Numerical and analytical solutions of Stokes theory of couple stress uid under the e ects of constant, space, and variable viscosity in the INCLINED CHANNEL are discussed in this paper. The considered couple stress uid is described mathematically with the de nition of the stress tensor. The dimensional form of the boundary value problem is transformed into dimensionless form by de ning dimensionless quantities, and is then solved with the aid of the perturbation technique. The analytical expressions of velocity and temperature of all cases based on the viscosity of the couple stress uid are presented. For the validity of the perturbation solution, the Pseudo-Spectral collocation method is employed for each case of the viscosity model including constant, space, and Vogel's models, respectively. The solution of the perturbation method and Pseudo-Spectral methods are shown together in the graphs. The e ects of couple stress parameters on velocity and temperature distributions are also elaborated with physical reasoning in the results and discussion section. It is observed that the velocity and temperature of uid escalate via the pressure gradient parameter and Brinkman number, while decelerating via the couple stress parameter.

The present work is a numerical study of simultaneous heat and mass transfer with phase change in an INCLINED CHANNEL formed by two parallel plates. The lower one is covered by a thin liquid water film and the upper one is considered impermeable. The plates are maintained at a constant temperature. The liquid film is assumed to be extremely thin and its temperature is uniform and equal to that of the wall. Thermo-physical properties are considered constant and the Boussinesq assumption is adopted. Results show that the effects of the buoyancy forces on the hydrodynamic, thermal and mass fraction fields are important. These effects depend on the CHANNEL inclination and may result on flow reversal when the CHANNEL approaches the vertical position. This phenomenon is addressed and a flow reversal chart, as well as the corresponding correlations, for different CHANNEL inclinations is given. These correlations give the values of Grashof numbers, which induce flow reversal for a given Reynolds number and inclination angle.

Scope and Background: Dissipating high kinetic energy of supercritical flows to protect downstream structures has always been a concern of hydraulic structure engineers. One of the approaches to tackle this problem is the utilization of hydraulic jump phenomena in which a great amount of kinetic energy is dissipated through turbulence which is more pronounced in roller part and conversion to potential energy in terms of depth increase at downstream end and turbulence. A hydraulic jump may occur in prismatic or non-prismatic, converged or diverged and horizontal or INCLINED CHANNELs. However, oblique shock waves initiating at the start of a contracted CHANNEL, interact with each other and sidewalls and may create a complex flow pattern which is detrimental to the CHANNEL itself and downstream facilities. The present research aims at studying hydraulic jumps taken place in a converging INCLINED CHANNEL. The main parameters of a hydraulic jump such as its location, initial depth, ratio of conjugate depths, jump length and energy dissipation are studied for various inclination and convergence ratios and inflow conditions. Methodology: The experiments were conducted in a CHANNEL with different bed slopes of 0, 5, 10, and 15 percent, and convergence angles of 3. 66 and 5. 4 degrees. The end sills of 0. 75 to 11 cm high were installed at the end, depending on the bed slope, to fix the jump location in the CHANNEL. The entrance was set carefully to produce the least disturbance due to sharp edges and protruding elements appeared in the flow,hence, a symmetric hydraulic jump may be observed all over a cross-section. To double-check the accuracy of measurements, clips of various hydraulic jumps were shot through sidewalls, converted into the images and digitized using Grapher TM. Discussion and Conclusion: The location of hydraulic jump approached downstream CHANNEL due to gravity forces as the CHANNEL slope increased. Therefore, in the converged CHANNEL by reducing the width, primary Froude number enhanced and the jump location tended upstream by increasing the convergence angle. Jump position was more sensitive to the sill depth variation in lower slopes. Specifying a unique initial depth in the converged CHANNEL was challenging. There were oblique waves originated from the concave corners and coincided at the centerline of the CHANNEL. In cases where the hydraulic jump occurred before the coincidence of the oblique waves, there were different depths at the front of the jump. In this work, the centerline depth was selected as the reference depth in the development of regressive equation. The ratio of conjugate depths increased directly with the primary Froude number and bed slope. By enhancing the bed slope, the initial depth decreased and the conjugate depth increased, accordingly. The length of a hydraulic jump, was more a function of bed slope and Froude number and less a function of convergence. It enhanced with the increase of primary Froude number, however, with a constant Fr it was longer in higher slopes. The energy dissipation increased by both the bed slope and convergence ratio. By increasing the primary Froude number, the difference between energy dissipation in various bed slopes approached that of a horizontal bed. Using regressive models and statistical analysis, empirical explicit equations were developed for the location, initial depth, conjugate depths ratio, estimation of length, and energy dissipation of a hydraulic jump in an INCLINED converging CHANNEL.

The influence of slight inclination, ‘α’ (i. e., 10°, 15°, and 20°) of the CHANNEL comprised of shrouded vertical rectangular non-isothermal fin array has been computationally investigated. Simulations are performed to obtain the convective coefficient of heat transfer for the different dimensionless fin spacing (S*= 0. 2, 0. 3, 0. 5), non-dimensional fin tip to shroud clearances (C*= 0. 1, 0. 2 and 0. 3), Grashof numbers (Gr= 1. 08×105, 4. 42×105 and 11. 5×105), fin lengths (L= 0. 25m and 0. 5m) and fin heights (H= 0. 025m, 0. 04m, and 0. 055m). Hydrodynamic behavior of the fluid indicates that a significant amount of flow reversal occurs near the entrance of the CHANNEL at very low inclination which vanishes with the increase in inclination. Further, at higher length, reverse flow is found to occur only in the clearance zone. An increase in the value of ‘α’ from 10° to 20°, results in enhancement of convective coefficient up to a maximum of 161%. An increase in the value of fin spacing from 0. 2 to 0. 5 results in the enhancement of the convective heat transfer coefficient. At α= 15° and 20°, the heat transfer is enhanced by 84. 1% and 101. 6%, respectively, while at α=10°, the same is reduced by 33. 3%. At lower fin spacing (S*<0. 3) increase in C* tends to reduce the efficiency. Further, the efficiency of the finite conductive fin is found to reduce by almost 8% with an increase in Grashof number.

The peristaltic motion of a third order fluid due to asymmetric waves propagating on the sidewalls of a INCLINED asymmetric CHANNEL is discussed. The key features of the problem includes long-wavelength and low-Reynolds number assumptions. A mathematical analysis has been carried out to investigate the effect of slip condition, variable viscosity and magnetohydrodynamics (MHD).Followed by the nondimensionalization of the nonlinear governing equations along with the non-linear boundary conditions, a perturbation analysis is made. For the validity of the approximate solution, a numerical solution is obtained using the iterative collocation technique.

Peristaltic transport of an incompressible electrically conducting viscous fluid in an INCLINED planar asymmetric CHANNEL is studied.The asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitude and phase. The closed form solutions of momentum and energy equation in presence of viscous dissipation term are obtained for long wave length and low Reynolds number approximations.The effects of different parameters entering into the problem are discussed numerically and explained graphically.