06_viscous_flow - 1 2.25 Equation of Motion for Viscous...

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1 2.25 Equation of Motion for Viscous Flow A inA .Son in Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 2003 (9th edition) Contents 1. Surface stress …………………………………………………………. 2 2. The stress tensor ……………………………………………………… 3 3. Symmetry of the stress tensor ………………………………………… 7 4. Equation of motion in terms of the stress tensor ……………………… 9 5. Stress tensor for Newtonian fluids ……………………………………. 12 The shear stresses and ordinary viscosity …………………………. 12 The normal stresses ………………………………………………. . 13 General form of the stress tensor; the second viscosity …………… 18 6. The Navier-Stokes equation …………………………………………… 21 7. Boundary conditions …………………………………………………. . 23 Appendix A: The Navier-Stokes and mass conservation equations in cylindrical coordinates, for incompressible flow ………………….24 Appendix B: Properties of selected fluids …………………………. .……… 26 Ain A. Sonin 2002
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2 1 Surface Stress Quantities like density, velocity, and pressure are defined by a value at every point in the fluid at every time t . The density r ( v r , t ) and pressure p ( v r , t )a re scalar fields . They have a numerical value at every point in space at any instant in time. The velocity v v ( v r , t )i sa vector field ; it is defined by a direction as well as a magnitude at every point. Fig. 1: A surface element at a point in a continuum. The surface stress is a more complicated type of quantity. One cannot talk of the stress at a point without first defining the particular surface through that point on which the stress acts. A small fluid surface element centered at the point v r is defined by its area d A (the prefix indicates a very small but finite quantity) and by its outward unit normal vector v n . The stress exerted by the fluid on the side toward which v n points on the surface element is defined as v s = lim A Æ 0 v F A (1) where v F is the force exerted on the surface by the fluid on that side (only one side is involved). In the limit A Æ 0 the stress is independent of the magnitude of the area, but will in general depend on the orientation of the surface element, which is specified by v n .In other words, v = v ( v x , t , v n ). (2) The fact that v depends on v n as well as x, y, z and t appears at first sight to complicate matters considerably. One apparently has to deal with a quantity that depends on six independent variables ( x, y, z, t , and the two that specify the orientation v n ) rather than four.
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3 Fortunately, nature comes to our rescue. We find that because v s is a stress, it must depend on v n in a relatively simple way.
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06_viscous_flow - 1 2.25 Equation of Motion for Viscous...

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