554 XI. HYDRODYNAMICS. 



n*A\ T^ /x*y*da\ / \ , / r xti d(o\ , N 



254) Y n = ii (^L ^j cos (nx) + (- # + 2jt -^ ^J cos (wy), 



Z n = p cos (w#) = 0. 



This shows that there is a normal pressure p, together with a tangential 

 stress which we obtain by resolving along the tangent, 



255) T = Y n cos (nx) - X n cos (ny) 



dofo; 2 w 2 , 9 9 \ , xy / \ / >1 



~ ^dr \ r~ ( COS "~ COS ^' "^ -- COS v 1 *) cos ( w ^) I 



and since cos (nx) = ^ cos (ny) = ^ 



The moment of the tangential stress on the cylinder of radius r and 

 unit length, is accordingly 



257) 



We may accordingly use this method to determine the viscosity, as 

 is in fact done in apparatus for the testing of lubricants. We see 

 that if the linear dimensions are multiplied in a certain ratio, the 

 moment is increased in the square of that ratio. We also see that 

 the moment of the force required to twist the cylinder is independent 

 of the pressure p, which contains an arbitrary constant, not given 

 by the equation 253), but depending on the hydrostatic pressure 

 applied at the ends. 



Let us now consider some simple cases where the flow is not 

 steady, limiting ourselves to the case of small velocities, so that the 

 terms in 233) involving the first space derivatives, being of the second 

 order, are negligible. Let us once more consider laminar flow, defined 

 by equations 234), 235). Let us also put p = const. Instead of 236) 

 we now have for the first of equations 233), 



258) ^ ==v d ^. 



dt fly* 



This equation is the same as that which represents the conduction 

 of heat in one direction. Let us first consider a solution periodic in 

 the time, such as may be realized physically by the harmonic small 

 oscillation in its own plane of a material lamina constituting the 

 plane y = 0, along which the liquid does not slip. We may take as 

 a particular solution 



which inserted in 258) gives 



n = vm 



