Analysis of a Thin Layer Formation of Third-Grade Fluid

: In present learning, surface protection layer progression of a third-grade ﬂuid (TGF) is examined. Fluid transport within the micro passage made by the ﬁrm bladehas beenpresented. Main system of equations of ﬂuidity have been narrated and streamlined by means of lubrication approximation theory (LAT). Here, approximate solutions of velocity, pressure gradient, and coating depth have been presented. Results of coating and layer forming have been tabulated and discussed as well. It is observed that the transport properties of third-order ﬂuid delivers an instrument to regulate ﬂow velocity, pressure, and a ﬀ ect the ﬁnal coated region.


Introduction
Third-grade fluids fit into the category of well-ordered flowing-particles. These have thermoviscoelastic properties and are amongst the non-Newtonian fluids (NNF) originated from the viscous constituents and elastic materials. Some of their specimens are polymeric-paints, DNA fluids, bio-organic solutions, and other synthetic materials. Polymeric fluids are practically ubiquitously exist and are used as thin layer deposition materials. Although these organic solutions and colloids demonstrate thermo-viscoelastic behavior. For these coating systems, applied stress takes into the mathematical relationship that is not simply existing in a single equation as described in [1][2][3][4][5]. In this work, Carapau et al. [2] based constitutive model for a third-order fluids is presented. In the present order, beta (β) is taken as a third-order type material factor. Phenomenaof shear thickening or shear thinning are largely governed by its mathematical assessment. If material factor beta is larger than zero, the physical system performs similar to a shear thickening substance. In caseswhere thematerial factor beta is a smaller than zero, the physical system acts similar to shear thinning LAT is manilydesignated for this flow based field. An NNF and incompressible TGF with elastic properties crawed in voids originated within narrow route with in unmovable blade and the movable substrate, and hence carved a homogeneous coating of width A on non-stationary surface. Principal models which administrate fluidity of NNF. Principle models which administrate stream of NNF involve the velocity profile

= [ ( , ), ( , )]
where is the velocity vector. This study begins with the LAT based approach. Least gap at the nip from the web and the surface is insignificant as matched to web measurement. it would be expedient to presume a parallel flow. All-purpose liquid drive is principally in −track, although the liquid speed in s-direction is minor. Here, it is rational to adopt << and << . The fact that the divergence of , i.e., ▽⋅ = 0 implies = 0 which implies = [ ( ), 0], fulfilling continuity equation, acceleration portion of the momentum and new form is where ρ denotes the density, is the pressure, and τ represents the extra tensor for the third grade fluid which is where μ is viscosity and α is the plasiticity, α is cross viscosity and β , β , β are material constants. Also , , and are Rivilin Erickson tensors. Here The Equation (1) clues to momentum equation in constituent formula as LAT is manilydesignated for this flow based field. An NNF and incompressible TGF with elastic properties crawed in voids originated within narrow route with in unmovable blade and the movable substrate, and hence carved a homogeneous coating of width A on non-stationary surface. Principal models which administrate fluidity of NNF. Principle models which administrate stream of NNF involve the velocity profile where V b is the velocity vector. This study begins with the LAT based approach. Least gap at the nip from the web and the surface is insignificant as matched to web measurement. it would be expedient to presume a parallel flow. All-purpose liquid drive is principally in r−track, although the liquid speed in s-direction is minor. Here, it is rational to adopt v << u and ∂ ∂r << ∂ ∂s . The fact that the divergence of V b , i.e., · V b = 0 implies ∂u ∂r = 0 which implies V b = [u(s), 0], fulfilling continuity equation, acceleration portion of the momentum and new form is where ρ denotes the density, p is the pressure, and τ represents the extra tensor for the third grade fluid which is where µ is viscosity and α 1 is the plasiticity, α 2 is cross viscosity and β 1 , β 2 , β 3 are material constants. Also B 1 , B 2 , and B 3 are Rivilin Erickson tensors. Here The Equation (1) clues to momentum equation in constituent formula as dτ rs ds where τ rs = τ sr = du ds and Now the generalized pressure P is given Using Equations (4)-(6), Equations (2) and (3) take the form ∂P ∂s = 0.
Equation (8) depicts that P depends on r alone. Thus, Equation (7)is written where β = β 2 + β 3 . In light of Physics, the boundary conditions are For the governing equations which are dimensionless for the analysis of blade coating, consider the following dimensionless variables The dimensional form of the volumetric flow rate Q b is where W is thickness of web. Dimensionless represntation is From above variables by neglecting the asterisks signs using Equation (11), the equation of motion (9) with the boundary condition (10) is where P r = dP dr .

OHAM Formulation
In the light of OHAM [22][23][24][25][26][27][28][29][30], the differential equation has the form where Ω refers to domain. Now in Equation (14), the operator D(v) is chosen as The construction in light of OHAM of an optimal homotopy is following where parameter q ∈ [0, 1] is called an embedding parameter, and is called an auxiliary function in optimal homotopy Equation (15), with properties that H(q) 0 for q 0, H(0) = 0. Here the constants C 1 , C 2, . . . are to be determined. Taylor's series about parameter q for expanding φ(s; q, C i ) to show estimated results are It ca be observed that the series convergence in Equation (16) depends mainly upon the constants C 1 , C 2, . . .. If at q = 1, the series is convergent, then Substitution of Equation (17) into (14) gives following residual expression give the exact solution. It does nothappen in general mostly in case of nonlinear problems. Using the method as mentioned in [20][21][22][23][24][25][26][27][28]. One can determine the values of constants C i , i = 1, 2, . . . , m.

Solution and Main Results
In this section, we will apply the OHAM to nonlinear ordinary differential Equation (13). According to the OHAM, we can construct homotopy of Equation (13) as We consider u(s) as u(s) = u 0 (s) + qu 1 (s) + q 2 u 2 (s) + q 3 u 3 (s). Substituting u(s) from Equation (19) into Equation (18), and some simplifications and rearranging based on powers of q−terms, we have (20) Solving the Equations (20)-(23)with boundary conditions, we have With q = 1, Equation (19) becomes Substituting values from Equations (24)- (26) in Equation (27), we get the first-order approximate solution of (13) as follows For finding value of the constant C 1 shown in Equation (28), using the method of least squares as described in [17][18][19] implies that setting gives the values of constant C 1 , where and here R for the Equation (13) of motion is Thus with the choice of β = 0.03 and P r = 2, the Equation (30) gives Finally using Equation (29), we get the following values of C 1 Choosing the real value of C 1 , i.e., C 1 = −0.6027727875127079; similarly for different values of β, the values of constant C 1 are shown in the Table 1.   Figure 2 shows the values of u at different values of β. Also, Figure 3 gives the nature of u at different values of β and s.    Table 2.
For fixed value of β = 0.03 and for different values of P r , the values of constant C 1 are shown in the Table 2.  Corresponding to these values, the values of λ are calculated as shown in the Equations (36)- (39).
(48) Figure 8 shows the values of stress p at different values of α. Also the Figure 9 gives the nature of p at different values of α and s. Stratagems the normal stress properties at altered locations of TGF coating progression in dissimilar standards it is perceived that strain upsurges with growing α for constant β. These results are in accordance with [29][30][31][32][33][34][35][36][37][38].     5 and 6, velocity contours reduce with enhancing NNF parameter. Upsurge in the NNF factor β resembles the shear condensing consequence that rises the liquid viscidness and declines liquid speed as supported by [37][38][39][40][41][42][43]. Figure 8 shows behavior of normal stresses at dissimilar values of α. Figure 9 shows behavior of shear stresses at varying values of αand . Results of Figures 2-5 obviously display β upsurges the NNF character upsurges, i.e., the shear thickening escalates that decreases the liquid flow rate.    5 and 6, velocity contours reduce with enhancing NNF parameter. Upsurge in the NNF factor β resembles the shear condensing consequence that rises the liquid viscidness and declines liquid speed as supported by [37][38][39][40][41][42][43]. Figure 8 shows behavior of normal stresses at dissimilar values of α. Figure 9 shows behavior of shear stresses at varying values of α and s. Results of Figures 2-5 obviously display β upsurges the NNF character upsurges, i.e., the shear thickening escalates that decreases the liquid flow rate.

Summary and Conclusions
In this work, TGF based coating model is investigated and its tranport behavior on the blade thin film where the stream is lying within the inflexible edge and the movable web. This effort examines the blade surface coating procedure for TGF. Lubrication approximation theory is employed to progress the main mathematical model for the TGF in the thin and slim conduit. Estimated results based on OHAM for velocities, pressure, and volumetric current rate. The thin film width, maximum pressure, and normal stresses are also been studied comprehensively. Our results strongly show that a third-order fluid performs as the surface coatings where the TGF transport is within inflexible blade and non-stationary system. Lubrication theory is employed to mature the major equation for the TGF in a thin conduit.