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Shear flow over a protrusion from a plane wall: Addendum
LETTER TO THE EDITORS SHEAR
FLOW OVER A PROTRUSION PLANE WALL: ADDENDUM
FROM
A
BY W. A. HYMAN* THE
SOLUTION presented in a recent paper (Hyman. 1972) does not completely satisfy the boundary conditions on the plane wall in the vicinity of the hemispherical mound. Examination of the velocity distribution given by equation (6) shows that there is a distribution of normal velocity:
where r, a, /3 is a spherical coordinate system in which a is measured from the y axis and p is measured from the x axis toward the negative y axis (see Fig. 1 of Hyman. 1972). There is a corresponding shear stress on the mound T,~= - yp sin a.
t’
(1)
In order to correct this error it is necessary to add an additional solution to the equations of slow viscous flow satisfying the following boundary conditions: V=O
on
r=tr
(2)
V-*0
as
r-x
(3)
(6)
The additional drag is computed from the expression D=_lF?l; T,@cos /3 sin a a” dz dp. -5::
0
(7)
The result is D = ypna:.
(8)
The additional torque is computed from 7”=/iz
and on the plane wall
J= ~~~sin’ UP da d/3.
(9)
-7:’ I, The result is v, = v: = 0.
(4)
The required solution is not readily obtained but certain of its characteristics can be deduced. The principle characteristic is that it will be a flow which is generally clockwise when viewed along the y axis. The additional drag and torque on the mound will therefore be of the same sign as that computed in the original paper. Thus the conclusion in the original paper that the drag is significantly greater than has sometimes been assumed is further ehhanced. In order to estimate the magnitude of the correction an approximate velocity distribution is assumed which is axisymmetric with respect to the y axis and whose velocity vector lies in planes parallel to the x-z plane. This velocity distribution is (5) * Receiaed 7 June 1972.
T, = + yqn”.
(10)
Comparing equation (8) here with the original computation of the drag the additional term is seen to represent a 25 per cent increase. The increase in the torque indicated in the approximate solution given above is 66 per cent. REFERESCE Hyman, W. A. (1972) Shear flow over a protrusion from a plane wall, J. Biomechanics 5.45-48. W. A. HYMAN Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge Massachusetts 02 139 u&4.
FLOW OVER A PROTRUSION PLANE WALL: ADDENDUM
FROM
A
BY W. A. HYMAN* THE
SOLUTION presented in a recent paper (Hyman. 1972) does not completely satisfy the boundary conditions on the plane wall in the vicinity of the hemispherical mound. Examination of the velocity distribution given by equation (6) shows that there is a distribution of normal velocity:
where r, a, /3 is a spherical coordinate system in which a is measured from the y axis and p is measured from the x axis toward the negative y axis (see Fig. 1 of Hyman. 1972). There is a corresponding shear stress on the mound T,~= - yp sin a.
t’
(1)
In order to correct this error it is necessary to add an additional solution to the equations of slow viscous flow satisfying the following boundary conditions: V=O
on
r=tr
(2)
V-*0
as
r-x
(3)
(6)
The additional drag is computed from the expression D=_lF?l; T,@cos /3 sin a a” dz dp. -5::
0
(7)
The result is D = ypna:.
(8)
The additional torque is computed from 7”=/iz
and on the plane wall
J= ~~~sin’ UP da d/3.
(9)
-7:’ I, The result is v, = v: = 0.
(4)
The required solution is not readily obtained but certain of its characteristics can be deduced. The principle characteristic is that it will be a flow which is generally clockwise when viewed along the y axis. The additional drag and torque on the mound will therefore be of the same sign as that computed in the original paper. Thus the conclusion in the original paper that the drag is significantly greater than has sometimes been assumed is further ehhanced. In order to estimate the magnitude of the correction an approximate velocity distribution is assumed which is axisymmetric with respect to the y axis and whose velocity vector lies in planes parallel to the x-z plane. This velocity distribution is (5) * Receiaed 7 June 1972.
T, = + yqn”.
(10)
Comparing equation (8) here with the original computation of the drag the additional term is seen to represent a 25 per cent increase. The increase in the torque indicated in the approximate solution given above is 66 per cent. REFERESCE Hyman, W. A. (1972) Shear flow over a protrusion from a plane wall, J. Biomechanics 5.45-48. W. A. HYMAN Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge Massachusetts 02 139 u&4.