By J. A. Fox (auth.)
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Extra resources for An Introduction to Engineering Fluid Mechanics
4) Also ilu au ax au ay au az = -ilx + -ily + -ilz when the flow is steady. 4) gives p =- p [f u du dx dx + gy + P + wrY + p ( f u du dy dy + u 2 + u2 +w2) 2 P V2 Wr 2g = = V2 w dw dz dz ] constant - + - +Y r constan t as u 2 + u2 + w 2 equation. 5) where V is the local velocity. 3 Normal strain and deformation of a fluid element It is necessary to describe, mathematically, the displacements, distortions and rotations of any fluid element. Consider an element that is a right parallelepiped. It can deform in two ways as illustrated in Figs.
A source is a point in space from which fluid is emitted at constant rate. This concept cannot of course represent a real situation. In this book we will consider only two dimensional flows. A source in this case will be a line from which fluid is emitted uniformly in all directions in the plane normal to the line. This is equivalent to saying that in the plane perpendicular to the source the two dimensional flow created by the source is radiating from a point (see Fig. 15). At a distance r from such a source the fluid velocity will be directed away from the source in 3 radial direction so U Q r =21tr and ul} =0 47 Hydrodynamics Fig.
4 Rotation of a fluid element The rotation of a fluid element is illustrated in Fig. 7. e. through an angle au - ay 'Oy 'Ot/'Oy au ay 'Ot that is - aa" y Sy S, l -- B'~-------r~ B \ \ o u - -- ~ 8x 8, a" A Fig. 7 Note that for line OB to rotate as shown in Fig. 7 the value of au/ay must be negative (point B moving in the x direction at a smaller velocity than point 0). The diagonal OC will rotate through an angle of ~ (au _ au) 'Ot 2 ax ay and its rate of rotation will be ~ (au _ au) 2 ax ay Thus Q, the mean angular velocity is Q= ~ (au _ au) 2 ax ay 38 An Introduction to Engineering Fluid Mechanics Vorticity (which will be dealt with later) is usually denoted by the symbol (zeta) and has a value of twice the angular velocity so ~ au au ~ = ax - ay Thus a fluid for which the vorticity is zero is one in which all the fluid elements have zero angular velocity, that is they are not rotating.
An Introduction to Engineering Fluid Mechanics by J. A. Fox (auth.)