Hidrológiai Közlöny 1976 (56. évfolyam)
1. szám - Dr. Bogárdi János: Elmélet, oktatás és gyakorlat a hidraulikában és hidromechanikában
Dr. Bogárdi, L. J.: Theory, education and practice in hydraulics Hidrológiai Közlöny 1976. 1. sz. 3 Considering further the change of any extensive quantity Xi within the elementary volume, the following facts must be borne in mind : a) Within the elementary volume there are sources Qi and b) the current of the extensive quantity Ii flows across the surface bounding the elementary volume. With the foregoing in mind the balance equation of the extensive quantity xi is as follows : d Xi ~7l7 Qi-h (i) It should be noted here that if the extensive quantity under consideration is volume [L 3], then Eq. (1) states that the temporal change in volume equals the difference of the volume source [L 3T _ 1] and the volume current [L 3T1].* In the case of a fluid volume, if the fluid is water, the volume current is named the discharge. When considering the processes in hydraulics it is further necessary to determine the local values at particular points within the elementary volume. The changeover to local values is possible by taking the values related to unit volume of the quantities involved in the balance equation. These are referred to as densities. If the extensive quantity Xi has the density vi, then the extensive quantity within the volume V becomes (2) xi = J v.-d All sources within the volume become / W W (3) where qi is the source density. The extensive current Ii can be expressed by the surface integral of the surface current density or introducing the theorem of Gauss —Ostrogradskij, by the volume integral of the divergence of the surface current density.** Thus the balance equation (1) expressed in terms of densities will be J (4^ + divJ;-,,) dJ7=0 4) The value of the integral is unrelated to the limits of integration and may assume zero value only if the integrand itself is zero. With this in mind the general form of the balance equation becomes: d v* J- I ——+ div Ji = q ( at (5) The individual terms involved in the above balance equation are preferably examined separately. Concerning vi the only specification so far was that it denoted the density of an extensive quantity. Thus in considerations of fluid dynamics this may indicate both the mass density g[ML~ 3], the energy density e[EL" 3], the momentum density pv[ML2T1], or even the specific gravity of the fluid y[FL~ 3]. Ji is thedensity of some extensive current, for instance pv[ML~ 2T _ 1] of mass, ovor [ML 'T" 2= FL 2] the surface current density of momentum, while the current density of energy is [EL 2T '], etc. The source density qi is the rate of generation (or decay) of a particular extensive quantity related to unit volume. For instance, in dimensional form the source density of mass is [ML _ 3T1], that of energy is * The dimensions are given the following notations: L — length, T — time, M — mass, E — energy, work, F — force, weight. The dimensions are included in angular brackets. ** Vectorial quantities will be denoted by solid block characters. [EL-'T" 1] while that of momentum is [FL" 3], etc. The current density Ji is in general composed of two terms, namely the conductive and the convective current densities. According to the theorem of Onsager, the conductive current density is determined by the inhomogeneity of not only the "corresponding", but of all intensive quantites in combination. The extent of inhomogeneity is obtained by applying the "nabla" operator. If the relevant intensive quantify is a scalar physical quantity, the application of the nabla operator yields the gradient of this particular quantity. The theorem of Onsager states that the current density of the i-th extensive quantity can be defined as the product sum of the gradient of the individual intensive quantities yi and of the conductivity coefficient La pertaining thereto. Accordingly, the conductive current density of the i-th extensive quantity is ; cond = ^ La grad y t (6) 1 = 1 Tn case where the center of the system is also displaced, the extensive quantities (which are the material properties) also follow this movement, so that a convective current also occurs. Convective current densities can be expressed in a very simple form. If at any point the density of the i-th extensive quantity is vi and the velocity of this point is v, then the density of the convective current is J t con v = J'{ v (7) The total current density J,; is the sum of conveetiveand conductive current densities, thus J; = Vi V + ^ Lil grad yi (8) 1=1 Introducing the total current density into Eq. (5) the final form of the general balance equation is obtained as Ovi 1)7 - + div m I ,,< v + ^ Li i g ra d VII = H 1=1 When solving particular problems in hydraulics it is necessary to determine : a) the relevant extensive and relevant intensive quantities, b ) the numerical values of the conductivities, and c) the actual form of the source densities. It should be noted that for solving particular problems in hydraulics, besides the balance equations, the so-called unambiguity criteria must also be specified. These are as follows: J°. The domain of interpretation —the interval within which the variables may assume values. 2°. The boundary conditions —specification of the interaction between the system and its environment. 3°. The initial conditions —definition of the state in which the system is at the instant selected for starting the examination. (Logically, in steady processes no initial conditions exist.) 4°. State equations —specification of the physical properties of the operating medium of the system. All these conditions can be formulated precisely for particular problems only. The balance equation expressed by Eq. (9) h;us been demonstrated to apply without any restriction to various engineering phenomena and processes. Consequently, all processes in hydraulics can be described thereby. This statement has been verified completely by the most recent theoretical investigations presently under way. On the strength of the foregoing considerations we are fully justified in suggesting the transport
