Numerical investigation of dusty tri-hybrid Ellis rotating nanofluid flow and thermal transportation over a stretchable Riga plate

Due to high-ultra thermic significances, the nanosize materials are used in various chemical and mechanical engineering, modern technology and thermic engineering eras. For industrial growth of a country, one of the biggest challenges for engineers and scientists is improvement in thermal production and resources. In this study we analyzed the momentum and thermic aspects of MHD Ellis ternary nano material embedded with dust particles via stretchable Riga plate including volume concentration of dust material. The flow generating PDE’s for two phase models are minimized into dimensionless nonlinear ODE’s by using the right modification. To acquire the graphical results the BVP4c method was adopted in MATLAB software. Fundamental aspects affecting velocity and temperature have investigated through graphs. Additionally Nusselt number and skin friction have also been evaluated. Compared it with previous literature to check the validity of results. Finding reveals that as compared to dusty phase the performance of trihybrid nano phase thermal transport is improved. Moreover, the temperature profile increases for rotational and volume fraction dust particles parameter. Dusty fluids are used in numerous manufacturing and engineering sectors, like petroleum transport, car smoke emissions, caustic granules in mining and power plant pipes.


List of symbols x, y, z
and fluid were assumed to be stable.The fluid is incompressible , therefore the dust particles density is constant and between the dust particles energy is prevent.Volume fraction of dusty particles has also been into account.Further, The plate having stretched velocity U w along x-axis.Due to tri-hybrid nanofluid is considered a stable mixture, therefore nano size particles agglomeration is ignored.zero velocity is assumed at ambient surafce.T w and T ∞ are the wall and ambient temperature.The model is sketched in Fig. 1.Considering the aloft conditions, the conservation of momentum and temperature equations can be mentioned as 33,34 : The appropriate boundary conditions are 35,36 : Here, ( u 1 , u 2 , u 3 ) are velocities component in (x, y, z) directions.Dust particles velocity components are rep- resented by ( u 1 p, u 2 p, u 3 p ), denotes the constant velocity, U w denotes the stretchable velocity component in x-direction, a is stretchable constant rate (a is positive), ρ Thnf is density tri-hybrid nanoliquid, ρ p is dust particles density, C p is dust particles concentration, T is liquid temperature, T p is dust particles temperature, c p is specific thermal capacity of liquid, k Thnf is thermic conductivity of tri-hybrid, τ T is thermic stability time, K is constant of Stoke's drag and L is micro-rotation factor.

Rheological and thermophysical characteristics
The thermophysical characteristics of TiO

Similarity conversion
We assumed the following appropriate transformation 38 .
Equation (1) is identically satisfied.By utilizing the aloft mentioned transformations in Eqs.(2), (3), (4), ( 5), ( 6), ( 7), ( 8), ( 9) and (10), we get Here, C is modified Hartman parameter, d is non-dimensional parameter, β is rotation parameter, φ d is concen- tration of dust particles, Pr is Prandtl factor, B 1 is fluid parameter, β t is thermal dust factor, γ t is specified thermal ratio, β v is velocity of fluid particles, γ v is dust particles mass concentration, Mathematically, The physical quantities are Nusselt number and skin friction coefficient are defined as: Where Cf x , Cf y are skin friction coefficients along x and y-axis, Nu is Nusselt number.The non-dimensional form of Nusselt number and skin friction coefficient are as follows: Where Re x = xU w ν f is the Reynolds number.

Results and discussion
The non-dimensional ODE's are solved by utilizing BVP4c technique.In Table 1 the thermo-physical characteristics of base fluid and nanosize particles are mentioned.For validation the present results are compared with existing literature, the results comparison is shown in Table 2.An excellent agreement is observed with the literature.The outcomes of this investigation are explained via Figs.2, 3, 4, 5, 6, 7, 8, 9 and 10. Figure 2a,b depicts the fluctuation in H 1 , H 2 w.r.t modified Hartmann number C. The excessing strength of C is due to the incre- ment of outward electric field.In this scheme the wall parallel force (Lorentz force) restrain the boundary layer growth.Since the magnetic range decreases rapidly, therefore velocity profile increased.Physically the magnetic range generates the Lorentz force that's in turn resisting the fluid flow.However in the present circumstance, the magnetic range decreases therefore the Lorentz force also decreases, as a result velocity profile increased.The magnitude of H 2 is decreases for higher values of C. It is ratified that the application of electro-magnetic field constructed as a Riga plate setting comfort to stable the rotatable flow.Figure 3a,b shows the impact of rotation parameter β on Primary velocity H 1 and secondary velocity H 2 .It is noted that with amplifying values of β there is retardation in H 1 .In case of β = 0 (pure stretchable case) the velocity attains its highest values.Due to Coriolis forces, the fluid motion slows down.For higher values of β the secondary velocity H 2 has the inverse behavior.Figure 4a,b demonstrate the influence of β on the dusty phase fluid velocities.Here, H 1p and H 2p denote the MBL (momentum boundary layer) for dusty case in x-axis and y-axis.In dusty case of fluid the axial velocity decrease due to rising strength of rotation parameter and transverse velocity shows the opposite behavior against

Pr
Ref. 39 Ref. 40 Present results 1.0 1.0000 1.0000 1.0000 this parameter.Figure 5a,b indicates the fluid velocities for β v .It reveals that the axial velocity of ternary fluid phase is depressed with higher input of β v .Physically an increasing the dust particles mass concentration the dust particles weight is increased which decreases the fluid velocity.On the other hand transvers velocity shows the opposite behavior for increasing trend of β v .Figure 6a,b demonstrate the effect of β v on the dusty phase fluid velocities.In dusty case of fluid the axial velocity increase due to rising values of dust particles mass concentration and against this parameter transverse velocity shows the opposite behavior.Figure 7a,b portrays the influences of dusty volume fraction variation on axial and transverse velocities.It is observed that by increasing the concentration of dust particles, the liquid becomes thick and creates more resistance, therefore axial velocity decreased.Due to rotation an opposite trend is noticed in transverse velocity.Figure 8a,b illustrates the impact of rotation parameter β on fluid temperature and dusty phase of ternary fluid.It is observed that in dusty and ternary fluid phase, temperature increased with higher values of β .Basically, the energy development is satisfied on the base of a diffusion procedure because of increased rotation.Figure 9a,b show the thermal dusty parameter influence on θ and θ p .For amplifying values of β t the fluid flow is slow down therefore temperature is decreased.On the other hand higher values of β t , in suspended debris enhance the friction force.Therefore dusty fluid temperature is increased.Figure 10a,b demonstrate the dusty volume fraction impact on temperature.For higher inputs, fluid temperature and dusty fluid temperature increases.Basically by increasing the dusty volume fraction thermal conductivity increased therefore temperature boost up.Figure 11a,b reveals the skin friction coefficient distinct values of rotating parameter and dust particles concentration.It is noted that both primary and secondary velocities shows decreasing trend for higher input of rotating parameter.For increasing values of dust concentration the primary velocity decreases and secondary velocity shows the opposite behaviour.Figure 12a,b portray the Nusselt number against thermal dust factor, rotating parameter and dust particles concentration.Nusselt number shows the decreasing trend for higher values of dust particles concentration.

Table 1 .
Thermophysical properties of nano size particles and water base fluid.

Table 2 .
Comparing the present numerical Nusselt number for Pr when all other parameters are zeros.