A mathematical analysis has been performed for heat and mass transfer of a time-dependent MHD flow of an electrically conducting viscoelastic fluid in nonuniform vertical channel with convective boundary condition. The fluid flow is considered between a vertical long wavy wall and a parallel flat wall saturated with the porous medium. The effects of thermal radiation, heat absorption, chemical reaction, and Hall current are taken into account. The prevailing nonlinear partial differential equations are derived by considering Boussinesq approximation, and the same equations are solved analytically using perturbation technique. Further the expressions for skin friction, Nusselt number, and Sherwood number are presented. The effects of various pertinent parameters on different flow fields are analyzed graphically and tabularly. It is found that effects of Hall parameter and Biot number are unfavorable on velocity profiles, but this trend is reverse for the effect of thermal and solutal Grashof numbers. The expressions of different flow fields satisfy the imposed boundary conditions, which is shown in all graphs; this implies accuracy of the solution.
The study of viscoelastic fluid has become important in the last few years. Qualitative analysis of these studies has significant bearing on several industrial applications such as polymer sheet extrusion from a dye and drawing of plastic firms. When manufacturing processes at high temperature need cooling, the flow may need viscoelastic fluid to produce a good effect or reduce the temperature. On the other hand, the flow and heat transfer of a viscoelastic fluid between parallel plates have significant role in many engineering fields such as petroleum production, chemical catalytic reactors, and solar power collectors. Boundary layer treatment for an idealized viscoelastic fluid was introduced by Beard and Walters [
Convective flow with simultaneous heat and mass transfer under the influence of a magnetic field and chemical reaction has attracted considerable attention of many researchers because of its applications in various branches of science and technology. On the other hand, radiative heat transfer has many applications in nuclear power plants, in gas turbines, and in the various propulsion devices for space vehicles, missiles, and aircrafts. Motivated by these applications very recently, Pal and Talukdar [
The study of fluid flow with subject to convective boundary condition plays an important role in several engineering and industrial processes like transpiration cooling process and material drying. Therefore several authors [
Previous studies of the flow with heat and mass transfer have focused mainly on a flat wall or a regular channel. It is necessary to study the flow, heat and mass transfer in an irregular channel because such flows find applications in different areas such as transpiration cooling of reentry vehicles and rocket boosters, crosshatching on ablative surfaces, and film vaporization in combustion chambers. In view of these applications, Das [
Keeping all these facts in mind, in this work the effect of radiation on unsteady laminar flow with heat and mass transfer of an electrically conducting, chemically reactive viscoelastic fluid in irregular channel with subject to convective boundary condition has been investigated. The perturbation technique is employed to solve governing coupled nonlinear partial differential equations.
The constitutive equations for the rheological equation of the state for the visco-elastic fluid (Walters liquid B) are
Consider an unsteady flow, heat and mass transfer of an electrically conducting, incompressible, chemically reacting viscoelastic fluid between a vertical long wavy wall under the influence of uniform transverse magnetic field of strength
The geometry of the problem.
The flow field is exposed to the influence of thermal and mass buoyancy effects, Hall effect, thermal radiation, heat absorption, and first order chemically reactive species. The magnetic Reynolds number is assumed to be small enough so that the induced magnetic field is negligible. The wavy wall
The equations governing the flow are in vector form.
The suitable initial and boundary conditions for the present problem are
Using the relation in [
The permeability of the porous medium is assumed to be of the following form:
Now we introduce the following non dimensional quantities:
The corresponding initial and boundary conditions (
Equations (
Let us assume the pressure gradient is of the form
The solutions of (
A hydromagnetic flow, heat and mass transfer of a chemically reacting viscoelastic fluid in an irregular channel with convective boundary condition are investigated in the presence of heat absorption, thermal radiation, and Hall effect. The governing partial differential equations are solved analytically using perturbation technique. In order to get a physical insight of the problem, we have written a MATLAB program to perform a parametric study on different flow fields versus
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of axial velocity profile for the different values of
Variation of transverse velocity profile for the different values of
Variation of temperature profile for the different values of
Variation of temperature profile for the different values of
Variation of temperature profile for the different values of
Variation of temperature profile for the different values of
Variation of concentration profile for the different values of
Variation of concentration profile for the different values of
Variation of skin friction profile for the different values of
Variation of skin friction profile for the different values of
Figures
In Figure
Figures
One can see that in Figures
Figures
Figure
The effect of the chemical reaction parameter on the species concentration profile is shown in Figure
Figures
Table
Effects of
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Gr < 0 | Gr < 0 | Gr > 0 | Gr > 0 | |||||
2.0 | 0.1 | 0.1 | 2.0 | 2.0 | 0.5139 | 0.1411 | 0.4591 | 0.1244 |
4.0 | 0.4200 | 0.2202 | 0.3767 | 0.1941 | ||||
6.0 | 0.3289 | 0.2437 | 0.2967 | 0.2153 | ||||
1.0 | 0.0 | 0.1 | 2.0 | 2.0 | 0.5690 | 0.0780 | 0.4471 | 0.0596 |
2.0 | 0.5699 | 0.0787 | 0.4476 | 0.0601 | ||||
4.0 | 0.5735 | 0.0818 | 0.4515 | 0.0631 | ||||
1.0 | 0.1 | 0.1 | 2.0 | 2.0 | 0.5470 | 0.0784 | 0.4882 | 0.0694 |
0.3 | 0.6121 | 0.0885 | 0.4231 | 0.0593 | ||||
0.5 | 0.6888 | 0.1007 | 0.3464 | 0.0472 | ||||
1.0 | 0.1 | 0.1 | 1.0 | 2.0 | 0.5558 | 0.0776 | 0.4970 | 0.0687 |
5.0 | 0.5200 | 0.0766 | 0.4612 | 0.0676 | ||||
9.0 | 0.5011 | 0.0697 | 0.4423 | 0.0607 | ||||
1.0 | 0.1 | 0.1 | 2.0 | 1.0 | 0.5731 | 0.0815 | 0.4510 | 0.0628 |
2.0 | 0.6792 | 0.1038 | 0.5571 | 0.0851 | ||||
3.0 | 0.7853 | 0.1262 | 0.6633 | 0.1075 |
Effects of
Parameter | Values |
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0.0 | 2.6676 |
0.72 | 2.5993 | |
3.0 | 2.5613 | |
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0.5 | 2.6033 |
1.5 | 2.5767 | |
2.5 | 2.5647 | |
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0.0 | 1.1447 |
1.0 | 1.4854 | |
2.0 | 2.5993 |
Effects of Kr,
Parameter | Values |
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1.0 | 1.8162 |
3.0 | 1.4499 | |
5.0 | 1.1138 | |
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0.0 | 0.6650 |
1.0 | 0.7933 | |
2.0 | 0.8405 | |
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0.22 | 0.9990 |
0.30 | 0.9599 | |
0.78 | 0.7409 |
The present investigation dealt with the combined effect of thermal radiation and Hall current on unsteady MHD visco-elastic fluid flow in an irregular channel subject to convective boundary condition. The study is also concerned with the free convective flow and mass transfer with heat absorption and chemical reaction effects. The perturbation method is used to solve the problem, and the results are evaluated numerically and displayed graphically using MATLAB package. In the light of the present investigation, the following conclusions can be summarized. The velocity profiles are parabolic in nature, and the velocity field is higher in heating ( The Hall effect plays an important role in controlling momentum of the fluid. The strength of the applied magnetic field should be as low as possible to realize. The temperature profile decreases for increasing values of heat absorption parameter. The effect of increasing values of Prandtl number decreases temperature distributions. An increase in the dimensionless time enhances the skin friction profile at the wavy wall, and this manner is opposite for increasing values of chemical reaction parameter. The rate of heat transfer decreases for increasing values of heat absorption parameter at the wavy wall. Combined effect of chemical reaction parameter and Schmidt number reduces the rate of mass transfer at the wavy wall, but this trend is reverse for the increasing in dimensionless time.
Thus the present study will serve as a good scientific tool for understanding more complex flow problems concerning the various physical parameters.
The authors wish to express their deep sense of gratitude to the editor and referee for their encouraging suggestions that significantly improved the paper.