 Ch 5. Fundamental Laws (Differential) Multimedia Engineering Fluids ConservationMass Navier-Stokesand Euler's ConservationEnergy
 Chapter 1. Basics 2. Fluid Statics 3. Kinematics 4. Laws (Integral) 5. Laws (Diff.) 6. Modeling/Similitude 7. Inviscid 8. Viscous 9. External Flow 10. Open-Channel Appendix Basic Math Units Basic Equations Water/Air Tables Sections Search eBooks Dynamics Fluids Math Mechanics Statics Thermodynamics Author(s): Chean Chin Ngo Kurt Gramoll ©Kurt Gramoll FLUID MECHANICS - CASE STUDY SOLUTION Problem Description Sign Convention for the Gravitational Acceleration (a) For steady flow, Euler's equation in vector form is given by: If one assumes the z-axis is positive upward, then the gravitational acceleration g can be written as (see figure): Now substitute the vector identity to write Euler's equation as  Differential Length on a Streamline In order to obtain Bernoulli's equation, one needs to integrate the above equation along a streamline. Hence, take the dot product of Euler's equation with a differential length ds along the streamline as follows: Recognize that the differential length ds can be expressed in terms of the Cartesian coordinates as      ds = dx i + dy j + dz k Each term on the left hand side of Euler's equation can be evaluated as follows:   Since streamlines are constructed such that they are tangent to the velocity field, the differential length ds is parallel to the velocity V. The vector is then perpendicular to V, which gives Euler's equation thus becomes which can be readily integrated to yield For constant density, the above equation reduces to the famous Bernoulli's equation  Streamline between Point A and B (b) Apply Bernoulli's equation between the inlet and outlet of the nozzle (i.e., along the streamline between points A and B), Practice Homework and Test problems now available in the 'Eng Fluids' mobile app
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