On inverse solutions of unsteady Riabouchinsky flows of second grade fluid.Abstract This investigation concerns the analytical solutions for the Riabouchinsky time-dependent flows of an incompressible in·com·press·i·ble adj. Impossible to compress; resisting compression: mounds of incompressible garbage. in second-grade fluid. Semi-inverse method has been used for the solutions of highly non-linear differential equations differential equation Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. . 1. Introduction The inadequacy of the classical Navier-Stokes theory to describe rheological rhe·ol·o·gy n. The study of the deformation and flow of matter. rhe  o·log complex fluids such as polymer solutions, blood, paints,
certain oils and greases, has led to the development of several theories
of non-Newtonian fluids. Amongst these the differential type have
received special attention. One particular subclass In programming, to add custom processing to an existing function or subroutine by hooking into the routine at a predefined point and adding additional lines of code. subclass - derived class of differential type fluids for which one can reasonably hope to get the analytic solution is the second-grade fluid. Motivated with the above facts, the analytical solutions for unsteady Riabouchinsky flows of second-grade fluids are constructed in this note. The analytical solutions of the governing non-linear equations are obtained using inverse methods The inverse method can refer to:
In studying Riabouchinsky flows [4] the streamfunction is taken to be linear in one of the space dimensions. In this way the fifth order compatibility partial differential equation partial differential equation In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. in terms of stream functions can readily be written by two coupled fifth order ordinary differential equations ordinary differential equation Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). . Solutions are then obtained by a suitable assumption of one equation. Actually, Riabouchinsky considered [psi PSI - Portable Scheme Interpreter ](x, y) = yf (x); the plane steady flow which represents the flow where flow is separated in the two symmetrical symmetrical equally on both sides. symmetrical multifocal encephalopathy inherited disease in two forms: Limousin form appears at about a month old with blindness, forelimb hypermetria, hyperesthesia, nystagmus, aggression, weight regions by a vertical or horizontal plane horizontal plane n. A plane crossing the body at right angles to the coronal and sagittal planes. Also called transverse plane. horizontal plane . Recently, Hamdan [2] gave an alternate approach to find exact solutions of Riabouchinsky flows. He after choosing [psi] (x, y) = yf (x) + g (x) obtained two coupled fourth order equations in f and g; and choose a solution of g and then find the function f. The advantage of this approach is that; the constant appearing in Riabouchinsky flows can easily be obtained. However, the solutions assumed by Hamdan [2] cannot be all applied in second grade fluid and accordingly the alternate approach will not be feasible. In the present work we consider unsteady flows which have the stream function, [psi] (x, y, t) = x[xi] (z, t), z = y + Kx. The Riabouchinsky's [4] and Hamdan's [2] solutions can be recovered from the present investigations. In terms of the stream function, the governing equation for unsteady flow is [3] [rho] [[[partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential ]/[partial derivative]t][[nabla].sup.2][psi] - {[psi], [[nabla].sup.2][psi]}] = ([mu] + [[alpha].sub.1][[partial derivative]/[partial derivative]t])[[nabla].sup.4][psi] - [[alpha].sub.1] {[psi], [[nabla].sup.4], [psi]}, (1) in which [[alpha].sub.1] is the viscoelasticity Viscoelasticity, also known as anelasticity, is the study of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. of the second-grade fluid, [rho] is the density, [mu] is the dynamic viscosity dynamic viscosity n. Symbol  A measure of the molecular frictional resistance of a fluid as calculated using Newton's law. , [psi] is the
stream function and
{[psi], [[nabla].sup.2]} = [[partial derivative][psi]/[partial derivative]x][[[partial derivative]([[nabla].sup.2][psi])]/[partial derivative]y] - [[partial derivative][psi]/[partial derivative]y] [[[partial derivative]([[nabla].sup.2][psi])]/[partial derivative]x]. It is worthmentioning to note that Eq. (1) is highly non-linear. The exact solutions in closed form is impossible. Even, the closed form solution in case of Newtonian fluid ([[alpha].sub.1] = 0) is not possible. Similar to several previous studies, our interest lies in finding some analytical solutions of Eq. (1) with appropriate form of the stream function. Thus we consider the stream function as [2] [psi] (x, y, t) = x[xi] (z, t), z = y + Kx, (2) [psi] (x, y, t) = x[xi] (z, t) + [eta] (z, t). (3) 2. Solutions of Specific Flows 2.1 Flow where [psi] (x, y, t) = x[xi] (z, t), z = y + Kx Substitution of Eq. (2) into Eq. (1) gives [[xi]".sub.t] + [xi]'[xi]" - [xi][xi]'" = [nu] (1 + [K.sup.2])[[xi].sup.IV] + [alpha] (1 + [K.sup.2]) [[[xi].sub.t.sup.IV] + [xi]'[[xi].sup.IV] - [xi][[xi].sup.V]], (4) where [xi] (z, t) is an arbitrary function See under Arbitrary. See also: Function of z, t, subscript (1) In word processing and scientific notation, a digit or symbol that appears below the line; for example, H2O, the symbol for water. Contrast with superscript. (2) In programming, a method for referencing data in a table. t indicate the derivative with respect to time, primes (', IV) denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the derivatives with respect to z and [nu] = [mu]/[rho], [alpha] = [[alpha].sub.1]/[rho], (5) where [nu] is the kinematic viscosity kin·e·mat·ic viscosity n. Symbol  A measure used in fluid flow studies, usually expressed as the dynamic viscosity divided by the density of the fluid. . For the solution of Eq. (4)
we write
[xi] (z, t) = V + [lambda] (z + Vt) = V + [lambda](s), s = z + Vt, (6) where V is an arbitrary constant (Math.) a quantity of function that is introduced into the solution of a problem, and to which any value or form may at will be given, so that the solution may be made to meet special requirements. and [lambda] satisfies the following non-linear ordinary differential equation [d[lambda]/ds][[[d.sup.2][lambda]]/[d[s.sup.2]]] - [lambda][[[d.sup.3][lambda]]/[d[s.sup.3]]] = [nu] (1 + [K.sup.2])[[[d.sup.4][lambda]]/[d[s.sup.4]]] + [alpha](1 + [K.sup.2])[d[lambda]/ds][[[d.sup.4][lambda]]/[d[s.sup.4]]] - [lambda][[[d.sup.5][lambda]]/[d[s.sup.5]]]. (7) The first integral of Eq. (7) for zero constant of integration is (d[lambda]/ds)[.sup.2] - [lambda][[[d.sup.2][lambda]]/[d[s.sup.2]]] = [nu] (1 + [K.sup.2])[[[d.sup.3][lambda]]/[d[s.sup.3]]] + [alpha](1 + [K.sup.2])[2[d[lambda]/ds][[d.sup.3][lambda]/d[s.sup.3]] - ([d.sup.2][lambda]/[d[s.sup.2]])[.sup.2] - [lambda][[[d.sup.4][lambda]]/d[s.sup.4]]]. (8) Taking [lambda](s) = A(1 + [ce.sup.as]) (9) into Eq. (8) we can write A = [[nu] (1 + [K.sup.2])a]/[[alpha] (1 + [K.sup.2]) [a.sup.2] - 1], (10) where A, c and a are arbitrary real constants. From Eq. (2) we have [psi](x, y, t) = Vx + [[[nu] (1 + [K.sup.2])a]/[[alpha] (1 + [K.sup.2]) [a.sup.2] - 1]] (1 + c[e.sup.a(y + Kx + Vt)]) (11) and the corresponding velocity fields are u = [[[a.sup.2]c[nu] (1 + [K.sup.2])]/[[alpha](1 + [K.sup.2]) [a.sup.2] - 1]] [e.sup.a(y + Kx + Vt)], (12) v = -V - [[cK[a.sup.2][nu] (1 + [K.sup.2])]/[[alpha] (1 + [K.sup.2]) [a.sup.2] - 1]] [e.sup.a(y+Kx+Vt)]. (13) 2.2 Flow where [psi] (x, y, t) = x[xi] (z, t) + [eta] (z, t), z = y + Kx Here, we use Eq. (3) into Eq. (1) and obtain the following non-linear partial differential equations [[xi]".sub.t] + [xi]'[xi]" - [xi][xi]'" = [nu] (1 + [K.sup.2])[[xi].sup.IV] + [alpha](1 + [K.sup.2])[[[xi].sub.t.sup.IV] + [xi]'[[xi].sup.IV] - [xi][[xi].sup.V]], (14) [[eta]".sub.t] + [eta]'[xi]" - [xi][eta]'" = [2K/[1 + [K.sup.2]]] ([xi][xi]" - [[xi]'.sub.t]) + [nu] [4K[xi]'" + (1 + [K.sup.2])[[eta].sup.IV]] + [alpha][4K ([[xi]'".sub.t] - [xi][[xi].sup.IV]) + (1 + [K.sup.2]){[[eta].sub.t.sup.IV] + [eta]'[[xi].sup.IV] - [xi][[eta].sup.V]}]. (15) We note that the partial differential equation for [xi] is the same as in section 2.1. When [xi] is known, the second equation is a linear partial differential equation for the determination of [eta] (z, t). In order to get the time independent equations, we introduce the following transformations [xi] (z, t) = V + [lambda] (z + Vt) = V + [lambda] (s), s = z + Vt, (16) eta (z, t) = V + [theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ] (z + Vt) = V + [theta] (s), (17) and get the non-linear ordinary differential equations in terms of [lambda] and [theta]. The differential equation for [lambda] is same as in subsection subsection Noun any of the smaller parts into which a section may be divided Noun 1. subsection - a section of a section; a part of a part; i.e. 2.1 and the differential equation satisfied by [theta] is [d[theta]/ds] [[[d.sup.2][lambda]]/[d[s.sup.2]]] - [lambda] [[[d.sup.3][theta]]/[d[s.sub.3]]] = [2K/[1 + [K.sup.2]]][lambda][[[d.sup.2][lambda]]/[d[s.sup.2]]] + [nu] [4K [[[d.sup.3][lambda]]/[d[s.sub.3]]] + (1 + [K.sup.2])[[[d.sup.4][theta]]/[d[s.sup.4]]]] + [alpha] [(1 + [K.sup.2]){[d[theta]/ds][[[d.sup.4][lambda]]/[d[s.sup.4]]] - [lambda][[[d.sup.5][theta]]/[d[s.sup.5]]]} - 4K[lambda] [[[d.sup.4][theta]]/[d[s.sup.4]]]]. (18) The first integral of Eq. (18) is [d[theta]/ds] [d[lambda]/ds] - [lambda][[[d.sup.2][theta]]/[d[s.sup.2]]] = [2K/[1 + [K.sup.2]]] ([lambda][d[theta]/ds] - [integral] ([d[lambda]/ds])[.sup.2] ds)+ [nu] [4K [[[d.sup.2][lambda]]/[d[s.sup.2]]] + (1 + [K.sup.2])[[[d.sup.3][theta]]/[d[s.sup.3]]]] [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], (19) where the constant of integration is equal to zero. Using Eq. (9) into Eq. (19) and making lengthy algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind. [CACM 2(5):16 (May 1959)]. 2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. calculations we get -[alpha] (1 + [K.sup.2]) A(1 + c[e.sup.as])[PHI phi n. Symbol  The 21st letter of the Greek alphabet.PHI, n See health information, protected. ]'" + (1 + [K.sup.2])([nu] + [alpha]a Ac[e.sup.as])[PHI]" (20) +A[1 + c[e.sup.as] - (1 + [K.sup.2])[alpha][a.sup.2]c[e.sup.as]][PHI]' - a Ac[e.sup.as][PHI] = [OMEGA 1. (programming) Omega - A prototype-based object-oriented language from Austria. ["Type-Safe Object-Oriented Programming with Prototypes - The Concept of Omega", G. Blaschek, Structured Programming 12:217-225, 1991]. 2. ](s) where [PHI] = d[theta]/ds and [OMEGA](s) = 2Ka[A.sup.2]c[e.sup.as] (1 + [1/2]c[e.sup.as]) ([1/[1 + [K.sup.2]]] - 2[alpha][a.sup.2]) + 4K[nu][a.sup.2]Ac[e.sup.as]. (21) Consequently, we may reduce the order of the equation by means of the consecutive substitutions [PHI](s) = P (s) [e.sup.as] and P' (s) = R(s) to obtain -[alpha] (1 + [K.sup.2]) A(1 + c[e.sup.as])R" + (1 + [K.sup.2])[[nu] - [alpha]a A (3 + 2c[e.sup.as])]R' + [A(1 + c[e.sup.as]) + (1 + [K.sup.2])(2a[nu] - [alpha][a.sup.2]A(3 + 2c[e.sup.as]))]R = -[OMEGA](s[e.sup.-as]. (22) In order to find the solution of Eq. (22) we discuss few of its special cases: Case I. The solution of Eq. (22) for c = 0 is given by R(s) = [C.sub.1][e.sup.-[X.sub.7]s] + [C.sub.2][e.sup.[X.sub.8]s], (23) where [X.sub.7] = [[lambda].sub.1] + [square root of ([[lambda].sub.1.sup.2] - 4[[lambda].sub.2])], [X.sub.8] = -[[lambda].sub.1] + [square root of ([[lambda].sub.1.sup.2] - 4[[lambda].sub.2])], [[lambda].sub.1] = [3[alpha]aA - [nu]]/[alpha]A, [[lambda].sub.2] = [3[alpha][a.sup.2]A - A - 2a[nu]]/[alpha]A. Now, the expressions for [eta] (z, t) can be written as [eta] (z, t) = V + [C.sub.4] + [[[C.sub.3][e.sup.as]]/a] - [[C.sub.1]/[[X.sub.7] (a - [X.sub.7])]][e.sup.(a-[X.sub.7])s] + [[C.sub.2]/[[X.sub.8] (a + [X.sub.8])]][e.sup.(a+[X.sub.8])s], (24) in which [C.sub.r] (r = 1 to 4) are arbitrary constants. The stream function and the velocity components are respectively of the following form [psi] (x, y, t) = x (V + a) + V + [C.sub.4] + [[[C.sub.3][e.sup.a](y+Kx+Vt)]/a] -[[C.sub.1]/[[X.sub.7] (a - [X.sub.7])]] [e.sup.(a-[X.sub.7])(y+Kx+Vt)] + [[C.sub.2]/[[X.sub.8] (a + [X.sub.8])]] [e.sup.(a+[X.sub.8])(y+Kx+Vt)], (25) u = [C.sub.3][e.sup.a(y+Kx+Vt)] - [[C.sub.1]/[X.sub.7]][e.sup.(a-[X.sub.7])(y+Kx+Vt)] + [[C.sub.2]/[X.sub.8]] [e.sup.(a+[X.sub.8])(y+Kx+Vt)], (26) v = -(V + a) - [C.sub.3]K[e.sup.a(y+Kx+Vt)] (27) + [K[C.sub.1]/[X.sub.7]] [e.sup.(a-[X.sub.7])(y+Kx+Vt)] - [[K[C.sub.2]]/[X.sub.8]][e.sup.(a+[X.sub.8])(y+Kx+Vt)]. Case II. The Eq. (22) for K = 0 ([OMEGA](s) = 0) becomes -[alpha]A(1 + c[e.sup.as])R" + [[nu] - [alpha]aA (3 + 2c[e.sup.as])]R' +[A(1 + c[e.sup.as]) + 2a[nu] - [a.sup.2]A(3 + 2c[e.sup.as])]R = 0. (28) In order to solve Eq. (28) we put [sigma] = [e.sup.as] and get -[alpha] A[a.sup.2] (1 + c[sigma]) [[sigma].sup.2] [[[d.sup.2]R]/[d[[sigma].sup.2]]] + a [[nu] - [alpha]aA (4 + 3c[sigma])] [sigma][dR/d[sigma]] +[Ac (1 - 2[alpha][a.sup.2])[sigma] + 2a[nu] + A(1 - 3[alpha][a.sup.2]A)]R = 0. (29) The solution of Eq. (29) for a = c = 1, is found through MATHEMATICA 4.1 and is given as R([sigma]) = [C.sub.5][[sigma].sup.-[[delta].sub.1]][.sub.2.F.sub.1] ([X.sub.1], [X.sub.2]; [X.sub.3]; -[sigma]) + [C.sub.6][[sigma].sup.[[delta].sub.2]][.sub.2.F.sub.1] [[X.sub.4], [X.sub.5]; [X.sub.6]; -[sigma]], (30) where [C.sub.5] and [C.sub.6] are arbitrary constants and [[delta].sub.1] = (3[alpha][A.sub.1] - [nu] + [[delta].sub.3])/2[A.sub.1][alpha], [[delta].sub.2] = (-3[alpha][A.sub.1] + [nu] + [[delta].sub.3])/2[A.sub.1][alpha], [[delta].sub.3] = ([square root of ([A.sub.1.sup.2][alpha] (4 - 3[alpha]) + [nu] (2[A.sub.1][alpha] + [nu]))]), (31) [X.sub.1] = -[1/2] - [square root of ([1/[alpha]] - 1)] + [[[nu] - [[delta].sub.3]]/[2[A.sub.1][alpha]]], [X.sub.2] = -[1/2] + [square root of ([1/[alpha]] - 1)] + [[[nu] - [[delta].sub.3]]/[2[A.sub.1][alpha]]], [X.sub.3] = 1 - [[[delta].sub.3]/[[A.sub.1][alpha]]], [X.sub.4] = 1 + [[[delta].sub.3]/[[A.sub.1][alpha]]], [X.sub.5] = -[1/2] - [square root of ([1/[alpha]] - 1)] + [[[nu] + [[delta].sub.3]]/[2[A.sub.1][alpha]]], [X.sub.6] = -[1/2] + [square root of ([1/[alpha]] - 1)] + [[[nu] + [[delta].sub.3]]/[2[A.sub.1][alpha]]]. In Eq. (30), [.sub.2.F.sub.1] is the hypergeometric function In mathematics, a hypergeometric function can be:
** The hypergeometric function [.sub.2.F.sub.1] has the following series expansion: [.sub.2.F.sub.1](a, b; c; z) = [[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (k=0)] [[(a)[.sub.k] (b)[.sub.k]]/[(c)[.sub.k]]] [[z.sup.k]/k!]. (32) ** The hypergeometric [.sub.2.F.sub.1](a, b; c; z) has a branch cut discontinuity dis·con·ti·nu·i·ty n. pl. dis·con·ti·nu·i·ties 1. Lack of continuity, logical sequence, or cohesion. 2. A break or gap. 3. Geology A surface at which seismic wave velocities change. in the complex z-plane running from 0 to [infinity]. The stream function and the velocity fields are [psi] (x, y, t) = V + x (V + [A.sub.1] (1 + [e.sup.s])) + [integral] [[integral] R(s) ds][e.sup.s]ds, (33) u (x, y, t) = x[A.sub.1][e.sup.s] + [[partial derivative]/[partial derivative]y] [integral] [[integral] R(s) ds][e.sup.s]ds, (34) v (x, y, t) = -(V + [A.sub.1]) - [A.sub.1][e.sup.s] - [[partial derivative]/[partial derivative]x] [integral] [[integral] R(s) ds][e.sup.s]ds, (35) where [A.sub.1] = [[nu] - [phi]]/[[alpha] -1], ([alpha] [not equal to] 1). Case III. The solution of Eq. (29) for [alpha] = 0 is also found through MATHEMATICA 4.1 as R([[theta].sub.1]) = -[[X.sub.9]/[1 + [K.sup.2]]][e.sup.c[[theta].sub.1][X.sub.9]][[theta].sub.1.sup.-1+[X.sub.9]] (36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[theta].sub.1] = [e.sup.as], [A.sub.2] = [nu]a (1 + [K.sup.2]), (37) and ExpIntegral E[n, z] gives the exponential integral In mathematics, the exponential integral Ei(x) is defined as Since 1/t diverges at t function [E.sub.n] (z), which is defined by [E.sub.n] (z) = [[integral].sub.1.sup.[infinity]][e.sup.-zt]/[t.sup.n]dt. (38) ExpIntegral E[n, z] has a branch cut discontinuity in the complex z-plane running from 0 to [infinity]. The stream function and the velocity components in this case are respectively given by [psi] (x, y, t) = V + x (V + [A.sub.2] (1 + c[e.sup.as])) + [integral] [[integral] [R.sub.1] (s) ds][e.sup.as]ds, (39) u (x, y, t) = xca[A.sub.2][e.sup.as] + [[partial derivative]/[partial derivative]y] [integral] [[integral] [R.sub.1] (s) ds][e.sup.as]ds, (40) v (x, y, t) = -(V + [A.sub.2]) - [A.sub.2]c[e.sup.as] - [[partial derivative]/[partial derivative]x] [integral] [[integral] [R.sub.1] (s) ds][e.sup.as]ds. (41) 3. Concluding Remarks The four analytic solutions of the involved non-linear equation for unsteady flow of second-grade fluid are constructed. Almost all such studies in the literature are restricted to the analysis of steady flows. Expressions for stream function and velocity components are obtained in each case. It is noted that Eq. (1) for [[alpha].sub.1] = 0 reduces to the Newtonian case [1]. Moreover, the results in all the cases for K = 0 recovers the results of reference [3]. This tends confidence in the mathematical calculations of the present analysis. Thus, the analytical solutions of the non-linear equations for the unsteadiness of the second-grade fluid is a step forward and may add a significant contribution to the literature. References [1] R. Berker, Integration des equations du mouvemont d'un fluide visqueux incompressible, Handbuch der Physik VII, Springer-Verlag, Berlin, 1963. [2] M. H. Hamdan, An alternative approach to exact solutions of a special class of Navier-Stokes flows, Appl. Math. Comput. 93 (1998), 83-90. [3] T. Hayat, M. R. Mohyuddin, S. Asghar, Some inverse (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. solutions for unsteanian fluid, Tamsui Oxford J. Math. Sci. 21(1) (2005), 1-20. [4] D. Riabouchinsky, Some considerations regarding plane irrotational ir·ro·ta·tion·al adj. Not rotating or involving rotation. motion of a liquid, C. R. Acad. Sci. Paris 179 (1936), 1133-1136. S. Asghar* Department of Mathematics, COMSATS Institute of Information Technology COMSATS Institute of Information Technology (CIIT) is one of the leading institution in Pakistan which is working under the umbrella of COMSATS, with a well-established reputation and a wide range of interests and facilities. H-8 Islamabad 44000, Pakistan M. R. Mohyuddin ([dagger]), T. Hayat ([double dagger double dagger n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ]) Department of Mathematics, Quaid-i-Azam University Quaid-i-Azam University, (more correctly Qaid-i A'zam University) is located in Islamabad, Pakistan, and is one of largest public universities in Pakistan. Founded as the University of Islamabad in 1965, it was later renamed in 1976 as Quaid-i-Azam University 45320 Islamabad 44000, Pakistan A. M. Siddiqui ([section]) Pennsylvania State University Pennsylvania State University, main campus at University Park, State College; land-grant and state supported; coeducational; chartered 1855, opened 1859 as Farmers' High School. , York Campus York, Pennsylvania York, known as the White Rose City (after the Wars of the Roses), is a city located in South Central Pennsylvania. The population was 40,862 at the 2000 census. York is the county seat of York County,GR6 17403, U.S.A. Received August 25, 2005, Accepted November 9, 2005. * E-mail:s_asgharpk@yahoo.com ([dagger]) E-mail:m_raheel@yahoo.com (Corresponding Author) ([double dagger]) E-mail:t_pensy@hotmail.com ([section]) E-mail:E-Mail: ams5@psu.edu |
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