Slides for Chapter 6 FM 2 part 1.pdf - Chapter 6 Differential Analysis of Fluid Flow Learning Objectives After completing this chapter you should be

Slides for Chapter 6 FM 2 part 1.pdf - Chapter 6...

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22/10/19 1 Chapter 6 Differential Analysis of Fluid Flow Learning Objectives After completing this chapter, you should be able to: ■ determine various kinematic elements of the flow given the velocity field. ■ explain the conditions necessary for a velocity field to satisfy the continuity equation. ■ apply the concepts of stream function and velocity potential. ■ characterize simple potential flow fields. ■ analyze certain types of flows using the Navier–Stokes equations In control volume and system analysis only conditions at surface of the CV is needed, and thus problem can be solved without detailed knowledge of the flow field. Unfortunately, there are many situations that arise in which the details of the flow are important and the finite control volume approach will not yield the desired information. For example, we may need to know how the velocity varies over the cross section of a pipe, or how the pressure and shear stress vary along the surface of an airplane wing. To solve such problems , involves an infinitesimal control volume, as distinguished from a finite control volume, is commonly referred to as differential analysis, since the governing equations are differential equations. 1 Kinematic Description In fluid mechanics, an element  may undergo four fundamental  types of motion.  a) Translation b) Rotation c) Linear strain d) Shear strain Because fluids are in constant  motion, motion and deformation  is best described in terms of rates  a) velocity: rate of translation b) angular velocity: rate of rotation c) linear strain rate: rate of linear  strain d) shear strain rate:  rate of shear  strain Fluid Element Kinematics
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22/10/19 2 Types of Motion / Deformation of Fluid Elements During fluid flow, a small fluid element (e.g. in the shape of a cube) initially at one position moves to another during a short time interval. Because of the generally complex velocity variation within the field the element is expected not only to: (i) translate from one position but also to have its  (ii)   volume changed (linear deformation), to  (iii)   rotate, and to undergo a change in shape (angular deformation).  The element’s motion and deformation are related to the velocity and variation of  velocity throughout the flow. 3 The movements and deformations though occur simultaneously, each one however can separately be considered as illustrated in Figure. Linear Motion and Deformation The simplest type of motion that a fluid element can undergo is translation. Translation occurs when there are no velocity gradients. The element will simply shift from one position to another due to fluid flow.
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