Horizontal momentum equation

The geostrophic wind considers a balance between the horizontal pressure gradient and Coriolis forces. A less restrictive balance is one that includes the centripetal acceleration terms in the horizontal momentum equations. The balance that follows is not obtainable by a rigorous... 

On the /-plane, a third solution can also be obtained (7 = 0 or steady motion. The horizontal momentum equations indicate that this steady but nontrivial flow is geostrophically balanced by pressure gradients. The integrated continuity equation demonstrates that the flow is nondivergent. 

The transformation of the horizontal momentum equations (9.2.10) and (9.2.11) to the log-p system requires the pressure gradient terms to be expressed in appropriate forms. For example, the derivative of pressure with respect to longitude can be written... 

Here we have defined a reference density, po z) = p/RTr, and / = 2 2 cos 0 is called the Coriolis parameter. The vertical velocity is now defined as lu = Dz/Dt. In general w 7 however, for most purposes lu is an adequate approximation to the true vertical velocity. The continuity equation (9.2.23) is linear when written in this coordinate system, and the horizontal momentum equations, (9.2.20) and (9.2.21), are also simplified with the density no longer appearing explicitly. 

Theoretical approaches to the horizontal-plane hydraulic problem are often based upon momentum equations derived from the Navier—Stokes equations being averaged over the flow depth. This approach was developed by Rodi, Emtsev, Sherenkov, Beffa, and other authors [48, 171, 540], In the case where the vegetation is present, the resulting two-dimensional shallow-water equations for a time-dependent flow read as follows [540] ... 

Shaw and Schumann [576] were the first to apply LES to the canopy environment in attempts to reproduce characteristics of the flow through and above a deciduous forest. The code numerically solves the basic conservation equations for momentum and heat with options for additional scalars. For flow through a horizontally homogeneous canopy on uniform terrain, the momentum equation appears in the following manner ... 

For a generalized shear flow in the vicinity of a flat horizontal solid wall, the boundary layer flow can be described in Cartesian coordinates. The stress, —Oxy,eff, associated with direction y normal to the wall is apparently dominant, thus the stream-wise Reynolds averaged momentum equation yields ... 

Consider die simple case in which a low rate of mass transfer normal to die surface does not influence die velocity profile (vy 0), The velocity profile vj,y) in die film is determined by solving the z component of die eqnetion of motion for a Newtonian fluid in steady flow owing to iha ection of gravity. If the solid surface is inclined at an angle a with respect to die horizontal the momentum equation is... 

SHEAR-STRESS DISTRIBUTION IN A CYLINDRICAL TUBE. Consider the steady flow of a viscous fluid at constant density in fully developed flow through a horizontal tube. Visualize a disk-shaped element of fluid, concentric with the axis of the tube, of radius r and length dL, as shown in Fig. 5.1. Assume the element is isolated as a free body. Let the fluid pressure on the upstream and downstream faces of the disk be p and p + dp, respectively. Since the fluid possesses a viscosity, a shear force opposing flow will exist on the rim of the element. Apply the momentum equation (4.14) between the two faces of the disk. Since the flow is fully developed, j8j, = and Fj, = F , so that E F = 0. The quantities for substitution in Eq. (4.15) are... 

In order to obtain an expression for the total pressure drop along a section of transport line we will write down the momentum equation for a section of pipe. Consider a section of pipe of cross-sectional area A and length SL inclined to the horizontal at an angle 9 and carrying a suspension of voidage s (see Figure 8.3). 

Originally, the concept of fluid boundary layer was presented by Prandtl [123]. Prandtl s idea was that for flow next to a solid boundary a thin fluid layer (i.e., a boundary layer) develops in which friction is very important, but outside this layer the fluid behaves very much like a ffictionless fluid. To define a demarcation line between these two flow regions the thickness of the boundary layer, 6, is arbitrarily taken as the distance away from the surface where the velocity reaches 99 % of the free stream velocity (e.g., [55], p. 192 [107], p. 12 [114], p. 545). To proceed giving a thorough description of the equations used for turbulent flows, we need some results from the semi-empirical turbulent boundary layer flow analysis. For a generalized shear flow in the vicinity of a flat horizontal solid wall, the boundary layer flow can be described in Cartesian coordinates. The stress, —Ojty.eff, associated with direction y normal to the wall is apparently dominant, thus the stream-wise Reynolds averaged momentum equation yields ... 

Since the inlet flow has no horizontal component in y direction, the horizontal momentum of the inlet flow of a channel is assumed to have a zero value. The corresponding l dance equation in direction is... 

This equation is most applicable when vju exceeds 4. Since momentum plume rise occurs quite close to the source, the horizontal distance to the final plume rise is considered to be zero. 

One simplified method for determining stack height is a geometric method described in ASHRAE. The geometric method assumes an exhaust plume shape with a lower boundary having a 1 5 slope relative to the horizontal. The stack and plume are raised until the lower plume boundary is above rooftop penthouses, separation zones, and zones of high turbulence. ASHRAE provides equations for the sizes and locations of the separation and turbulence zones. A stack height reduction credit is provided to account for the vertical exhaust momentum. 

Equation 2.58 is the momentum balance for horizontal turbulent flow ... 

A momentum balance for the flow of a two-phase fluid through a horizontal pipe and an energy balance may be written in an expanded form of that applicable to single-phase fluid flow. These equations for two-phase flow cannot be used in practice since the individual phase velocities and local densities are not known. Some simplification is possible if it... 

To describe the flow in a horizontal heated capillary we use the mass, momentum and energy balance equations. At moderate velocity, the effects due to compressibility of liquid and vapor, as well as energy dissipation in gaseous and liquid phases are negligible. Assuming that thermal conductivity and viscosity of the vapor and the liquid are independent of temperature and pressure, we arrive at the following system of equations ... 

Great care has to be given to the physics of rotation and to the treatment of its interaction with mass loss. For differentially rotating stars, the structure equations need to be written differently [9] than for solid body rotation. For the transport of the chemical elements and angular momentum, we consider the effects of shear mixing, meridional circulation, horizontal turbulence and in the advanced stages the dynamical shear is also included. Caution has to be given that advection and diffusion are not the same physical effect. 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

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