By J. A. Fox (auth.)
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Extra resources for An Introduction to Engineering Fluid Mechanics
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.
An Introduction to Engineering Fluid Mechanics by J. A. Fox (auth.)