exact+solution+for+the+unsteady+flow+of+a+semi infinite+micropolar+fluid

Acta Mech. Sin. (2011) 27(3):354–359 DOI 10.1007 / s10409-011-0452-4 RESEARCH P APER Exact solution for the unsteady flow of a semi-infinite micropolar fluid H.H. Sherief · M.S. Faltas · E.A. Ashmawy Received: 27 October 2009 / Revised: 2 December 2010 / Accepted: 26 December 2010 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011 Abstract The unsteady motion of an incompressible micropolar fluid filling a half-space bounded by a horizontal infinite plate that started to move suddenly is considered. Lapla ce transfor m tech nique s are used . The solution in the Laplace transform domain is obtained by using a direct ap- proac h. The inverse Laplace transforms are obtained in an exact manner using the complex inversion formula of the transform together with contour integration techniques. The solution in the case of classical viscous fluids is recovered as a special case of this work when the micropolarity coefficient is assumed to be zero. Nume rical comput ation s are carri ed out and represented graphically. Keywords Unsteady motion · Laplace transform · Microp- olar fluid 1 Introducti on The theory of microfluids introduced by Eringen [1], deals with a class of fluids which exhibit certain microscopic ef- fects arising from the local structure and micro-motions of the fluid elements. The microfluids are categorized into three mai n sub -cl asses; the firs t is the the ory of mic romorphic fluids which has nine degrees of freedom, three for trans- latio n, three for micro rotat ion and three for micros tretc h of microele ments [2, 3]. The theory of microst retch fluid s is the second cat egory whi ch has seven de gre es of fre e- dom, three for translation, three for microrotation and one for micro stretc h [4]. The thir d sub-class of microfluid s is the theory of micropolar fluids which has only six degrees of freedom, three for translation and three for micro- H.H. Sherief · M.S. Faltas · E.A. Ashmawy ( ) Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt e-mail: emad [email protected] rotat ion of microelements. The micropolar fluids exhi bit micro-rotational eff ects and micro -rota tiona l inertia. Erin- gen’s micropolar model includes the classical Navier–Stokes equations as a special case, but can cover, both in theory and applications, many more phenomena than the classical model. Physically , the mathematical model underlying micropolar fluids may represent the behavior of polymeric ad- ditives, animal blood with rigid cells, lubricants, liquid crys- tals, dirty oils and solutions of colloidal suspensions [2,4,5]. In practice, the theory requires that one must solve an addi- tio nal tra nsp ort equ ati on repres ent ing the pri nci ple of con ser- vation of local angular momentum, as well as the usual trans- port equations for the conservation of mass and momentum. Additional constitutiv e parameters are also introduced. Ex- tensive reviews of the theory and applications can be found in the revie w articles by Arima n et al. [6, 7] and the recen t books by Łukasze wicz [8] and Eringen [3]. The steady micro polar Couette flow betwe en two parallel pl ates wa s prese nted in Ref . [9]. Cvetkovi´ e [10] dis- cussed the problem of steady micropolar flow between two parallel plates with couple stress boundary conditions. Study of MHD oscillatory flow of a micropolar fluid over a vertical porou s plate was stu died by Kim and Lee in Ref. [11]. El- Arabawy [12] studied the eff ect of suction / injection on the flow of a micropolar fluid past a continuously moving plate in the pre sen ce of rad iat ion . The shea r flow of a mic rop - olar fluid contained between two parallel plates was studied by Mizu kami in Ref. [13]. Łuk aszewicz [14] studi ed the long time behavio r of 2D micropol ar fluid flows. Ye et al. [15] reconst ructe d the gover ning equatio ns for lamin ar flows of micropolar fluid in rectangular microchannels and obtained some numerical results for the velocity and microrotation vectors using a procedure based on the Chebyshev collo catio n method. Wang and Zhu [16] hav e discusse d a numerical study of the non-Newtonian behavior for a finite journal bearing lubricated with micropolar fluids consider- ing both thermal and cavitating eff ects. They hav e deriv ed the modified Reynolds equation and energy equation based

exact+solution+for+the+unsteady+flow+of+a+semi infinite+micropolar+fluid

Documents