Asymptotic expansions for the Sturm-Liouville problem by homotopy perturbation method

The asymptotic formulae for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the Dirichlet boundary conditions when the potential is square integrable on [0, 1] are obtained by using homotopy perturbation method.     

An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study)

Transition to turbulence in axially symmetrical laminar pipe flows with periodic time dependence classified as pure oscillating and pulsatile (pulsating) ones is the concern of the paper. The current state of art on the transitional characteristics of pulsatile and oscillating pipe flows is introduced with a particular attention to the utilized terminology and methodology. Transition from laminar to turbulent regime is usually described by the presence of the disturbed flow with small amplitude perturbations followed by the growth of turbulent bursts. The visual treatment of velocity waveforms is therefore a preferred inspection method. The observation of turbulent bursts first in the decelerating phase and covering the whole cycle of oscillation are used to define the critical states of the start and end of transition, respectively. A correlation study referring to the available experimental data of the literature particularly at the start of transition are presented in terms of the governing periodic flow parameters. In this respect critical oscillating and time averaged Reynolds numbers at the start of transition; Re _os,crit and Re _ta,crit are expressed as a major function of Womersley number, $\sqrt {\omega ^\prime } $ defined as dimensionless frequency of oscillation, f. The correlation study indicates that in oscillating flows, an increase in Re _os,crit with increasing magnitudes of $\sqrt {\omega ^\prime } $ is observed in the covered range of $1<\sqrt {\omega ^\prime } <72$ . The proposed equation (Eq. 7), ${\rm{Re}}_{os,crit} ={\rm{Re}}_{os,crit} \left( {\sqrt {\omega ^\prime } } \right)$ , can be utilized to estimate the critical magnitude of $\sqrt {\omega ^\prime }$ at the start of transition with an accuracy of ±12?% in the range of $\sqrt {\omega ^\prime } <41$ . However in pulsatile flows, the influence of $\sqrt {\omega ^\prime }$ on Re _ta,crit seems to be different in the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ . Furthermore there is rather insufficient experimental data in pulsatile flows considering interactive influences of $\sqrt {\omega ^\prime } $ and velocity amplitude ratio, A ₁. For the purpose, the measurements conducted at the start of transition of a laminar sinusoidal pulsatile pipe flow test case covering the range of 0.21<?A ₁?<0.95 with $\sqrt {\omega ^\prime } <8$ are evaluated. In conformity with the literature, the start of transition corresponds to the observation of first turbulent bursts in the decelerating phase of oscillation. The measured data indicate that increase in $\sqrt {\omega ^\prime } $ is associated with an increase in Re _ta,crit up to $\sqrt {\omega ^\prime } =3.85$ while a decrease in Re _ta,crit is observed with an increase in $\sqrt {\omega ^\prime } $ for $\sqrt {{\omega }'} >3.85$ . Eventually updated portrait is pointing out the need for further measurements on i) the end of transition both in oscillating and pulsatile flows with the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ , and ii) the interactive influences of $\sqrt {\omega ^\prime } $ and A ₁ on Re _ta,crit in pulsatile flows with the range of $\sqrt {\omega ^\prime } >8$ .     

Purification of Acetylcholinesterase by 9-Amino-1,2,3,4-tetrahydroacridine from Human Erythrocytes

The acetylcholinesterase enzyme was purified from human erythrocyte membranes using a simple and effective method in a single step. Tacrine (9-amino-1,2,3,4-tetrahydroacridine) is a well-known drug for the treatment of Alzheimer's disease, which inhibits cholinesterase. We have developed a tacrine ligand affinity resin that is easy to synthesize, inexpensive and selective for acetylcholinesterase. The affinity resin was synthesized by coupling tacrine as the ligand and l-tyrosine as the spacer arm to CNBr-activated Sepharose 4B. Acetylcholinesterase was purified with a yield of 23.5 %, a specific activity of 9.22 EU/mg proteins and 658-fold purification using the affinity resin in a single step. During purification, the enzyme activity was measured using acetylthiocholine iodide as a substrate and 5,5′-dithiobis-(2-nitrobenzoicacid) as the chromogenic agent. The molecular weight of the enzyme was determined as about 70 kDa monomer upon disulphide reduction by sodium dodecyl sulphate polyacrylamide gel electrophoresis. K _m, V _max, optimum pH and optimum temperature for acetylcholinesterase were found by means of graphics for acetylthiocholine iodide as the substrate. The optimum pH and optimum temperature of the acetylcholinesterase were determined to be 7.4 and 25–35 °C. The Michaelis–Menten constant (K _m) for the hydrolysis of acetylthiocholine iodide was found to be 0.25 mM, and the V _max was 0.090 μmol/mL/min. Maximum binding was achieved at 2 °C with pH 7.4 and an ionic strength of approximately 0.1 M. The capacity for the optimum condition was 0.07 mg protein/g gel for acetylcholinesterase.