G-jitter induced mixed convection flow of a second grade fluid past an inclined stretching sheet

An analysis was carried out to study the mixed convection flow of a non-Newtonian second grade fluid past an inclined stretching sheet in the presence of a g-jitter effect. The transformed governing equation, consisting of coupled non-linear partial differential equations, was solved numerically using an implicit finite-difference scheme known as the Keller-box method. The flow and heat transfer characteristics in terms of velocity and temperature profiles as well as the skin friction and heat transfer coefficients influenced by the amplitude of modulation, frequency of oscillation, inclination angle and second grade parameters were presented graphically and analysed in detail. The findings revealed that the heat transfer coefficient is enhanced with an increase in the second grade parameter, whereas the opposite trend is observed for the skin friction coefficient.


Introduction
Second grade fluid, also known as viscoelastic fluid, is a non-Newtonian fluid due to the nonlinear relationship between stress and the rate of strain.In view of the increasing importance of non-Newtonian flow, especially in second grade fluids which exhibit both viscous and elastic characteristics, many researchers have studied convective heat transfer in second grade fluids using various conditions and geometries, for example: Mushtaq et al. [1], Hsiao [2], Hayat et al. [3], Hayat and Qasim [4], Olajuwon [5] and Turkyilmazoglu [6].Recently, Hayat et al. [7] investigated the three-dimensional mixed convection flow of viscoelastic fluid past an exponentially stretching surface by taking into account the effects of thermal radiation and convective conditions.
In all of the above-mentioned papers, the second grade fluid flow over an inclined plate was not considered.It is essential to analyse the effect of an inclined plate on second grade fluid because of its great importance in the engineering, technology and manufacturing industries.In addition, Jaluria [8] indicates that the importance of buoyancy force on an inclined, continuously moving sheet not only depends on the angle of orientation, but also on the mixed convection parameter which indicates the strength of natural and forced convection flow effects.Therefore, in this paper, the unsteady mixed convection flow of second grade fluid past an inclined stretching sheet in the presence of a g-jitter effect is investigated.g-Jitter can be defined as the inertia effect due to quasi-steady, oscillatory or transient accelerations arising from crew motions and machinery vibrations in parabolic aircrafts, space shuttles or other microgravity environments.A series study on the behaviour of g-jitter with different effects and type of fluids focusing on boundary layer flow was made by Sharidan et al. [9][10][11].For example, Sharidan et al. [10] studied the effect of g-jitter mixed convection on the flow and heat transfer characteristics associated with a vertical stretching sheet in a Newtonian fluid.Considering the effect of g-jitter in that paper, the time dependent gravitational field,   g t is defined as , where 0 g is the time-averaged value of the gravitational acceleration,  is a scaling parameter which gives the magnitude of the gravity modulation relative to 0 g , t is the time and  is the frequency of oscillation of the g-jitter driven flow.

Problem formulation
Consider the unsteady mixed convection flow of a viscous and incompressible second grade fluid past an inclined stretching sheet associated with the effect of g-jitter.In this problem, the x-axis extends along the sheet with an inclination angle of  to the vertical, and the y-axis is normal to the sheet.The sheet is assumed to have a linear velocity of Under the boundary layer and Boussinesq approximations, the basic governing equation of a second grade fluid moving past an inclined stretching sheet can be written as: subject to the initial and boundary conditions:   0 : 0, for any , , 0 : , 0 , at 0, 0 , 0, as .where u and v are the velocity components along the x-and y-axes,  is the density of the fluid,  is the effective viscosity,  is the thermal expansion of the fluid, 1  is the material parameter of the second grade fluid, T is the temperature of the fluid, p C is the specific heat of the fluid at a constant pressure, and k is the effective thermal conductivity of the fluid.The complexity of the problem is reduced by introducing the similarity transformations [10]: where  is the effective kinematic viscosity and   subjecting this to the boundary conditions (4), it becomes: 0, 1, 1 on 0, 0, 0, 0 as .
Dimensionless parameters of present problem are, where K is the dimensionless second grade parameter, Pr is the Prandtl number,  is the dimensionless frequency of oscillation and  is the mixed convection parameter with x Gr and Re x being the local Grashof number and Reynolds numbers respectively.In this study, the non-dimensional skin friction and heat transfer coefficients are obtained by: Re ,0 3 ,0 ,0 ,0 ,0 , Re ,0 .

Results and discussion
The system of equations ( 6) and ( 7) together with the boundary conditions (8) are solved numerically using the Keller-box method, an unconditionally stable implicit finite difference scheme.The effect of different values of parameters including amplitude of modulation,  , frequency of oscillation,  , second grade parameter, K, and inclination angle,  , on the variation of velocity,   under the influence of the second grade parameter, K, and an inclination angle,  .The problem of the mixed convection flow of second grade fluid past an inclined stretching sheet in the presence of a g-jitter effect has been investigated.Numerical results reveal that the inclination angle has a retarding effect on the flow field and enhances the temperature field, whereas the opposite effect is observed for the second grade parameter.Furthermore, the enhancement of heat transfer can be clearly seen under the influence of the second grade parameter.
x-direction of the flow, and the temperature of the sheet varies linearly with distance x along the sheet: the temperature of the sheet and T  being the uniform temperature of the ambient fluid.The continuous stretching sheet is assumed to have velocity and temperature in the form of where a and c are constants, and 0 c  .
is automatically satisfied.Substitution of equation (5) into equations (2) and (3) give the following transformed governing equations:

Figure 1 Fig. 1 .Fig. 2 . 4 MATEC
Figure1shows that an increase in K leads to an increase of

Table 1 .
Comparison results of the heat transfer rate,   [12]etail.The comparison with the previous published work of Sharidan et al.[10]and Freidoonimehr et al.[12], as shown in table 1, reveals a good agreement, which validates the use of the present scheme.