Equations of Motion of a Viscous Liquid. 43 



subjected to a shearing force exerted parallel to the planes of 

 the strata, e. (j. plane A TRB, such that the element takes 

 the form AM G R, the coefficient of viscosity is defined as 

 the ratio of the shearing stress to shear per unit of time. 

 Putting this definition in the form of an equation, we have 



shearing stress _ 

 shear per unit of time 



in which /j- is the coefficient of viscosity; or shearing stress 

 =// X shear per unit of time. Let u be the velocity in the direc- 

 tion of A B; and the point A be located by x, y, z; the sides 

 of the element be cZx, dy, dz. Then 



— =f\x, y, 2;) = rate of normal displacement 

 dy 



along y. This force acts upon the surface of the element 

 dx dz. Hence the shearing force on this surface is 



„ dn , , 



F=!i — dx dz. 

 dy 



The force on the opposite face is 



F^=!J. — dx dz-\-/i — dx dz dy. 

 dy dy' 



The difference of the forces on the opposite faces of the 

 element considered is the retarding force due to the viscosity 

 of the fluid. 



Derivation of the Equations. — Sufficient has now been said 

 to proceed to the derivation of the equations. We shall take 

 up this problem in much the same way that we did the prob- 

 lem of a perfect fluid. But this new element of viscosity 

 must be taken into account. It is evident that our equations, 

 when derived, must be the same as those already obtained 

 for a perfect fluid with terms attached to account for the 

 retardation due to viscosity. For if in our new equations .« 

 be equated to zero the equations of a perfect fluid must 

 remain. 



