Unsteady flat plate

Fig. 4. Migration contribution to the limiting current in acidified CuS04 solutions, expressed as the ratio of limiting current (iL) to limiting diffusion current (i ) r = h,so4/(( h,so, + cCuS(>4). "Sulfate refers to complete dissociation of HS04 ions. "bisulfate" to undissociated HS04 ions. Forced convection" refers to steady-state laminar boundary layers, as at a rotating disk or flat plate free convection refers to laminar free convection at a vertical electrode penetration to unsteady-state diffusion in a stagnant solution. [F rom Selman (S8).]...

Fig. 5.5 Temperature profiles for unsteady-state heat conduction in finite flat plates T(x, 0) = To, T( b, t) = 7j. [Reprinted by permission from H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, New York, 1973.]...

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 ... 

One-dimensional charts find an important application in the solution of multidimensional problems. It can be shown (see, for example, Section 5.2 of Ret 1) that the dimensionless unsteady temperature of an infinitely long rod of rectangular cross section 21 X 2L may be expressed as the product of the dimensionless temperature of an infinite flat plate of thickness 21 times the dimensionless temperature of an infinite flat plate of thickness 2L,... 

As a simple illustration, we wish to develop the unsteady one-dimensional finite-difference formulation for a flat plate of thickness i having an initial temperature difference AT = 7i — r2 between its surfaces. Assume the surface of the plate with temperature T is suddenly insulated. 

To illustrate an analytical method of solving this first case, we will derive the equation for unsteady-state conduction in the. x direction only in a flat plate of thickness 2H as shown in Fig. 5.3-1. The initial profile of the temperature in the plate at i = 0 is uniform at T = 7j>. At time t = 0, the ambient temperature is suddenly changed to T, and held there. Since there is no convection resistance, the temperature of the surface is also held constant at T,. Since this is conduction in the x direction, Eq. (5.1-10) holds. 

Figure 5.3-1. Unsteady-state conduction in a flat plate with negligible surface resistance.

C Unsteady-State Conduction in a Large Flat Plate... 

A geometry that often occurs in heat-conduction problems is a flat plate of thickness 2xj in the x direction and having large or infinite dimensions in the y and z directions, as shown in Fig. 5.3-4. Heat is being conducted only from the two flat and parallel surfaces in the x direction. The original uniform temperature of the plate is TJ, and at time t = 0, the solid is exposed to an environment at temperature Tj and unsteady-state conduction occurs. A surface resistance is present. 

The numerical results of this case are presented graphically in Figs. 5.3-5 and 5.3-6. Figure 5.3-5 by Gurney and Lurie (G2) is a convenient chart for determining the temperatures at any position in the plate and at any time t. The dimensionless parameters used in these and subsequent unsteady-state charts in this section are given in Table 5.3-1 (x is the distance from the center of the flat plate, cylinder, orsphere. x, is one half the thickness of the flat plate radius of cylinder, or radius of sphere, x = distance from the surface for a semiinfinite solid.)... 

Figure 5.3-5. Unsteady-state heat conduction in a large flat plate. [From H. P. Gurney and J. Lurie, Ind. Eng. Chem., 15, 1170 (/92i).]... 

Figure 5.3-6. Chan for delermin mg temperature at the center of a large flat plate for unsteady-state heat conduction. [From H. P. Heisler, Trans. A.S.M.E., 69, 227 ( 947). With permission.]...

Das, S.S., Tripathy, U.K., Das, J.K., Sahoo, S.K., Mishra, S., 2008. Magnetohydrodynamic unsteady flow of a viscous stratified fluid through a porous medium past a porous flat moving plate in the slip flow regime with heat source. Far East J. Math. Sci. 29, 71-88. 

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