HEAT TRANSFER ANALYSIS ON ROTATING FLOW OF A SECOND-GRADE FLUID PAST A POROUS PLATE WITH VARIABLE SUCTION

We deal with the study of momentum and heat transfer characteristics in a second-grade rotating flow past a porous plate. The analysis is performed when the suction velocity normal to the plate, as well as the external flow velocity, varies periodically with time. The plate is assumed at a higher temperature than the fluid. Analytic solutions for velocity, skin friction, and temperature are derived. The effects of various parameters of physical interest on the velocity, skin friction, and temperature are shown and discussed in detail.


Introduction
The study of non-Newtonian fluids has attracted much attention, because of their practical applications in engineering and industry particularly in extraction of crude oil from petroleum products, food processing, and construction engineering.Due to complexity of fluids, various models have been proposed.The equations of motion of non-Newtonian fluids are highly nonlinear and one order higher than the Navier-Stokes equations.Finding accurate analytic solutions to such equations is not easy.There is a particular class of non-Newtonian fluids namely the second-grade fluids for which one can reasonably hope to obtain an analytic solution.Important studies of second-grade fluids in various contexts have been given in the references [1,3,6,7,9,10,11,12,13,17,19,20,21,22,24].
Since the pioneering work of Lighthill [16] there has been a considerable amount of research undertaken on the time-dependent flow problems dealing with the response of the boundary layer to external unsteady fluctuations about a mean value.Important contributions to the topic with constant and variable suction include the work of Stuart [25], Messiha [18], Kelley [15], Soundalgekar and Puri [23], and Hayat et al. [8].
Despite the above studies, no attention has been given to the study of the simultaneous effects of the rotation and heat transfer characteristics on the non-Newtonian flow with variable suction.Such work seems to be important and useful for gaining our basic understanding of such flow and partly for possible applications to geophysical and astrophysical problems.Also, heat transfer plays an important role during the handling and processing of non-Newtonian fluids.The understanding of heat transfer in boundary

Mathematical formulation
Let us consider an incompressible second-grade fluid past a porous plate.The plate and the fluid rotate in unison with an angular velocity Ω about the z -axis normal to the plate.The plate is located at z = 0 having temperature T 0 .The flow far away from the plate is uniform and temperature of the fluid is T ∞ .
For the problem under question, we consider the velocity and temperature fields as in which u , v , and w are the velocity components in x , y , and z directions, respectively, and T indicates the temperature.The governing equations in absence of body forces and radiant heating are div V = 0, (2.3) ) (2.5) In above equations d/dt , ρ , e, L, and q are, respectively, the material derivative, density, the specific internal energy, the gradient of velocity, the heat flux vector, and the radial distance r 2 = x 2 + y 2 .The Cauchy stress T in an incompressible homogeneous fluid of second grade is of the form A 1 = (grad V) + (gradV) , (2.7) T. Hayat et al. 557 where µ, −pI, α j ( j = 1,2), A 1 , and A 2 are, respectively, the dynamic viscosity, spherical stress, normal stress moduli, and first two Rivlin-Ericksen tensors.The thermodynamic analysis of model (2.6) has been discussed in detail by Dunn and Fosdick [4].The Clausius-Duhem inequality and the assumption that the Helmholtz free energy is a minimum in equilbrium provide the following restrictions [5]: It is evident from (2.1) and (2.3) that ∂w ∂z = 0. (2.10) The above equation shows that w is a function of time.Following Messiha [18] and Soundalgekar and Puri [23] we take (2.11) In above equation W 0 is nonzero constant mean suction velocity, A is real positive constant, is small such that A ≤ 1, and negative sign indicates that suction velocity normal to the plate is directed towards the plate.From (2.1), (2.4), (2.6), (2.8), and (2.11) we get (2.12) ) subject to the boundary conditions where U (t ) is the free stream velocity and modified pressure (2.17) 558 Rotating second-grade flow with suction In view of (2.11) and (2.14), ∂ p/∂z is small in the boundary and hence can be ignored [8,18,23].The modified pressure p is assumed constant along any normal and is given by its value outside the boundary layer.Equations (2.12) and (2.13) for the free stream yields (2.20) where U is periodic free stream velocity given by where U 0 is the reference velocity.

Analytical solution
The solution of (2.27) subject to conditions (2.28) is written as Using above equation into (2.27) and separating the harmonic and nonharmonic terms we obtain where The corresponding boundary conditions are (3.4) 560 Rotating second-grade flow with suction It is worth emphasizing that (3.2) for second-grade fluid are third order (one order higher than the Navier-Stokes equation).Thus, one needs three conditions for the unique solution whereas two conditions are prescribed.One possible way to overcome this difficulty is to employ a perturbation analysis (as in Beard and Walters [2], Soundalgekar and Puri [23], Kaloni [14], and Hayat et al. [8]) and take the solution as follows (3.5) Substituting (3.5) into (3.2),(3.4), equating the coefficients of α, and then solving the corresponding problems we have for f 1 and f 2 and so from (3.1), which upon separating real and imaginary parts gives T. Hayat et al. 561 where , , The drag P xz and lateral stress P yz at the plate in nondimensional form can be written, respectively, as ( The above equations after using (3.8) and (3.9) give , (3.12) 564 Rotating second-grade flow with suction The above equations can also be written as where T. Hayat et al. 565 We now proceed to derive the energy equation appropriate for the problem under consideration.We start with the energy equation (2.5).It follows from (2.5), (2.6), (2.7), (2.8), and (2.9) and L = gradV that Following the thermodynamical considerations given in Dunn and Fosdick [4] for fluids of second grade and representing q by Fourier's law with a constant thermal conductivity, k, (2.5) reduces to where c is the specific heat.The boundary conditions for the temperature are Using in which We further assume that Substituting (3.24) into (3.20) and boundary conditions (3.21), and equating the coefficients of the harmonic and nonharmonic term after neglecting the coefficients of 2 , we get ) where , 568 Rotating second-grade flow with suction , 570 Rotating second-grade flow with suction , , T. Hayat et al. 571 , , , , e −(hr +gr )η . (3.33)

Discussion of results
In this paper, we consider the problem of heat transfer in rotating flow of an incompressible fluid of second grade.A perturbation procedure has been used to obtain the analytic solution.The effects of various parameters such as Ω, P r , and E c on the real and imaginary parts of velocity (u,v) and temperature (θ r ,θ i ) distributions are studied and the results have been presented by several graphs.
To study the effect of Ω on the velocity components, we have plotted u and v against η in Figures 4.1       the plate, and then at a distance of η = 1, the θ r begins to decrease.That is, the behavior of θ r is quite opposite for Newtonian and second-grade fluids near the plate.Figure 4.4(a) shows the variation of Ω on θ i .It can be seen that as Ω increases, the value of θ i decreases at a distance of approximately η = 0.8 and then increases.In order to illustrate the variation of P r on θ r and θ i , we have prepared Figures 4.5 25) and (3.26) along with the boundary conditions (3.27), we obtain θ 0 = 1 − 1 + d 7 e −Pr η + d 7 + d 8 η e −2hr η , (3.28) θ 1 = −m 16 e − f η + m 7 + m 9 + m 8 η e −2hr η + m 10 + m 14 + m 12 η e −(hr +gr )η cos h i − g i η + m 11 + m 15 + m 13 η e −(hr +gr )η sin h i − g i η, (3.29)

Figure 4 .
1(b)  indicates that u increases very near to the plate and then fluctuates through an increase in Ω.The comparison of these two figures reveals that u in case of second-grade fluid is greater than that of Newtonian fluid.Also, the velocity boundary layer thickness for second-grade fluid is larger than the Newtonian fluid.It is also seen from Figures 4.2 (a) and 4.2(b) that v increases near the plate and then decreases for large value of Ω.The fluctuations in second-grade fluid are more visible than that of Newtonian fluid.Also, the value of v for second-grade fluid is smaller than in the case of Newtonian fluid.Figures 4.3 and 4.4 show the effect of Ω on the real (θ r ) and imaginary (θ i ) parts of temperature distributions.
Figure 4.3(a) shows that with the increase of Ω, θ r decreases near the wall.As shown in Figure 4.3(b), we can see that as Ω increases, θ r increases near T. Hayat et al.

and 4 . 6 .
Figures 4.5(a)  and 4.6(a) explain the effect of P r on θ r and θ i , respectively, for Newtonian fluid case.From these figures, it is revealed that near the plate, θ r decreases and θ i increases for P r > 2. The thermal boundary layer thickness in θ r increases where as for θ i decreases.For second-grade fluid, we note that from Figures 4.5(b) and 4.6(b) that for P r > 2, θ r decreases near the wall and increases far away.Also θ i for P r > 2. Figures 4.7 and 4.8 show the effect of E c on θ r and θ i .From Figures 4.7(a) and 4.7(b), we observe that θ r decreases near the wall with the increase in E c and increases far away.The thermal boundary layer thickness increases for large E c .Moreover, it can be seen from Figure 4.8(a) that θ i increases for large values of E c .From Figure 4.8(b), it can be seen that with the increase in the values of E c , the temperature θ i decreases near the plate and increases far away.The thermal boundary layer thicknesses in both fluids increases. .18) 6r n 4r n 3i − n 4i n 3r + m 4r n 4r n 3r + n 4i n 3i 16 = m 16r + im 16i = m 7 + m 9 + m 10 + m 14 ,m 16r = m 7r + m 9r + m 10r + m 146r , m 16i = m 7i + m 9i + m 10i + m 146i , fr η m 16r cos f i η − m 16i sin f i η + m 7r + m 9r + m 8r η e −2hr η + m 10r + m 14r + m 12r η cos h i − g i η + m 11r + m 15r + m 13r η sin h i − g i ηe −(hr +gr )η ,(3.32)