Stokes flow around deformed sphere - Semantic Scholar

STEADY STOKES FLOW AROUND DEFORMED SPHERE. CLASS OF OBLATE AXISYMMETRIC BODIES D. K. Srivastava1, R. R. Yadav2, and S. Yadav3 1

Department of Mathematics, B.S.N.V. Post Graduate College(University of Lucknow, Lucknow), Lucknow(Uttar Pradesh)-226001, India Email: [email protected] 2 Department of Mathematics, University of Lucknow, Lucknow(U.P.), India Email: [email protected] 3 Department of Mathematics, University of Lucknow, Lucknow(U.P.), India Email: [email protected] Received 30 May 2011; accepted 24 January 2012

ABSTRACT In this paper, the problem of steady Stokes flow past deformed sphere has been dealt in both situations when uniform stream is along the axis of symmetry(axial flow) and is perpendicular to the axis of symmetry(transverse flow). The most general form of deformed sphere,    governed by polar equation, r  a1  ε  d P cos θ  , (where dk is shape factor and Pk is k k k 0   Legendre function of first kind) has been considered for the study. The method used here is based on geometry of axially symmetric bodies developed by author [Datta and Srivastava, 1999] which, in particular, holds good for sphere and class of spheroidal bodies. The general expressions for axial and transverse Stokes drag for deformed sphere has been derived up to the order of o(2). The class of oblate axisymmetric bodies is considered for the validation and further numerical discussions. All the expressions of drag are updated up to the order of o(2), where  is deformation parameter. In particular, up to the order of o(), the numerical values of drag coefficients have been evaluated for the various values of deformation parameters() and aspect ratio (b/a)for a class of oblate axially symmetric bodies including flat circular disk and compared with some known values already exist in the literature. Keywords. Stokes flow, deformed sphere, oblate axially symmetric bodies. AMS Subject Classification. 76D07

1

INTRODUCTION

In physical and biological sciences, and in engineering, there is a wide range of problems of interest like sedimentation problem, lubrication processes etc. concerning the flow of a viscous fluid in which a solitary or a large number of bodies of microscopic scale are moving, either being carried about passively by the flow, such as solid particles in sedimentation, or moving actively as in the locomotion of micro-organisms. In the case of suspensions Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.

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D. K. Srivastava et al.

containing small particles, the presence of the particles will influence the bulk properties of the suspension, which is a subject of general interest in Rheology. In the motion of microorganisms, the propulsion velocity depends critically on their body shapes and modes of motion, as evidenced in the flagellar and cilliary movements and their variations. A common feature of these flow phenomena is that the motion of the small objects relative to the surrounding fluid has a small characteristic Reynolds number Re. Typical values of Re may range from order unity, for sand particles settling in water, for example, down to 102 to 106, for various micro-organisms. In this low range of Reynolds numbers, the inertia of the surrounding fluid becomes insignificant compared with viscous effects and is generally neglected and the Navier-Stokes equations of motion reduce to the Stokes equations as a first approximation. The zero Reynolds number flow is called Stokes flow. There are many examples, where oblate and prolate spheroidal geometry plays a vital role as far as the existence and smooth movement of the obstacle whether it is placed in liquid medium or in gaseous medium. Among all, mentioning few of them are worth full here to justify the present study like motion of oblate jellyfish(McHenry, 2007), 8 foot long scorpion found to live in the sea just like oblate spheroid(Jenks 2007), bending instabilities in homogeneous oblate spheroidal galaxy models(Jessop et al., 1997), oblate spheroidal shape helps in finding the shape of human transferrin molecule(Martel et al., 1980), discoid(oblate ellipsoid) shape in modeling the vestibular membranes in animals(Pender, 2009), oblate and prolate spheroidal cells are used for successful human cell permeabilization(Wall et al., 1999). All these motions are characterized by low Reynolds numbers and are described by the solution of the Stokes equations. Although the Stokes equations are linear, to obtain exact solutions to them for arbitrary body shapes or complicated flow conditions is still a formidable task. There are only relatively few problems in which it is possible to solve exactly the creeping motion equations for flow around a single isolated solid body. Stokes(1851) calculated the flow around a solid sphere undergoing uniform translation through a viscous fluid whilst Oberbeck(1876) solved the problem in which an ellipsoid translates through liquid at a constant speed in an arbitrary direction. Edwards(1892), applying the same technique, obtained the solution for the steady motion of a viscous fluid in which an ellipsoid is constrained to rotate about a principal axis. The motion of an ellipsoidal particle in a general linear flow of viscous fluid at low Reynolds number has been solved by Jeffery(1922), whose solution was also built up using ellipsoidal harmonics. The analysis described by Jeffery extended further by Taylor(1923). Goldstein(1929a) obtained a force on a solid body moving through viscous fluid. Goldstein(1929b) studied the problem of steady flow of viscous fluid past a fixed spherical obstacle at small Reynolds numbers. In this paper, Goldstein obtained the expression of drag on fixed sphere. Lamb(1932) gave the solution for the general ellipsoid. Lighthill(1952)studied the problem of squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds number. Aoi(1955) computed the drag experienced by a spheroid and obtained the general formula for the drag. He concluded that the pressure drag and frictional drag experienced by a spheroid contribute to the total drag in a definite ratio which is independent of the Reynolds number. The prolate and oblate spheroid has been treated by him as a detailed case study. Saffman(1956) observed the problem of small moving spheroidal particles in a viscous liquid and shown that the rate of orientation of a particle would then be independent of its size, and verified the prediction experimentally.

Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.

Steady Stokes Flow Around Deformed Sphere

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Stokes flow of an arbitrary body is of interest in biological phenomena and chemical engineering. In fact, the body with simple form such as sphere or ellipsoid is less encountered in practice. The body, which is presented in science and technology, often takes a complex arbitrary form. For example, under normal condition, the erythrocyte(red blood cell) is a biconcave disk in shape, which can easily change its form and present different contour in blood motion due to its deformability. In second half of twentieth century, a considerable progress has been made in treating the Stokes flow of an arbitrary body. Payne and Pell(960) used the methods of generalized axially symmetric potential theory to calculate the flow past a class of axisymmetric bodies, including the lens, ellipsoid of revolution, spindle, and two separated spheres. Breach(1961) tackled the problem of slow flow past ellipsoids of revolution. Brenner(1963) gave the general expression for Stokes resistance over an arbitrary particle. Brenner and Cox(1963) obtained the expression of resistance to a particle of arbitrary shape in translational motion at small Reynolds number. Tuck(1964, 1970) developed a method for a simple problem in potential theory and is applied to a problem in Stokes flow, yielding a procedure for obtaining the Stokes drag on a blunt slender body of arbitrary shape. Acrivos and Taylor(1964) presented the general solution of the creeping flow equations for the motion of an arbitrary particle in an unbounded fluid in terms of spherical coordinates. They derived the force exerted on the particle, and the particular case of a slightly but otherwise arbitrarily deformed sphere was treated by them. Brenner(1964a,b,c, 1966 a, b)further presented a theoretical calculation of the low Reynolds number resistance of a rigid, slightly deformed sphere to translational and rotational motions in an unbounded fluid. In which, he derived explicit expressions, to the first-order in the small parameter characterizations, which relates the Stokes resistance dyadic with the torque dyadic and the location of the centre of hydrodynamic stress of the particle to its geometry. Cox(1965) generalized the results given by Brenner and Cox(1963). Shi(1965) generalized the results of Proudman and Pearson(1957) and Kaplun and Lagerstrom(1957) for a sphere and a cylinder to study an ellipsoid of revolution of large aspect ratio with its axis of revolution perpendicular to the uniform flow at infinity. Matunobu(1966) used the Stokes equations for creeping flow to obtain steady flow of an incompressible, viscous fluid of infinite extent past a liquid drop which deviates slightly from a sphere and includes fully circulating flow. The expression of drag force experienced by the drop was derived by him. Chester et al.(1969) obtained the approximate expression of drag on sphere up to higher order beyond the first term given by Stokes(1851) in an incompressible viscous fluid at low Reynolds number. Batchelor(1970) has studied Stokes flow past a slender body of arbitrary(not necessarily circular) cross-section. Gautesen and Lin(1971) studied the problem of creeping flow past a radially deforming sphere within the framework of the Stokes approximation. They have shown that, independent of the deformation, the viscous drag equals twice the pressure drag. Lin and Gautesen(1972) studied the problem of creeping flow of an incompressible viscous fluid past a deforming sphere for all values of Reynolds number. They found the expression of drag up to the order of second power of Reynolds number Re. Takagi(1973) employed tangent-sphere coordinates to obtain the slow viscous flow due to the translational motion of torus without central opening along the axis of symmetry. The drag exerted on it is shown to be given by D = 5.6 a, where ‘a’ is the diameter of its generating circle. Naruse(1975) studied the low Reynolds number flow of an incompressible fluid past a body by solving the Navier-Stokes equations, on the basis of the method of matched asymptotic expansions. It is shown that, when the shape of the body is symmetric with respect to a point, the force on the body is determined to the order of Re squared times log Re, where Re denotes the Reynolds number.


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Stokes flow past slender, prolate and oblate bodies of revolution has been studied by several investigators. For example, Cox(1970), Keller and Rubinow(1976), Geer(1976), Johnson and Wu(1979, 1980) and Sellier(1999) and many others have presented general theories for creeping motion of long slender bodies and slender particles in a viscous fluid. Chwang and Wu(part 1, 1974; part 2, 1975; part 4, 1976), Chwang(part 3, 1975), Jhonson and Wu(part 5, 1979) and Huang and Chwang(part 6, 1986), in a series of work over low Reynolds number hydromechanics, have explored the fundamental singular solution of the Stokes equation to obtain solutions for several specific body shapes translating and rotating in a viscous fluid. Regarding the distribution range of the singularities, it was pointed out that some results for plane-symmetric bodies in a potential flow may also be valid in all types of Stokes flow. By providing the exact solution of the Stokes equation in an elegant, closed form, the singularity method proved to be a useful alternative to the more standard methods of solution. Unfortunately, one cannot, in a straight forward manner, generalize this approach to the systems of many particles or to particles in the vicinity of a wall. Usha and Nigam(1976) obtained the expression of Stokes drag over deformed sphere(up to the order of o(), where  is deformation parameter) by the help of integral equation simulation. Alawneh and Kanwal(1977) obtained closed form solutions for various boundary value problems in mathematical physics by considering suitable distributions of the Dirac delta function and its derivatives on lines and curves. Sthapit and Datta(1977) studied the problem of Stokes flow past a radially deforming sphere incorporating the effect of the rate of deformation through the non-linear inertia terms. They obtained the expression of drag for small values of time. In a series of paper, Gluckman et al.(1971, 1972) developed a new numerical method for treating the slow viscous motion past finite assembles of particles of arbitrary shape, termed the multipole representation technique. The approach is based on the theory that the solution for any object conforming to a natural coordinate system in a particle assemblage can be approximated by a truncated series of multi-lobular disturbances in which the accuracy of the representation is systematically improved by the addition of higher-order multipoles. For example, for a system of spherical particles the solution is found in terms of Legendre functions. Youngren and Acrivos(1975) used the boundary-element method to calculate hydrodynamic forces and torques acting on spheroidal and cylindrical particles in a uniform and simple shear flow. They expressed the solution of Stokes equations in the form of linear integral equations for the Stokeslet distribution over the particle surface. The required density of the Stokeslets, identical with the surface stress forces can be obtained numerically by reducing the integral equations to a system of linear algebraic equations. The technique has been successfully tested against the analytical solutions for spheroidal particles in a shear flow. Rallison and Acrivos(1978) applied a similar method in order to determine the deformation and condition of break-up in shear of a liquid drop suspended in another liquid of different viscosity. This very general method, which can be used in the case of bodies of arbitrary shape, so far has not been used to analyze flow fields around systems of particles. Harper(1983) derived few theorems for the hydrodynamic image of an axially symmetric slow viscous(Stokes) flow in a sphere which is impermeable and free of shear stress. He establishes a second theorem in a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with centre on the axis of symmetry, a flow past the rigid or shear-free inverse of the surface or sphere. Barshinger and Geer(1984) have tackled the problem of Stokes flow past a thin oblate body of revolution for the special case of an axially incident uniform flow. They have solved the Stokes equations asymptotically as the thinness ratio  of the body approaches zero. The total net force experienced by the body and the point wise stress on the body surface are computed Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.


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and discussed by them. Fischer et al.(1984) calculated the total force exerted on the isolated rigid obstacle in three dimensional space and in two dimensions placed in the stationary flow of an incompressible viscous fluid with the help of matched asymptotic expansions. Dabros(1985) attempted to find the hydrodynamic forces and velocities of arbitrary shaped particles, placed in an arbitrary flow field, particularly in the vicinity of the wall, using a singular point solution as the base function. Zhu and Wu(1985) used the method of continuous distribution of singularities to treat the Stokes flow of the arbitrary oblate axisymmetric body. Ramkissoon(1986) examined the Stokes flow past a fluid spheroid whose shape deviates slightly from that of a sphere. To the first order in the small parameter characterizing the deformation, an exact solution has been obtained. As an application, he discussed the drag experienced by a fluid oblate spheroid and deduced the some well known cases. Leith(1987) extended the Stokes law on a sphere to a nonspherical object by allocating the interaction of the fluid with the object into its interaction with two analogous spheres, one with the same projected area and one with the same surface area as the object. He used this approach to characterized dynamic shape factor for objects whose shape factors are already exists in the literature. He reported the shape factor for a sphere, cylinders, prisms, spheroids and double conicals. Weinheimer(1987) presented the comparative analysis between measured drag forces on cylinders and disks with those computed for Stokes flow around equivalent spheroids in both axial and transverse flow situations. With the help of Stokes drag expression, he calculated the terminal velocities of ice crystals falling in the atmosphere with major dimensions of up to a few tens of microns. Yuan and Wu(1987) obtained the analytic expressions in closed form for flow field by distributing continuously the image Sampsonlets with respect to the plane and by applying the constant density, the linear and the parabolic approximation. They calculated the drag factor of the prolate spheroid and the Cassini oval for different slender ratios and different distances between the body and the plane. Power and Miranda(1987) have successfully given the Fredholm integral equation representation of second kind for Stokes resistance problems i.e. when the velocity of particle is known, and the forces and moments are to be found. They represented the velocity as a double layer integral to which they added a Stokeslet and a Rotlet, both located at the centre of the body. Equating the representation to the given velocity resulted in a Fredholm integral equation of the second kind in the double layer density, and a numerical solution became possible after relating the Stokeslet and Rotlet strengths(force and moment) to the unknown double-layer density. Lawrence and Weinbaum(1988) presented the more general analysis of the unsteady Stokes equations for the axisymmetric flow past a spheroidal body to elucidate the behaviour of the force at arbitrary aspect ratio. These results are used to propose an approximate functional form for the force on an arbitrary body in unsteady motion at low Reynolds number. Karrila and Kim(1989) showed the completion of the double layer representation by Power and Miranda(1987) to be one of many possible completions. They suggested the same representation for the velocity as did by Power and Miranda(1987) and discuss various completions, suggesting one which is advantageous to an iterative numerical process for multiparticle systems. Both these completions are successful because, as observed by Power and Miranda(1987) and previously by Ladyzhenskaya(1963), the double layer representation alone is able to represent flow fields that correspond to the total force and total moment equal to zero. Authors like Hsu and Ganatos(1989) and Tran-Cong and PhanThien(1989) solved the Fredholm integral equation of first kind which is known to be illposed problem without evaluating the eigen function as formulated by Ladyzhenskaya(1963). A much more extensive review over numerical methods may be Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.

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found in the paper of Weinbaum and Ganatos(1990) and to the paper of Karrila and Kim(1989). Chester(1990) considered the motion of a body through a viscous fluid at low Reynolds number. He derived general formulae for the force and couple acting on a body of arbitrary shape and implemented over to reduce some special cases. Liron and Barta(1992) have presented a new singular boundary-integral equation of the second kind for the stresses on a rigid particle in motion in Stokes flow. They also produced the forces and moments on the particle with the help of generalized Faxen law. Chang et al.(1992) have studied axisymmetric viscous flow around ellipsoids of circular section in detail by the method of matched asymptotic expansion and a deterministic hybrid vortex method. Lowenberg(1993) computed the Stokes resistance, added mass, and Basset force numerically for finite-length, circular cross-section cylinders using a boundary integral formulation. In this study, he found analytical formulas for the Stokes force, added mass, Basset force of spheroids which contrasted with the numerical results for cylinders of the aspect ratio in the range: 0.01  a/b  100. He concluded that for some of these parameters, significant differences persist for disk and rod shaped particles. Palaniappan(1994) investigated the problem of slow streaming flow of a viscous incompressible fluid past a spheroid which departs but little in shape from a sphere with mixed slip-stick boundary conditions. He obtained the explicit expression for the stream function to the first order in the small parameter characterizing the deformation. For validation, he considered the oblate spheroid and evaluated the drag on this non-spherical body. He concluded that the drag in the present case is less than that of the Stokes resistance for a slightly oblate spheroid. Tanzosh and Stone(1996) have developed a concise analytic method to investigate the arbitrary motion of a circular disk through unbounded fluid satisfying Stokes equations. Four elementary motions are considered by them: broad side translation, edge wise translation, in-plane rotation and out-of-plane rotation of a disk. They reduced the Stokes equations to a set of dual integral expressions relating to the velocity and traction in the plane of the disk. They solved the dual integral equations exactly for each motion and lead, in turn, to closed-form analytical expressions for the velocity and pressure fields. Ramkissoon(1997) investigated the creeping axi-symmetric slip flow past an approximate spheroid whose shape deviates from that of a sphere. He obtained exact solution to the first order in the small parameter characterizing the deformation. As an application, the case of flow past an oblate spheroid has been considered and the drag experienced by it is evaluated and some special well-known cases are deduced as a validation. Datta and Srivastava(1999) developed a new approach to evaluate the Stokes drag force in a simple way on a axially symmetric body with some geometrical constraints placed in axial flow and transverse flow under the no-slip boundary conditions. The results of drag on both the flow situations were successfully tested not only for sphere, prolate and oblate spheroid but also for other bodies like deformed sphere, cycloidal and egg-shaped bodies of revolution with acceptable limit of error. This method has been described in the section 2 as the same is exploited here to study the problem of Stokes flow around deformed sphere. Alassar and Badr(1999) have solved the problem of uniform steady viscous flow over an oblate spheroid in the low-Reynolds-number range 0.1  Re  1.0 . They have written the full Navier-Stokes equations in the stream function-vorticity form and solved numerically by means of the series truncation method.They considered spheroids having axis ratio ranging from 0.245 to 0.905. They obtained the drag coefficients for oblate spheroid and compared with previous analytical formulae which were based on the solution of the linearized Stokes equations. Datta and Srivastava(2002) obtained the optimum drag profile in axi-symmetric Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.


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Stokes flow under the restrictions of constant volume and constant cross section area by exploiting DS conjecture given by Datta and Srivastava(1999). Zhuang, et al.(2002) proposed a new three-dimensional fundamental solution to the Stokes flow by transforming the solid harmonic functions in Lamb’s solution into expression in terms of oblate spheroidal coordinates. They used the oblate spheroid as model of a variety of particle shapes between a circular disks and a sphere. The effect of various geometric factors on the forces and torques exerted on two oblate spheroids were systematically studied for the first time by them using the proposed fundamental solution. Benard et al.(2004) have presented an analytic solution for the motion of a slightly deformed sphere in creeping flows with the assumption of slip on the particle surface. They obtained the explicit expressions for the hydrodynamic force and torque exerted by the fluid on the deformed sphere with the help of perturbation method used previously by Brenner(1964 a, b, c) and Lamb(1932). Palaniappan and Ramkissoon(2005) provided a complete survey of drag formula over axi-symmetric particle in Stokes flow. Senchenko and Keh(2006) attempted first to obtain analytical approximations for the resistance relations for slightly deformed slip sphere in an unbounded Stokes flow. To the first order in the small parameter characterization of the deformation, they derive expressions for the hydrodynamic force and torque exerted on the particle. They checked the obtained results of axial and transverse drag for the spheroid. All these expressions of drag have been used by us for the validation purposes in the situation of no-slip boundary condition. Srivastava(2007) obtained the optimum volume profile in axi-symmetric Stokes flow by exploiting the DS conjecture given by Datta and Srivastava(1999). Keh and Chang(2008) presented a combined analytical and numerical study of the slip Stokes flow caused by a rigid spheroidal particle translating along its axis of revolution in a viscous fluid. The drag force exerted on the spheroidal particle by the fluid is evaluated by them with good convergence behaviour for various values of the slip parameter and aspect ratio of the particle. They proved that, for a spheroid with a fixed aspect ratio, its drag force is monotonically decreasing function of the slip coefficient of the particle. Chang and Keh(2009) analyzed the steady translation and rotation of a rigid, slightly deformed colloidal sphere in arbitrary directions in a slip viscous fluid in the limit of small Reynolds number. They solved Stokes equations asymptotically using a method of perturbed expansions. They presented the expression of drag and torque for spheroid up to the order of o(2), where  is the deformation parameter. Bowen and Masliyah(2009) obtained an approximate solution to the equation of motion governing Stokes flow past a number of isolated closed bodies of revolution by the least square fitting of a truncated series expression for the stream function to known boundary conditions. They found reasonably accurate (  5% ) estimate for the Stokes resistance on body shapes, such as cylinders and cones, for which the solutions are exceedingly difficult. They applied the computed drag values in determining the limitations of the various empirical expressions used to predict the drag resistance of these geometrically simple bodies. Pratibha and Jeffrey(2010) calculated the mobility functions for two unequal spheres at low Reynolds number by using the method of twin multipole expansion. Datta and Singhal(2011) studied the uniform viscous flow with slip boundary condition under Stokes approximation at low Reynolds number past a pervious sphere with a source at its centre by using the method of matched asymptotic expansions. For the detailed study over the concerned topic, reader is advised to go through the books of Lamb(1932), Batchelor(1967), Ladyzhenskaya(1963), Happel and Brenner(1964), Langlois(1964), Kim and Karrila(1991), Pozrikidis(1992, 1997), Kohr and Pop(2004).



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In most of these investigations the main result of physical interest is the drag formula for the particular obstacle under consideration. In this paper, which is the first part of the analysis, we targeted to study the salient features of class of oblate axi-symmetric bodies as a case of deformed sphere by applying the method of SD conjecture proposed by Datta and Srivastava(1999) described in the section 2.

2

BODY GEOMETRY AND METHOD

Axial flow Let us consider the axially symmetric body of characteristic length L placed along its axis(x-axis, say) in a uniform stream U of viscous fluid of density ρ1 and kinematic viscosity . When Reynolds number UL/ is small, the steady motion is governed by Stokes equations(Happel and Brenner, 1964),

1 0   grad p  ν  2 u , divu= 0 ,  ρ1 

(2.1)

subject to the no-slip boundary condition.

y

dy

dF/2

P y

A'

U A

x

dF/2

Fig. 1(a)

Elemental force system on the sphere



25

U

(==/2)B

P(x, y) or (r, ) b r = =



O

C

R

U

y 



M

A

Q

N ==0

am=xmax

D Fig. 1(b) Geometry of axially symmetric body For the case of a sphere of radius R, the solution is easily obtained and on evaluating the stress, the drag force F comes out as (Happel and Brenner1964) π

F

9 π μ U  R sin 3 α dα  λR , 2 0

(2.2)

where  = 6U.

(2.3)

This shows that the drag force increases linearly with the radius of the sphere. In other words, the difference between drag force on two spheres of radii y and y + dy is given by dF =  dy.

(2.4)

A sphere of radius ‘b’ is obtained by rotating the curve x = b cos t, y = b sin t(0  t ) about b

the x-axis and the force F = b is obtained from (2.4) as

 λ dy exhibiting that the force 0

system dF may be considered as lying in the xy plane. The element force dF may be decomposed into two parts (1/2) dF, each acting over the upper half and lower half; (1/2) dF on the upper half acts at a height y(say) above the x-axis. The total force F/2 on the upper half, may be considered as made up of these differential forces dF/2 acting over elements corresponding to a system of half spheres of radii increasing from 0 to b and spread over from A to A(figure 1(a)). The moment of this force system(taken to be in the xy plane) about O, provides



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1 1 1  F h   M   y dF  λ  y dy  λb 2 , 20 2 0 4 2 b

b

or

F

1 λb 2 , 2 h

(2.5)

Where ‘h’ is the height of centroid of the force system. In the case of a sphere of radius ‘b’ we have F = b, and so we get from (2.5), h = b/2, as it should be. Next, we can express (2.2) also as π

 df ,

(2.6)

3 λR sin 3 α dα , 4

(2.7)

F

α 0

where df 

is the elemental force on a circular ring element at P(figure 1(a)) (Happel and Brenner 1964, eq. (4-17.23), p. 122). For the purpose of calculating F/2, the force on upper half, (1/2) df may be taken to be acting at height (say), above x-axis, given by  df 

h π



 0

 η 2  df 2 3  η λR sin 3 α dα  4 



π

 0

3 λR sin 3 α dα 4

π



3 η sin 3 α dα . 4 0

Taking = R/2, the result is seen to correspond to the value h = b/2 confirmed earlier. Thus, we have π

3 h   R sin 3 α dα . 80

(2.8)

It is proposed that the formula (2.8) holds good for an axially symmetric body also, when R is interpreted as the normal distance PM between the point P on the body and the point of intersection M of the normal at P with axis of symmetry and  as its slope(figure 1(b)). On Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.


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inserting the value of h from (2.8) in (2.5), we finally obtained the expression of drag on axially symmetric body in axial flow F 

1 λb 2 4  2 h  3

λb  2

π

 R sin α dα

, where λ  6 π μ U  ,

(2.9)

3

0

where the suffix ‘’ has been introduced to assert that the force is in the axial direction. While using (2.9), it should be kept in mind that ‘b’ denotes intercept between the meridian curve and the axis of the normal perpendicular to the axis i.e., b = R at  = /2. Sometimes it will be convenient to work in Cartesian co-ordinates. Therefore, referring to the figure 1(b), for the profile geometry, we have  dy  y = R Sin, tan =     dx 

1  

dx   x . dy

(2.10)

Using above transformation, we may express (2.3) as

h  

3 a  4 0

yy



1  y2



2

dx , (2.11)

where 2am represents the axial length of the body and dashes represents derivatives with respect to x. In the sequel, it will be found simpler to work with y as the independent variable. Thus, hx assumes the form

h  

3 b yx  2 x  dy ,  2 4 0 2 1  x





(2.12)

where dashes represents derivatives with respect to y. Transverse flow We set up a polar coordinate system (R, , ) with  as the polar angle with y-axis and  the azimuthal angle in zx plane. Since y-axis is not the axis of symmetry for the body we can not make use of circular ring elemental force (3/4) R sin3 d corresponding to (2.7). But we can easily write down the elemental force on the element R2 sin d d as δf 

3λ R sin 3β dβ dγ . 8π

Transforming the above to the polar coordinate (R, , ) with the x-axis as the polar axis, we have Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.


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δf 





3λ R 1  sin 3 α cos2 φ sin α dα dφ , 8π

As the force on the element R2 sin  d d. On integrating over  from 0 to 2, we get df  





3λ R 2  sin 3 α sin α dα , 8

(2.13)

where the suffix ‘’ has been placed to designate the force due to the external flow along the y-axis, the transverse direction. Integrating df  over the surface of the sphere, we get π





3λ F  2 sin α  sin 3 α dα  λR ,  8 0

(2.14)

agreeing with the correct value. This suggests we can take the force dfas given by (2.13) as the element force on the circular ring element at P. Although the force F is along the y direction, we have reduced it to elemental forces on a system of spheres centered on the xaxis. Since F and df themselves are scalar quantities, on comparing (2.13) and (2.7), we can use the analysis as in the axial flow case with ‘h’ replaced by π

h 





3 R 2 sin α  sin 3 α dα .  16 0

(2.15)

Thus, we get from (2.5)

F 

1 λb 2 , where λ  6 π μ U  . 2 h

(2.16)

we have taken up the class of those axially symmetric bodies which possesses continuously turning tangent, placed in a uniform stream U along the axis of symmetry (which is x-axis), as well as constant radius ‘b’ of maximum circular cross-section at the middle of the body. In the same manner as we did in axial flow, equation (2.8) may also be written in Cartesian form as (in both cases having x and y treated as independent)





(2.17)



(2.18)

2 3 a yy  1  2y  h    dx , 2 2 8 0 1  y 





and



2 3 b yx  2  x  h    dy , 2 2 8 0 1  x  





In (2.17) and (2.18), the dashes represents derivative with respect to x and y respectively. Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.


29

This axi-symmetric body is obtained by the revolution of meridional plane curve (depicted in figure 1(b)) about axis of symmetry which obeys the following limitations: i. Tangents at the points A, on the x-axis, must be vertical, ii. Tangents at the points B, on the y-axis, must be horizontal, iii.The semi-transverse axis length ‘b’ must be fixed. The point P on the curve may be represented by the Cartesian coordinates (x,y) or polar coordinates (r, ) respectively, PN and PM are the length of tangent and normal at the point P. The symbol R stands for the intercepting length of normal between the point on the curve and point on axis of symmetry and symbol  is the slope of normal PM which can be vary from 0 to . The proposed drag formulae is, of course, subject to restrictions on the geometry of the meridional body profile y(x) of continuously turning tangent implying that y(x) is continuous together with y(x)  0, thereby avoiding corners or sharp edges or other kind of nodes and straight line portions, y = ax + b, x1  x  x2. If such type of cases arises in the body, the contribution of drag corresponding to those parts will be zero and true drag value experienced by the body may not be achieved. Also, it should be noted here that the method holds good for convex axially symmetric bodies which possesses fore-aft symmetry about the equatorial axis perpendicular to the axis of symmetry(polar axis). Apart from this argument, It is interesting to note here that the proposed conjecture is applicable also to those axi-symmetric bodies which fulfills the condition of continuously turning tangent but does not possesses fore-aft symmetry like egg shaped body(Datta and Srivastava 1999). This conjecture is much simpler to evaluate the numerical values of drag than other existing numerical methods like Boundary Element Method(BEM), Finite Element Analysis(FEA) etc. as it can be applied to a large set of convex axi-symmetric bodies possessing fore-aft symmetry about maximal radius situated in the middle of the body for which analytical solution is not available or impossible to evaluate. Since both axial and transverse flows have been considered in a free stream results of the force at an oblique angle of attack may be resolved into its components to get the required result. The present analysis can be extended to generate a drag formula for axi- symmetric bodies for more complex flows like paraboloidal flow for which free stream may be represented by average velocity(Chwang and Wu, part 2 1974). Authors are working in this direction and also searching the avenues of this analysis for non-linear Stokes flow. The proposed analysis can be extended to calculate the couple on a body rotating about its axis of symmetry and about axis perpendicular to the axis of symmetry. Authors are working over it and the corresponding study will appear soon.

3

FORMULATION OF THE PROBLEM

Consider the axially symmetric body defined by    r  a 1  ε  d k Pk μ  , μ  cos θ , k 0  


(3.1)


30

where (r, ) are spherical polar coordinates,  is the small deformation p1arameter, dk’s are the design or shape factors and Pk() are Legendre function of first kind. For a small values of parameter , equation (3.1) represents the deformed sphere. Now, our prime task is to calculate the expressions of drag on the deformed sphere and other physical features related to the body in both axial and transverse flow situations corrected up to the order of O(2) with the help of proposed conjecture stated in the section 2.

4

SOLUTION

According to the body geometry(fig. 1) and with the aid of calculus and trigonometry, we have following values  dr  aε  d k Pk ' μ  sin θ , dθ k 0

(4.1)

 1  ε  d P μ  dθ k 0 k k tan φ  r  ,  dr  ε  d P ' μ  sin θ k 0 k k

(4.2)

2

1    sin φ  1  ε 2   d k Pk ' μ  sin 2 θ  O ε 3 , 2  k 0 

 

(4.3)

      cos φ  ε  d k Pk ' μ  sin θ  ε 2   d k Pk μ   d k Pk ' μ  sin θ  O ε 3 . k 0  k 0  k 0 

 

(4.4)

From fig. 1, we can write ψ

π π α,ψ θφ  α  θφ, 2 2

so, we have

π  cos  α   cosθ  φ  , therefore we can further write 2  

sin α  sin θ  ε  d k Pk ' μ  sinθ cosθ k 0

2    1       ε   d k Pk μ   d k Pk ' μ  sinθ cosθ    d k Pk ' μ  sin 3θ   O ε 3 , (4.5) 2  k 0  k 0   k 0   π  also, by using, sin  α   sinθ  φ  , we have 2  2


 


31

cos α  sin θ cosφ  cos θ sin φ 

 cosθ  ε  d k Pk ' 2 μ  sin 2 θ k 0

2   1     2         ε   d k Pk μ   d k Pk ' μ sin θ    d k Pk ' μ  sin 2 θ cosθ   O ε 3 , 2  k 0  k 0   k 0  

 



2

(4.6)

from triangle OPM in fig. 1, sine rule provides R r r sinθ ,  R  sinθ sinα sinα

(4.7)

   b  r θ  π  a 1  ε  d k Pk μ  , μ  cosθ , 2 k 0   θπ 2

(4.8)

by using the properties of Legendre function, we have P2k 1 0   0 i.e. P1 0   P3 0   P5 0   ......  0 , for k  0, k   1 2k  P2k 0   2 k , for k  0, 1, 2, 3, 4,...... 2 2 k

(4.9)

1 3 5 35   etc.  P0 0   1, P2 0    , P4 0   , P6 0    , P8 0   2 8 16 128  

Now, by using (4.8 and (4.9), cross-section radius ‘b’ may be expressed in series expansion form as b  a1  εd 0 P0 0  d 2 P2 0  d 4 P4 0  d 6 P6 0  d 8 P8 0  ......  ,  1 3 5 35   b  a 1  ε d 0  d 2  d 4  d 6  d 8  ......  . 2 8 16 128   

(4.10)

General expression of drag in axial flow For axial flow configuration, the expression of Stokes drag(2.2) required the value of hx 3 π 3 π 3 R sinα sin α  sin α dα  dθ , R sin α dα    α  0 θ  0 8 8 dθ   where h  

 



sin α  sinθ  ε  d k Pk ' μ  sinθ cosθ  O ε 2 ,

(4.11)

(4.12)

k 0



 

cos α  cos θ  ε  d k Pk ' μ  sin 2 θ  O ε 2 , k 0


(4.13)


32

on differentiating with respect to ‘’, we get sin α



  

 dα  sin θ  ε  d k sin θ 2Pk ' μ  cos θ  Pk ' ' μ  sin 2 θ  O ε 2 . dθ k 0

(4.14)

The value of h may be obtained from equation(4.11) upon utilization of (4.7), (4.12) and (4.14) together with the help of following relations of Legendre’s Polynomial[Abramowitz and Stegun, 1972]

1  μ P ' μ   k  1μP 2

k

and

k

 Pk 1  , (Beltrami Result)

1  μ P ' ' μ   k  2μP 2

k 1

k 1

 Pk  2  ,

(4.15) (4.16)

it is to be noted here that the dashes related to Legendre function indicates the derivative with respect to variable  = cos ,

h 

a 4   1  ε d 0  d 2   .  2 5  

(4.17)

It is interesting to note that only design factors up to k=2 occurs in the expression due to the fact of orthogonality feature of the Legendre polynomial which plays the role and the other terms in h  vanishes. The other important part of the axial Stokes drag is b2, which is the square of cross-section radius b described in (4.10), given by

  1 3   1  2ε d 0  2 d 2  8 d 4 .......      2 2 . b a  2  2 1 2 9 2  1 3 3       ε  d 0  d 2  d 4 ......   2d 0   d 2  d 4 .....   2d 2  d 4 ......   4 64 8   2  8    

(4.18)

Now, finally, by utilizing the equations (4.17) and (4.18), the expression for axial Stokes drag[given in (2.9)] comes out to be   1 3   1  ε d 0  5 d 2  8 d 4 .....         3   21  2 89 2 9 2 F   6 π μ U   a   .(4.19)  2d 0  100 d 2  64 d 4  ....   2d 0   10 d 2  8 d 4  .....        2 3    ε    O ε     3    2d  d  ......   .....    2 4     8   



33

This general expression of drag on deformed sphere immediately reduces to classical Stokes drag on sphere of radius ‘a’ i.e. 6Ua on taking the limit as  0. It is interesting to note here that the expression has been written up to the order of O(2) and even indexed shape factors are appearing in the coefficient of deformation parameter  which is in contrary to that given by Happel and Brenner(1964, page no. 208) that the term corresponding to k=0 and k=2 only contributes to the expression of drag. Beyond this statement, our result seems to be more general for deformed sphere.

General expression of drag in transverse flow In order to calculate the Stokes drag over deformed sphere placed in transverse flow, we have to calculate h given by (2.15) as

h 

a 11 3 3   1  ε d 0  d 2  d 4  d 6  .......    O ε 2 , (4.20)  2 10 2 2  

 

Substituting h  in (2.16), the expression of drag comes out to be   1 9   1  ε d 0  10 d 2  4 d 4  .....       . F  6 π μ U  a  2  9 2 225 2   1 3 51      d 4  .....   2d 0   d 2  d 4  ...   2d 2  d 4  ...   O ε 3  ε  d 2  64 8   2   40    25  (4.21)

 

Which on taking limit as  0, matches with the well known Stokes drag i.e. 6Ua. It is interesting to note here again that the expression has been written up to the order of O(2) and even indexed shape factors are appearing in the coefficient of deformation parameter  has never appeared in the literature and seems to be new. It is to be noted here that the expressions of drag (4.19) and (4.21) are valid only for the class of axisymmetric deformed sphere governed by polar equation (3.1) in which second order terms in deformation parameter are absent.



34

5

OBLATE SPHEROID 

(Transverse Flow) U

Equatorial Axis

a

Polar Axis a(1-)

O

U(axial flow) z

Fig. 2. Oblate spheroid in meridional two-dimensional plane (z, ) We consider the oblate spheroid, as it belongs to the class of axi-symmetric deformed sphere whose Cartesian equation is

x 2  y2 z2  1 , 2 a2 a 2 1  ε 

(5.1)

Where equatorial radius is ‘a’ and polar radius is a(1 ), in which deformation parameter is positive and sufficiently small that squares and higher powers of it may be neglected. Its polar equation, [by using z = r cos ,  = r sin ,  = (x2 + y2)1/2, up to the order of o()] is r = a (1   cos2 ) ,

(5.2)

it can be written in linear combination of Legendre functions of first kind(Happel and Brenner(1964), Senchenko and Keh(2009))

 2 1  r  a 1  ε  P0 μ   P2 μ  , μ  cosθ , 3 3  

(5.3)

On comparing this equation with the polar equation of deformed sphere (3.1), the appropriate design factors for this oblate spheroid are

with

1 2 d 0   , d1  0, d 2   , d k  0 for k  3 , 3 3

(5.4)

3μ 2  1 P0 μ   1 , P1 μ   μ and P2 μ   , μ  cosθ . 2

(5.5)



35

Also, this polar equation (5.3) of oblate spheroid may be written in terms of Gegenbauer functions of first kind by using the following well known relation

Ik μ  

r  a1  ε  2ε I2 μ  , = cos

as or, if we put we have

Pk- 2 μ   Pk μ  , k  2 , μ  cosθ , 2k - 1

(5.6) (5.7)

d=a(1),

r  d1  2ε I2 μ  .

(5.8)

Now, we find the expression Stokes drag on this axi-symmetric oblate body placed in both axial flow(in which uniform stream is parallel to polar axis or axis of symmetry) and transverse flow(in which uniform stream is perpendicular to polar axis or axis of symmetry) situations with the aid of general expressions of drag (4.19) and (4.21). Axial flow With the aid of equations (5.4) and (5.5), the expression of axial Stokes drag can be written up to the order of O() with the help of (4.19) and comes out to be

 

 ε  F   6 π μ U   a 1   O ε 2  ,  5 

(5.9)

which matches with that already exist in Happel and Brenner(1964, page 215, eq. 5-9.53), Usha and Nigam(1976, page 13, eq. 20) and Sechencho and Keh(2006, page 088103, eq.38). In the paper of Sechenscho and Keh, the validation took place after removing the slip effect on the vicinity of oblate body i.e., 1 0 or  , where  is slip parameter, which is the clear cut case of no-slip boundary condition as it should be. This force is less than would be exerted on a sphere of radius equal to the equatorial radius ‘a’ of the oblate spheroid. It is due to the fact that the surface area and volume of the oblate spheroid are less than that of the sphere. The relative smallness of this resistance is not surprising since the polar regions of the sphere contributes least to its resistance; hence, their removal does not have a profound effect on its resistance. A more appropriate comparison might be made between equation (5.9) and the resistance of a sphere of equal volume or surface area. The volume of the spheroid (5.1) is 4 2 4 π a d  π a 3 1  ε  . 3 3

(5.10)

Hence, a sphere of equal volume would have a radius of a(1  /3) and its resistance would be, due to well known Stokes law(Happel and Brenner(1964), Stokes(1851))  ε F  6 π μU  a 1   .  3 Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.

(5.11)


36

A sphere of equivalent volume therefore has a smaller resistance than the oblate spheroid. Similarly, since the surface area of the oblate spheroid is 4a2(1  2/3) , a sphere of equal surface area has a radius of a(1  /3). Therefore, the low resistance F [given in (5.11)] of the sphere holds also on the basis of equal surface area. It has been mentioned in the introduction that the conjectured expression for axial and transverse Stokes drag on axially symmetric bodies proposed in (Datta and Srivastava(1999)) holds good for spheroid and provide the closed form solution. For the convenience, we write both expressions of drag on oblate spheroid









1

F   8 π μ U   a e 3 e 1  e 2  1  2e 2 sin 1 e ,







F  16 π μ U  a e 3  e 1  e 2  1  2e 2 sin 1 e

(5.12)



1

,

(5.13)

where ‘e’ is the eccentricity of the spheroid. The expression of Stokes drag in axial flow (5.9) on oblate spheroid may also be deduced from the exact expression axial Stokes drag(5.12) on a oblate spheroid for flow parallel to its axis of revolution by utilizing the appropriate relation between eccentricity ‘e’ and deformation parameter ‘’. 2

b For the considered oblate spheroid, eccentricity is e  1    , with b  a 1  ε  , which a further gives us the relation between eccentricity ‘e’ and deformation parameter ‘’ as

e  1  1  ε   2ε  ε 2  2ε (leaving the square term) . 2

(5.14)

On re-writing the expression of drag F as   e3 F   6 π μ U   a   3 e 1  e 2  1  2e 2 sin 1 e  4







    6 π μ U a K ,  



(5.15)

where K is called correction to Stokes law due to Happel and Brenner(1964, page 148, eq. 4-26.38). Using the expression of eccentricity (5.14) in terms of deformation parameter , up to the second power, this correction factor K can be expressed in the powers of deformation parameter  as 2 2  1  K  1  ε  ε .........  . 175  5 

(5.16)

Now, the revised expression of drag F, in powers of deformation parameter , would be of the form



2 2  1  F  6 π μ U  a 1  ε  ε .........  . 175  5 

37

(5.17)

This expression is in good agreement to that presented by Chang and Keh(2009, page 205, eq. 35(a)), but in Happel and Brenner(1964, page 145, eq. 4-25.29), the third term -332/175 is in error which seems to be algebraic error and should be modified to 22/175 with same first two terms 1  /5. Transverse flow By using the general expression (4.21), with the aid of equations (5.4) and (5.5), the expression for drag on oblate spheroid placed in transverse flow would be obtained as(up to the order of O())

 

 2  F  6 π μ U  a 1  ε  O ε 2  .  5 

(5.18)

Also, the complete expression of Stokes drag on oblate spheroid in transverse flow by conjectured method(Datta and Srivastava, 1999) is given by (5.13)







F  16 π μ U  a e 3  e 1  e 2  1  2e 2 sin 1 e



1

,

where ‘e’ is the eccentricity of oblate spheroid. The expression of drag (5.18) on oblate spheroid (5.1) may also be deduced from this exact expression (5.13) for the transverse flow or flow perpendicular to its axis of revolution with correctness up to the order of O(2). On rewriting the above exact expression of drag (5.13) as

F 

16 π μ U  a e 3

 e 1  e

2





 1  2e 2 sin 1 e



  3   e  6 π μ U a ,  3  e 1  e 2  1  2e 2 sin 1 e   8 

(5.19)

 6 π μ U a K ,

(5.20)









where K is called correction to Stokes law due to Happel and Brenner(1964, page 148, eq. 4-26.38). Using the expression of eccentricity (5.14) in terms of deformation parameter , up to the second power, this correction factor K can be expressed after some algebraic manipulations in the powers of deformation parameter  as 9 2  2  F  6 π μ U  a 1  ε  ε .......  . 350  5  Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.

(5.21)


38

Up to the order of O(), this gives us the drag (5.18) on oblate spheroid. Also, this expression of drag in transverse flow situation matches with Chang and Keh(2009, page 205, eq. (35b)). It is to be noted here that the third terms of the eq.(5.17) and eq.(5.21) are in disagreement with those obtained by the general expressions of drag presented in eq. (4.19) and (4.21). In the paper of Chang and Keh(2009), the polar equation of deformed sphere was considered to be





r  a 1  ε f 1 θ   ε 2 f 2 θ  ,

(5.22)

while in this paper, the polar equation of deformed sphere is considered to be    r  a 1  ε  d k Pk μ  , μ  cos θ , k 0  

In which the terms of second order in  is absent. This might be the main reason for the disagreement between Chang and Keh(2009) and those presented in this paper at the level of second order in . Apart from this fact, we can achieve the same expressions of drag on oblate spheroid given in equations (5.17) and (5.21) by applying Datta and Srivastava(1999) conjecture independently over deformed sphere whose polar equation is (5.22) with proper choice of f1() and f2(). The expressions of axial drag (5.9) and transverse drag (5.18) are presumably valid only for small values of , they are in fact, surprisingly accurate for even large departures from spheroid shape. We can explain it by considering flat circular disk as a special case of oblate spheroid.

6

FLAT CIRCULAR DISK

The eccentricity ‘e’ of an oblate spheroid is

e  1

d2 , d  a 1  ε  a2



e  1  1  ε 



ε  1  1  e2 .

2

(6.1)

The eccentricity of a flat circular disk of radius ‘a’ is unity, whence by (5.22), the value of deformation parameter ‘’ is also unity. For this case, the expressions for axial drag(5.9) and transverse drag(5.18) due to Stokes are F  6 π μ U a

 0.8 broadside  on case of flat circular disk  ,


(6.2)


and

F  0.6 edge  on case of flat circular disk  . 6 π μ U a

39

(6.3)

These results are in agreement with those given in Happel and Brenner(1964, page 215) which are in error under six percent for any oblate spheroid on corresponding with the exact results given by Lamb(1932) F  6 π μ U a

and

 0.8492 broadside  on case of flat circular disk  ,

F  0.5658 edge  on case of flat circular disk  , 6 π μ U a

(6.4)

(6.5)

these errors decrease rapidly with decreasing eccentricity. For eccentricity e = 0.8, b d b 2   1  0.8  0.6 , ε  1  1  e 2  1   0.4 , a a a

The discrepancy is less than 0.5 percent. Which clearly indicates that the general expressions(axial and transverse both) (4.19) and (4.21) for drag on deformed sphere provide good approximations up to the order of O() for a class of axially symmetric bodies like oblate and prolate spheroids. 7

NUMERICAL DISCUSSION

First of all we find numerical values of parameters like eccentricity ‘e’, aspect ratio b/a, axial F  F Stokes drag coefficient C Fz  , transverse Stokes drag coefficient C Fρ  6 π μ U  a 6 π μ U a for different values of deformation parameter ‘’ so that numerical discussion for a class of oblate axi-symmetric bodies (generated from deformed sphere between sphere and flat CF circular disk) could be compiled. These variations together with drag ratio   z are C Fρ displayed in table 1 for the first order and in table 2 for the second order in deformation parameter ‘’. The corresponding error estimates between the first and second order values of drags are presented in table 3.



40

Table 1 [Approximate numerical values of drag up to first order of ] 

b/a=1-

e =(1-b2/a2)1/2

C Fz 

F  6 π μ U a

 1

ε 5

F 6 π μ U a 2  1 ε 5

C Fρ 



C Fz C Fρ

0.0 0.1 0.2 0.3 0.4 0.5

1.0 0.9 0.8 0.7 0.6 0.5

0.0000 0.4358 0.6000 0.7141 0.8000 0.8660

1.0000 0.9800 0.9600 0.9400 0.9200 0.9000

1.0000 0.9600 0.9200 0.8800 0.8400 0.8000

1.0000 1.0208 1.0434 1.0682 1.0952 1.1250

0.6 0.7 0.8 0.9 1.0

0.4 0.3 0.2 0.1 0.0

0.9165 0.9539 0.9792 0.9949 1.0000

0.8800 0.8600 0.8400 0.8200 0.8000

0.7600 0.7200 0.6800 0.6400 0.6000

1.1578 1.1944 1.2352 1.2812 1.3333

Axisymmetric bodies

Sphere

Oblate body zone

Flat circular disk

Table 2 [Approximate numerical values of drag up to second order of ] 

b/a=1-

e =(1-b2/a2)1/2

C Fz   1

F  6 π μ U a ε 2 2  ε 5 175

F 6 π μ U a 2 9 2  1 ε  ε 5 350

C Fρ 



C Fz C Fρ

0.0 0.1 0.2 0.3 0.4 0.5

1.0 0.9 0.8 0.7 0.6 0.5

0.0000 0.4358 0.6000 0.7141 0.8000 0.8660

1.0000 0.9801 0.9604 0.9410 0.9218 0.9028

1.0000 0.9597 0.9189 0.8776 0.8358 0.7935

1.0000 1.0208 1.0434 1.0682 1.0952 1.1250

0.6 0.7 0.8 0.9 1.0

0.4 0.3 0.2 0.1 0.0

0.9165 0.9539 0.9792 0.9949 1.0000

0.8841 0.8656 0.8473 0.8292 0.8114

0.7507 0.7074 0.6635 0.6191 0.5742

1.1578 1.1944 1.2352 1.2812 1.3333


Axisymmetric bodies

Sphere

Oblate body zone

Flat circular disk


41

Table 3 [Error estimates between first and second order drag values] Deformation parameter

C Fz



O()

O(2)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0000 0.9800 0.9600 0.9400 0.9200 0.9000 0.8800 0.8600 0.8400 0.8200 0.8000

1.0000 0.9801 0.9604 0.9410 0.9218 0.9028 0.8841 0.8656 0.8473 0.8292 0.8114

Error with Error % 0.0000(0.00%) 0.0001(0.01%) 0.0004(0.04%) 0.0010(0.10%) 0.0018(0.18%) 0.0028(0.28%) 0.0041(0.41%) 0.0056(0.56%) 0.0073(0.73%) 0.0092(0.92%) 0.0114(1.14%) 

C Fρ O()

O(2)

1.0000 0.9600 0.9200 0.8800 0.8400 0.8000 0.7600 0.7200 0.6800 0.6400 0.6000

1.0000 0.9597 0.9189 0.8776 0.8358 0.7935 0.7507 0.7074 0.6635 0.6191 0.5742

Error with Error % 0.0000(0.00%) 0.0003(0.03%) 0.0011(0.11%) 0.0024(0.24%) 0.0042(0.42%) 0.0065(0.65%) 0.0093(0.93%) 0.0126(1.26%) 0.0165(1.65%) 0.0209(2.09%) 0.0258(2.58%)

e=0, sphere e=.60, .80, .91, .98 oblate axisymmetric bodies e=1.0, flat circular disk z

Fig. 3 The class of oblate axially symmetric bodies between sphere and flat circular disk for various deformation parameter . From the table 1, it is clear that axial Stokes drag coefficient C Fz decreases steadily and slowly from 1.0 to 0.8 as deformation parameter ‘’ increases from 0 to 1, aspect ratio b/a decreases from 1.0 to 0.0 and eccentricity ‘e’ increases from 0 to 1. The transverse Stokes drag coefficient C Fρ decreases also from 1.0 to 0.6 as deformation parameter ‘’ and eccentricity ‘e’ increases from 0 to 1 and aspect ratio b/a decreases from 1 to 0. It is clear that all the numerical values of drag, in both longitudinal and transverse flow situations are scaled with respect to the drag value of sphere having radius ‘a’. Also, corresponding to each deformation parameter ‘’, longitudinal drag value is always greater then transverse drag



42

value. That’s why their ratio  

C Fz C Fρ

increases from 1.0 to 1.3333, the case of flat circular

disk. The corresponding drag coefficients and their ratio with respect to deformation parameter ‘’, aspect ratio ‘b/a’ and eccentricity ‘e’ are depicted graphically in figures 4, 5 and 6. From the table 2, it is clear that axial Stokes drag coefficient C Fz decreases steadily and slowly from 1.0 to 0.8114 as deformation parameter ‘’ increases from 0 to 1, aspect ratio b/a decreases from 1.0 to 0.0 and eccentricity ‘e’ increases from 0 to 1. The transverse Stokes drag coefficient C Fρ decreases also from 1.0 to 0.5742 as deformation parameter ‘’ and eccentricity ‘e’ increases from 0 to 1 and aspect ratio b/a decreases from 1 to 0. It is clear that all the numerical values of drag, in both longitudinal and transverse flow situations are scaled with respect to the drag value of sphere having radius ‘a’. Also, corresponding to each deformation parameter ‘’, longitudinal drag value is always greater then transverse drag CF value. That’s why their ratio   z increases from 1.0 to 1.4130, the case of flat circular C Fρ disk. The corresponding drag coefficients and their ratio with respect to deformation parameter ‘’, aspect ratio ‘b/a’ and eccentricity ‘e’ are depicted graphically in figures 7, 8 and 9. The corresponding axi-symmetric oblate bodies for various deformation parameters are shown in figure 3. According to table 3, the difference between the non-dimensional numerical values for longitudinal and transverse Stokes drag for both first and second orders are very low with respect to increasing values of deformation parameter from 0 to 1. For the increasing values of deformation parameter ‘’, the error percentage of C Fz increases slightly from 0%(=0, case of sphere) to 1.14%(=1, case of flat circular disk) and remain below 1% up to those oblate bodies for which =0.9. Similarly, the error percentage of C Fρ increases slightly from 0%(=0, case of sphere) to 2.58%(=1, case of flat circular disk) and remain below 1% up to those oblate bodies for which =0.6. It is interesting to note here that second order term in ‘’, increases the numerical values of drag in axial flow situation while in the transverse flow situation, the numerical values of drag decreases slightly. Also, for each values of ‘’, error percentage in transverse case is more than double with respect to axial case. 8

CONCLUSION

From the graphical representation, we can clearly visualize the deviation of second order drag values from first order drag values in both longitudinal and transverse flow geometries. All the numerical values of drag are calculated here with respect to the drag value of sphere (6Ua) having radius ‘a’. It can be seen that the numerical values of drag on deformed sphere comes out to be less than the numerical value of drag on sphere. The authors employed a lot of analytical skills to evaluate these values without the use of readymade software available and all the corresponding graphs are plotted with the help of microcal origin. These values of drag on class of oblate bodies plays a vital role in finding the optimum shape of moving microorganisms, under water projectiles, colloidal particles , red blood cell(erythrocytes) in hematocrit tube during precipitation process in medicine etc. By knowing the value of drag, we can easily get the information about those body shapes which are Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.


43

suitable for slow or swift movement through the fluid medium. The analysis used in this paper is very simple, as far as its applicability over axi-symmetric body is concerned, in comparison to the other available analytical methods like separation of variables, singularity method etc. and numerical techniques like Finite Element Analysis(FEA), Boundary Element Method(BEM) etc. There are many streams like chemical engineering science, naval engineering science and biology where the proposed analysis can play a vital role in new findings. First author is heavily engaged during the last ten years or so to extend and exploit the conjecture for new problems. Optimum drag profile under constant volume and constant cross section area (Datta and Srivastava, 2002), Optimum volume profile under constant drag(Srivastava, 2007) are few of them. 1.5

1.5

1.4

1.4

1.3

1.3

CF /CF

1.2

Drag coefficients and their ratio

Z

1.2



1.1

1.1

1.0

1.0

CF

0.9

0.9

Z

0.8

0.8

CF

0.7



0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 0.0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Deformation parameter ()

0.8

0.9

0.0 1.0

Fig.(4) Variation of drag coefficients and thir ratio w.r.t. deformation parameter 

1.5

1.5

1.4

1.4

1.3

CF /CF


1.2

Z

1.3 1.2



1.1

1.1

1.0

1.0

CF

0.9

0.9

Z

0.8

0.8

CF

0.7

0.7



0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 1.0

Aspect ratio b/a Fig.(5) Variation of drag coefficient and their ratio w.r.t. aspect ratio b/a



44

1.5

1.5

1.4

1.4

1.3

1.3

CF /CF


1.2

Z

1.1 1.0

1.2 

1.1

CF

Z

0.9 0.8

CF

0.7

1.0 0.9 0.8



0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 1.0


eccentricity of spheroid 'e' Fig.(6) Variation of drag coefficients and their ratio w.r.t. eccentricity of spheroid 'e'

2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

CF /CF z



CF



CF

z

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0

Deformation parameter () Fig. (7) Variation of drag coefficients and their ratio w.r.t. deformation parameter




2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

45

CF /CF z



CF

z

CF



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0

Aspect ratio b/a


Fig. (8) Variation of drag coefficients and their ratio w.r.t. aspect ratio

2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

CF /CF z



CF

z

CF



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0

Eccentricity 'e' Fig. (9) Variation of drag coefficients and their ratio w.r.t. eccentricity

9

ACKNOWLEDGMENTS

Authors are thankful to the referees for their invaluable comments over the manuscript and convey warm regards to all of them. Authors are indebted to the authorities of B.S.N.V.Post Graduate College, Lucknow, for providing the basic infrastructure facilities in the department of mathematics. Int. J. of Appl. Math and Mech. 8 (9): 17-53, 2012.

46


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