Unsteady Viscous Dissipative Dusty Nanofluid Flow Over a Vertical Plate

Abstract— The flow past an infinite vertical isothermal plate started impulsively in its own plane in a viscous incompressible two-phase nanofluid has been considered by taking into account the viscous dissipative heat. Two nano particles Copper (Cu) and Alumina (Al2O3) are submerged in a base fluid, Water (H20). The coupled non-linear partial differential equations which govern the flow are solved for nanofluid and dust particle phases by finite difference method. The velocity and temperature fields have been shown graphically for various parameters. Here Grashof number, (Gr) being positive (cooling of the plate) for dusty air. Also the effects of Eckert number on heat transfer and skin friction coefficient for various parameters are represented graphically. It is observed that dusty nanofluid enhances both skin friction and heat transfer rate in the case of cooling. Keyword-Free convection, Nanofluid, Viscosity, Two-phase flow, Dust particles

0   t and the temperature is instantaneously raised or lowered to a constant temperature w T  . It is also taken into account that all the fluid properties are assumed to be constant except that the influence of density variation with temperature has been considered only in the body force term. The buoyancy force on the dust particles is neglected. It is further neglected the volume fraction and viscosity of the pseudo fluid of dust particles. The set of governing equations of motion modelled in [6] for dusty fluid is extended to nanofluid as follows. Equation of Continuity: where u , p u are velocities of the fluid and dust particles respectively, g is acceleration due to gravity,  is coefficient of volume expansion. The temperature of fluid and dust particles are denoted by T  , p T  respectively.
T are wall and ambient temperatures respectively. Also, one can write the relaxation time during which the velocity of the particle phase relative to the fluid is reduced ( e / 1 ) times its initial value, p p r m   6 /  and thermal relaxation time of particle phase, where m is mass of each dust particle, s c is specific heat of dust particle, 0 N is number density of dust particle, K is Stokes resistance coefficient. Further if  is the solid nanoparticles volume fraction, then the effective dynamic viscosity, density, heat capacitance, thermal conductivity of the nanofluid are defined respectively as The initial and boundary conditions of the problem may be defined as Using non dimensional variables, One can transform the governing equations of motion given in (2) -(5) as the following non-dimensional partial differential equations ) ( . Also from the equations (6) and (7), one can express the initial and boundary conditions in terms of non-dimensional quantities as follows.
SOLUTION PROCEDURE The transformed equations of motion (8) -(11) along with the initial and boundary conditions (12a)and (12b) are solved by finite difference method. The partial differential equation coefficients are replaced by their finite difference quotients forming the following finite difference equations. , Pr .
where the index i refers to y and j refers to t. y  is taken to be 0.1. From the condition (12a), According to the numerical solutions [14], it may be taken y =4. 1 (13) and (14) in terms of velocities and temperatures at points of the earlier time step. Similarly (15) and (16). The procedure is repeated till t=1 (j=400). The computations are carried out and the curves of velocity and temperature against y are traced using MATLAB software.
IV. RESULTS AND DISCUSSION The velocity and temperature profiles are drawn for Prandtl number 6.785(Water) and time t=0.2. Thermo physical properties of nanoparticles [15] are given in Table1. The velocity profiles of both fluid and dust particle phases for different values of Grashof number are depicted in fig.1and fig.2. In the case of the plate being cooled by free convection currents we observe that due to presence of viscous dissipative heat, there is a rise not only in velocity of fluid(nanofluid) but also in dust particle phase. Also it follows from the figures that greater cooling of the plate causes a rise in velocity. Fig.3 and fig.4 explain how the Eckert number along with solid volume fraction influences the temperature profiles of both fluid and dust particle phases. In both the figures it is observed that there is a decrease in the thermal boundary layer with respect to increase in the volume fraction. Also in the case of cooling (Gr > 0), greater viscous dissipative heat causes a rise in temperature not only in fluid but also in dust particles phase. The velocity profiles of both the phases are displayed in fig.5 for both the nanoparticles at different Grashof number. It is noted that compared to fluid velocity, dust particle velocity is less in both the nanoparticles. Also Copper nanoparticles move with more velocity than that of Alumina. From fig.6, it is observed that in the case of cooling, Alumina particles have high temperature profile than Copper. This variation is significant at low viscous dissipation. The variation of velocity profiles of both fluid and dust particle phase can be seen in fig.7 and fig.8 for two different values of a slightly higher mass concentration parameter. It is noticed that velocity increases with the increase of concentration parameter. We see a significant change in the base fluid ( 0   ). The same is observed in both velocity profiles. Fig.9 shows the impact of Eckert number on heat transfer rate. The curves are drawn for two different values of mass concentration of dust particle's parameter. We notice that heat transfer insignificantly enhances with a small increase in the Eckert number. At higher mass concentration, the variation in the heat transfer rate is more. This is due to the fact that increasing dust particles will enhance the heat transfer in the fluid. In the similar way a plot is drawn for coefficient of skin friction against Eckert number which is shown in fig.(10). With the increase in time, the skin friction coefficient decreases. Also with small increase in Eckert number there is no much difference in the profile.

CONCLUSION
A two-phase nanofluid viscous flow past an infinite vertical isothermal plate is considered. Copper (Cu) and Alumina (Al 2 O 3 ) are submerged in a base fluid, Water (H 2 0). The coupled non-linear governing differential equations are solved by finite difference method. It is observed that due to viscous dissipative heat, there is a rise not only in velocity of fluid (nanofluid) but also in dust particle phase. Volume fraction increases the thermal boundary layer. With respect to Grashof number, it is noted that compared to fluid velocity, dust particle velocity is less in both the nanoparticles. Eckert number enhances the heat transfer rate in both phases.