Rotationally symmetric flow of Reiner-Rivlin fluid over a heated porous wall using numerical approach

This research features one parameter family of solutions representing rotationally symmetric flow of non-Newtonian fluid obeying Reiner-Rivlin model. Such flows take place over a revolving plane permeable surface through origin such that fluid at infinity also undergoes uniform rotation about the vertical axis. Heat transfer accompanied with viscous heating effect is also analyzed. A similarity solution is proposed that results into a coupled non-linear system with four unknowns. Boundary layer structure is characterized by a parameter [Formula: see text] that compares angular velocity of external flow with that of the rotating surface. Solutions are developed by a well-known package bvp4c of MATLAB for full range of [Formula: see text]. Flow pattern for different choices of [Formula: see text] is scrutinized by computing both 2 D and 3 D streamlines. It is further noted that value of suction velocity is important with regards to the sign of axial velocity component. Boundary layer suppresses considerably whenever the surface is permeable. For sufficiently high suction velocity with [Formula: see text], entire fluid undergoes rigid body rotation. In no suction case, axially upward flow accelerates for increasing values of parameter [Formula: see text] in the range [Formula: see text], whereas opposite trend is apparent in the case [Formula: see text]. Results for normalized wall shear and Nusselt number are scrutinized for various choices of embedded parameters.

Quasi-Stationary Turbulent Flow in Cylinder Channel with Irregular Walls

A simple asymptotic model of a stationary rotationally symmetric flow in an infinite cylindrical channel with irregular walls is constructed and investigated. We assume that kinematic viscosity is depends on the radial and axial coordinates. The kinematic condition corresponding to the impermeability of the boundary for the liquid is chosen. It is assumed that the classical non-slip condition is completed only in straight segments of the boundary. For parts of the boundary that are not straight, any additional terms, in addition to the kinematic conditions, not required. It is shown that at any fluid flow rate in domains where the curvature of the cylinder boundary is negative, there are toroidal vortices in the stationary flow. The solution of the problem is constructed in analytical form. This solution is valid in any domain with sufficiently smooth cylinder boundaries. As an example, we consider the case of turbulent kinematic viscosity vanishing at the cylinder boundary. To prevent the velocity singularities the roughness of the wall is introduced. The proposed model can simulate the blood flow through the vessels and, in particular, be used to study the quasi-stationary motion of impurities, for example, erythrocytes, in the flow with the known structure.

Concentration Distribution of Photosensitive Liquid in a Droplet Under Ultraviolet Light

A semi-analytical solution for the concentration of photosensitive suspension is developed in a hemispherical droplet illuminated with ultraviolet (UV) laser. A biharmonic equation in stream function is analytically solved using toroidal coordinates, which is used to solve the transport equation for concentration. Flow pattern and photosensitive material concentration are affected by the peak location of the UV light intensity, which corresponds to the surface tension profile. When the laser beam is moved from the droplet center to its edge, a rotationally symmetric flow pattern changes from a single counter clockwise circulation to a circulation pair and finally to a single clockwise circulation. This modulation in the orientation of circulation modifies the concentration distribution of the photosensitive material. The distribution depends on both diffusion from the droplet surface and the Marangoni convection. The region beneath the droplet surface away from the UV light intensity peak has low concentration, while the region near the downward dividing streamline has the highest concentration. When the UV light peak reaches the droplet edge, the concentration is high everywhere in the droplet.

Concentration Distribution of Photosensitive Liquid in a Droplet Under UV Light

A semi-analytical solution for the concentration of photosensitive suspension is developed in a hemispherical droplet illuminated with UV laser. A biharmonic equation in stream function is analytically solved using toroidal coordinates and the velocity is then used to solve the mass transport equation for concentration. Flow pattern and photosensitive material concentration are affected by the peak location of the UV light intensity, which corresponds to a surface tension profile. When the laser beam is moved from the droplet center to its edge, a rotationally symmetric flow pattern changes from a single counter clockwise circulation to a circulation pair and finally to a single clockwise circulation. This modulation in the orientation of circulation modifies the concentration distribution of the photosensitive material. The distribution depends on both diffusion from the droplet surface as well as Marangoni convection. The region beneath the droplet surface away from the UV light intensity peak has low concentration, while the region near the downward dividing streamline has the highest concentration. When the UV light peak reaches the droplet edge, the concentration is high everywhere in the droplet.

Unsteady flow in a rotating torus after a sudden change in rotation rate

We consider the temporal evolution of a viscous incompressible fluid in a torus of finite curvature; a problem first investigated by Madden & Mullin (J. Fluid Mech., vol. 265, 1994, pp. 265–217). The system is initially in a state of rigid-body rotation (about the axis of rotational symmetry) and the container’s rotation rate is then changed impulsively. We describe the transient flow that is induced at small values of the Ekman number, over a time scale that is comparable to one complete rotation of the container. We show that (rotationally symmetric) eruptive singularities (of the boundary layer) occur at the inner or outer bend of the pipe for a decrease or an increase in rotation rate respectively. Moreover, on allowing for a change in direction of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which eruptive singularities can occur at both the inner and outer bend simultaneously. We also demonstrate that the flow is susceptible to a combination of axisymmetric centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability arises as a consequence of the developing eruption and is shown to be in qualitative agreement with the experimental observations of Madden & Mullin (1994). Throughout our work, detailed quantitative comparisons are made between asymptotic predictions and finite- (but small-) Ekman-number Navier–Stokes computations using a finite-element method. We find that the boundary-layer results correctly capture the (finite-Ekman-number) rotationally symmetric flow and its global stability to linearised perturbations.