Infinite Wall Research Articles

Interplay of non-conducting and conducting walls on magnetohydrodynamic (MHD) natural convection flow in vertical micro-channel in the presence of induced magnetic field

In this research paper, exact solution for fully developed magnetohydrodynamic (MHD) natural convection flow of viscous, incompressible, electrically conducting fluid in vertical micro-channel formed by electrically non-conducting and conducting infinite vertical parallel walls in the presence of velocity slip and temperature jump at the micro-channel walls is investigated. The influence of induced magnetic field (I.M.F) arising due to the motion of an electrically conducting fluid in the presence of transverse magnetic field is taken into consideration. Due to the presence of induced magnetic field, the momentum and induction equations are coupled. The exact solutions in dimensionless form have been obtained under relevant boundary conditions. The expressions for velocity field, the induced magnetic field, temperature field, induced current density and skin friction have been obtained. The effects of the various physical parameters appearing into the model such as rarefaction, fluid wall interaction, Hartmann number and the magnetic Prandtl number are demonstrated through graphs and table. The results indicate that, increase in Hartmann number and magnetic Prandtl number causes a pronounced reduction in volume flow rate.

Effect of Thermal Radiation, Chemical Reaction and Viscous Dissipation on MHD Flow

This study examines the effect of thermal radiation, chemical reaction and viscous dissipation on a magnetohydro- dynamic flow in between a pair of infinite vertical Couette channel walls. The momentum equation accounts the effects of both the thermal and the concentration buoyancy forces of the flow. The energy equation addresses the effects of the thermal radiation and viscous dissipation of the flow. Also, the concentration equation includes the effects of molecular diffusivity and chemical reaction parameters. The gray colored fluid considered in this study is a non-scattering medium and has the property of absorbing and emitting radiation. The Roseland approximation is used to describe the radiative heat flux in the energy equation. The velocity of flow transforms kinetic energy into heat energy. The increment of the velocity due to internal energy results in heating up of the fluid and consequently it causes increment of the thermal buoyancy force. The Eckert number being the ratio of the kinetic energy of the flow to the temperature difference of the channel walls is directly proportional to the thermal energy dissipation. It can be observed that increasing the Eckert number results in increasing velocity. A uniform magnetic field is applied perpendicular to the channel walls. The temperature of the moving wall is high enough due to the presence of thermal radiation. The solution of the governing equations is obtained using regular perturbation techniques. These techniques help to convert partial differential equations to a set of ordinary differential equations in dimensionless form and thus they are solved analytically. The following results are obtained: from the simulation study it is observed that the flow pattern of the fluid is affected due to the influence of the thermal radiation, the chemical reaction and viscous dissipation. The increment in the Hartmann number results in the increment of the Lorentz force but a decrement in velocity of the flow. An increment in the radiative parameter results in a decrement in temperature. An increment in the Prandtl number results in a decrement in thermal diffusivity. An increment in both the chemical reaction parameter and molecular diffusivity results in a decrement in concentration.

A k · p treatment of edge states in narrow 2D topological insulators, with standard boundary conditions for the wave function and its derivative

For 2D topological insulators with strong electron–hole hybridization, such as HgTe/CdTe quantum wells, the widely used 4 × 4 k · p Hamiltonian based on the first electron and heavy hole sub-bands yields an equal number of physical and spurious solutions, for both the bulk states and the edge states. For symmetric bands and zero wave vector parallel to the sample edge, the mid-gap bulk solutions are identical to the edge solutions. In all cases, the physical edge solution is exponentially localized to the boundary and has been shown previously to satisfy standard boundary conditions for the wave function and its derivative, even in the limit of an infinite wall potential. The same treatment is now extended to the case of narrow sample widths, where for each spin direction, a gap appears in the edge state dispersions. For widths greater than 200 nm, this gap is less than half of the value reported for open boundary conditions, which are called into question because they include a spurious wave function component. The gap in the edge state dispersions is also calculated for weakly hybridized quantum wells such as InAs/GaSb/AlSb. In contrast to the strongly hybridized case, the edge states at the zone center only have pure exponential character when the bands are symmetric and when the sample has certain characteristic width values.

A note on Stokes’ problem in dense granular media using the -rheology

The classical Stokes’ problem describing the fluid motion due to a steadily moving infinite wall is revisited in the context of dense granular flows of mono-dispersed beads using the recently proposed$\unicode[STIX]{x1D707}(I)$-rheology. In Newtonian fluids, molecular diffusion brings about a self-similar velocity profile and the boundary layer in which the fluid motion takes place increases indefinitely with time$t$as$\sqrt{\unicode[STIX]{x1D708}t}$, where$\unicode[STIX]{x1D708}$is the kinematic viscosity. For a dense granular viscoplastic liquid, it is shown that the local shear stress, when properly rescaled, exhibits self-similar behaviour at short time scales and it then rapidly evolves towards a steady-state solution. The resulting shear layer increases in thickness as$\sqrt{\unicode[STIX]{x1D708}_{g}t}$analogous to a Newtonian fluid where$\unicode[STIX]{x1D708}_{g}$is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular medium, such as grain diameter$d$, density$\unicode[STIX]{x1D70C}$and friction coefficients, but also on the applied pressure$p_{w}$at the moving wall and the solid fraction$\unicode[STIX]{x1D719}$(constant). In addition, the$\unicode[STIX]{x1D707}(I)$-rheology indicates that this growth continues until reaching the steady-state boundary layer thickness$\unicode[STIX]{x1D6FF}_{s}=\unicode[STIX]{x1D6FD}_{w}(p_{w}/\unicode[STIX]{x1D719}\unicode[STIX]{x1D70C}g)$, independent of the grain size, at approximately a finite time proportional to$\unicode[STIX]{x1D6FD}_{w}^{2}(p_{w}/\unicode[STIX]{x1D70C}gd)^{3/2}\sqrt{d/g}$, where$g$is the acceleration due to gravity and$\unicode[STIX]{x1D6FD}_{w}=(\unicode[STIX]{x1D70F}_{w}-\unicode[STIX]{x1D70F}_{s})/\unicode[STIX]{x1D70F}_{s}$is the relative surplus of the steady-state wall shear stress$\unicode[STIX]{x1D70F}_{w}$over the critical wall shear stress$\unicode[STIX]{x1D70F}_{s}$(yield stress) that is needed to bring the granular medium into motion. For the case of Stokes’ first problem when the wall shear stress$\unicode[STIX]{x1D70F}_{w}$is imposed externally, the$\unicode[STIX]{x1D707}(I)$-rheology suggests that the wall velocity simply grows as$\sqrt{t}$before saturating to a constant value whereby the internal resistance of the granular medium balances out the applied stresses. In contrast, for the case with an externally imposed wall speed$u_{w}$, the dense granular medium near the wall initially maintains a shear stress very close to$\unicode[STIX]{x1D70F}_{d}$which is the maximum internal resistance via grain–grain contact friction within the context of the$\unicode[STIX]{x1D707}(I)$-rheology. Then the wall shear stress$\unicode[STIX]{x1D70F}_{w}$decreases as$1/\sqrt{t}$until ultimately saturating to a constant value so that it gives precisely the same steady-state solution as for the imposed shear-stress case. Thereby, the steady-state wall velocity, wall shear stress and the applied wall pressure are related as$u_{w}\sim (g\unicode[STIX]{x1D6FF}_{s}^{2}/\unicode[STIX]{x1D708}_{g})f(\unicode[STIX]{x1D6FD}_{w})$where$f(\unicode[STIX]{x1D6FD}_{w})$is either$O(1)$if$\unicode[STIX]{x1D70F}_{w}\sim \unicode[STIX]{x1D70F}_{s}$or logarithmically large as$\unicode[STIX]{x1D70F}_{w}$approaches$\unicode[STIX]{x1D70F}_{d}$.

Including diverging electrostatic potential in 3D-RISM theory: The charged wall case.

Although three-dimensional site-site molecular integral equations of liquids are a powerful tool of the modern theoretical chemistry, their applications to the problem of characterizing the electrical double layer originating at the solid-liquid interface with a macroscopic substrate are severely limited by the fact that an infinitely extended charged plane generates a divergent electrostatic potential. Such potentials cannot be treated within the standard 3D-Reference Interaction Site Model equation solution framework since it leads to functions that are not Fourier transformable. In this paper, we apply a renormalization procedure to overcome this obstacle. We then check the validity and numerical accuracy of the proposed computational scheme on the prototypical gold (111) surface in contact with water/alkali chloride solution. We observe that despite the proposed method requires, to achieve converged charge densities, a higher spatial resolution than that suited to the estimation of biomolecular solvation with either 3D-RISM or continuum electrostatics approaches, it still is computationally efficient. Introducing the electrostatic potential of an infinite wall, which is periodic in 2 dimensions, we avoid edge effects, permit a robust integration of Poisson's equation, and obtain the 3D electrostatic potential profile for the first time in such calculations. We show that the potential within the electrical double layer presents oscillations which are not grasped by the Debye-Hückel and Gouy-Chapman theories. This electrostatic potential deviates from its average of up to 1-2 V at small distances from the substrate along the lateral directions. Applications of this theoretical development are relevant, for example, for liquid scanning tunneling microscopy imaging.

On the force–velocity relationship of a bundle of rigid bio-filaments

In various cellular processes, bio-filaments like F-actin and F-tubulin are able to exploit chemical energy associated with polymerization to perform mechanical work against an obstacle loaded with an external force. The force–velocity relationship quantitatively summarizes the nature of this process. By a stochastic dynamical model, we give, together with the evolution of a staggered bundle of Nf rigid living filaments facing a loaded wall, the corresponding force–velocity relationship. We compute the evolution of the model in the infinite wall diffusion limit and in supercritical conditions (monomer density reduced by critical density ρ^1>1), and we show that this solution remains valid for moderate non-zero values of the ratio between the wall diffusion and the chemical time scales. We consider two classical protocols: the bundle is opposed either to a constant load or to an optical trap setup, characterized by a harmonic restoring force. The constant load case leads, for each F value, to a stationary velocity Vstat(F;Nf,ρ^1) after a relaxation with characteristic time τmicro(F). When the bundle (initially taken as an assembly of filament seeds) is subjected to a harmonic restoring force (optical trap load), the bundle elongates and the load increases up to stalling over a characteristic time τOT. Extracted from this single experiment, the force–velocity VOT(F;Nf,ρ^1) curve is found to coincide with Vstat(F;Nf,ρ^1), except at low loads. We show that this result follows from the adiabatic separation between τmicro and τOT, i.e., τmicro ≪ τOT.

Confining annealed branched polymers inside spherical capsids.

The Lifshitz equation for the confinement of a linear polymer in a spherical cavity of radius R has the form of the Schrödinger equation for a quantum particle trapped in a potential well with flat bottom and infinite walls at radius R. We show that the Lifshitz equation of a confined annealed branched polymer has the form of the Schrödinger equation for a quantum harmonic oscillator. The resulting confinement energy has a 1/R4 dependence on the confinement radius R, in contrast to the case of confined linear polymers, which have a 1/R2 dependence. We discuss the application of this result to the problem of the confinement of single-stranded RNA molecules inside spherical capsids.

Self-propelled colloidal particle near a planar wall: A Brownian dynamics study

Miniaturized, self-propelled locomotors use chemo-mechanical transduction mechanisms to convert fuel in the environment to autonomous motion. Recent experimental and theoretical studies demonstrate that these autonomous engines can passively follow the contours of solid boundaries they encounter. Boundary guidance, however, is not necessarily stable: Mechanical disturbances can cause the motor to hydrodynamically depart from the passively guided pathway. Furthermore, given the scaled-down size of micromotors (typically 100 nm -10 $\mu$m), Brownian thermal fluctuation forces are necessarily important and these stochastic forces can randomize passively-steered trajectories. Here we examine theoretically the stability of boundary guided motion of micromotors along infinite planar walls to mechanical disturbances and to Brownian forces. Our aim is to understand under what conditions this passively guided motion is stable. We choose a locomotor design in which spherical colloids are partially coated with a catalytic cap that reacts with solute to produce a product. The product is repelled from the particle surface, causing the particle to move with the inert face at the front (autonomous motion via self-diffusiophoresis). When propelled towards a planar wall, deterministic hydrodynamic studies demonstrate that these locomotors can exhibit, for large enough cap sizes, steady trajectories in which the particle either skims unidirectionally along the surface at a constant distance from the wall, or becomes stationary. We first investigate the linear hydrodynamic stability of these states by expanding the equations of motion about the states, and find that linear perturbations decay exponentially in time. We then study the effects of thermal fluctuations by formulating a Langevin equation for the particle motion which includes the Brownian stochastic force...

Growth mechanisms of perturbations in boundary layers over a compliant wall

The temporal modal and nonmodal growth of three-dimensional perturbations in the boundary-layer flow over an infinite compliant flat wall is considered. Using a wall-normal velocity/wall-normal vorticity formalism, the dynamic boundary condition at the compliant wall admits a linear dependence on the eigenvalue parameter, as compared to a quadratic one in the canonical formulation of the problem. This greatly simplifies the accurate calculation of the continuous spectrum by means of a spectral method, thereby yielding a very effective filtering of the pseudospectra as well as a clear identification of instability regions. The regime of global instability is found to be matching the regime of the favorable phase of the forcing by the flow on the compliant wall so as to enhance the amplitude of the wall. An energy-budget analysis for the least-decaying hydroelastic (static-divergence, traveling-wave-flutter and near-stationary transitional) and Tollmien--Schlichting modes in the parameter space reveals the primary routes of energy flow. Moreover, the flow exhibits a slower transient growth for the maximum growth rate of a superposition of streamwise-independent modes due to a complex dependence of the wall-boundary condition with the Reynolds number. The initial and optimal perturbations are compared with the boundary-layer flow over a solid wall; differences and similarities are discussed. Unlike the solid-wall case, viscosity plays a pivotal role in the transient growth. A slowdown of the maximum growth rate with the Reynolds number is uncovered and found to originate in the transition of the fluid-solid interaction from a two-way to a one-way coupling. Finally, a term-by-term energy budget analysis is performed to identify the key contributors to the transient growth mechanism.

A cellular automaton-lattice Boltzmann method for modeling growth and settlement of the dendrites for Al-4.7%Cu solidification

A coupled model of CA-LBM for simulating the simultaneous growth and motion of equiaxed dendrites in solidifying melt is presented. The model uses the CA method for the growth of the dendrite, the LBM for the momentum, mass, and heat transfer of the domain, the Ladd method for handling the rigid solid–liquid interfaces and the equations of motion for tracking the translational and rotational motion of the solid phase. The simulation of a circular particle between two infinite and parallel walls proved the accuracy of boundary treatment. Two-dimensional computations of dendritic translation and rotation show that the shape preservation conditions are satisfied in this method. Then, the present method is used to calculate the growth behavior of a binary alloy dendrite in the process of its dropping motion in the melt. Initially, the dendrite is standard and only translation happened, but later the dendrite is inclined and simultaneous translational and rotational motion happened. The simulation result shows that during its falling process the dendrite settles and grows in a specific morphology, which differs from that of a dendrite dropping vertically without rotation.

Wall effect on the motion of a rigid body immersed in a free molecular flow

Motion of a rigid body immersed in a semi-infinite expanse of gas in a $d$-dimensional region bounded by an infinite plane wall is studied for free molecular flow on the basis of the free Vlasov equation under the specular boundary condition. We show that the velocity $V(t)$ of the body approaches its terminal velocity $V_{\infty}$ according to a power law $V_{\infty}-V(t)\approx Ct^{-(d-1)}$ by carefully analyzing the pre-collisions due to the presence of the wall. The exponent $d-1$ is smaller than $d+2$ for the case without the wall found in the classical work by Caprino, Marchioro and Pulvirenti~[Comm. Math. Phys., \textbf{264} (2006), pp. 167--189] and thus slower convergence rate results from the presence of the wall.

Guidance of microswimmers by wall and flow: Thigmotaxis and rheotaxis of unsteady squirmers in two and three dimensions.

The motions of an unsteady circular-disk squirmer and a spherical squirmer have been investigated in the presence of a no-slip infinite wall and a background shear flow in order to clarify the similarities and differences between two- and three-dimensional motions. Despite the similar bifurcation structure of the dynamical system, the stability of the fixed points differs due to the Hamiltonian structure of the disk squirmer. Once the unsteady oscillating surface velocity profile is considered, the disk squirmer can behave in a chaotic manner and cease to be confined in a near-wall region. In contrast, in an unsteady spherical squirmer, the dynamics is well attracted by a stable fixed point. Additional wall contact interactions lead to stable fixed points for the disk squirmer, and, in turn, the surface entrapment of the disk squirmer can be stabilized, regardless of the existence of the background flow. Finally, we consider spherical motion under a background flow. The separated time scales of the surface entrapment (thigmotaxis) and the turning toward the flow direction (rheotaxis) enable us to reduce the dynamics to two-dimensional phase space, and simple weather-vane mechanics can predict squirmer rheotaxis. The analogous structure of the phase plane with the wall contact in two and three dimensions implies that the two-dimensional disk swimmer successfully captures the nonlinear interactions, and thus two-dimensional approximation could be useful in designing microfluidic devices for the guidance of microswimmers and for clarifying the locomotions in a complex geometry.

Effect of rotation on the stability of side-heated buoyant convection between infinite horizontal walls

The Hadley circulation has been widely studied because of its involvement in various natural and industrial situations. In this work, we examine the effect of the Coriolis force on this type of flow. A basic flow is derived to include the rotation and then its linear stability is presented.

Initial development of a free-surface wall jet at moderate Reynolds number

The steady laminar flow of a moderately inertial wall jet is examined theoretically near the exit of a channel. The free-surface jet emerges asymmetrically from the channel as it adheres to an infinite (upper) wall subject to a pressure gradient. The problem is solved using the method of matched asymptotic expansions. The small parameter involved in the expansions is the inverse cubic power of the Reynolds number. The flow field is obtained by matching the inviscid rotational core flow separately with the free-surface and the two wall layers. The upstream influence is examined as well as the break in the symmetry between the two wall layers. The wall jet exhibits a contraction near the channel exit that is independent of inertia, and eventually expands for any Reynolds number. Unlike the flow of a wall jet emerging into the same ambient fluid, the free-surface jet experiences a limited weakening in shear stress along the infinite wall, suggesting the possibility of separation for a jet with relatively low inertia. Significant shearing and elongation ensue at the exit, accompanied by flattening of the velocity profile near the upper wall.

Motion of a spherical liquid drop in simple shear flow next to an infinite plane wall or fluid interface

The velocity of translation of a spherical liquid drop under the influence of simple shear flow near an infinite plane wall or fluid interface is computed with high accuracy using a boundary-integral method for Stokes flow based on Fourier decomposition in the absence of significant gravitational effects. Accurate results are obtained for very small gaps between the drop surface and the wall or interface, several orders of magnitude smaller than the drop diameter. As the gap decreases, a liquid drop tends to stick to a solid wall but is able to slide over an interface irrespective of the drop and ambient fluid viscosities. In the case of flow near an interface, the dependence of the drop velocity on the drop position and viscosity ratio is not necessarily monotonic. The interfacial deformation is computed to leading order with respect to the capillary number and is shown to exhibit an asymmetry due to the presence of the wall or interface. The predictions of the asymptotic analysis for nearly spherical drops are in agreement with boundary-element simulations of three-dimensional flow.

Three-dimensional flow of Jeffrey fluid along an infinite plane wall with periodic suction

The laminar flow of an incompressible Jeffrey fluid past an infinite wall is modelled and analyzed analytically. The suction velocity distribution consisting of a basic steady distribution with a superimposed weak transversely varying distribution is assumed. The problem turns out to be three-dimensional because of variation of suction velocity in transverse direction on the wall. A series expansion technique is employed to obtain approximate solutions for velocity field, skin friction, and pressure. The effects of various non-dimensional parameters emerging in model on main flow velocity component and wall shear stress in main flow direction and perpendicular to it are presented graphically. Strongly dependence of components of wall shear stress on elastic parameter is noted. Moreover, it is noted that Jeffrey fluid parameters are controlling parameters of drag force.

Traction reveals mechanisms of wall effects for microswimmers near boundaries.

The influence of a plane boundary on low-Reynolds-number swimmers has frequently been studied using image systems for flow singularities. However, the boundary effect can also be expressed using a boundary integral representation over the traction on the boundary. We show that examining the traction pattern on the boundary caused by a swimmer can yield physical insights into determining when far-field multipole models are accurate. We investigate the swimming velocities and the traction of a three-sphere swimmer initially placed parallel to an infinite planar wall. In the far field, the instantaneous effect of the wall on the swimmer is well approximated by that of a multipole expansion consisting of a force dipole and a force quadrupole. On the other hand, the swimmer close to the wall must be described by a system of singularities reflecting its internal structure. We show that these limits and the transition between them can be independently identified by examining the traction pattern on the wall, either using a quantitative correlation coefficient or by visual inspection. Last, we find that for nonconstant propulsion, correlations between swimming stroke motions and internal positions are important and not captured by time-averaged traction on the wall, indicating that care must be taken when applying multipole expansions to study boundary effects in cases of nonconstant propulsion.

Analytical solution of the problem of heat transfer in rarefied gas between two coaxial cylinders

The method of characteristics was used within the framework of the kinetic approach to construct an analytical solution of the problem of heat transfer in a channel whose walls were formed by two coaxial cylinders. The main equation was the Williams kinetic equation, and the boundary condition on the channel walls was the diffusion reflection model. The vector field of the heat flux in the channel was determined, and the specific heat flux through the cross section of the channel was calculated. It was shown that the results obtained for a limiting case, in which the cylinder radii were significantly greater than the mean length of free path of gas molecules, were in good agreement with the results obtained for a plane channel with infinite parallel walls.

Combined experimental and numerical investigation of a vortex dipole interaction with a deformable plate

The impact of a vortex dipole with a cantilevered plate is studied experimentally in a quasi-two-dimensional environment, using a combination of flow visualization and particle image velocimetry to elucidate the flow kinematics and estimate the pressure loading on the plate. The experimental results are complemented by numerical simulations that further illuminate the attendant fluid-structure interaction and vortex dynamics. Both deformable and rigid plates are investigated at a dipole Reynolds number of 1030. Upon impact, the dipole halves split, with each half joining with surface-generated vorticity of opposite sign to form secondary structures. One half forms a secondary dipole at the plate tip, while the other advects along the plate surface and exhibits a rebounding effect, similar to the dipole impact on an infinite wall. The rate of vorticity generation is lower along the deformable plate versus the rigid one, resulting in altered secondary dipole trajectories. The experimental and numerical results are compared in terms of vortex dynamics, plate response, and plate loading, all of which are in good agreement to within experimental uncertainty. The most significant discrepancy in tip deflection occurs around the time of the dipole impact with the plate. The sensitivity of the plate response to the initial distribution of vorticity in the dipole core is also discussed.

The effect of slip condition on the three-dimensional flow of Jeffrey fluid along a plane wall with periodic suction

The laminar flow of an incompressible Jeffrey fluid past an infinite wall is modeled and analyzed analytically considering slip effect. The suction velocity distribution consisting of a basic steady distribution with a super-imposed weak transversely varying distribution is assumed. The problem turns out to be three dimension because of variation of suction velocity in transverse direction on the wall. A series expansion technique is employed to obtain approximate solutions for velocity field, skin friction, and pressure. The effects of non-Newtonian Jeffrey fluid parameter $$\lambda _{1},$$ elastic parameter K, and slip parameter $$\gamma $$ on velocity field and skin friction are presented graphically. Strongly dependence of skin friction factor in the direction of main flow on slip parameter is noted.