Therefore, certain specific cases will be discussed regarding the practical importance. Such specific applications are exercises in mathematics rather than fluid mechanics. Two-dimensional problems can be especially easily treated by applying complex variable functions. The equation system contains five unknown functions (q x, q y, q z, p, ρ), thus the five equations can be solved. (3.68).ĭarcy's law together with continuity equation and equation of state is sufficient to determine the flow pattern in a porous medium for given boundary conditions. The fact is that the empirically verified relative permeability equations are originated in Eq. Relative permeability is actually variable through the porous medium during a flow experiment and therefore a distinction between Eq. If Darcy's law is extended to immiscible multiple phase flow, it leads to the relative permeability concept.

On the other hand, there is an indirect experimental indication of the correctness of Eq. This is allowed only for constant permeability, density, and viscosity as the preceding show. The Coriolis force must also be taken into account when computing long-range missile or artillery trajectories. Thus, the Coriolis force is not important for the dynamics of individual cumulus clouds but is essential for an understanding of longer time scale phenomena such as synoptic scale systems. The Coriolis force is negligible for motions with time scales that are very short compared to the period of Earth's rotation (a point that is illustrated by several problems at the end of the chapter). The vertical component of the Coriolis force in (1.18) is ordinarily much smaller than the gravitational force so that its only effect is to cause a very minor change in the apparent weight of an object depending on whether the object is moving eastward or westward. In either case the deflection is to the right of the direction of motion in the Northern Hemisphere and to the left in the Southern Hemisphere. Thus, for example, an object moving eastward in the horizontal is deflected equatorward by the Coriolis force, whereas a westward moving object is deflected poleward. Where f ≡ 2 Ω sin ϕ is the Coriolis parameter. The best way to remove the geometric effect of a surface force acting on a body is to normalize the force by the area over which it acts, which leads to the concept of stress. Therefore, having a common way of quantifying the effect of a surface force regardless of the size of the area over which it acts is useful. Because the influence of a surface force is proportional to the size of the area over which it acts, an inherent geometric effect influences the changes in a body caused by that force. We can also envision that a box having a small footprint might be easier to push than one having a large footprint, because the magnitude of the tangential force transmitted to the base of the small box is focused across a smaller area and can more easily overcome the frictional force resisting sliding. As long as the magnitude of the tangential force exceeds the force of friction resisting sliding the box will slide. As the box is pushed, there is not only a force acting perpendicular to the face being pushed, but also a component of that force acting tangentially along the surface of the box in contact with the table. An example of surface forces acting on a body can be illustrated by envisioning a person pushing a box across a table. Unlike a body force, the influence of a surface force is proportional to the size of the area over which it acts furthermore, it acts in a specific direction and in a specific position (e.g., Middleton and Wilcock, 1994 Turcotte and Schubert, 2002). Surface, or contact, forces, on the other hand, act on the bounding surface of a body. An example of a body force is the force of gravity the weight of an element is the product of its mass and the acceleration of gravity, g. Body forces act equally on every element of mass within a body and are proportional to its mass or volume ( Middleton and Wilcock, 1994 Turcotte and Schubert, 2002). In mechanics, two classes of forces can be defined: body forces and surface forces. From the above expression, we see that force has a unit of kg m s −2, which is called a newton (N).

Second law of newton