By J. A. Fox (auth.)

ISBN-10: 0333231503

ISBN-13: 9780333231500

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**Additional resources for An Introduction to Engineering Fluid Mechanics**

**Example text**

5 The Navier-Stokes equations When the Euler equations were developed it was assumed that there were no frictional forces acting upon the fluid element. In any real fluid such forces must exist and the Euler equations cannot give an accurate description of the behaviour of the fluid especially if its viscosity is large. Shear stresses must be exerted on all faces of the parallelepiped shown in Fig. 2 in addition to the pressure and body forces illustrated. If the relationship between these shear stresses and the shear deformations occurring is given by Newton's law of viscosity (this assumes laminar flow) it is possible to include the frictional effects in the analysis by adding another set of terms to the Euler equations.

10 Surface tension The molecules comprising any substance attract one another and it is these attractive forces which make the material cohesive. A steel bar breaks when placed under sufficiently large tension because the tension is larger than the cohesive stress generated by the steel. In a fluid, cohesive forces are present and account for rhe development of surface tension. Surrounding any molecule within sz Fig. 10 the fluid is a zone within which another molecule will attract it. Such a molecule is shown at A in Fig.

Before continuing to describe how these concepts of stream function, velocity potential, circulation and vorticity can be used to analyse some flow situations it is necessary to give the expressions for velocities in terms of the derivatives of the velocity potential and stream function expressed in cylindrical coordinates. The velocity V can either be resolved into two velocity components u and u in the x and y directions or into two velocities in a cylindrical coordinate system (see Fig. 14), one in the direction of r, un and the other in the direction of e,uo, Now a'{J al/l ax ay u = - = -- and a'{J ay a1/1 ax u=-=-- 46 An Introduction to Engineering Fluid Mechanics v y x Fig.