Flat plate laminar boundary layer flow

Take into consideration two-dimensional, rectilinear, steady, incompressible, constant-property, laminar boundary layer flow in the x direction along a flat plate. Assume that viscous energy dissipation may be neglected. Write the continuity, momentum and energy equations. 

Laminar Boundary Layer Flow Along a Flat Plate with Radiation Boundary Condition... 

Consider a steady, laminar boundary layer flow of incompressible, transparent fluid along a flat plate, with a constant applied heat flux qw Btu/(hr ft2) at the wall surface. The properties of the fluid are assumed constant. The main considerations are conduction to the fluid, and radiation from the plate to the environment at Te. Surface of the plate is opaque and gray, and the uniform emissivity is 8. The fluid which is at a temperature of T,, flows at a uniform velocity of Uo. Flow velocities are sufficiently small so that viscous dissipation may be neglected. 

VISCOUS DISSIPATION EFFECTS ON LAMINAR BOUNDARY LAYER FLOW OVER A FLAT PLATE... 

In order to measure the velocity of a stream of air, a flat plate of length 2 cm in the flow direction is placed in the flow. This plate is electrically heated, the heat dissipation rate being uniform over the plate surface. The plate is wide so a two-dimensional laminar boundary layer flow can be assumed to exist The velocity is to be deduced by measuring the temperature of the plate at its trailing edge. If this temperature is to be at least 40°C when the air temperature is 20 C and the air velocity is 3 m/s, find the required rate of teat dissipation in the plate per unit surface area. 

A vertical flat plate is maintained at a uniform surface temperature and is exposed to air at standard ambient pressure. At a distance of 10 cm from the leading edge of the plate the boundary layer thickness is 2 cm. Estimate the thickness of the boundary layer at a distance of 25 cm from the leading edge. Assume a laminar boundary layer flow. 

Consider mixed convective laminar boundary layer flow over a horizontal flat plate that is heated to a uniform surface temperature. In such a flow there will be a pressure change across the boundary induced by the buoyancy forces, i.e. ... 

It is noted that the integral method gives only approximate values for the mass transfer coefficient as the model derivation is based on several simplifying assumptions regarding the concentration and velocity profiles. Nevertheless, the given relation has been confirmed by experiments for laminar boundary layer flows over a flat plate (e.g., [134], p 80 and p 201 [27], p 345). 

Problem 10-9. Translating Flat Plate. Consider the high-Reynolds-number laminar boundary-layer flow over a semi-infinite flat plate that is moving parallel to its surface at a constant speed (7 in an otherwise quiescent fluid. Obtain the boundary-layer equations and the similarity transformation for f (r ). Is the solution the same as for uniform flow past a semi-infinite stationary plate Why or why not Obtain the solution for f (this must be done numerically). If the plate were truly semi-infinite, would there be a steady solution at any finite time (Hint. If you go far downstream from the leading edge of the flat plate, the problem looks like the Rayleigh problem from Chap. 3). For an arbitrarily chosen time T, what is the regime of validity of the boundary-layer solution ... 

Obtain expressions for the local and mean values of the wall shear stress and friction factor (or drag coefficient) for the laminar boundary layer flow of an incompressible power-law fluid over a flat plate Compare these results with the predictions presented in Table 7.1 for different values of the power-law index. 

With such a flat plate, the boundary layer will increase in thickness indefinitely, if slowly (see Fig. 1.11(c)). On the other hand, if the flow is in a restricted channel, e.g. a circular tube or a parallel-plate cell, the boundary layers at the two walls must merge at some point and beyond, a steady-state situation or fully developed laminar... 

Continuous Flat Surface Boundaiy layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48 7 shows the continuous flat surface (Saldadis, AIChE J., 7, 26—28, 221-225, 467-472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. FluidMech., 26,145—161 [1966]). For a laminar boundary layer, the thickness is given by... 

Laminar boundary layer theory assumes that a uniform flow (V = constant) approaches a flat plate. A laminar flow region develops near the plate where the thickness of the laminar boundary layer increases with thickness along the plate, as developed in Example 4.2. If we assign 5 to be the boundary layer thickness, or the distance from the plate where the velocity is equal to 0.99 times the velocity that approached the plate, and 5c to be the concentration boundary layer thickness, then we can see that both 5 and 5c are functions of distance, x, from the leading edge, as shown in Figure 8.11. 

J0. Show how die numerical method for solving die laminar boundary layer equations discussed in this chapter can be modified to allow for viscous dissipation. Use a computer program based on this modified procedure to estimate the importance of this dissipation on the heat transfer rate along an isothermal flat plate in low speed flow. 

The conditions under which transition occurs depend on the geometrical situation being considered, on the Reynolds number, and on the level of unsteadiness in the flow well away from the surface over which the flow is occurring [2], [30]. For example, in the case of flow over a flat plate as shown in Figure 5.6, if the level of unsteadiness in the freestream flow ahead of the plate is very low, transition from laminar to turbulent boundary layer flow occurs approximately when ... 

Consider transition in the boundary layer flow over a flat plate. Using the expression for the thickness of a laminar boundary layer on a flat plate given in Chapter 3, find the value of the Reynolds number based on the boundary layer thickness at which transition begins. 

Consider laminar free-conveqtive flow over a vertical flat plate at whose surface the heat transfer rate per unit area, qw, is constant. Show that a similarity solution to the two-dimensional laminar boundary layer equations can be derived for this case. 

This is the momentum equation of the laminar boundary layer with constant properties. The equation may be solved exactly for many boundary conditions, and the reader is referred to the treatise by Schlichting ll] for details of the various methods employed in the solutions. In Appendix B we have included the classical method for obtaining an exact solution to Eq. (5-13) for laminar flow over a flat plate. For the development in this chapter we shall be satisfied with an approximate analysis which furnishes an easier solution without a loss in physical understanding of the processes involved. The approximate method is due to von Karman [2],... 

Derive an expression for the heat transfer in a laminar boundary layer on a flat plate under the condition = w, = constant. Assume that the temperature distribution is given by the cubic-parabola relation in Eq. (5-30). This solution approximates the condition observed in the flow of a liquid metal over a flat plate. 

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

Knowledge of the temperature field in the fluid is a prerequisite for the calculation of the heat transfer coefficient using (1.25). This, in turn, can only be determined when the velocity field is known. Only in relatively simple cases, exact values for the heat transfer coefficient can be found by solving the fundamental partial differential equations for the temperature and velocity. Examples of this include heat transfer in fully developed, laminar flow in tubes and parallel flow over a flat plate with a laminar boundary layer. Simplified models are required for turbulent... 

Structure of the flow. Velocity profile. The flow in the boundary layer on a flat plate is laminar until Rex = U[X/v 3 x 105. On a longer plate, the boundary layer becomes turbulent, that is, its thickness increases sharply and the longitudinal velocity profile alters. 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

The thickness of the laminar boundary layer on a flat plate Z, is approximately given by the equation = 5.5Qtx/u py. Show that at the transition to the turbulent flow the Reynolds number based on this thickness, instead of on x as in Eq. (3.21), is close to the transition Reynolds number for flow in a pipe,... 

FIGURE 1.6 Laminar, transition, and turbulent boundary layer flow regimes in flow over a flat plate. 

FIGURE 6.2 Temperature distributions in the laminar boundary layer on a flat plate at uniform temperature—constant property, low-speed flow. 

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