In many figures the curves which are streamlines when is the potential will be marked 

 with arrows. For the conjugate flow the arrows are then to be supposed transferred to the other 

 set of curves. Curves are always drawn for equally spaced values of qS or of i/r, and in flow 

 nets the same spacing is used for both <^ and i/r. Sometimes, however, an intermediate curve, 

 for a value midway between those for the adjacent curves, may be shown as a broken line. 



Physical cases can be constructed as desired by inserting a rigid boundary along any 

 streamline; this does not disturb the flow, since friction is assumed to be absent. If the bound- 

 ary extends to infinity so as to divide the field completely, the flow can be assumed to occur 

 only on one side of it, or to differ by a constant factor on the two sides. Special cases cor- 

 responding to different possible positions of such a boundary are not usually illustrated. 



The positive direction for angles, and for tracing closed curves, is taken as usual to be 

 counterclockwise. Thus, in tracing a closed curve positively, its interior lies on the left. 

 This direction is understood in the symbol ^, denoting the line integral around a closed ci ' v'e, 

 and in the fundamental definition of the circulation. 



The circulation F around any closed curve is also equal to the negative of the algebraic 

 change in the velocity potential on going once around the curve in the positive direction. 



Many types of two-dimensional flow possess one or more planes of symmetry, which are 

 represented on the xy-plnne by a line of symmetry. Two types of sym,metry may be distinguished. 



In one type, which will be called symmetry of flow, the actual motion on one side of the 

 plane is the mirror image of that on the other side. At points symmetrically located relative to 

 the plane of symmetry, the values of q and (^ are the same, also those of the pressure p, and 

 of the component of velocity parallel to the plane; whereas the component of velocity perpendic- 

 ular to tlie plane is oppositely directed. The difference between the value of i/r and its value 

 on the plane, which is necessarily composed of streamlines, is equal and opposite at the two 

 points. 



In the other type of symmetry, the flow net is again geometrically symmetrical, but the 

 motions have a different relation; p, tl/, and the vector component of velocity perpendicular to 

 the plane of symmetry have equal values at corresponding points, whereas the component of 

 velocity parallel to the plane, and also the algebraic excess of (f, above its value on the plane, 

 have equal and opposite values. 



Many examples of the two types of symmetry may be found in succeeding sections. The 

 contrast is specifically mentioned, for example, in Sections 41 and 55. 



The kinetic energy of the mass of fluid that is contained between two planes parallel 

 to the flow and unit distance apart will be denoted by T,. Its dimensions are those of kinetic 

 energy divided by distance or ml/fi. 



Formulas for the pressure p will not usually be given. When the boundaries are station- 

 ary and the motion of the fluid is steady, the pressure is given by the Bernoulli equation, 



■ p= lp(f;2_<^2)^p^ [34h] 



72 



