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fr the falling fm . Lerive anl vcloci Pey o 42) assumin 5 usinte equatienmtion (6.5-3), niam ity, average velocity, or force on solid surfaces. tion appear, and In the integrations mentioned above, several constants of integration a the velocit stress at the boundaries of the system. The most commonly used boundae are as follows: using "boundary conditions"-that is, statements about a. At solid-fluid interfaces the fluid velocity equals the velocity with which surface is moving: this statement is applied to both the tangential and th component of the velocity vector. The equality of the tangential corm referred to as the "no-slip condition." See Problem 2D.3 for further ideas b. At a liquid-liquid interfacial plane of constant x, the tangential velocity nents t, and o, are continuous through the interface (the "no-slip condit are also the molecular stress-tensor components p+Taay Tre and c. At a liquid-gas interfacial plane of constant x, the stress-tensor compons and r, are taken to be zero, provided that the gas-side velocity gradient is t large. This is reasonable, since the viscosities of gases are much less than tho a liquids In all of these boundary conditions it is presumed that there is no material Passizg through the interface; that is, there is no adsorption, absorption, dissolution, evan tion, melting, or chemical reaction at the surface between the two phases. Boundaryc ditions incorporating such phenomena appear in Problems 3C 5 and 11C.6, and $181 In this section we have presented some guidelines for solving simple viscous o problems. For some problems slight variations on these guidelines may prove tob 2.2 FLOW OF A FALLING FILM The first example we discuss is that of the flow of a liquid down an inclined flat plate of length L and width W, as shown in Fig. 22-1. Such films have been studied in connecticn with wetted-wall towers, evaporation and gas-absorption experiments, and applications of coatings. We consider the viscosity and density of the fluid to be constant. A complete description of the liquid flow is difficult because of the disturbances a the edges of the system (:-0, z-L, y o, y-w. An adequate description can often be Entrance disturbance Liquid film Liquid in Reservoir Exit disturbance Fig. 2.2-1 Schematic diagram of the falling film experiment, show- ing end effects Direction of gravity $22 Flow of a Falling Film43 obtained by neglecting such disturbances, particularly if W and L are large compared to the film thickness δ. For small flow rates we expect that the viscous forces will prevent continued acceleration of the liquid down the wall, so that p, will become independent that of z in a short distance down the plate. Therefore it seems reasonable to postulate only nonvanishing components of τ are then,- -μ(do/ds). lx),, -0, and ,-0, and further that p p(x). From Table B.1 it is seen that the We now select as the "system" a thin shell perpendicular to the x direction (sece Fig. 2.2-2). Then we set up a 2-momentum balance over this shell, which is a region of thickness Ax, bounded by the planes z 0 and : -L, and extending a distance W in the y direction. The various contributions to the momentum balance are then obtained with the help of the quantities in the "z-component columns of Tables 1.2-1 and 1.7-1. By using the components of the "combined momentum-flux tensor" ф defined in Tables 17-1 and 2, we can include all the possible mechanisms for momentum transport at once rate of z-momentum in across surface at z-0 rate of z-momentum out across surface at z-L rate of z-momentum in across surface at x rate of z-momentum out across surface at x + Ax (22-1) (2.2-2) (2.2-3) (2.2-4) (2.2-5) (LW gravity force acting on fluid in the z direction (LWAx)US cos β) By using the quantities and фи we account for the 2-momentum transport by all mechanisms, convective and molecular. Note that we take the "in" and "out" directions in the direction of the positive r- and z-axes (in this problem these happen to coincide with the directions of z-momentum transport). The notation l.àr means "evaluated at x+ Ar," and g is the gravitational acceleration. When these terms are substituted into the z-momentum balance of Eq. 2.1-1, we get Direction of gravity of thickness Ax over which a z-momentum balance is made. Arrows show the fluxes associated with the surfaces of the shell. Since o, and o, are both zero, pu,u, Fig 22-2 Shell and po,p, are zero. Since o, and a and po,p, are the same at z-0 and z o,p, are zero. Since o, does not depend on y and z, it follows from Table B.1 that 0 0. Therefore, the dashed-underlined fluxes do not need to be considered. Both p L, and therefore do not appear in the final equation for the balance of z momentum, Eq. 2.2-10 artesian coordinates (x, y, z): ди hese equations have been written without making the assumption that τ is symmetric. This means, for mple, that when the usual assumption is made that the stress tensor is symmetric, Ty and Ty may be erchanged

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