Download e-book for kindle: An Introduction to Engineering Fluid Mechanics by J. A. Fox (auth.)

By J. A. Fox (auth.)

ISBN-10: 0333231503

ISBN-13: 9780333231500

ISBN-10: 1349158356

ISBN-13: 9781349158355

ISBN-10: 1761761781

ISBN-13: 9781761761782

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Extra resources for An Introduction to Engineering Fluid Mechanics

Example text

If the x and z axes are located in a horizontal plane the only body force that exists is the weight acting in the y direction but if the coordinate system has no particular orientation relative to the earth's gravitational fieldthe vertical-then weight body-forces can exist in all three directions (see Fig. 3). x y Fig. 3 These body forces are usually denoted by X for the body force, that is force/unit volume, in the x direction y for the body force, that is force/unit volume, in the y direction Z for the body force, that is force/unit volume, in the z direction Then the force in the x direction due to pressure forces and body forces is " ax 3x ) 3y 3z + X 3x 3y 15z (- ap This must cause an acceleration of the element of du/dt.

By this defmition the flow velocity is tangential to the streamline (see Fig. 8) so that v dy -=- u dx u 8y - v8x = 0 Hydrodynamics 41 Fig. 8 Substituting for u and u gives a1/l 8y _ (_ a1/l) 8x = 0 ay so a1/1 ax a1/1 -8y+-3x=0 ay ax This, from the definition of a total differential, gives 01/1 = O. So along a streamline 01/1 = 0 and therefore I/; = constant. Thus the function 1/1 defines the streamlines of a fluid motion. 6) au au -=- ax ay substituting for u and u gives a2 1/1 a2 1/; -+--=0 ax 2 ay2 so the stream function must also satisfy the Laplace equation.

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.

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An Introduction to Engineering Fluid Mechanics by J. A. Fox (auth.)

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