The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Nature And Properties Of Soils

저작근 근전도에 관한 정상교합자와 ii급 부정교합자의 비교 연구

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

Hydrodynamic and hydromagnetic stability

Motivated from numerous practical applications, the present theoretical and numerical work investigates the nonlinear magnetohydrodynamic (MHD) laminar boundary layer flow of an incompressible, viscous fluid over a porous stretching sheet in the presence of suction/injection (mass transpiration). The flow characteristics are obtained by solving the underlying highly nonlinear ordinary differential equation using homotopy analysis method. The effect of parameters corresponding to suction/injection (mass transpiration), applied magnetic field, and porous stretching sheet parameters on the nonlinear flow is investigated. The asymptotic limits of the parameters regarding the flow characteristics are obtained mathematically, which compare very well with those obtained using the homotopy analysis technique. A detailed numerical study of the laminar boundary layer flow in the vicinity of the porous stretching sheet in MHD and offers a particular choice of the parametric values to be taken in order to practically model a particular type of the event among suction and injection at the sheet surface.

/pdf/mass-transpiration-in-nonlinear-mhd-flow-due-to-porous-blf8na77p3.pdf

Mass Transpiration in Nonlinear MHD Flow Due to Porous Stretching Sheet.

Convective instability is suppressed by sinusoidal variation in temperature of horizontal boundaries of a ferrofluid layer subjected to vertical magnetic field. On increasing the amplitude of modulation, instability may arise in the form of oscillations which may have time period equal to that of the temperature modulation or double of it. The effect of frequency of modulation, applied magnetic field, and Prandtl number on the onset of a periodic flow in the ferrofluid layer has been investigated numerically using the Floquet theory. Some theoretical results have also been obtained to discuss the limiting behavior of the underlying instability with the temperature modulation. Depending upon the parameters, the flow patterns at the onset of instability have been found to consist of time-periodically oscillating vertical magnetoconvective rolls.

Temperature modulation in ferrofluid convection

Couette flow in ferrofluids with magnetic field

The present investigation is focused on the study of a micropolar flow in a porous medium subject to inclined magnetic field, mass transpiration and internal radiation. The analytical solutions of the flow and temperature fields were derived in closed forms for the first time here by using similarity transformations. The transformed velocity, microrotation and temperature fields were plotted and analyzed for a number of relative dimensionless parameters, while skin friction coefficient and Nusselt number were also calculated and discussed. The results showed that the micropolar flow may accelerate or decelerate depending on the effect of porous medium, the mass transpiration, the radiation and the inclined applied magnetic field. Moreover, heat transfer may enhanced or diminished depending on the same phenomena. The present analytical results are anticipated to be of great significance regarding the effect of important MHD and heat transfer parameters on micropolar fluids and they can be used in a variety of industrial applications.

Combined Effect of Radiation and Inclined MHD Flow of a Micropolar Fluid Over a Porous Stretching/Shrinking Sheet with Mass Transpiration

https://iopscience.iop.org/article/10.1088/0169-5983/43/2/025502/pdf

Jitender Singh

Papers

Mass Transpiration in Nonlinear MHD Flow Due to Porous Stretching Sheet.

Temperature modulation in ferrofluid convection

Couette flow in ferrofluids with magnetic field

Combined Effect of Radiation and Inclined MHD Flow of a Micropolar Fluid Over a Porous Stretching/Shrinking Sheet with Mass Transpiration

Convective instability in a ferrofluid layer with temperature-modulated rigid boundaries