Two-Dimensional Laminar Boundary Layer

Consider two-dimensional laminar boundary layer flow over a flat isothermal surface. Very close to the surface, the velocity components are very small. If the pressure changes are assumed to be negligible in the flow being considered, derive an expression for the temperature distribution near the wall. Viscous dissipation effects should be included in the analysis. 

By numerically solving the two-dimensional laminar boundary layer equations, determine how the local heat transfer rate in W/m2 varies along the plate. 

Numerically determine the heat transfer rate variation with two-dimensional laminar boundary layer air flow over a fiat plate with a uniform heat flux at the surface. Compare the numerical results with those given by the similarity solution. 

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. 

The boundary layer equations for free convective flow will be deduced using essentially the same approach as was adopted in forced convective flow. Attention will, as discussed above, be restricted to the case of two-dimensional laminar boundary layer flow. Attention will initially be focused on a plane surface that is at an angle, 4>, the vertical as shown in Fig. 8.4. The x-axis is chosen to be parallel to this surface as shown in Fig. 8.4. 

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. 

Similar solutions for Prandtl numbers other than unity may be obtained from Eqs. 6.117 and 6.118 or their equivalent. A major simplification is the independence of the momentum equation (Eq. 6.117), from the energy equation (Eq. 6.118), which makes/independent of /. Also, the linear form of the energy equation in / permits handling arbitrary surface temperature distributions as in the case of the flat plate. (See the section on the two-dimensional laminar boundary layer.)... 

Surface With Mass Transfer. The advantages of mass transfer cooling systems in certain applications were discussed in the section on two-dimensional laminar boundary layers beginning on page 6.19. Reference 109 should be consulted for a complete description of the per-... 

In laminar flow with low mass-transfer rates and constant physical properties past a solid surface, as for the two-dimensional laminar boundary layer of Fig. 3.10, the momentum balance or equation of motion (Navier-Stokes equation) for the X direction becomes [7]... 

One concludes from (12-17a) and (12-17c) that neither 4> nor Vp is a function of the Reynolds number because Re does not appear in either equation. Consequently, dynamic pressure and its gradient in the x direction are not functions of the Reynolds number because Re does not appear in the dimensionless potential flow equation of motion, given by (12-16), from which /dx is calculated. In summary, two-dimensional momentum boundary layer problems in the laminar flow regime (1) focus on the component of the equation of motion in the primary flow direction, (2) use the equation of continuity to calculate the other velocity component transverse to the primary flow direction, (3) use potential flow theory far from a fluid-solid interface to calculate the important component of the dynamic pressure gradient, and (4) impose this pressme gradient across the momentum boundary layer. The following set of dimensionless equations must be solved for Vp, IP, u, and v in sequential order. The first three equations below are solved separately, but the last two equations are coupled ... 

The heat transfer rate from a flush-mounted shear stress tensor depends on the near-wall flow, i.e., the magnitude of the velocity gradient. For a laminar two-dimensional thermal boundary layer developing over the heated sensor with an approaching linear velocity profile (Fig. 2) and negligible free convectimi effect, the heat loss from the thermal element can be derived from the thermal boundary layer equation as... 

LAMINAR BOUNDARY LAYERS, edited by L. Rosenhead. Engineering classic covers steady boundary layers in two- and three-dimensional flow, unsteady boundary layers, stability, observational techniques, much more. 708pp. 5k x 8k. 

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. 

SOLAN, A. WINOGRAD, Y. 1969. Boundary-layer analysis of polarization in electrodialysis in a two-dimensional laminar flow. Phys. Fluids 12, 1372-1377. 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

Consider the locally flat description of heat transfer by convection and conduction from a hot plate to an incompressible fluid at high Peclet numbers with two-dimensional laminar flow in the heat transfer boundary layer adjacent to the hot surface. The tangential fluid velocity component Vx is only a function of position x parallel to the interface. 

If terms of the order (8/L)2 and less are again neglected, it will be seen that the energy equation for laminar two-dimensional boundary layer flow becomes ... 

If terms of the order (SIL) and less are again neglected and itf it is assumed that the Prandtl number, Pr, has order 1 or greater, i.e., is not small,j it follows that the x-wise diffusion term, i.e., d2T/dx2. is negligible compared to the other terms. Hence, the energy equation for free convective laminar two-dimensional boundary layer flow becomes ... 

A solution that was accurate to first order in the buoyancy parameter, Gr. for near-forced convective laminar two-dimensional boundary layer flow over an isothermal vertical plate was discussed in this chapter. Derive the equations that would allow a solution that was second order accurate in Gx to be obtained. Clearly state the boundary conditions on the solution. 

The Reynolds number at observed transition location (defined as a location where the intermittency factor is about 0.1 i.e. the flow is 10 % of time turbulent and rest of the time it is laminar) for zero pressure gradient flat plate boundary layer is of the order of 3.5 X 10 . This corresponds to Re = 950. The distance between the point of instability and the point of transition depends on the degree of amplification and the kind of disturbance present with the oncoming flow. This calls for a study of local and total amplification of disturbances. The following description is as developed in Arnal (1984) for two-dimensional incompressible flows. 

Consider a vertical hot flat plate immersed in a quiescent fluid body. We assume the natural convection flow to be steady, laminar, and two-dimensional, and the fluid to be Newtonian with constant properties, including density, with one exception the density difference p — is to be considered since it is this density difference between the inside and the outside of the boundary layer that gives rise to buoyancy force and sustains flow. (This is known as the Boussines.q approximation.) We take the upward direction along the plate to be X, and the direction normal to surface to be y, as shown in Fig. 9-6. Therefore, gravelly acts in the —.t-direclion. Noting that the flow is steady and two-dimensional, the.t- andy-compoijents of velocity within boundary layer are II - u(x, y) and v — t/(.Y, y), respectively. 

Transition length for laminar and turbulent flow. The length of the entrance region of the tube necessary for the boundary layer to reach the center of the tube and for fully developed flow to be established is called the transition length. Since the velocity varies not only with length of tube but with radial distance from the center of the tube, flow in the entrance region is two dimensional. 

Dhir and Lienhard [118] studied laminar film condensation on two-dimensional isothermal surfaces for which boundary layer similarity solutions exist and found that a similarity solution exists for body shapes that give g(x) = x". Nakayama and Koyama [119] extended the analysis of arbitrarily shaped bodies to include turbulent film condensation. 

Consider two-dimensional steady-state mass transfer in the liquid phase external to a solid sphere at high Schmidt numbers. The particle, which contains mobile reactant A, dissolves into the passing fluid stream, where A undergoes nth-order irreversible homogeneous chemical reaction with another reactant in the liquid phase. The flow regime is laminar, and heat effects associated with the reaction are very weak. Boundary layer approximations are invoked to obtain a locally flat description of this problem. 

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with... 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

In two-dimensional Cartesian coordinate system x,y) we consider magneto-convection, steady, laminar, electrically conduction, boundary layer flow of a viscoelastic fluid caused by a stretching surface in the presence of a uniform transverse magnetic field and a heat source. The x -axis is taken in the direction of the main flow along the plate and the y -axis is normal to the plate with velocity components u,v in these directions. 

Streefer, V. F., ed. 1961. Handbook of Fluid Dynamics. New York McGraw-Hill. A classic handbook on fluid dynamics wifh confributions from distinguished experts. Written for engineers and scientists in the field. Deals wifh bofh fundamenfal concepts and applications. Covers fluid flow (one-dimensional, ideal, laminar, compressible, two phase, open channel, stratified), turbulence, boundary layers, sedimentation, turbomachinery, fluid transients, and magnetohydrodynamics. Includes many formulas, equations, tables, graphs, and illustrations. Each chapter has a bibliography and the volume has subject and author indexes. 

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