International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 05 Issue: 06 | June-2018 www.irjet.net p-ISSN: 2395-0072 © 2018, IRJET | Impact Factor value: 6.171 | ISO 9001:2008 Certified Journal | Page 408 Deriving one dimensional shallow water equations from mass and momentum balance laws Amir Dehghanghadikolaei 1 , Navid Namdari 2 , Behrouz Mohammadian 3 , Seyed Reza Ghoreishi 4 1 Mechanical, Industrial and Manufacturing Engineering Department, Oregon State University, Corvallis, Oregon 97331, USA 2,3 Mechanical, Industrial and Manufacturing Engineering Department, University of Toledo, Toledo, Ohio 43606, USA 4 Mechanical Engineering Department, Semnan University, Semnan, Iran -----------------------------------------------------------------------***------------------------------------------------------------------------ Abstract - The Short Wave Equations (SWE) is a system of hyperbolic/parabolic Partial Differential Equations (PDE) that governing fluid flow in the oceans, coastal regions, estuaries, rivers and channels. In shallow water, flows there would be a relatively smaller vertical flow compared to horizontal one and in order to get rid of the vertical parameter, we can average over the depth. The SWE can be used to predict tides, storm surge levels and coastline changes like hurricanes and ocean currents. SWE also arise in atmospheric flows and debris flows. The SWE are derived from the Navier-Stokes equations, which describe the motion of fluids. The Navier-Stokes equations are themselves derived from the equations for conservation of mass and linear momentum. Keywords: continuum mechanics, hydraulic jump, Navier-Stokes, Hydraulics. 1. Introduction Continuum mechanics is a branch of mechanics in which we study and analyze the kinematics and mechanical behavior of materials [1]. Continuum mechanics are a continuous mass that deals with the physical properties of solids and liquids. This technique could be used in many fields such as simulation of corrosion of additively manufactured parts [2], simulation of precision finishing processes [3], vibration [4], heat transfer [5], or fluid mechanics [6]. Physical properties are independent of the coordinates they take place. In this paper we will focus conservation equations on fluids and the shallow water equations [7]. The shallow water equations explains the behavior of a thin layer of fluid with a constant density that has boundary conditions from below by the bed of the flow and from above by a free surface of water. There are so many different features inside of these flows due to the fact that their behavior is based on lots of conservation laws [7]. We consider this type of flow with a simple vertical structure and we assume that fluid system is the flow of a thin layer of water over terrain, which varies in elevation. Friction is ignored and the flow velocity is assumed to be uniform with elevation [7]. Besides, the slope of terrain is assumed much less than unity, like the slope of the fluid surface. These assumptions allow the vertical pressure profile of the fluid to be determined by the hydrostatic equation [8, 9]. Although the derived shallow water equations are idealized, there are still some common essential characteristics with more complex flows such as Tsunami, deep in ocean or near shore. Near shore, a more complicated model is required [8]. Fig. 1 illustrates a schematic view of shallow water flow [1]. Fig.1, schematic view of a shallow water flow [1]. Hydraulic jump occurs whenever we have a flow changes from supercritical to subcritical. In phenomenon the water surface rises suddenly and we have surface rollers while an intense mixing occurs and air is entrained followed by a large amount of energy dissipation [10]. In other words, a hydraulic jump happens when a supercritical upstream meets by a subcritical downstream flow [10]. There are also artificial hydraulic jumps, which are created by devices like sluice gates. Totally, a hydraulic jump can be used as an energy dissipater, chemical mixer or to act as an aeration device [11]. Since we have unknown loss of energy in hydraulic jumps, one should conservation of momentum to derive jump equations [10]. To develop this equation, they generally consider a situation with or without loss of energy between upstream and downstream. Besides, that situation may come with or without some obstacles which may cause a drag force of Pf [11]. Fig.2 shows a schematic hydraulic jump in a flow.