The flow of flow and heat on a continuous flat surface moving in a parallel free flow of a power flow
Boundary cover solutions are provided to incorporate the features of stable movement and continuous horizontal surface moving heat transfer in a parallel free flow of power-law flow. Movement equations are reduced to different orderly nonlinear equations with similar transformations. These equations are solved numerically with double precision, using a method based on finite difference estimates. The results are presented for circulating images of velocity and temperature within the boundary layer. The effects of the power-law index on shear stress at the wall and the rate of heat transfer is studied. For the same general numbers of Reynolds and Prandtl, the parameter of fluid-law, and the speed difference of Iu, - u-1, the coefficient result of greater skin-movement and heat transfer from u,> u, than from u, < and,. 0 1996 by Elsevier Science Inc.
Keywords: boundary cover, power-law flow, finite difference estimates
1. Introduction:
From a technological point of view, the study of boundary cover flow on a continuous mobile hard surface is always important. Such flow analysis finds applications in various fields, such as aerodynamic extraction of plastic sheets, the boundary phase associated with material handling behavior, infinite metal plate cooling in a cooling tank, and the boundary phase to the side of melting film in thick processes. Despite these claims, Sakaidis ’study of the flow of boundary cover over a continuous hard surface moving at a constant speed began to be studied. Due to the introduction of environmental flow, this boundary cover flow is very different from a semicircular horizontal plate. Erickson et al. * This problem are to monitor temperature circulation in the boundary phase when the leaf is kept at a constant temperature by juice or blowing. Experimental and theoretical treatment for the mobile continuous flat surface was carried out by Tsou et al.3 These studies have applications in the polymer industry, where a polymer sheet is continuously extracted from death, by the supportive assumption that the page is unstable. However, in real situations, one encounters the flow of a boundary cover over a stretched page.
For example, in the process of melt spinning, the extrudate is stretched into a filament or sheet while being pulled from the die. This page eventually solidifies as it passes through the controlled cooling system. Carne4 and McCormack and Came ’studied the boundary cover flow of a Newtonian current caused by stretching an elastic flat sheet that moves in its own plane with a speed varying linearly with the distance from a fixed point due to uniform pressure. The movement of heat and mass on a stretched sheet with juice or puff was studied by Gupta and Gupta.6 They treated the isothermal moving plate and obtained the circulating temperature and density. Chen and Char ’studied the heat transfer properties over a continuous stretched surface with variable surface temperature.
Because the physical effects of the environmental current have an effective effect on boundary Layer properties, the study of the flow of non-Newtonian fluid over a moving sheet has been of paramount importance. Fox et al. ’Studied the flow of power-law on a moving surface. Hassanien and Gorla have studied the flow of micropolar fluid past a stretched sheet with variable surface temperature. ‘The flow of second-order fluid over a stretched sheet with and without heat transfer was studied by Rajagopal et al.” and Dandapat and Gupta, ”respectively.
All of the above studies limited their analysis when the moving surface was placed in a relatively intermittent moving medium, where the speed of free flow was zero. The author I2 has recently studied the flow and heat transfer from a continuous surface by introducing the effect of a parallel free flow of second-order viscoelastic liquid. In the present analysis, we examine the boundary cover flow of a power model (Ostwald-de Waele fluid) on a continuous plane sheet coming out of a slit with a constant surface speed u, in free-flow parallel CL,. The continuously moving flat surface rubs against the surrounding stream and is pumped back into the bottom. Thus the transverse distance part in the boundary phase is directed toward the plate when u,> u,. The change in the direction of the transverse distance part leads to significantly different characteristics of fracture factor and surface heat transfer rates for u,> u, compared to u, <u, for the same inter- speed difference lu, - u, I, Reynolds and Prandtl numbers, and power-index. The speeds reported above are unmeasured. The method of making them unmeasured is discussed in the next section.
2. Boundary cover analysis:
The non-Newtonian fluid model used in this study in the power-law model (Ostwald-de Waele fluid), with parameters defined:
i3 where A is the symmetric degree of tensor deformation, K is the consistency coefficient, and n is the power-to-power index. The two-parameter astronomy is called the Ostwald-de Waele model or, more commonly,
The power-law model. Equation 12 # 1 (1) represents Newtonian fluid with the dynamic coefficient of viscosity K. Thus the movement of n from unity reflects the degree of the shift from Newtonian behavior (see, e.g., Anderson and Irgens ) .14 With n = 1 an established equation (1) represents shear thinning (n <1) and shear thinning (n> 1). However, unlike the second-order viscoelastic fluid studied in Refs. 10-12, the unstable power-law model (1) does not show typical differences. The model of the law of power, equation (l), has been shown to be valid for a large class of non-Newtonian fluid.'5T'6 From equation (1) it must be assumed that Ju / day is much greater than all other distance gradients. This is the same hypothesis used in driving the momentum equation of the boundary surface, which emphasizes the shear as (2) Now we are considering model compliance of the flow field (1) over a continuous moving horizontal surface with a constant velocity u, in its non-Newtonian moving center with a free flow velocity of u,. It is assumed that the plate and the liquid move in the same direction. Figure 1 shows the coordinate system (x, y) based on space and the flow model for the boundary level on a moving continuous surface. A coordinate origin is placed at the point where the plate is drawn into the moving center and the y-axis is normally measured to the plates. Distances are normalized by the standard length z between the slit and the curved roll, or the portion of this length called the laminar flow (the bars indicate dimensions). Speed is normalized by the maximum speed in the problem as reference speed E.
Closing remarks:
This work investigates the movement and transfer of heat in a power-law flow on a continuous flat surface moving in a parallel free flow. This type of flow represents a new class of boundary coating problems, with a very different solution from those for boundary coating on a flat surface of the finished length. That is, the solution of the problems of flow past a flat plate depends on the relative distance between the flat plate and the free flow, but the solutions of the problems of flow over a continuous flat surface are dependent not. not only on the speed difference but also on the speed ratio. Numerical results are provided for the properties of flow rate and heat transfer. The required wall values of speed and temperature function are given for a range of the current-power index of the stream. For the same values of the standard speed difference, power-law index, Re, and Pr, the case represents u,> u.
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