Stokes drag on axially symmetric bodies: A new approach
The paper deals with the problem of external steady Stokes flow past axisymmetric impermeable rigid bodies. Using an elementary, but refined, analysis of the drag force formula for a sphere in uniform far-field flow, the authors obtain a simple formula, the evaluation of which renders correct values for the drag force not only on a sphere, but also on a spheroidal body in a uniform far-field flow parallel to its axis of symmetry. In the same way of reasoning, similar and correct formulas are also found for the drag force in the case of uniform transverse far-field flow and for the moment in the case of rotation about the axis of symmetry of a spherical or spheroidal body. These formulas, so far, are very interesting, but, unfortunately, of no practical use, since the final exact solutions for these body shapes and flow situations are already well known. However, the authors conjecture, and that is the main point of the paper, that the same formulas are also valid in general for every axisymmetric body with continuously varying tangent. The validity of such a conjecture would be, of course, of considerable practical importance, since we are interested, in the first place, in the drag force and the moment acting on a body in Stokes flow, and so we would not have to solve the problem as a whole for every new body form of the aforementioned kind. However, knowing the complexity of the final Stokes flow integral equation system for determining axial and radial stress distributions along the meridional contour of a strictly convex axisymmetric body in a uniform far-field flow parallel to the symmetry axis of the body, one must first doubt the general validity of such a conjecture. The two integral equations of the said system are of the first kind and involve complete elliptical integrals of the first and second kind which makes the problem quite difficult for an analytical treatment. For the cases of a spherical or spheroidal (prolate or oblate) body (and this reviewer conjectures in these cases only) the radial stress component vanishes, and the integral equation system reduces to only one integral equation of the first kind for the axial stress component, which is a less refractory mathematical problem. This may be an indication why such a simple formula, as found in the paper under review, may exist for spherical and spheroidal bodies. To analytically prove or disprove the general validity of the hypothesis made in the paper turns out to be a rather difficult mathematical task. Also, no further exact solutions are available for such body forms which could be used as pro or counter examples in proving the conjecture. As only means which remains for testing the validity or invalidity of the above conjecture, is to use reliable numerical solutions for body forms other than spherical or spheroidal ones. In the following discussion on that basis, only the Stokes flow past an axisymmetric body in uniform far-field flow parallel to its symmetry axis will be examined. For an explicitly given meridional body contour, \(y( x)\), equation (2.9) for the drag force can be written as \[ F_x=-8\pi\mu = Uy_{\max} \int_{x_1}^{x_2} {yy''dx\over(1+y'{^2})^2}, \tag{A} \] where \(x_1,x_2\), are the intersection points of the profile with the symmetry \(x\)-axis of the body. Corresponding formulas can be also written down for the cases where the body contour is given in parametric form or in polar coordinates. From equation (A) we first see at once (because of the second derivative of \(y( x)\) in the integrand in the numerator) that a straight line segment \(y=ax+b\), \(x_I<x<x_{II}\), \(x_1\leq x_I\leq x_{II}\leq x_2\), as a possible part of the meridional body contour with continuously varying tangent, would not contribute to the total drag force acting on the body. This does not have any theoretical justification, contradicts numerical experience, and so cannot be accepted to be true. In addition, it seems to this reviewer to be an odd situation that formula (A) should be valid only for a curve with continuously varying tangent, but not for a curve with edges or other kind of nodes. Next, the drag force results for a prolate (equation (5.4)) and an oblate cycloidal body (equation (5.11)) are correct in the sense of the conjecture in the paper. However, they differ from the carefully obtained numerical values of the drag for the same body contours. For instance, for \(a=1/\pi\) the numerical value of the drag force on a prolate cycloidal body is \(F_{Cx}=13.103\mu U\), whereas equation (5.4) of the paper yields a value \(F_{Cx}=13.588\mu U\) that gives a relative error of 3.7\%. For \(a=1/\pi\) for oblate cycloidal body this reviewer obtains numerical value of \(F_{Cx}=26.9503\mu U\), whereas equation (5.11) renders \(F_{Cx}=32.6319\mu U\) which makes an unacceptable relative error of about 21.1\%. In addition, equation (2.9) in the paper cannot be valid for cycloidal body shapes. The drag force value obtained in the paper for the flow past a deformed sphere seems to be also incorrect because the radius of maximum cross-sectional area is not given as in the paper by equation (4.5), namely \(b=a[1+\varepsilon (d_0-{1\over 2}d_2)]\), but it should be \[ b=a\Biggl[1+\varepsilon \biggl(d_0-{1\over 2}d_2+{3\over 8}d_4- {5\over 16}d_6+{35\over 128}d_8-\cdots\biggr)\Biggr]. \tag{B} \] We cannot justify the omission of \(d_4,d_6,\ldots\) in (B) by saying these coefficients are all of \(O(1)\) with respect to \(\varepsilon\ll 1\), because such an assumption was made for all the coefficients \(d_0,d_1,d_2,\ldots\) and not only for \(d_4,d_6,\ldots\)\ . The drag obtained in the paper for a body named egg-shaped body, the right part of which has the shape of a semi-prolate spheroid and the left part is a hemisphere, may or may not be the true one. This example was not numerically checked by the reviewer. Two further examples that were numerically solved by this reviewer are the following: 1) A profile in the form of a Cassinian curve given by the function \(y=\sqrt{-{1\over 3}-x^2+{2\over 3}\sqrt{1+3x^2}}\). The numerical value of the drag force for this case is \(F_{C_x}=13.1248\mu U\). The conjectured value obtained using equation (A) is \(F_{C_x}=11.489\mu U\). This yields an error of 12.5\%. 2) A hypocycloid-like profile given by the function \(y=\sqrt{-3x^2+\sqrt{1+8x^4}}\). Numerical drag force value: \(F_{C_x}=17.599\mu U\), conjectured value obtained using equation (A) \(F_{C_x}=19.296\mu U\), relative error 9.64\%. Note that in the case of a sphere or spheroidal body, the same computer code as used for obtaining the above numerical values yields results which are identical, in the range of the computer precision, with the exact solution for these body shapes. Based on the above analysis and numerical comparisons, the reviewer must conclude that the drag force and moment formulas as presented in the paper cannot have the expected general validity. Without this general validity of the proposed formulas, the contribution of the paper to fluid mechanics is of a rather limited extent. Remark: Acting as a referee for a European journal, this reviewer has already reported on the original manuscript to this paper. The above review text is a slightly adapted version of that report. New in comparison with the original manuscript are the two last paragraphs of section 2, a short but still not satisfactory extension of section 4, the whole section 8 ``Conclusion'', and the acknowledgement. Note also that the sentence before eq. (4.6) is incomplete and should be read: ``\dots{} we get the result derived by Usha and Nigam [6] and mentioned in [4], viz.,\dots''.
scientific article; zbMATH DE number 1394551