Oer — Stability or Instahility of Motions of a Viscous Liquid. 81 



denote the kinematic viscosity, or quotient of \dscosity by density, by v, the 

 fundamental equations are 



duldt + ^yd'ajdx + 13^ = vVht - p''^dpjdx, 



dvjdt + jSydvldx = vV^v - fr'^dpjd.y, 



dwjdt + jiydw/dx = vVHa - p~'^d2j/dz ( 



dujdx + dvldy + diojdz = / 



and from these we obtain, by elimination, 



{djdt + ^yd/dx - vV") a = 0, (2) 



where 



(T = V-v. (3) 



Ignoring, for the sake of brevity, any further reference to u, w, it is 



desired to obtain an expression for v, satisfying (2) and also the following 



initial and boundary conditions : — 



when z^ = 0, v to be a given arbitrary function of x, y, z ; (4) 



when y = 0, and when y = h, for all values of o:, z, t, both v 



and dv/dy to vanish. (5) 



Lord Kelvin first proceeds to find a particular solution, v, of (2) which 

 satisfies the initial conditions (4) irrespectively of the boundary conditions (5), 

 except as follows : — 



V = when t = 0, and y = oi y = h. (6) 



He next finds another particular solution, t>, satisfying the following 



initial and boundary conditions : — 



i) = 0, d^/dy = 0, when t = 0, (7) 



i> = - V, d'o/dy = - dv/dy, when y = 0, y = h. (8) 



The required complete solution will then be 



V = Y + ^. (9) 



To find V, Lord Kelvin remarks, that if v were zero, the complete integral 



of (2) would be 



<T=f(x-(iyt,y,z), (10) 



where / is a perfectly arbitrary function, and takes therefore as a trial for a 

 type of solution with v not zero, 



where T is a function of t. Substituting in (2), one obtains 

 and hence, from (3), 



y ^_ J± (13) 



