- Non Newtonian Fluids
- Heat Transfer in a Micropolar Fluid over a Stretching Sheet with Newtonian Heating
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In many practical problems, the heat transfer from the plate surface is proportional to the local surface temperature, known as Newtonian heating NH effect Because of the interesting and practical applications of NH problems, scientists are attracted to take the NH condition in their studies.
Non Newtonian Fluids
Merkin 22 took NH condition and investigated the boundary layer natural convection flow over a vertical surface. Hussanan et al. In another article, Hussanan et al. They solved these problems analytically by using the Laplace transform method.
- Mass transfer from small capillaries with wall resistance in the laminar flow regime;
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Recently, the influence of thermal radiation and heat transfer for unsteady Casson fluid with NH boundary condition is observed by Hussanan et al. They obtained the exact solution and discussed various parameters graphically. Furthermore, Vieru et al. Recently, Fetecau et al. However, in all the above studies and previously published other works, the force of magnetic lines are fixed to the fluid.
Lately, Narahari and Debnath 28 investigated the MHD, free convection flow over a vertical plate with heat flux, absorption or generation when the force of the magnetic line is fixed to the fluid or plate. That is the magnetic field was taken in relative status. When magnetic field is fixed to the fluid or plate, play an important role in manufacturing industries for the design of fins, gas turbines nuclear, power plants, steel rolling and various propulsion devices for aircraft, satellites. The solutions are obtained exactly and exponentially or constantly accelerating plate.
Tokis 32 was the first who obtained the exact solution of this type of problem. He discussed the solution by uniform, constantly acceleration or decaying of the plate. Onyango et al. From the above discussion, it is crystal clear that the effect of NH condition for Casson fluid moving over a vertical plate when the magnetic field show relative behavior with additional consequence of thermal radiation, porous medium, chemical reaction and heat generation is not investigated so far.
Therefore, the objective of the current work is to make such an attempt. Exact solution of the problem is obtained for general velocity condition at the wall using the Laplace transform method with some interesting cases of particular solutions. Let suppose an electrically conducting, unsteady free convection incompressible flow of Casson fluid over an infinite perpendicular plate.
The magnetic field shows a relative behavior, i. The effects of chemical reaction and thermal radiation are also taken. The magnetic field B is taken normal to V.
The Reynolds number is so small that flow is laminar. The rheological equation for an incompressible flow of a Casson fluid can be written as, see 24 , Under these conditions and the concentration, temperature and velocity are the functions of y , t only, then the governing equations are reduced to the set of partial differential equations.
Equation 6 is valid if the magnetic field is fixed while for the relative magnetic field the Eq. The concentration and temperature fields corresponding to this problem are obtained easily from the previous work, here we are interested in the solution of the fluid velocity field. By using the Laplace transform of Eqs 12 — 15 , we developed the solutions:.
In order to determine the skin fraction or shear on the plate, introducing Eq. It is clear from Eq. The influence of mass and heat transfer is significant with some possible engineering applications. The dimensional velocities for sine and cosine oscillations describe the fluid motion for unsteady state solution.
After that the transients disappear, they reduce to steady state permanent solutions:. Though, they satisfy the boundary condition and the governing equation. Equations 40 and 41 Can be written using simple but beautiful forms by using Eqs A8 and A9 from appendix,. Exact solution of MHD natural convection radiative flow of Casson fluid over a moving vertical plate with chemical reaction, concentration, and Newtonian heating are developed by using the Laplace transform method.
The magnetic field is taken relative i. Velocity plot against permeability of porous medium k for the variable and oscillating plate. In all three Figs, time t shows an increasing effects on all the three profiles. Moreover, Fig. An identical behavior is noticed for all the three cases. All the three cases have the same effects. The influence of permeability of the porous medium k on velocity profile, for a variable and oscillating sine and cosine plate is highlighted in Fig.
It is clear that for decreasing of velocity, we need to decrease the permeability of a porous medium. It is indeed due to decreasing drag force, which leads to the decreasing of velocity. In other words, the resistance of the porous medium is depressed which increases the momentum development of the flow regime, ultimately accelerate the velocity of the fluid.
Magnetic parameter M leads to a decrease in the velocity profile as shown in Fig. It is due to Lorentz force. Lorentz force works in the opposite direction of the fluid which makes a decreasing effect on velocity. Furthermore, increasing value of M increases resistive forces which resist the fluid flow and thus decreasing the fluid velocity.
For both profiles, the increase of Pr leads to falling in velocity and temperature. The low rate of thermal diffusion causes arising in velocity boundary layer thickness. Pr controls the relative thickness of the thermal and momentum boundary layers in the problems of heat transfer. Consequently Pr can be utilized to expand the percentage of cooling. Growth in Schmidt number declines the concentration boundary layer thickness.
As Schmidt number is the ratio between viscous forces and mass diffusivity, therefore, it reduced the consecration as well as velocity profile. Physically, if the increase occurs in viscous forces, it leads to a decrease in the velocity of the fluid. This shows the enhancement of temperature profiles by increasing heat generation parameter, which is because of a thickness of the thermal boundary layer. The variation of the ratio of buoyancy force N is highlighted in Fig.
Heat Transfer in a Micropolar Fluid over a Stretching Sheet with Newtonian Heating
This shows the increasing effect on the velocity profile. Increase in N means, the increase of concentration gradients which leads the significant of buoyancy forces and hence observed a small rising near the plate. Physically, it is true because higher values of v make fall in the chemical molecular diffusivity, i. It is observed that MFFRP have greater velocity profile as compare to MFFRF, and the same result is analyzed for all three cases which is plotted in a , b and c for variable and oscillating sine and cosine oscillations conditions.
The comparison of present solution is made in Fig. It is observed that in the absence of Casson, Newtonian heating and chemical reaction parameters i. This shows the validity of our obtained solutions. Comparison plot of the present solution with Kumaresana et al. An exact analysis of unsteady, incompressible, Casson fluid of free convection flow ended moving vertical plate set in a pours medium, the constant magnetic field is vertically applied to the plate with the MFFRF and MFFRP, and our interest to focus on the motion of the fluid. The influence of temperature and concentration is considered with the presence of thermal radiation and Newtonian-heating.
The method of Laplace transformation is used to determine the closed form of concentration, velocity and temperature and the corresponding skin fraction are established in the term of error and complementary error functions.
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The velocity is obtained for a variable and oscillating plate. Both the velocity and skin fraction are presented in the sum of their mechanical, thermal and concentration components. For comparison, the concentration, velocity and temperature profile have schemed graphically and the effect of the various parameter is studied. The main findings are as follows:. Contributions of concentration, thermal and mechanical component of velocity and skin fraction on the motion of fluid are significant and they cannot be ignored. Bejan, A. The thermodynamic design of heat and mass transfer processes and devices.
International Journal of Heat and Fluid Flow 8 4 , — Khan, I. Magnetohydrodynamic free convection flow past an oscillating plate embedded in a porous medium.
Journal of Heat and Mass Transfer Research (JHMTR)
Journal of the Physical Society of Japan 80 10 , Sandeep, N. Impact of cross diffusion on MHD viscoelastic fluid flow past a melting surface with an exponential heat source. Multi discipline Modeling in Materials and Structures. Radiation and inclined magnetic field effects on unsteady hydromagnetic free convection flow past an impulsively moving vertical plate in a porous medium.
Journal of Applied Fluid Mechanics 7 2 , — Kempannagari, A. Impact of frictional heating on MHD radiative ferrofluid past a convective shrinking surface. In Defect and Diffusion Forum Vol. Ramandevi, B. Alexandria Engineering Journal.