Reservoir routing is discussed in dealing with storage-discharge relations, mehods of routing, modified Puls method, numerical methods of flood routing through reservoirs, and accuracy of calculations, existing and proposed reservoirs, reservoir routing in HEC-HMS, and storage methods.

The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.

In this chapter, the basic equations of fluid dynamics are derived and their physical significances are discussed in depth and in examples. Both integral and differential forms of the continuity equation, momentum equation, and energy equation are derived. In addition, Bernoulli’s equation, angular momentum equation, enthalpy equation and entropy equation are also introduced. Finally, several analytical solutions of these governing equations are shown, and the mathematical properties of the equations are discussed. Besides the fundamental equations, some important concepts are explained in this chapter, such as the shaft work in integral energy equation and its origin in differential equations, the viscous dissipation term in the differential energy equation and its relation with stress and deformation, and the method to increase total enthalpy of a fluid isentropically.

We complete the basic equations of cosmology by introducing the second Friedmann and continuity equations. We next introduce the equation of state parameter, and describe the evolution of the universe in the simplest cosmological models: matter-only, radiation-only, and lambda-only. We introduce the concept of a cosmological horizon, and explain how to calculate the age of the universe in a given model. Along the way, we establish the fiducial cosmological model -- a set of cosmological parameters that we will use for all our results in the remainder of the book. We end by introducing two observationally accessible distance measures -- luminosity distance and angular-diameter distance.

Hydrology deals mainly with the cycling of water in the natural environment, as related specifically to the continental water processes, at all scales. Globally, precipitation on the land surfaces averages roughly 0.8 m per year, of which around 60% is evaporated back into the atmosphere and 40% runs off into the oceans. Beside statistical analysis of direct measurements, several methodologies are available to describe the various components of the hydrologic cycle. These can be distinguished from one another primarily by the spatial and temporal scales at which the particular hydrologic phenomena are parameterized.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

Physically based approaches to hydraulic geometry relations for width, depth, velocity, and slope require equations of continuity of water, roughness, and sediment transport. Different methods have been employed for different expressions of roughness and sediment transport. Without delving into their underlying theories, this chapter briefly outlines these expressions as they will be invoked in subsequent chapters. Also, unit stream power, stream power as well as entropy have been employed, which are also briefly discussed.

Empirical equations of downstream hydraulic geometry, entailing width, depth, velocity, and bed slope, can be derived using the scaling theory. The theory employs the momentum equation, a flow resistance formula, and continuity equation for gradually varied open channel flow. The scaling equations are expressed as power functions of water discharge and bed sediment size, and are applicable to alluvial, ice, and bedrock channels. These equations are valid for any value of water discharge as opposed to just mean or bank-full values that are used in empirical equations. This chapter discusses the use of scaling theory for the derivation of downstream hydraulic geometry. The scaling theory-based hydraulic geometry equations are also compared with those derived using the regime theory, threshold theory, and stability index theory, and the equations are found to be consistent.

Hydraulic geometry, described by depth, width, velocity, slope, and friction, is determined using three equations of continuity, resistance, and sediment transport and by satisfying the condition of minimum production of entropy in the channel system. This chapter discusses the methodology based on this condition.

Design of a stable alluvial channel is based on the hypothesis that the equilibrium state of a channel corresponds to maximum flow. The channel design can then be accomplished by employing the continuity equation, resistance law, sediment transport equation, and the channel cross-section shape. This chapter derives the channel hydraulic geometry for primarily three cross-sections, namely trapezoidal, rectangular, and triangular.

Rivers tend to follow the path of least action for transporting the sediment and water loads imposed on them. Because regime hydraulic geometry relations entail more unknowns than the equations of continuity, resistance, and sediment transport, optimization is utilized to determine the preferred cross-section from among many possible cross-sections and this cross-section satisfies the path of least action. This chapter discusses this principle and derives the hydraulic geometry based on this principle.

Fluid dynamic principles that are fundamental to understanding the motion of fluids in radial compressors are highlighted. These include the continuity and the momentum equations in various forms. These equations are then used to delineate the effect of the fluid motion on pressure gradients on the flow. The simple radial equilibrium equation for a circumferentially averaged flow is introduced. Special features of the flow in radial compressors due to the radial motion are considered, such as the effects of the Coriolis and centrifugal forces. The relative eddy, which gives rise to the slip factor of a radial impeller, is explained. A short overview of boundary layer flows of relevance to radial compressors is provided. The flow in radial compressor impellers is strongly affected by secondary flows and tip clearance flows, and an outline is provided of the current understanding of the physics related to these. The phenomenon of jet-wake flow in compressors is described.

1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation.

2. Introduce the Gross-Pitaevskii equation and its applicable conditions.

3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap.

4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior.

5. Introduce the criterion for superfluidity and determine the superfluid critical velocity.

6. Introduce two-fluid hydrodynamics.

7. Introduce the Bogoliubov theory.

8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation.

9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate.

10. Summarize the difference between an interacting and non-interacting Bose condensate.

11. Introduce the Bragg spectroscopy.

12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence.

13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit.

14. Introduce scale invariant quantum gases.

15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence.

16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state.

17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state.

18. Discuss interference effects in density and density-density correlations.

19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.