# Uniplanar Sentence Examples

**Uniplanar**Motion.- - In the
**uniplanar**motion of a homogeneous liquid the equation of continuity reduces to du dv dx' dy-O' u= -d,y/dy, v = d i t/dx, (2) surface containing so that we can put _ (6) (9) we have (I) (2) (5) (I) where 4 is a function of x, y, called the streamor current-function; interpreted physically, 4-4c, the difference of the value of 4, at a fixed point A and a variable point P is the flow, in ft. - In the equations of
**uniplanar**motion = dx - du = dx + dy = -v 2 ?, suppose, so that in steady motion dx I +v24 ' x = ?' - The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of
**uniplanar**motion, in which the velocity at every point is turned through a right angle without alteration of magnitude. - For instance, in a
**uniplanar**flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 27rm, the velocity must be mfr, and 0=m log r,, y=m0, +4,i =m log re ie, w=m log z. **Uniplanar**Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.**Uniplanar**motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q.- The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with
**uniplanar**motion, and denoting by the resultant axial linear momentum of the circulation.