hapter- Steay Stoes flow aroun eforme sphere. class of oblate axi-symmetric boies. General In physical an biological sciences, an in engineering, there is a wie range of problems of interest lie seimentation problem, lubrication processes etc. concerning the flow of a viscous flui in which a solitary or a large number of boies of microscopic scale are moving, either being carrie about passively by the flow, such as soli particles in seimentation, or moving actively as in the locomotion of micro-organisms. In the case of suspensions containing small particles, the presence of the particles will influence the bul properties of the suspension, which is a subject of general interest in Rheology. In the motion of micro-organisms, the propulsion velocity epens critically on their boy shapes an moes of motion, as evience in the flagellar an cilliary movements an their variations. A common feature of these flow phenomena is that the motion of the small objects relative to the surrouning flui has a small characteristic Reynols number R. Typical values of R may range from orer unity, for san particles settling in water, for example, own to to 6, for various microorganisms. In this low range of Reynols numbers, the inertia of the surrouning flui becomes insignificant compare with viscous effects an is generally neglecte an the Navier-Stoes equations of motion reuce to the Stoes equations as a first approximation. There are many examples, where oblate spheroial geometry plays a vital role as far as the existence an smooth movement of the obstacle whether it is place in liqui meium or in gaseous meium. Among all, mentioning few of them are worth full here to justify the present stuy lie motion of oblate jellyfish(mchenry 7), 8 foot long scorpion foun to live in the sea :: oblate spheroi(jens 7), bening instabilities in homogeneous oblate spheroial galaxy moels (Jessop et al. 997), oblate spheroial shape helps in fining the shape of human transferrin molecule(martel et al. 98), iscoi(oblate ellipsoi) shape in moeling the vestibular membranes in animals(ener 9), oblate an prolate spheroial cells are use for successful human cell permeabilization(wall et al. 999). All these motions are characterize by low Reynols numbers an are escribe by the solution of the Stoes equations. Although the Stoes equations are linear, to obtain exact solutions to them for arbitrary boy shapes or complicate flow conitions is 4
still a formiable tas. There are only relatively few problems in which it is possible to solve exactly the creeping motion equations for flow aroun a single isolate soli boy. Stoes flow of an arbitrary boy is of interest in biological phenomena an chemical engineering. In fact, the boy with simple form such as sphere or ellipsoi is less encountere in practice. The boy, which is presente in science an technology, often taes a complex arbitrary form. or example, uner normal conition, the erythrocyte(re bloo cell) is a biconcave is in shape, which can easily change its form an present ifferent contour in bloo motion ue to its eformability. In secon half of twentieth century, a consierable progress has been mae in treating the Stoes flow of an arbitrary boy. Datta an Srivastava(999) evelope a new approach to evaluate the Stoes rag force in a simple way on a single isolate axially symmetric boy with some geometrical constraints place in axial flow an transverse flow uner the no-slip bounary conitions. The results of rag on both the flow situations were successfully teste not only for sphere, prolate an oblate spheroi but also for other boies lie eforme sphere, cycloial an egg-shape boies of revolution with acceptable limit of error. This metho has been escribe in the chapter, section.5(e) as the same is exploite here to stuy the problem of Stoes flow aroun eforme sphere. In most of these investigations the main result of physical interest is the rag formula for the particular obstacle uner consieration. In this chapter, which is the first part of the whole analysis, we targete to stuy the salient features of class of oblate axi-symmetric boies as a case of eforme sphere by applying the metho of DS conjecture propose by Datta an Srivastava(999) escribe briefly in the section.. Metho Let us consier the axially symmetric boy(see figure b) of characteristic length a place in a uniform stream U of viscous flui of ensity ρ an inematic viscosity. When Reynols number Ua/ is small, the steay motion is governe by Stoes equations[happel an Brenner (964)], gra p ν u, ivu =, (..) ρ 44
subject to the no-slip bounary conition. Axial flow The expression of Stoes rag on such axially symmetric boy place in a uniform stream U parallel to axis of symmetry(i.e. axis of x ) is given by DSconjecture[by using eq. (.5.9) an (.5.)] λb h 4 λb π,where R sin α α λ 6 π μ U. (..) Transverse flow The expression of Stoes rag on such axially symmetric boy place in a uniform stream U perpenicular to axis of symmetry(i.e. axis of x ) is given by DSconjecture[by using eq. (.5.9) an (.5.)] λb h λb 8 π R sin sin α,where α λ 6 π μ U. (..) Since both axial an transverse flows have been consiere in a free stream, results of the force at an oblique angle of attac may be resolve into its components to get the require result. The present analysis can be extene to generate a rag formula for axi- symmetric boies for more complex flows lie paraboloial flow for which free stream velocity may be represente by average velocity(hwang an Wu, part, 975). We are woring in this irection an also searching the avenues of this analysis for non-linear Stoes flow.. ormulation of the problem onsier the axially symmetric boy efine by r a ε μ, μ cosθ, (..) 45
where (r, ) are spherical polar coorinates, is the small eformation parameter, s are the esign or shape factors an () are Legenre function of first in. or small values of parameter, equation (..) represents the eforme sphere. In this chapter, the problem of steay Stoes flow past eforme sphere will be stuie in both situations when uniform stream is along the axis of symmetry(axial or longituinal flow) an is perpenicular to the axis of symmetry(transverse flow). The general expressions for axial an transverse Stoes rag for eforme sphere are targete an erive up to the orer of O( ). The class of oblate axi-symmetric boies will be consiere for the valiation an further numerical iscussions. All the expressions of rag will be upate up to the orer of O( ), where is eformation parameter. In particular, up to the orer of O( ), the numerical values of rag coefficients will be evaluate for the various values of eformation parameter, aspect ratio b/a an eccentricity e for a class of oblate axially symmetric boies incluing flat circular is as a special case an compare with some nown values alreay exist in the literature. Now, in the next section.4, our first prime tas is to calculate the general expressions of rag on the eforme sphere in both axial an transverse flow situations, up to the orer of O( ), with the help of propose conjecture state in the section...4 Solution Accoring to the boy geometry(figure b) an with the ai of calculus an trigonometry, we have following values r θ aε ' μ sin θ, (.4.) ε μ θ tan φ r, r (.4.) ε ' μ sin θ ε sin φ ' μ sin θ Oε, (.4.) 46
47. ε O sin θ μ ' μ ε sin θ μ ' ε φ cos (.4.4) rom figure b, we can write φ, θ α π φ θ ψ α, π ψ so, we have φ cos θ α π cos, therefore we can further write (.4.5), ε O sin θ μ ' sinθ cosθ μ ' μ ε sinθ cosθ μ ' ε sin θ sin α also, by using, φ θ sin α π sin, we have cosθ sin φ sin θ cosφ α cos (.4.6), ε O sin θ cosθ μ ' sin θ μ ' μ ε sin θ μ ' ε cosθ from triangle OM in figure b, sine rule provies sinα r sinθ R sinα r sinθ R, (.4.7)
b θπ r a ε μ, μ cosθ θπ, (.4.8) by using the properties of Legenre function, we have i.e., for 8... 5 6,,,, etc. 4 5,,,, 6 4,..., for, 8 5 8 (.4.9) Now, by using (.4.8) an (.4.9), cross-section raius b may be expresse in series expansion form as b a ε 4 4 6 6 8 8..., b a ε 8 4 5 6 6 5 8 8.... (.4.) General expression of rag in axial flow or axial flow configuration, the expression of Stoes rag (..) require the value of h x or h [given in eq. (.5.)] π π α R sin R sinα sin α sin α θ α h α α 8 8, (.4.) θ θ where sin α sinθ ε ' μ sinθ cosθ Oε, (.4.) 48
cos α cosθ ε ' μ sin θ Oε, (.4.) on ifferentiating with respect to, we get α sin α θ sin θ ε sin θ ' μ cosθ '' μ sin θ Oε. (.4.4) The value of h may be obtaine from equation (.4.) upon utilization of (.4.7), (.4.) an (.4.4) together with the help of following relations of Legenre s olynomial[abramowitz an Stegun, 97] μ ', (Beltrami Result) (.4.5) μ μ, (.4.6) an μ '' μ μ it is to be note here that the ashes relate to Legenre function inicates the erivative with respect to variable = cos, a 4 h ε 5. (.4.7) It is interesting to note that only esign factors up to = occurs in the expression ue to the fact of orthogonality feature of the Legenre polynomial which plays the role an the other terms in h vanishes. The other important part of the axial Stoes rag(..) is b, which is the square of cross-section raius b [ escribe in (.4.)], given by b ε a ε 4... 8 4 9... 64 4 8... 4... 8 4. (.4.8) 49
Now, finally, by utilizing the equations (.4.7) an (.4.8), the expression for axial Stoes rag[given in eq. (..)], up to the secon orer in eformation parameter, comes out to be 6 π μ U ε a ε 5 89... 8 4 9 64 4...... 8 4....... 8 4 O ε (.4.9) This general expression of rag on eforme sphere immeiately reuces to classical Stoes rag on sphere of raius a i.e. 6Ua on taing the limit as eformation parameter. It is interesting to note here that the expression has been written up to the orer of O( ) an even inexe shape factors are appearing in the coefficient of eformation parameter which is in contrary to that given by Happel an Brenner(964, page no. 8) that the terms corresponing to = an = only contributes to the expression of rag. Beyon this statement, our result seems to be more general for eforme sphere. General expression of rag in transverse flow In orer to calculate the Stoes rag[efine in eq. (..)] over eforme sphere place in transverse flow, we have to calculate h [given in eq. (.5.9)] as a h ε... Oε, (.4.) 4 6 Substituting the value of h from (.4.) in (..), together with the value of b from (.4.8), the expression of rag comes out to be 5
6 π μ U ε a 9 ε 5 9... 4 4. 5 5......... O ε 4 4 4 64 8 4 (.4.) Which on taing limit as, matches with the well nown Stoes rag on sphere having raius a i.e. 6Ua. It is interesting to note here again that the expression has been written up to the orer of O( ) an even inexe shape factors are appearing in the coefficient of eformation parameter has never appeare in the literature an seems to be new. It is to be note here that the expressions of rag (.4.9) an (.4.) are vali only for the class of axi-symmetric eforme sphere governe by polar equation (..) in which secon orer term in eformation parameter is absent. The general expressions of rag (.4.9) an (.4.) are applie to the class of oblate eforme spheroi in section.5..5 Oblate spheroi We consier the oblate spheroi(see figure ), as it belongs to the class of axisymmetric eforme sphere whose artesian equation is x y a z a ε, (.5.) where equatorial raius is a an polar raius is a( ), in which eformation parameter is positive an sufficiently small that squares an higher powers of it may be neglecte. Its polar equation, [ by using z = r cos, = r sin, = (x + y ) /, up to the orer of O()] is r = a ( cos ), (.5.) it can be written in linear combination of Legenre functions of first in[happel an Brenner(964), Sencheno an Keh(9)] 5
r a ε μ μ, μ cosθ, (.5.) On comparing this equation with the polar equation of eforme sphere (..), the appropriate esign factors for this oblate spheroi are,,, for, (.5.4) μ, μ cosθ. (.5.5) with μ, μ μ an μ Also, this polar equation (.5.) of oblate spheroi may be written in terms of Gegenbauer functions of first in by using the following well nown relation As μ μ I μ -,, μ cosθ, (.5.6) - μ r a ε ε I, = cos (.5.7) or if we put = a ( ), we have μ r ε I. (.5.8) Now, we fin the expression of Stoes rag on this axi-symmetric oblate boy place in both axial flow(in which uniform stream is parallel to polar axis or axis of symmetry) an transverse flow(in which uniform stream is perpenicular to polar axis or axis of symmetry) situations with the ai of general expressions of rag (.4.9) an (.4.). Axial flow With the ai of equations (.5.4) an (.5.5), the expression of axial Stoes rag can be written up to the orer of O() with the help of (.4.9) an comes out to be ε 6 π μ U a Oε 5, (.5.9) 5
which matches with that alreay exist in Happel an Brenner(964, page 5, eq. 5-9.5), Usha an Nigam(976, page, eq. ) an Sencheno an Keh (6, page 88, eq. 8). In the paper of Sencheno an Keh, the valiation too place after removing the slip effect in the vicinity of oblate boy i.e., or, where is slip parameter, which is the clear cut case of no-slip bounary conition as it shoul be. This force is less than woul be exerte on a sphere of raius equal to the equatorial raius a of the oblate spheroi. It is ue to the fact that the surface area an volume of the oblate spheroi 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 oes not have a profoun effect on its resistance. A more appropriate comparison might be mae between equation (.5.9) an the resistance of a sphere of equal volume or surface area. The volume of the spheroi (.5.) is 4 π a 4 π a ε. (.5.) Hence, a sphere of equal volume woul have a raius of a( /) an its resistance woul be, ue to well nown Stoes law[happel an Brenner(964), Stoes(85)] ε 6 π μu a. (.5.) A sphere of equivalent volume therefore has a smaller resistance than the oblate spheroi. Similarly, since the surface area of the oblate spheroi is 4a ( /), a sphere of equal surface area has a raius of a( /). Therefore, the low resistance [given in eq. (.5.)] of the sphere hols also on the basis of equal surface area. It has been mentione in the introuction that the conjecture expression for axial an transverse Stoes rag on axially symmetric boies propose in [Datta an Srivastava(999)] hols goo for spheroi an provie the close form solution. or the convenience, we write both expressions of rag on oblate spheroi a e e e e 8 π μ U sin e, (.5.) 5
a e e e e 6 π μ U sin e, (.5.) where e is the eccentricity of the spheroi. The expression of Stoes rag in axial flow (.5.9) on oblate spheroi may also be euce from the exact expression of axial Stoes rag (.5.) on a oblate spheroi for flow parallel to its axis of revolution by utilizing the appropriate relation between eccentricity e an eformation parameter. b or the consiere oblate spheroi, eccentricity is e, with b a ε, a which further gives us the relation between eccentricity e an eformation parameter as e ε ε ε ε (leaving the square term). (.5.4) On re-writing the expression of rag (.5.) as 6 π μ U a 4 e e e e sin e 6 π μ U a K, (.5.5) where K is calle correction to Stoes law ue to Happel an Brenner(964, page 48, eq. 4-6.8). Using the expression of eccentricity (.5.4) in terms of eformation parameter, up to the secon power, this correction factor K can be expresse in the powers of eformation parameter as ε ε K... 5 75. (.5.6) Now, the revise expression of rag, in powers of eformation parameter, will be of the form 54
ε ε 6 π μ U a... 5 75. (.5.7) This expression is in goo agreement to that presente by hang an Keh(9, page 5, eq. 5(a)), but in Happel an Brenner(964, page 45, eq. 4-5.9), the thir term /75 is in error which seems to be algebraic error an shoul be moifie to /75 with same first two terms /5 in braces. Transverse flow By using the general expression (.4.) of transverse Stoes rag, with the ai of equations (.5.4) an (.5.5), the expression for rag on oblate spheroi place in transverse flow woul be obtaine as(up to the orer of O()) 6 π μ U a ε Oε. (.5.8) 5 Also, the complete expression of Stoes rag on oblate spheroi in transverse flow by conjecture metho(datta an Srivastava, 999) is given by (.5.) 6 π μ U a e e e e sin e, where e is the eccentricity of oblate spheroi. The expression of rag (.5.8) on oblate spheroi (.5.) may also be euce from this exact expression (.5.) for the transverse flow or flow perpenicular to its axis of revolution with correctness up to the orer of O( ). On rewriting the above exact expression of rag (.5.) as e 6 π μ U e a e e sin e e 6 π μ U a e e, (.5.9) e sin e 8 55
6 π μ U a K, (.5.) where K is calle correction to Stoes law ue to Happel an Brenner(964, page 48, eq. 4-6.8). Using the expression of eccentricity (.5.4) in terms of eformation parameter, up to the secon power, this correction factor K can be expresse after some algebraic manipulations in the powers of eformation parameter as 9 6 π μ U a ε ε... 5 5. (.5.) Up to the orer of O(), this gives us the rag (.5.8) on oblate spheroi. Also, this expression of rag in transverse flow situation matches with hang an Keh(9, page 5, eq. (5b)). It is to be note here that the thir terms of the eq.(.5.7) an eq.(.5.) are in isagreement with those obtaine by the general expressions of rag presente in eq. (.4.9) an (.4.). In the paper of hang an Keh(9), the polar equation of eforme sphere was consiere to be θ θ r a ε f f, (.5.) ε while in this paper, the polar equation(..) of eforme sphere is consiere to be r a ε μ, μ cosθ, in which the terms of secon orer in is absent. This might be the main reason for the isagreement between hang an Keh(9) an those presente in this paper at the level of secon orer in. Apart from this fact, we can achieve the same expressions of rag, correcte up to secon orer in eformation parameter, on oblate spheroi given in equations (.5.7) an (.5.) by applying Datta an Srivastava(999) conjecture (..) an (..) inepenently over eforme sphere whose polar equation is (.5.) with proper choice of f () = cos an f () = (/) cos sin. Verification for the same is provie below for valiation purposes. 56
onsier the polar equation of eforme sphere as r a ε cos ε cos sin. (.5.) It is to be note here that this polar equation (.5.) consists of secon egree term in eformation parameter while the polar equation (.5.) of oblate eforme sphere contains only first two terms i.e. up to first egree in eformation parameter. Axial flow The expression of Stoes rag experience by axi-symmetric boy(.5.) can be evaluate on inepenent application of formula propose in eq. (.5.) an eq. (..) which requires the value of mi cross-section raius b [see figure b an efine in eq. (.4.8)] an h [efine in eq. (.5.)]. After careful calculation, these quantities are comes out to be b r a, (.5.4) h π 8 α a R sin α α 5 O. 5 (.5.5) By substituting the values of b (.5.4) an h (.5.5) in eq.(..), the expression of Stoes rag on axi-symmetric boy (.5.) place in axial(longituinal) flow comes out to be 6 π μ U 57 (.5.6) This value of rag matches with that given in eq. (.5.7) obtaine from exact form solution given in eq.(.5.). Transverse flow a ε 5 ε... 75 The expression of Stoes rag experience by axi-symmetric boy(.5.) can be evaluate on inepenent application of formula propose in eq. (.5.) an eq.
(..) which requires the value of mi cross-section raius b [see figure b an efine in eq. (.4.8)] an h [efine in eq. (.5.9)]. After careful calculation, these quantities are comes out to be b r a, (.5.7) h π 6 α a R sin sin α α O. 5 7 (.5.8) By substituting the values of b (.5.7) an h (.5.8) in eq.(..), the expression of Stoes rag on axi-symmetric boy (.5.) place in transverse flow comes out to be 6 π μ U a ε 5 9 5 ε.... (.5.9) This value of rag matches with that given in eq. (.5.) obtaine from exact form solution given in eq.(.5.). The confirmation of rag expressions for oblate eforme boy (.5.) in both axial an transverse cases, written up to the secon orer in eformation parameter, justifies the application of DS conjecture (..) an (..) for evaluation of general expressions of rag presente in (.4.9) an (.4.) for eforme axially symmetric boy (..) consist the term up to first orer in eformation parameter. The expressions of axial rag (.5.9) an transverse rag (.5.8) are presumably vali only for small values of, they are in fact, surprisingly accurate for even large epartures from spheroi shape. We can explain it by consiering flat circular is as a special case of oblate spheroi..6 lat circular is The eccentricity e of an oblate spheroi is 58
e a, a ε ε e, ε e. (.6.) The eccentricity of a flat circular is of raius a is unity, whence by (.6.), the value of eformation parameter is also unity. or this case, the expressions for axial rag (.5.9) an transverse rag (.5.8) ue to Stoes are 6 π μ U a.8 broasie on caseof flat circular is, (.6.). (.6.) 6 π μ U a an.6 ege on caseof flat circular is These results are in agreement with those given in Happel an Brenner(964, page 5) which are in error uner six percent for any oblate spheroi on corresponing with the exact results given by Lamb(9) 6 π μ U a.849 broasie on caseof flat circular is, (.6.4), (.6.5) 6 π μ U a an.5658 ege on caseof flat circular is these errors ecrease rapily with ecreasing eccentricity. or eccentricity e =.8, b b.8.6, ε e. 4, a a a The iscrepancy is less than.5 percent. Which clearly inicates that the general expressions(axial an transverse both) (.4.9) an (.4.) for rag on eforme 59
sphere provie goo approximations up to the orer of O() for a class of axially symmetric boies lie oblate spheroi..7 Numerical Discussion irst of all we fin numerical values of parameters lie eccentricity e, aspect ratio b/a, axial Stoes rag coefficient z, transverse Stoes rag 6 π μ U a coefficient ρ for ifferent values of eformation parameter so 6 π μ U a that numerical iscussion for a class of oblate axi-symmetric boies (generate from eforme sphere between sphere an flat circular is) coul be compile. These variations together with rag ratio z are isplaye in table for the first orer an in table for the secon orer in eformation parameter. The corresponing error estimates between the first an secon orer values of rags are presente in table. rom the table, it is clear that axial Stoes rag coefficient ecreases steaily an slowly from. to.8 as eformation z parameter increases from to, aspect ratio b/a ecreases from. to. an eccentricity e increases from to. The transverse Stoes rag coefficient ecreases also from. to.6 as eformation parameter an eccentricity e increases from to an aspect ratio b/a ecreases from to. It is clear that all the numerical values of rag, in both longituinal an transverse flow situations are scale with respect to the rag value of sphere having raius a. Also, corresponing to each eformation parameter, longituinal rag value is always greater then transverse rag value. That s why their ratio z increases from. to., the case of flat circular is. The corresponing rag coefficients an their ratio with respect to eformation parameter, aspect ratio b/a an eccentricity e are epicte graphically in figures 4, 5 an 6. rom the table, it is clear that axial Stoes rag coefficient ecreases steaily an slowly from. to.84 as eformation parameter increases from to, aspect ratio b/a ecreases from. to. an eccentricity e increases from to. The transverse Stoes rag coefficient ecreases also from. to.574 as eformation ρ parameter an eccentricity e increases from to an aspect ratio b/a 6 z ρ ρ ρ
ecreases from to. It is clear that all the numerical values of rag, in both longituinal an transverse flow situations are scale with respect to the rag value of sphere having raius a. Also, corresponing to each eformation parameter, longituinal rag value is always greater then transverse rag value. That s why their ratio z increases from. to.4, the case of flat circular is. The ρ corresponing rag coefficients an their ratio with respect to eformation parameter, aspect ratio b/a an eccentricity e are epicte graphically in figures 7, 8 an 9. The corresponing axi-symmetric oblate boies for various eformation parameters are shown in figure. Accoring to table, the ifference between the non-imensional numerical values for longituinal an transverse Stoes rag for both first an secon orers are very low with respect to increasing values of eformation parameter from to. or the increasing values of eformation parameter, the error percentage of increases slightly from %(=, case of sphere) to.4%(=, case of flat circular is) an remain below % up to those oblate boies for which =.9. Similarly, the error percentage of increases slightly from %(=, case of sphere) to.58%(=, case of flat ρ circular is) an remain below % up to those oblate boies for which =.6. It is interesting to note here that secon orer term in, increases the numerical values of rag in axial flow situation while in the transverse flow situation, the numerical values of rag ecreases slightly. Also, for each values of, error percentage in transverse case is more than ouble with respect to axial case. 8. onclusion rom the graphical representation, we can clearly visualize the eviation of secon orer rag values from first orer rag values in both longituinal an transverse flow geometries. All the numerical values of rag are calculate here with respect to the rag value of sphere (6Ua) having raius a. It can be seen that the numerical values of rag on eforme sphere comes out to be less than the numerical value of rag on sphere. We employe a lot of analytical sills to evaluate these values without the use of reaymae software available an all the corresponing graphs are plotte with the help of microcal origin. These values of rag on class of oblate boies plays a vital role in fining the optimum shape of moving microorganisms, uner water projectiles, colloial particles, re bloo cell(erythrocytes) in hematocrit tube uring precipitation process in meicine etc. By nowing the value of rag, we can easily get the information about those boy 6 z
shapes which are suitable for slow or swift movement through the flui meium. The analysis use in this paper is very simple, as far as its applicability over axisymmetric boy is concerne, in comparison to the other available analytical methos lie separation of variables, singularity metho etc. an numerical techniques lie inite Element Analysis(EA), Bounary Element Metho(BEM) etc. There are many branches of science an engineering lie chemical engineering science, naval engineering science an biology where the propose analysis can play a vital role in new finings. Optimum rag profile uner constant volume an constant cross section area [Datta an Srivastava, ], Optimum volume profile in axi-symmetric Stoes flow [Srivastava, 7] an optimum cross-section profile in axi-symmetric Stoes flow[srivastava, ] are few very important applications of DS conjecture which are highly applicable in to stuy the optimum boy profiles of minute/micro organisms an naval war heas moving through the flui. The application of general expressions of rag (.4.9) an (.4.) for class of prolate axi-symmetric boies are presente in chapter. 6
Table Numerical values of rag coefficients with respect to eformation parameter (up to first orer of ), aspect ratio b/a, eccentricity e b/a=- e =(-b /a ) / z 6 π μ U ε 5 a ρ 6 π μ U ε 5 a z ρ Axi-symmetric boies...... Sphere..9.458.98.96.8..8.6.96.9.44..7.74.94.88.68.4.6.8.9.84.95.5.5.866.9.8.5 Oblate boy.6.4.965.88.76.578 zone.7..959.86.7.944.8..979.84.68.5.9..9949.8.64.8....8.6. lat circular is 6
Table Numerical values of rag coefficients with respect to eformation parameter (up to secon orer of ), aspect ratio b/a, eccentricity e b/a=- e =(-b /a ) / z 6 π μ U ε 5 75 ε a ρ 6 π μ U a 9 ε ε 5 5 z ρ Axi-symmetric boies...... Sphere..9.458.98.9597.8..8.6.964.989.44..7.74.94.8776.68.4.6.8.98.858.95.5.5.866.98.795.5 Oblate boy.6.4.965.884.757.578 zone.7..959.8656.774.944.8..979.847.665.5.9..9949.89.69.8....84.574. lat circular is Table Error estimates between first an secon orer rag values Deformation z Error with ρ Error with parameter O() O( ) Error % O() O( ) Error %....(.%)...(.%)..98.98.(.%).96.9597.(.%)..96.964.4(.4%).9.989.(.%)..94.94.(.%).88.8776.4(.4%).4.9.98.8(.8%).84.858.4(.4%).5.9.98.8(.8%).8.795.65(.65%).6.88.884.4(.4%).76.757.9(.9%).7.86.8656.56(.56%).7.774.6(.6%).8.84.847.7(.7%).68.665.65(.65%).9.8.89.9(.9%).64.69.9(.9%)..8.84.4(.4%).6.574.58(.58%) 64
(Transverse low) U Equatorial Axis a O a(-) olar Axis z U (axial flow) ig. Oblate spheroi in meriional two-imensional plane (z, ) e=, sphere e=.6,.8,.9,.98 oblate axisymmetric boy e=., flat circular is z ig. The class of oblate axially symmetric boies between sphere an flat circular is for various eformation parameter. 65
Drag coefficients an their ratio Drag coefficients an their ratio.5.4.....9.8.7.6.5.4... Z / Z.......4.5.6.7.8.9. Deformation parameter '' ig. 4 Variation of rag coefficients an their ratio w.r.to eformation parameter ' '(upto first orer).5.4.....9.8.7.6.5.4....5.4.....9.8.7.6.5.4... Z Z /.5.4.....9.8.7.6.5.4..........4.5.6.7.8.9. Aspect ratio 'b/a' ig. 5 Variation of rag coefficient an their ratio w.r. to aspect ratio 'b/a' 66
Drag coefficients an their ratio Drag coefficients an their ratio.5.4.....9.8.7.6.5.4... Z / Z.5.4.....9.8.7.6.5.4..........4.5.6.7.8.9. Eccentricity of spheroi 'e' ig. 6 Variation of rag coefficients an their ratio w.r. to eccentricity of spheroi 'e'..9.8.7.6.5.4.....9.8.7.6.5.4.... z /.....4.5.6.7.8.9. Deformation parameter '' ig. 7 Variation of rag coefficients an their ratio w.r. to eformation parameter ''(upto secon orer) z..9.8.7.6.5.4.....9.8.7.6.5.4.... 67
Drag coefficients an their ratio Drag coefficients an their ratio..9.8.7.6.5.4.....9.8.7.6.5.4.... z z /.....4.5.6.7.8.9. Aspect ratio 'b/a' ig. 8 Variation of rag coefficients an their ratio w.r. to aspect ratio 'b/a'..9.8.7.6.5.4.....9.8.7.6.5.4......9.8.7.6.5.4.....9.8.7.6.5.4.... z /.....4.5.6.7.8.9. eccentricity 'e' ig. 9 Variation of rag coefficients an their ratio w.r. to eccentricity'e' z..9.8.7.6.5.4.....9.8.7.6.5.4.... 68