The effect of radiation on flow and heat transfer over a vertically oscillating porous flat plate embedded in porous medium with oscillating surface temperature is investigated. The analytic solutions of momentum and energy equations are obtained. The velocity and temperature profiles are computed. The frictional force at the plate due to viscosity of fluid is estimated in terms of non dimensional skin friction coefficient and heat convection at the plate is estimated in the form of Nusselt number. The effects of physical parameters Prandtl number Pr, Grashof number Gr, Suction parameter S and radiative parameter R on velocity and temperature profiles are analyzed through graphs. The effects of oscillation on the velocity and temperature profiles are shown through 3-D surface plot.

Radiation Suction Heat Flux Oscillating Porous Plate
1. Introduction

The radiative free convective flow has many important applications in countless industrial and environment processes e.g. heating and cooling chambers, fossil fuel combustion energy processes, evaporation from large open water reservoirs, astrophysical flows, solar power technology and space vehicle re-entry. The radiative heat transfer plays an important role in manufacturing industries for the design of reliable equipment. Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles and satellites are examples of such engineering applications.

The aim of the present study is to investigate the radiative effect on flow and heat transfer over a vertically oscillating porous flat plate with oscillating surface temperature.

2. Formulation and Solution2.1. Mathematical Model of Flow

To study free convective flow and heat transfer through a vertical porous flate plate in the influence of radiative heat flux is considered (Figure 1). The axis of x is taken along the vertical plate and the axis of y is normal to the plate.The plate is oscillating in its own plane with a frequency of oscillation and mean velocity. The temperature at the plate is also oscillating and the free stream temperature is constant. A constant suction velocity is applied at the oscillating porous plate. Since the plate is of semi infinite length therefore the

variation along x-axis will be negligible as compared to the variation along y-axis so

In view of the physical description the governing equations are defined as follows:

The equation of continuity

reduces into

Physical model of the problem.

Þv, is independent of y and (constant) is suction normal to the plate.

The momentum equation for the prescribed geometry is given by

Under usual Boussinesq’s approximation the Equation (2) becomes

The energy equation is given by

The associated boundary conditions are

where u the flow velocity component in the x-direction, ν the kinematic viscosity, g the acceleration due to gravity, β the volumetric expansion coefficient, α the thermal diffusivity, the thermal conductivity, the radiative heat flux, T the fluid temperature, the ambient temperature and the amplitude of oscillation of the plate.

The Roseland approximation for radiative heat flux [Brewster, 1992] is given by

where σ is Stefan ? Boltzmann constant and is the mean absorption coefficient.

Taking the Taylor series expansion of T4 and neglecting terms with higher powers, we have

2.2. Mathematical Formulation

Introducing following dimensionless quantities

Equations (3), (4) and the associated boundary conditions (5) reduces into

where, is suction parameter, the Grashof number, M the magnetic parameter, Da the Darcy number, R the radiative parameter, the Prandtl number, and

The corresponding boundary conditions are

There is no loss of generality in omitting the asterisk from (6) to (8).

2.3. Numerical Solution

To solve the coupled nonlinear partial differential Equation (6) and (7) along with the boundary conditions (8), the solution for u and (after dropping *) the following form will be suitable

where, , and are unknown to be determined.

Invoking the Equations (9) and (10) in the Equations (6) and (7) and equating harmonic and non harmonic terms, the set of ordinary differential equations are given as

where, primes denote derivative with respect to y

The corresponding boundary conditions are

The Equations (13) and (14) are ordinary differential equations with prescribed boundary conditions as given in (15) and (16), therefore their solutions are straight forward are given by

where

Now using expression of and in the Equations (11) and (12) we get second order ordinary differential equations in and given by

In view of the boundary conditions associated with and, the solution of the Equations (19) and (20) are known and given by

where, and are constant quantities.

Now substituting expression of, , and in the Equations (9) and (10)

3. Skin Friction

The skin-friction at the plate, which in the non-dimensional form is given by

and computed values are given in Table 1.

4. Nusselt Number

The non dimensional coefficient of heat transfer defined by Nusselt number is obtained and given by

and computed values are given in Table 2.

5. Results and Discussion

The velocity profiles for different parameter like Suction parameter, Grashof number, magnetic parameter, Darcy number, Prandtl number and radiation parameter are shown by Figures 2-7. Temperature profiles are also shown by Figures 8-10. In Figure 2, the positive values of S correspond to cooling of the plate decreases in the vicinity of the permeable plate while increases in region close to non permeable wall. In Figure 3, the positive values of Gr correspond to cooling of the plate increases in the vicinity of the permeable plate while decreases in region close to non permeable wall. In Figure 4 when magnetic parameter M is increased, keeping other parameters constant the velocity increases. In Figure 5 when Darcy number Da is increased, keeping other parameters constant the velocity decreases. In Figure 6 and Figure 7 when Prandtl number Pr and radiation parameter

Non dimensional coefficient of Skin Friction-C<sub>f</sub>
SGrMDaPrRCf
0.0550.11215.9195
0.2550.11216.2228
0.5550.11216.6869
−0.2550.11215.6222
−0.5550.11215.1854
0.5250.1123.9841
0.51050.11237.8582
0.5−250.112−12.9529
0.5−550.112−25.6557
0.5−1050.112−46.8270
0.5530.11211.2156
0.5580.11226.3903
0.5550.011235.8229
0.55511213.7920
0.5550.10.528.7570
0.5550.10.71211.0189
0.5550.11428.0977
0.5550.11639.5676
Non dimensional coefficient of Heat Transfer- Nu
SPrRNu
0.0120.0004
0.2120.0757
0.5120.2058
−0.212−0.0657
−0.512−0.1479
0.50.520.2174
0.50.7120.2110
0.5140.2028
0.5160.2018

R is increased, velocity increases in the vicinity of the permeable plate while decreases in region close to non permeable wall. In Figure 8, the positive values of S correspond to cooling of the plate increases temperature decreases and the negative values of S correspond to heating of the plate decreases temperature increases. Figure 9 and Figure 10 represent that temperature decreases when Prandtl number and radiation parameter R increases. The variation in fluid velocity and temperature in the porous medium through the one period of oscillation is demonstrated by Figure 11 & Figure 12. The fluid velocity and temperature oscillates up to a certain distance from the plate while this oscillation diminishes at large distance from the plate.

Table 1 represents the values of skin friction. The positive values of S correspond to cooling of the plate increases skin friction increases and the negative values of S correspond to heating of the plate decreases skin friction decreases. The positive values of Gr correspond to cooling of the plate increases skin friction increases and the negative values of Gr correspond to heating of the plate decreases skin friction decreases. When M is increased the skin friction increases and when Da is increased skin friction decreased. When Pr and R is increased the skin friction increases.

The values of Nusselt number is given in Table 2. The positive values of S correspond to cooling of the plate increases Nusselt number increases and the negative values of S correspond to heating of the plate decreases Nusselt number decreases. When R and Pr are increased the Nusselt number decreases.

6. Conclusions

・ Fluid flow slows down in the vicinity of the permeable plate while enhances in region close to nonpermeable

Variation in velocity with <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-2320275x90.png" xlink:type="simple"/></inline-formula> and distance from the plate at S = 0.5, Gr = 5, M = 5, Da = 0.1, Pr = 1, R = 2 Variation in temperature with <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-2320275x92.png" xlink:type="simple"/></inline-formula> and distance from the plate at S = 0.5, Pr = 1, R = 2

wall on increasing suction parameter correspond to cooling of the plate.

・ Fluid velocity profiles increase in the vicinity of the permeable plate while decrease in region close to non permeable wall with the increase in Grashof number correspond to cooling of the plate

・ Fluid velocity profiles increase in the vicinity of the permeable plate while decrease in region close to non permeable wall when Prandtl number and radiation parameter is increased.

・ Fluid velocity and temperature in the porous medium through the one period of oscillation oscillates up to a certain distance from the plate and this oscillation disappears far away from the plate.

・ The values of skin friction increase when magnetic parameter, Prandtl number and radiation parameter are increased while the values of skin friction decrease when Darcy number is increased.

・ Nusselt number decreases when Prandtl number and radiation parameter are increased.

Cite this paper

Monika Miglani,Net Ram Garg,Mukesh Kumar Sharma, (2016) Radiative Effect on Flow and Heat Transfer over a Vertically Oscillating Porous Flat Plate Embedded in Porous Medium with Oscillating Surface Temperature. Open Journal of Fluid Dynamics,06,119-129. doi: 10.4236/ojfd.2016.62010