[...]... the motion of a £uid One of these methods approaches the question from the molecular point of view That is, this method treats the £uid as consisting of molecules whose motion is governed by the laws of dynamics The macroscopic phenomena are assumed to arise from the molecular motion of the molecules, and the theory attempts to predict the macroscopic behavior of the £uid from the laws of mechanics. .. the conservation -of- momentum principle will utilize dynamics Basic Conservation Laws 15 FIGURE 1.3 Flow of a density-stratified fluid in which Dr=Dt ¼ 0 but for which @r=@x 6¼ 0 and @r=@y 6¼ 0 1.7 CONSERVATION OF MOMENTUM The principle of conservation of momentum is, in e¡ect, an application of Newton’s second law of motion to an element of the £uid That is, when considering a given mass of £uid in a lagrangian... changes in the basic ideas or principles, they will not be included in this treatment of the fundamentals In many practical cases of £uid £ow the variation of density of the £uid may be ignored, as for most cases of the £ow of liquids In such cases the £uid is said to be incompressible, which means that as a given mass of £uid is followed, not only will its mass be observed to remain constant but its... the rate of change of the total energy (intrinsic plus kinetic) of the £uid as it £ows is equal to the sum of the rate at which work is being done on the £uid by external forces and the rate at which heat is being added by conduction With this basic law in mind, we again consider any arbitrary mass of £uid of volume V and follow it in a lagrangian frame of reference as it £ows The total energy of this... location The choice of which coordinate system to employ is largely a matter of taste It is probably more convincing to apply the conservation laws to a control volume that always consists of the same £uid particles rather than one through which di¡erent £uid particles pass This is particularly true when invoking the law of conservation of energy, which consists of applying the ¢rst law of thermodynamics,... the resultant forces evaluated above is equal to the rate of change of momentum (or inertia force).The mass per unit volume is r and R its momentum is ru, so that the momentum contained in the volume V is V ru dV Then, if the mass of the arbitrarily chosen volume V is observed in the lagrangian frame of reference, the rate of change of momentum of the R mass contained with V will be ðD=DtÞ V ru dV Thus,... of momentum becomes r @uj @uj @sij þ ruk ¼ þ rfj @t @xk @xi ð1:4Þ It is useful to recall that this equation came from an application of Newton’s second law to an element of the £uid The left-hand side of Eq (1.4) represents the rate of change of momentum of a unit volume of the £uid (or the inertia force per unit volume) The ¢rst term is the familiar temporal acceleration term, while the second term... nonlinear, since the velocity appears quadratically On the right-hand side of Eq (1.4) are the forces causing the acceleration The ¢rst of these is due to the gradient of surface shear stresses while the second is due to body forces, such as gravity, which act on the mass of the £uid A clear understanding of the physical signi¢cance of each of the terms in Eq (1.4) is essential when approximations to the full... CONSERVATION OF ENERGY The principle of conservation of energy amounts to an application of the ¢rst law of thermodynamics to a £uid element as it £ows The ¢rst law of thermodynamics applies to a thermodynamic system that is originally at rest and, after some event, is ¢nally at rest again Under these conditions it is stated that the change in internal energy, due to the event, is equal to the sum of the... instantaneous energy of the £uid to consist of two parts: intrinsic or internal energy and kinetic energy That is, when applying the ¢rst law of thermodynamics, the energy referred to is considered to be the sum of the internal energy per unit mass e and the kinetic energy per unit mass 1 uÁu In this way the modi¢ed 2 form of the ¢rst law of thermodynamics that will be applied to an element of the £uid states . Graphics and Design, Daniel L. Ryan 3. Lubrication Fundamentals, J. George Wills 4. Solar Engineering for Domestic Buildings, William A. Himmelman 5. Applied Engineering Mechanics: Statics and. Nelik Gear Noise and Vibration: Second Edition, Revised and Expanded, J. Derek Smith Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, George A. Antaki Turbomachinery:. spe- cializing in fluid mechanics. This book is intended to remedy this situation. The book is divided into four parts. Part I, ‘‘Governing Equations,’’ deals with the derivation of the basic conservation