance normal to the stretching sheet

Greek Symbols

βc

coefficient of concentration expansion

βt

coefficient of thermal expansion

η

similarity variable

inline image

small parameter

γ

spin gradient viscosity

κ

thermal conductivity

inline image

kinematic viscosity

µ

coefficient of viscosity

φ

non-dimensional concentration

ρ

density of fluid

θ

non-dimensional temperature

σ

electrical conductivity of the fluid

Subscripts

w

condition at the wall

condition at free stream

REFERENCES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL FORMULATIONS
  5. NUMERICAL SOLUTIONS
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. NOMENCLATURE
  9. REFERENCES