66 



HYDKAULICS AND ITS APPLICATIONS 



this that with steady motion the resistance is independent of the nature 

 of the solid surface. With unsteady motion on the other hand, there 

 would appear to be an actual interchange, by the breaking down of 

 adhesion, of molecules in intimate contact with the boundaries (Arts. 17 

 and 67), and since any such interchange will be greater as the rough- 

 ness of the surface increases, and will vary with the material of the pipe, 

 it might be inferred that resistance to unsteady motion will depend 

 on the state and material of the surface, and will increase as its roughness 

 increases. 



ART. 24. STEADY FLOW BETWEEN HORIZONTAL PARALLEL PLATES. 



Take OX, OY, OZ, co-ordinate axes in the fluid (Fig. 32). Let the 

 direction of motion be parallel to the axis of x, and the plates perpen- 

 dicular to that of y. Sup- 

 Y pose the boundaries in the 



direction of OZ to be in- 

 finitely distant, so as not to 

 affect the motion, and neg- 

 lect the effect of gravity. 

 Let 2 h = distance between 

 plates, and let OX bisect 

 this distance. 



Let u, v, and w denote 

 velocities of flow in direc- 

 is the intensity of normal 



FIG. 32. 



tions OX, OY, OZ. Then v = 0, .-. if p 



pressure on any plane perpendicular to OX we have 



- 

 d y 



= 0. The 



variations of velocity and pressure in the direction OZ may be neglected, 

 since the boundaries in this direction and therefore the points at which 

 the pressure may be zero, are at an infinite distance. 



It follows that *- = 0, so that on any plane perpendicular to OX the 



intensity of normal pressure is uniform. 



The tractive or shear force on any plane perpendicular to OY and of 



area A is i*. j~ . A, where /u = coefficient of viscosity (Art. 4). 



i/ 



The difference of tractive force on the two faces of a stratum of thick- 

 ness y, and of area A (Fig. 32 b), is thus given by 

 d du d 2 u . 



