Steady laminar natural convection flow over a semiinfinite moving vertical plate with internal heat generation and convective surface boundary condition in the presence of thermal radiation, viscous dissipation, and chemical reaction is examined in this paper. In the analysis, we assumed that the left surface of the plate is in contact with a hot fluid while the cold fluid on the right surface of the plate contains a heat source that decays exponentially with the classical similarity variable. We utilized similarity variable to transform the governing nonlinear partial differential equations into a system of ordinary differential equations, which are solved numerically by applying shooting iteration technique along fourthorder RungeKutta method. The effects of the local Biot number, Prandtl number, buoyancy forces, the internal heat generation, the thermal radiation, Eckert number, viscous dissipation, and chemical reaction on the velocity, temperature, and concentration profiles are illustrated and interpreted in physical terms. A comparison with previously published results on the similar special cases showed an excellent agreement. Finally, numerical values of physical quantities, such as the local skinfriction coefficient, the local Nusselt number, and the local Sherwood number, are presented in tabular form.
Convective flows with simultaneous heat and mass transfer under the influence of the chemical reaction arise in many transport processes both naturally and artificially in many branches of science and engineering applications. This phenomenon plays an important role in the chemical industry, power and cooling industry for drying, chemical vapour deposition on surfaces, cooling of nuclear reactors, and petroleum industries.
Natural convection flow occurs frequently in nature. It occurs due to temperature differences, as well as due to concentration differences or the combination of these two; for example, in atmospheric flows, there exist differences in water concentration, and hence the flow is influenced by such concentration difference.
Changes in fluid density gradients may be caused by nonreversible chemical reaction in the system as well as by the differences in the molecular weight between values of the reactants and the products. Chemical reactions can be modeled as either homogenous or heterogeneous processes. This depends on whether they occur at an interface or as a single phase value reaction. A homogeneous reaction is one that occurs uniformly throughout a given phase. On the other hand, a heterogeneous reaction takes place in a restricted area or within the boundary of a phase. In most cases of chemical reaction, the reaction rate depends on the concentration of the species itself. A reaction is said to be of first order if the rate of reaction is directly proportional to the concentration itself (Cussler [
The study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Heat generation effects may alter the temperature distribution, and these in turn can affect the particle deposition rate in nuclear reactors, electronic chips, and semiconductor’s wafers. Although exact modeling of internal heat generation or absorption is quite difficult, some simple mathematical models can be used to express its general behavior for most physical situations. Heat generation or absorption can be assumed to be constant, space dependent, or temperature dependent. Crepeau and Clarksean [
Convective heat transfer studies are very important in processes involving high temperatures, such as gas turbines, nuclear plants, and thermal energy storage. Ishak [
Viscous dissipation changes the temperature distributions by playing a role like an energy source, which leads to the affected heat transfer rates. The merit of the effect of viscous dissipation depends on whether the plate is being cooled or heated. Heat transfer analysis over porous surface is of much practical interest due to its abundant applications. To be more specific, heattreated materials traveling between a feed roll and windup roll or materials manufactured by extrusion, glassfiber and paper production, cooling of metallic sheets or electronic chips, and crystal growing, just to name a few. In these cases, the final product of desired characteristics depends on the rate of cooling in the process and the process of stretching. The work of Sonth et al. [
In many new engineering areas processes such as fossil fuel combustion energy processes, solar power technology, astrophysical flows, gas turbines, and the various propulsion devices for aircraft, missiles, satellites, and space vehicle reentry occur at high temperatures so knowledge of radiation heat transfer beside the convective heat transfer plays a very important role and hence its effect cannot be neglected. Also thermal radiation is of major importance in many processes in engineering areas which occur at a very high temperature for the design of many advanced energy conversion systems and pertinent equipment. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. Pal and Mondal [
The objective of this paper was to explore the effects of thermal radiation, heat generation, viscous dissipation, and chemical reaction on the similarity solution for natural convection from a moving vertical plate under a convective boundary condition, which is an extension of Makinde [
We consider the steady laminar incompressible natural convection boundary layer flows over the right surface of a vertical flat plate moving with uniform velocity
Flow configuration and coordinate system.
The boundary conditions at the plate surface and for the cold fluid may be written as
For the momentum and energy equations to have a similarity solution, the parameters
The coupled nonlinear boundary value problems represented by (
To analyze the results, numerical computation has been carried out using the method described in the previous paragraph for various governing parameters, namely, thermal Grashof number Gr, modified Grashof number Gc, Prandtl number Pr, thermal radiation parameter
Table
Computations showing comparison with Makinde [
Bi  Gr  Pr 








0.1  0.1  0.72  1.0  −0.2000518  0.076578477  1.76578477  −0.253226  0.0353022  1.35302 
1.0  0.1  0.72  1.0  −0.2459676  0.281651449  1.28165144  −0.279242  0.128217  1.12822 
10  0.1  0.72  1.0  −0.2695171  0.382952717  1.03829527  −0.280211  0.173623  1.01736 
0.1  0.5  0.72  1.0  0.4221216  0.048257030  1.48257030  0.250851  0.016076  1.16076 
0.1  1.0  0.72  1.0  0.9895493  0.034011263  1.34011263  0.717892  0.00569792  1.05698 
0.1  0.1  3.0  1.0  −0.3748695  −0.023814576  0.76185423  −0.26024  0.100968  2.00969 
0.1  0.1  7.10  1.0  −0.4138825  −0.057164001  0.42835998  −0.258586  0.153863  2.53863 
0.1  0.1  0.72  5  0.3741286  0.576670381  6.76670381  0.198617  0.41344  5.13444 
0.1  0.1  0.72  10  0.9010790  1.106605802  12.0660580  0.620946  0.818126  9.18126 
Computation showing
Bi  Gr  Gc  Pr 


Ec  Sc 






0.1  1.0  1.0  1.0  0.5  0.1  0.1  0.6  0.5  0.617935  0.0690068  0.309932  0.690904 
1.0  1.0  1.0  1.0  0.5  0.1  0.1  0.6  0.5  0.995463  0.24145  0.75855  0.705335 
10.0  1.0  1.0  1.0  0.5  0.1  0.1  0.6  0.5  1.16294  0.325667  0.967433  0.71137 
0.1  2.0  1.0  1.0  0.5  0.1  0.1  0.6  0.5  0.914867  0.0688259  0.311741  0.703017 
0.1  3.0  1.0  1.0  0.5  0.1  0.1  0.6  0.5  1.21966  0.0680544  0.319456  0.714747 
0.1  1.0  2.0  1.0  0.5  0.1  0.1  0.6  0.5  1.27296  0.067557  0.32443  0.711755 
0.1  1.0  3.0  1.0  0.5  0.1  0.1  0.6  0.5  1.8969  0.064555  0.35445  0.729823 
0.1  1.0  1.0  1.0  0.5  0.1  0.1  0.6  0.5  0.598557  0.0696409  0.303591  0.689679 
0.1  1.0  1.0  3.0  0.5  0.1  0.1  0.6  0.5  0.537444  0.070888  0.29112  0.685736 
0.1  1.0  1.0  1.0  1.0  0.1  0.1  0.6  0.5  0.635229  0.0684205  0.315795  0.691995 
0.1  1.0  1.0  1.0  1.5  0.1  0.1  0.6  0.5  0.645997  0.0680532  0.319468  0.692676 
0.1  1.0  1.0  1.0  0.5  0.2  0.1  0.6  0.5  0.694807  0.0621171  0.378829  0.694271 
0.1  1.0  1.0  1.0  0.5  0.3  0.1  0.6  0.5  0.769222  0.0553443  0.446557  0.697471 
0.1  1.0  1.0  1.0  0.5  0.1  0.2  0.6  0.5  0.654059  0.0662619  0.337381  0.692653 
0.1  1.0  1.0  1.0  0.5  0.1  0.3  0.6  0.5  0.693151  0.0632311  0.367689  0.694521 
0.1  1.0  1.0  1.0  0.5  0.1  0.1  0.78  0.5  0.562746  0.0687271  0.312729  0.786616 
0.1  1.0  1.0  1.0  0.5  0.1  0.1  1.0  0.5  0.511741  0.068638  0.311362  0.889928 
0.1  1.0  1.0  1.0  0.5  0.1  0.1  0.6  1.0  0.514718  0.0688634  0.311366  0.877011 
0.1  1.0  1.0  1.0  0.5  0.1  0.1  0.6  1.5  0.488687  0.0687467  0.312533  1.03195 
Figures
Effects of local Grashof number on velocity profile.
Effects of local modified Grashof number on velocity profile.
Effects of Prandtl number on velocity profile.
Effects of radiation parameter on velocity profile.
Effects of internal heat generation on velocity profile.
Effects of Eckert number on velocity profile.
Effects of Schmidt number on velocity profile.
Effects of chemical reaction parameter on velocity profile.
: Effects of local Biot number on velocity profile.
Figures
Effects of Prandtl number on temperature profile.
Effects of radiation parameter on temperature profile.
Effects of internal heat generation parameter on temperature profile.
Effects of Eckert number on temperature profile.
Effects of local Biot number on temperature profile.
Figures
Effects of Schmidt number on concentration profile.
: Effects of chemical reaction parameter on concentration profile.
Table
The similarity solution for natural convection from a moving vertical plate with internal heat generation and a convective boundary condition in the presence of thermal radiation, viscous dissipation, and chemical reaction is studied. A set of nonlinear coupled differential equations governing the fluid velocity, temperature, and concentration is solved numerically for various material parameters. A comprehensive set of graphical results for the velocity, temperature, and concentration is presented and discussed. Our results reveal, among others, that the internal heat generation, thermal radiation, and the Eckert number prevent the flow of heat from the left surface to the right surface of the plate unless the local Grashof number is strong enough to convert away both the internally generated heat in the fluid. Generally, the fluid velocity increases gradually away from the plate, attains its peak value within the boundary layer, and decreases to the free stream zero value satisfying the boundary conditions. It is interesting to note that the fluid velocity within the boundary layer increases with increasing values of the exponentially decaying internal heat generation, thermal radiation, and the Eckert number little away from the wall plate and attains its peak before obeying the boundary conditions. The velocity and concentration both decrease with an in increase in the Schmidt number and the chemical reaction parameter.