International Engineering Mathematics Volume 04, Article ID 46593, 7 pages http://x.oi.org/0.55/04/46593 Research Article Invisci Uniform Shear Flow past a Smooth Concave Boy Abullah Mura Department of Mathematics, University of Chittagong, Chittagong 433, Banglaesh Corresponence shoul be aresse to Abullah Mura; mura-math@cu.ac.b Receive February 04; Accepte 30 June 04; Publishe 3 July 04 Acaemic Eitor: Shouming Zhong Copyright 04 Abullah Mura. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Uniform shear flow of an incompressible invisci flui past a two-imensional smooth concave boy is stuie; a stream function for resulting flow is obtaine.results for the same flow past a circular cyliner or a circular arc or a kiney-shape boy are presente as special cases of the main result. Also, a stream function for resulting flow aroun the same boy is presente for an oncoming flow which is the combination of a uniform stream an a uniform shear flow. Possible fiels of applications of this stuy inclue water flows past river islans, the shapes of which eviate from circular or elliptical shape an have a concave region, or past circular arc-shape river islans an air flows past concave or circular arc-shape obstacles near the groun.. Introuction Shear flow is a common type of flow that is encountere in many practical situations. Milne-Thomson has iscusse invisci uniform shear flow past a circular cyliner. It shoul be note here that in practical situations, objects past which flows occur eviate from regular geometric shapes. In the present paper, we have examine two-imensional incompressible invisci uniform shear flow past a smooth concave cyliner.wehaveobtaineastreamfunctionfortheresulting flow.itisfounthatthestreamfunctiongivenin(obtaine by using Milne-Thomson s secon circle theorem for the resulting flow ue to insertion of a circular cyliner in a uniform shear flow of an invisci flui is a special case of that oftheresultingflowpasttheconcaveboypresenteinthis paper. Moreover, flow aroun the same boy has been stuie for an oncoming flow that is a combination of uniform stream an uniform shear flow. The stream function for each of shear flow past a circular arc or a kiney-shape two-imensional boy has been calculate from the main result as special cases. The mathematical results for invisci flui flows hol gooforflowsofcommonfluislikewateranair;theresult is vali for the whole region of a flow fiel except in the thin layer, calle bounary layer, ajacent to the boy aroun which the flow occurs. The results of the present stuy may have applications in many areas of science, engineering, an technology. Here we woul like to mention a few particular areas for applications of the present theoretical work, in water flows aroun river islans, the shapes of which eviate from circular or elliptical shape an have a concave region, noting that rivers have shear flow across an from riverbe to surface; it shoul be mentione here that the present results for oncoming flow parallel to line of symmetry of the concave boy (i.e., parallel to horizontal axis can easily be extene tothecaseswhereoncomingflowsmakearbitraryangles with the line of symmetry. The result for invisci shear flow past circular arc, obtaine here as special case, may fin its application in flows past circular arc-shape islans in rivers. As air flow near the groun is shear flow, the present stuy may also have applications in scientific investigation of air flow past concave or circular arc-shape obstacles near the groun.. The Shape of the Boy The shape of the boy forms owing to inversion of the oblate ellipse by transformation : where x= x c R, y = y R, R =x +y, ( R = R = ((x c +y / (
International Engineering Mathematics which efines geometrical inversion with respect to the unit circle centere at the point (c, 0 in the z (x,y plane. Transformation ( can be conveniently expresse in complex form as z = z c. (3 The two-imensional boy in the inverse plane (z (x, y plane in general has a concave region facing the flui. Here the bounary in the z planeisanoblateellipsegiven by z +λ z =(+λ (λ, 0 λ. (4 A parametric form of (4canbewrittenas z =x +iy =ζ λ ζ, (5 where ζ=e iφ, 0 φ π,intheζ-plane. The inverse transformation of (5 is expresse by ζ= z +(z +4λ /, (6 in orer that the exterior of the unit circle ζ = maps onto the exterior of the ellipse. Moreover, for c>λ an z ( = λ =c, > (the raius of the unit circle in ζ-plane, the exterior of the ellipse inverts into the exterior of the close curve in the z plane an vice versa. The equation, obtaine from (4 by using transformation (3, that represents the close curve in the z plane is z+λ z zz (7 +c(+λ =(+λ (λ. (8 Since corresponing to ifferent values of the parameter λ (4 will represent istinct oblate ellipses, consequently, (8 will also yiel istinct smooth close two-imensional objects in the z plane for a given fixe c. It is mentione in Ranger that λ = 0, λ = /, an λ = correspon to, in the same orer, a circle, a concave boy, an a circular arc in z plane; a figure of an object which is a smooth close curve with a concavity is given in, without mentioning any mathematical equation for it. We note that a kiney-shape boy may be obtaine for λ=/ an c=3(figure. 3. Mathematical Formulation an Solution A uniform shear flow parallel to the x-axis in the z plane, in absence of any bounary, may be expresse by the stream function: Ψ (z, z 8 ω(zz as z, (9 where ω is the constant vorticity. The stream function (9canbemappeontothez -plane using transformation (3, which yiels Ψ (z, z 8 ω( z c z c as z c. (0 Again, the mapping of the stream function (0, by utilizing transformation (5, onto the ζ-plane leas to Ψ (ζ, ζ 8 ω( z ( ( ζ ζ as ζ, ( where z is given by (5 anprime( stansforifferentiation with respect to ζ. Now we insert a circular cyliner of raius unity with its centre at the origin, represente by ζζ =.Sincethestream function ( oes not have constant vorticity, we cannot usecircletheorem or secon circle theorem in orer to obtain the resulting flow. In this situation, we propose a formula that will give the resulting flow, an it is Ψ R 0 (ζ, ζ = Ψ 0 (ζ, ζ Ψ 0 ( ζ, ζ +Ψ (ζ, ζ, ( where Ψ R 0 (ζ, ζ an Ψ 0(ζ, ζ are resulting an basic stream functions, respectively, an Ψ (ζ, ζ is a perturbation stream function. Sinceonthebounaryofthecircleζζ =, therefore, Ψ 0 (ζ, ζ Ψ 0 (/ζ, /ζ becomes zero on the bounary of the unit circle; moreover, Ψ 0 (ζ, ζ Ψ 0 (/ζ, /ζ becomes the same as Ψ 0 (ζ, ζ as ζ. If we assume that all the singularities of Ψ 0 (ζ, ζ lie at a istance greater than unity from the origin, then all the singularities of Ψ 0 (/ζ, /ζ lie insie the circle of raius unity. Regaring Ψ (ζ, ζ, we assume that all the singularities lie insie the unit circle an Ψ (ζ, ζ 0 or a constant on ζ = an for ζ. Thus, the stream function Ψ R 0 (ζ, ζ in ( possessesallthe properties to represent the resulting flow. In the light of ( the resulting flow for the present case may be written as Ψ R (ζ, ζ 8 ω( z ( ( ζ ζ ( ζ ζ ζ ζ +Ψ (ζ, ζ. (3 The function Ψ will be evaluate afterwars in this paper, an we will show that the function satisfies all the conitions that the function must fulfill in accorance with the propose formula (.
International Engineering Mathematics 3.5 0.5 0.5 y 0 y 0 0.5 0.5.5 0.4 0. 0 x 0. 0.4.5 x 0.5 0 (a (b Figure : (a Oblate ellipse in z x +iy plane an (b kiney-shape boy in z x+iyplane (foun by putting λ =/an c=in (8. The flow (3 aroun the circular bounary can be mappe, by using transformation (6, onto the region outsie the oblate ellipse in the z -plane, which yiels Ψ R (z, z 8 ω( z ( (/ (z +(z +4λ / (/ (z +(z +4λ / (/ (z +(z +4λ / (/ (z +(z +4λ / (/ (z +(z +4λ / (/ (z +(z +4λ / +Ψ ( z +(z +4λ /, z +(z +4λ /. (4 Again, the flow, given by (4, aroun oblate ellipse can be mappe onto the region outsie the smooth concave boy, given by (8, in the z plane by using transformation (3, which leas to Ψ R (z, z 8 ω( z ( (/ ((/z + c +(/z + c +4λ / (/ ((/z+c +(/z+c +4λ / (/ ((/z+c +(/z+c +4λ / (/ ((/z+c +(/z+c +4λ / (/ ((/z + c +(/z + c +4λ / (/ ((/z + c +(/z + c +4λ / +Ψ ( ( / z +c+(( z +c +4λ, ( / z +c+(( z +c +4λ. (5
4 International Engineering Mathematics 4. Uniform Shear Flow aroun a Fixe Circular Cyliner or a Circular Arc or a Kiney-Shape Cyliner 4.. Uniform Shear Flow past a Circular Cyliner. For λ=0, (8 represents a circle in the z plane;thecircleisgivenby z+ =, since for λ=0we have c=. (6 We put λ=0in the stream function (5 to obtain the stream function for flow aroun the circle (6as Ψ R 3 (z, z 8 ω(zz ( + z ( z+ ( + z ( z+ +Ψ ( z +, z +. The transformation (7 Z=z( (8 givesustheequationofthecircle(6as ZZ = (. (9 Uner transformation (8thestreamfunction(7takesthe form Ψ R 4 (Z, Z 8 ω (Z Z Z( Z( +Ψ ( Z/( +, Z/( +. (0 Sincetherecanbenochangeinthevalueofvorticitynear the cyliner, therefore 4 Ψ R 4 (Z, Z =ω. ( Z Z Utilizing (, on calculation, it is foun that in (0 Ψ ( Z/( +, Z/( + = 8 ω((/ ( 4. ZZ ( Therefore, the result (0 representsuniformshearflow past a circular cyliner, which is in agreement with the known result forthesameflow. The relation (impliesthat Ψ (ζ, ζ = Ψ (ζ, ζ where = 8 ω( z ( ( (ζ (ζ, (ζ (ζ (3 z ( =+λ. (4 It is clear from (3 thatallthesingularitiesofψ (ζ, ζ lie insie the unit circle in ζ-plane (since >, an Ψ (ζ, ζ 0 as ζ an Ψ (ζ, ζ (/4ω(/z ( (/( (a constant on the circle ζ =.Thus,thefunctionΨ satisfies alltheassumptionsthatwehavemaeinproposingformula (, which, therefore, effectively gives the resulting flow ue to insertion of a circular cyliner in the flow (, of which vorticity is not constant. 4.. Uniform Shear Flow past a Circular Arc. The stream function for uniform shear flow past a circular arc can be obtaine by putting λ=(an when λ=, c=( / in the stream function (5, which yiels Ψ R 5 (z, z 8 ω( + ( (( z + +( z + / +4 ( (( z + +( z + / +4
International Engineering Mathematics 5 (( z + +( z + / +4 ( (( z + +( z + / +4 (( z + +( z + / +4 ( (( z + +Ψ ( z + z + +( z + / +4 +(( z + +(( z + +4/, +4/. (5 4.3. Uniform Shear Flow past a Kiney-Shape Two- Dimensional Boy. The stream function for the uniform shear flow past a kiney-shape cyliner can be obtaine by putting λ =/an c=in the stream function (5, which yiels Ψ R 6 (z, z 8 ω(+ 3 3+ 3 ( (( z ++( z + + ( + 3 / ( (( / z ++( z + + ( + 3 (( z ++( z + ( + 3 4 / + (( / z ++( z + + (( / z ++( z + + ( + 3 4 (( / z ++( z + + +Ψ ( / z ++(( z + +, / z ++(( z + +. 5. Flow Consisting of a Uniform Stream of Constant Velocity V Parallel to x-axis an a Uniform Shear Flow Parallel to the Same Axis with Constant Vorticity ω past a Concave Boy Here the basic flow in the z plane is (6 Ψ 7 (z, z iv (zz 8 ω(zz as z. (7 Now if we insert the two-imensional concave boy given by (8 into the flow(7, the resulting flow, following an analogous proceure that we have aopte to obtain stream function (5, may be expresse as Ψ R 7 (z, z iv ( z ( (( ( / z +c+(( z +c +4λ ( ( / z +c+(( z +c +4λ ( ( / z +c+(( z +c +4λ
6 International Engineering Mathematics +Ψ R (z, z, ( ( / z +c+(( z +c +4λ ( / z +c+(( z +c +4λ ( ( / z +c+(( z +c +4λ (8 where Ψ R (z, z is given by (5. The resulting flow ue to insertion of a circular cyliner represente by the circle (6into the oncomingflow(7can be obtaine by putting λ=0(an when λ=0, c=in(8 as Ψ R 8 (z, z +z iv (zz ( ( z+ +z ( z+ +z 8 ω(zz ( ( z+ +z ( z+ +Ψ ( z +, z +. (9 Using transformation (8 (i.e., consiering the centre of thecircle(6 as the origin of the coorinate system an with the help of relation (, the stream function (9 can be written as Ψ R 9 (Z, Z iv (Z Z + ( (/ ( (/ ( Z Z 8 ω (Z Z ( (/ ( (/ ( Z Z + 8 ω((/ ( 4. ZZ (30 The resulting flow ue to inclusion of the circular arc into the basic flow (7 can be obtaine by putting λ = (an when λ=, c=( /in(8as Ψ R 0 (z, z iv ( + (( ( z + +(( z + ( ( z + +(( z + +4/ +4/ ( ( z + +(( z + +4/ ( ( z + +(( z + +4/ ( z + +(( z + +4/ ( ( z + +(( z + +4/ +Ψ R 5 (z, z, (3 where Ψ R 5 (z, z is given by (5. The stream function for resulting flow ue to insertion of the kiney-shape two-imensional boy into the oncoming
International Engineering Mathematics 7 flow (7 can be obtaine by putting λ =/an c=in (8, which leas to Ψ R (z, z iv (+ 3 3+ 3 (( ( / z ++(( z + + ( + 3 ( ( / z ++(( z + + ( + 3 ( ( / z ++(( z + + (( + 3 4 ( / z ++(( z + + ( / z ++(( z + + 6. Conclusions In this work, we have obtaine a stream function in complex variables for invisci uniform shear flow past a twoimensional smooth boy which has a concavity facing the flui flow. It is foun that the result for the same flow past a circle that is euce from the central result as a special case agrees completely with the known result. Also, stream functions for the same flow past a circular arc or a twoimensional kiney-shape boy are evaluate. Conflict of Interests The author eclares that there is no conflict of interests regaring the publication of this paper. Acknowlegment TheauthorwoulliketothankProfessorDr.S.K.Sen (retire of Jamal Nazrul Islam Research Centre for Mathematical an Physical Sciences, University of Chittagong, Banglaesh, for valuable iscussions uring preparation of this paper. References L. M. Milne-Thomson, Theoretical Hyroynamics, Macmillan, Lonon, UK, 5th eition, 97. K. B. Ranger, The Stokes flow roun a smooth boy with an attache vortex, Engineering Mathematics, vol., no., pp. 8 88, 977. 3 J.M.Dorrepaal, Stokesflowpastasmoothcyliner, Engineering Mathematics,vol.,no.,pp.77 85,978. (( + 3 4 ( / z ++(( z + + +Ψ R 6 (z, z, (3 where Ψ R 6 (z, z is given by (6. The function Ψ is given by (3. Therefore, the stream functions (5, (0, (5, an (6 explicitly give resulting flow for the uniform shear flow past a concave boy, a circular cyliner, a circular arc, an a kiney-shape boy, respectively. An stream functions (8, (9, (3, an (3 explicitly give the resulting flow aroun a concave boy, a circle, a circular arc, an a kiney-shape boy, respectively, where in each of the cases the oncoming flow is the combination of the uniform stream an the uniform shear flow.
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