Boundary-layer flow body forces

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

The boundary layer problem is difficult to solve exactly. There are several approximate methods to solve the problem. This chapter looks at external forced convection, that is, flow outside and around a solid body like a plate. The next chapter discusses flows inside a solid structure such as a pipe, or between two plates. 

If the Darcy assumptions are used then with forced convective flow over a surface in a porous medium, because the velocity is not assumed to be 0 at the surface, there is no velocity change induced by viscosity near the surface and there is therefore no velocity boundary layer in the flow over the surface. There will, however, be a region adjacent to the surface in which heat transfer is important and in which there are significant temperature changes in the direction normal to the surface. Under many circumstances, the normal distance over which such significant temperature changes occur is relatively small, i.e., a thermal boundary layer can be assumed to exist around the surface as shown in Fig. 10.9, the ratio of the boundary layer thickness, 67, to the size of the body as measured by some dimension, L, being small [15],[16]. 

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. 

In the following we will assume the velocities, temperatures and concentrations in the outer region to be known, and consider a steady-state, two-dimensional flow. The body forces are negligible. Flow along a curved wall can be taken to be two-dimensional as long as the radius of curvature of the wall is much bigger compared to the thickness of the boundary layer. The curvature is then insignificant for the thin boundary layer, and it develops just as if it was on a flat wall. The curvature of the wall is merely of influence on the outer flow and its pressure distribution. 

In this section we will focus on the heat and mass transfer from or to the surface of a body with external flow. Neighbouring bodies should not be present or should be so far away that the boundary layers on the bodies over which the fluid is flowing can develop freely. Velocities, temperatures and concentrations shall only change in the boundary layer and be constant in the flow outside of the boundary layer. A forced flow, which we will consider here, is obtained from a pump or blower. Local heat and mass transfer coefficients are yielded from equations of the form... 

The separated boundary layer and wake displace the outside streamline pattern, which causes the pressure distribution to be significantly altered. Boundary layer separation causes a force on the body called drag force. The drag coefficient is defined as the ratio of total profile drag force divided by the flow pressure and projected area of an object and is expressed as [49]... 

In spite of this, we shall see that potential-flow theory plays an important role in the development of asymptotic solutions for Re i>> 1. Indeed, if we compare the assumptions and analysis leading to (10-9) and then to (10-12) with the early steps in analysis of heat transfer at high Peclet number, it is clear that the solution to = 0 is a valid first approximation lor Re y> 1 everywhere except in the immediate vicinity of the body surface. There the body dimension, a, that was used to nondimensionalize (10-1) is not a relevant characteristic length scale. In this region, we shall see that the flow develops a boundary layer in which viscous forces remain important even as Re i>> 1, and this allows the no-shp condition to be satisfied. 

An alternative point of view is that vorticity accumulates at the rear of the body, which leads to a large recirculating eddy structure, and as a consequence, the flow in the vicinity of the body surface is forced to detach from the surface. This is quite a different mechanism from the first one because it assumes that the primary process leading to separation is the production and accumulation of vorticity rather than the local dynamics within the boundary layer.25 However, viscosity still plays a critical role for a solid body in the production of vorticity. In fact, for any finite Reynolds number, there is probably some element of truth in both explanations. Furthermore, it is unlikely that experimental evidence (or evidence based on numerical solutions of the complete Navier Stokes equations) will be able to distinguish between these ideas, because such evidence for steady flows will inevitably be limited to moderate Reynolds numbers. 

In this regard, it is of interest to contrast the two problems of the streaming motion of a fluid at large Reynolds number past a solid sphere and a spherical bubble. In the case of a solid sphere, the potential-flow solution (10 155)—(10—156) does not satisfy the no-slip condition at the sphere surface, and the necessity for a boundary layer in which viscous forces are important is transparent. For the spherical bubble, on the other hand, the noslip condition is replaced with the condition of zero tangential stress, Tr = 0, and it may not be immediately obvious that a boundary layer is needed. However, in this case, the potential-flow solution does not satisfy the zero-tangential-stress condition (as we shall see shortly), and a boundary-layer in which viscous forces are important still must exist. We shall see that the detailed features of the boundary layer are different from those of a no-shp, sohd body. However, in both cases, the surface of the body acts as a source of vorticity, and this vorticity is confined at high Reynolds number to a thin 0(Re x/2) region near the surface. 

An obvious question that may occur to the reader is why the very simple method of integrating the viscous dissipation function has not been used earlier for calculation of the force on a solid body. The answer is that the method provides no real advantage except for the motion of a shear-stress-free bubble because the easily attained inviscid or potential-flow solution does not generally yield a correct first approximation to the dissipation. For the bubble, Vu T=0(l) everywhere to leading order, including the viscous boundary layer where the deviation from the inviscid solution yields only a correction of 0(Re x 2). For bodies with no-slip boundaries, on the other hand, Vu T is still 0(1) outside the boundary layer, but inside the boundary layer Vu T = O(Re). When integrated over the boundary layer, which is G(Re k2) in radial thickness, this produces an ()( / Re) contribution to the total dissipation,... 

In the preceding section, we have examined a variety of steady thermocapillary and diffusocapillary flows. Not all such flows are stable and in fact surface tension variations at an interface can be sufficient to cause an instability. We consider here the cellular patterns that arise with liquid layers where one boundary is a free surface along which there is a variation in surface tension. It is well known that an unstable buoyancy driven cellular convective motion can result when a density gradient is parallel to but opposite in direction to a body force, such as gravity. An example of this type of instability was discussed in Section 5.5 in connection with density gradient centrifugation. 

For many real flows, the streamlines separate from the body around which they are flowing. This results in the formation of eddying wakes, low pressure behind the body, and large drag forces. This is not predictable by perfect-fluid theory, but it can be approached through boundary-layer theory. 

An important parameter is the Reynolds number. At Re 1 the viscous term in (5.107) is small in comparison with the inertial one. Neglecting it, one obtains the equations of motion of an ideal liquid (Euler s equations). These equations describe flow of liquid in a volume, with the exception of small regions, adjoining the surface of an immersed body. Near such surfaces, the viscosity force can be comparable with inertial force, which results in formation of a viscous boundary layer with thickness S I/(Re), where L is the characteristic size of the body. Approximation Re 1 leads to an inertialess flow described by Stokes equations. These equations follow from (5.107), in which the inertial terms are omitted. Such equations describe the problems of micro-hydrodynamics, for example, problems of the small particles motion in a liquid. 

At very high Reynolds numbers the viscous forces are quite small compared to the inertia forces and the viscosity can be assumed as zero. These equations are useful in calculating pressure distribution at the outer edge of the thin boundary layer in flow past immersed bodies. Away from the surface outside the boundary layer this assumption of an ideal fluid is often valid. 

As discussed earlier, there are two spiral flow patterns existing in the hydrocyclone. Only particles existing in the outer spiral flow will be separated by the centrifugal force. Any particles in the inner spiral flow will pass upward to the overflow outlet. It should be noted that there are two important stages in the process of particle separation. One is the separation of the solids from the main body of the flow into the boundary layer adjacent to the inner wall of the hydrocyclone by centrifugal forces. The other is the removal of the separated solids from the boundary layer by downward fluid flow (not by gravity) to the apex of the cone and out of the hydrocyclone. 

The above electroneutrality assumption holds everywhere except in the thin Debye screening layer next to the solid surface. The potential drop across the Debye layer can be significant even though it is only a few nanometers in thickness. It may be noted that ionic mass transport affects the current density, J, which influences the flow field by Lorentz body force. Therefore, one needs to solve simultaneously the full mathematical model consisting of continuity, N-S equation, Nernst-Planck equation, and the local electroneutrality conditions with the appropriate boundary condition. 

The above theoretical approaches apply estimating MTCs for the forced convection of fluid flowing parallel to a surface. Typically, the fluid forcing process is external to the fluid body. In the case of natural or free convection fluid motion occurs because of gravitational forces (i.e., g = 9.81 m/s ) acting upon fluid density differences within (i.e., internal to) regions of the fluid. Temperature differences across fluid boundary layers are a major factor enhancing chemical mass transport in these locales. Concentration differences may be present as well, and these produce density differences that also drive internal fluid motion (i.e., free convection). 

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