LITMIR.BIZ

Figure 1.9 Relationship between velocity, pressure, elevation, and density for a stream tube.

For a compressible fluid, the integration of Eq. (1.64) can only be completed to provide the following:

      (1.66)

      Note that the relationship between 𝜌 and p needs to be known for the given case, and that for gases the relationship can be in the form p𝜌ⁿ = constant, varying from adiabatic to isothermal conditions, while for a liquid, 𝜌(dp/d𝜌) = K, which is an adiabatic modulus.

      (d) Bernoulli's Equation

      This equation can be written for both incompressible and compressible flows. Under certain flow conditions, Bernoulli's equation for incompressible flow is often referred to as a mechanical‐energy equation because of the fact that it is similar to the steady‐flow energy equation obtained from the FLT for an inviscid fluid with no external heat transfer and no external work. It is necessary to point out that for inviscid fluids, viscous forces and surface tension forces are not taken into consideration, leading to negligible viscous effects. The Bernoulli equation is commonly used in a variety of practical applications, particularly in flows in which the losses are negligibly small, for example, in hydraulic systems. The following is the general Bernoulli equation per unit mass for inviscid fluids between any two points:

(1.67)

Here, each term has a dimension of a length or head scale. In this regard, u²/2g (kinetic energy per unit mass) is referred to as the velocity head, p/𝜌g (pressure energy per unit mass) as the pressure head, z (potential energy per unit mass) as the potential head (constant total head), and H (total energy per unit mass) as the total head in meters. Subscripts 1 and 2 denote where the variables are evaluated on the streamline.

The terms in Eq. (1.67) represent energy per unit mass and have the unit of length. Bernoulli's equation can be obtained by dividing each term in Eq. (1.65) by g. These terms, both individually and collectively, indicate the quantities that may be directly converted to produce mechanical energy.

      In summary, if we compare Eq. (1.67) with the general energy equation, we see that the Bernoulli equation contains even more restrictions than might first be realized, due to the following main assumptions:

      (e) Navier–Stokes Equations

      The Navier–Stokes equations are the differential expressions of Newton's second law of motion, and are known as constitutive equations for viscous fluids. These equations were named after C.L.M.H. Navier and Sir G.G. Stokes, who are credited with their derivation.

      For viscous fluids, two force aspects, namely, a body force and a pressure force on their surface, are taken into consideration. The solution of these equations is dependent upon what flow information is known. The solutions evolving now for such problems have become extremely useful. Recently numerical software packages have been developed in the field of fluid flow for many engineering applications.

      Exact solutions to the nonlinear Navier–Stokes equations are limited to a few cases, particularly for steady, uniform flows (either two‐dimensional or with radial symmetry) or for flows with simple geometries. However, approximate solutions may be undertaken for other one‐dimensional simple flow cases which require only the momentum and continuity equations in the flow direction for the solution of the flow field. Here, we present a few cases: uniform flow between parallel plates, uniform free surface flow down a plate, and uniform flow in a circular tube.

       Uniform Flow Between Parallel Plates

Consider a two‐dimensional uniform, steady flow between parallel plates, which extend infinitely in the z direction, as shown in Figure 1.10. We also consider the plates to be oriented at an angle θ to the horizontal plane, which results in a body force per unit mass as the gravitational term g sin θ. Therefore, the pressure gradient is specified as (𝜕p/𝜕y) = constant = P*. After making necessary simplifications and integrations, we find that the total flow rate per unit width and the velocity profile as follows:

      (1.68)

      (1.69)

Figure 1.10 Uniform flow between two stationary parallel plates.