Laminar Boundary Layer on a Flat Plate

Consider the elemental control volume shown in Fig. 5-4. We derive the equation of motion for the boundary layer by making a force-and-momentum balance on this element. To simplify the analysis we assume  

There are no pressure variations in the direction perpendicular to the plate. 

The above form of Newton s second law of motion applies to a system of constant mass. In fluid dynamics it is not usually convenient to work with elements of mass rather, we deal with elemental control volumes such as that shown in Fig. 5-4, where mass may flow in or out of the different sides of the 

The momentum flux in the x direction is the product of the mass flow through a particular side of the control volume and the x component of velocity at that point. 

The mass entering the left face of the element per unit time is 

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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. 

What is the momentum equation for the laminar boundary layer on a flat plate What assumptions are involved in the derivation of this equation ... 

Using the linear-velocity profile in Prob. 5-2 and a cubic-parabola temperature distribution [Eq. (5-30)], obtain an expression for heat-transfer coefficient as a function of the Reynolds number for a laminar boundary layer on a flat plate. 

Show that d3u/dy3 = 0 at y = 0 for an incompressible laminar boundary layer on a flat plate with zero-pressure gradient. 

We further observed that the ratio via, the Prandtl number, was the connecting link between the velocity and temperature field and was thus an important parameter in all convection heat-transfer problems. If we considered a laminar boundary layer on a flat plate in which diffusion was occurring as a result of... 

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 6.4 Influence of Prandtl number on the recovery factor and modified Reynolds analogy for a laminar boundary layer on a flat plate. 

M. W. Rubesin and H. A. Johnson, A Summary of Skin Friction and Heat Transfer Solutions of the Laminar Boundary Layer on a Flat Plate, Proc. 1948 Heat Transfer Fluid Mech. Inst. also Trans. ASME (71) 383-388,1949. 

In a few limited situations mass-transfer coefficients can be deduced from theoretical principles. One very important case in which an analytical solution of the equations of momentum transfer, heat transfer, and mass transfer has been achieved is that for the laminar boundary layer on a flat plate in steady flow. 

Blasius solution for the laminar boundary layer on a flat plate, shown in Fig. 11.3, rests on a considerable string of assumptions and simplifications. However, it has been tested by numerous investigators and found to represent the experimental data very well (note that Fig. 11.3 shows the comparison between Blasius solution and Nikuradse s experimental data). Thus, these assumptions and simplifications seem to be justified. 

Blasius solution for the laminar boundary layer on a flat plate and Nikuradse s experimental tests of same. [From J. Nikuradse, Laminar Reibungsschichten an der laengsangestroemten Platte (Laminar friction layers on plates with parallel flow), Monograph Zentralefuer Wiss. Berichtwesen, Berlin (1942). 

In addition to the boundary-layer thickness 5, two other thicknesses occur frequently in the boundary-layer literature the displacement thickness S and the momentum thickness 6. To see the meaning of the displaicement thickness, consider the streamlines for the laminar boundary layer on a flat plate, as sketched in Fig. 11.5. 

By assuming that the velocity was a function of y yj xv)X Blasius was able to solve Prandtl s equations for the steady-flow laminar boundary layer on a flat plate. He found that the laminar boundary-layer thickness is proportional to the square root of the length down the plate. 

Assuming that the transition from a laminar to a turbulent boundary layer takes place at a Reynolds number of 10 what is the maximum thickness for the laminar boundary layer on a flat plate for ( ) air flowing at 10 ft/s, (b) water flowing at jlOft/s, and (c) glycerin flowing at 10 ft/s = 8.07 x 10 ft /s) ... 

For boundary layers on curved surfaces, the pressure will change with distance. This greatly complicates the solution of the boundary-layer equations compared with that on a flat plate (in which dPIdx was zero), and so very few exact solutions are known for such boundary layers. Some estimate of the behavior of such boundary layers is given by several methods. To illustrate, we apply them to the laminar boundary layer on a flat plate, where we can compare the results with Blasius exact solution. These methods begin by assuming a velocity profile of the form V tV where S is the boundary-layer thickness. 

This is an example of rapid chemical reaction in the laminar boundary layer on a flat plate. Consider the steady-state dissolution of a slightly soluble solid in a flowing dilute solution, and suppose the solid is acidic whereas the flowing solution is basic [23]. The geometry under consideration is a flat plate consisting of acid, located at zero incidence to the flow. The process may be assumed isothermal, and the fluid properties are to be treated as independent of position. 

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