In this article we consider the linear stability of the two-dimensional flow induced by the linear stretching of a surface in the streamwise direction. The basic flow is a rare example of an exact analytical solution of the Navier-Stokes equations. Using results from a large Reynolds number asymptotic study and a highly accurate spectral numerical method we show that this flow is linearly unstable to disturbances in the form of Tollmien-Schlichting waves. Previous studies have shown this flow is linearly stable. However, our results show that this is only true for Görtler-type disturbances.
Comment on Conjugate heat transfer of mixed convection for viscoelastic fluid past a stretching sheet, by Hsiao and Chen, Mathematical Problems in Engineering
The equations governing the magnetohydrodynamic stagnation point flow toward a non-conducting, thermally insulated, nonporous, linearly stretching sheet are cast in a self similar form. Consistent boundary conditions on the velocity, magnetic field and temperature are invoked. The flow problem involves three parameters- the magnetic Prandtl number, the magnetic interaction number, and the ratio of the stretching rate to the strength of the stagnation point flow. The energy equation includes viscous dissipation and Joule heating, and introduces the Prandtl number as a fourth parameter. Numerica…
The solution for the MHD flow, due to a linearly stretching sheet, has a simple form for the velocity field, with a companion simple form for the induced magnetic field. The associated thermal problem, including viscous dissipation and Joule heating, involves three temperature constituents, the solutions for two of which are obtained in terms of Kummer's function. The solution for the third temperature constituent is obtained in a convergent series form.
The full MHD equations, governing the flow due to the axisymmetric stretching of a sheet in the presence of a transverse magnetic field, can be cast in a self similar form. This allows evaluation of the induced magnetic field and its effect on the flow and heat transfer. The problem involves three parameters- the magnetic Prandtl number, the magnetic interaction number, and the Prandtl number. Numerical solutions are obtained for the velocity field, the magnetic field, and the temperature, at different values of the magnetic Prandtl number and the magnetic interaction number. The contributions…
The full MHD problem of the flow and heat transfer due to a linearly stretching sheet in the presence of a transverse magnetic field is put in a self-similar form. Traditionally ignored physical processes such as induced magnetic field, viscous dissipation, Joule heating, and work shear are included and their importance is established. Cases of prescribed surface temperature, prescribed heat flux, surface feed (injection or suction), velocity slip, and thermal slip are also considered. The problem is shown to admit self similarity. Sample numerical solutions are obtained for chosen combination…
The current study is a pioneering work in presenting the boundary layer equations for the two-dimensional flow and heat transfer of the Cross fluid over a linearly stretching sheet. The system of partial differential equations is turned down into highly non-linear ordinary differential equations by applying suitable similarity transformations. The stretching sheet solutions are presented via. a numerical technique namely the shooting method and graphs are constructed for the shear-thinning as well as shear-thickening regime. The impact of the emerging parameters namely the power-law index , th…
An analysis is carried out to examine the effects of the steady flow of an electrically conducting viscous incompressible nanofluid in the existence of a uniform transverse magnetic field with chemical reaction over a stretching sheet considering suction or injection. The present model is demonstrated experimentally to reveal the effects of Thermophoresis and Brownian motion. Similarity solutions are investigated for the governing equations and solved numerically by using a shooting technique with fourth-order Runge-Kutta integration scheme. The impact of associated parameters on velocity prof…
In the present study, MHD boundary layer flow with heat and mass transfer of a nanofluid with viscous dissipation over a non-linear stretching sheet embedded in a porous medium is studied. The governing non-linear partial differential equations are converted into coupled non-linear ordinary differential equations which are solved by hybrid technique consisting of finite element method with symbolic computation. The effect of Viscous dissipation, magnetic influence parameter, permeability parameter, Brownian motion, Eckert number, Thermophoresis parameter, Prandtl number parameter, Dufour Lewis…
In this paper we present a mathematical analysis of thin film flow and heat transfer to a laminar liquid film from a horizontal stretching sheet. The flow of thin liquid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformations. Similarity transformations are used to convert unsteady boundary layer equations to a system of non-linear ordinary differential equations. The resulting non-linear differential equations are solved numerically using Runge-kutta-Fehlberg and Newton-Raphson schemes. A relationship between film thickness $β$ …
Comment on "Chebyshev finite difference method for the effects of variable viscosity and variable thermal conductivity on heat transfer to a micro-polar fluid from a non-isothemal stretching sheet with suction and blowing," by S.N. Odda and A.M. Farhan
The effects of MHD boundary layer flow of non-linear thermal radiation with convective heat transfer and non-uniform heat source/sink in presence of thermophortic velocity and chemical reaction investigated in this study. Suitable similarity transformation are used to solve the partial ordinary differential equation of considered governing flow. Runge-Kutta fourth fifth order Fehlberg method with shooting techniques are used to solved non-dimensional governing equations. The variation of different parameters such as thermophoretic parameter, chemical reaction parameter, non- uniform heat sourc…
The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics, and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Foppl-von Karman equations to derive closed-form analytical …
Motivated by the redrawing of hot glass into thin sheets, we investigate the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed non-uniform temperature field. We first determine the associated base flow by solving the long-timescale stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized stretching velocity $α$, and a dimensionless width of the heating zone $β$. This allows us to determine the conditions for the onset of an out-of-plane wrinkling instability stated in terms of an eigenvalue problem for a linear part…
In this paper we have studied the flow and heat transfer in a viscous fluid by a horizontal sheet. The stretching rate and temperature of the sheet vary with time. The governing equations for momentum and thermal energy are reduced to ordinary differential equations by means of similarity transformation. These equations are solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM) which provides us with a convenient way to control the convergence of approximation solutions and adjust convergence rigorous when necessary. Some examples are given and the results obtained reve…
We propose a simple model for the evolution of an inviscid vortex sheet in a potential flow in a channel with parallel walls. This model is obtained by augmenting the Birkhoff-Rott equation with a potential field representing the effect of the solid boundaries. Analysis of the stability of equilibria corresponding to flat sheets demonstrates that in this new model the growth rates of the unstable modes remain unchanged as compared to the case with no confinement. Thus, in the presence of solid boundaries the equilibrium solution of the Birkhoff-Rott equation retains its extreme form of instabi…
Many biological materials such as cervical mucus and collagen gel possess a fibrous micro-structure. This micro-structure affects the emergent mechanical properties of the material, and hence the functional behaviour of the system. We consider the canonical problem of stretching a thin sheet of transversely-isotropic viscous fluid as a simplified version of the spinnbarkeit test for cervical mucus. We propose a novel solution to the model constructed by Green & Friedman by manipulating the model to a form amenable to arbitrary Lagrangian-Eulerian techniques. The system of equations, reduced by…
The conformation of a long linear polymer dissolved in fluid and exposed to an extensional flow is well-known to exhibit a "coil-stretch" transition, which for sufficiently long chains can lead to bistability. The present work reports computations indicating that an analogous "compact-stretched" transition arises in the dynamics of a thin elastic sheet. Sheets of nominally circular, square or rectangular shape are simulated in planar and biaxial flows using a finite element method for the sheet conformations and a regularized Stokeslet method for the fluid flow. If a neo-Hookean constitutive m…
A 2D temporal incompressible stability analysis is carried out for lobed jets. The jet base flow is assumed to be parallel and of a vortex-sheet type. The eigenfunctions of this simplified stability problem are expanded using the eigenfunctions of a round jet. The original problem is then formulated as an innovative matrix eigenvalue problem, which can be solved in a very robust and efficient manner. The results show that the lobed geometry changes both the convection velocity and temporal growth rate of the instability waves. However, different modes are affected differently. In particular, m…
In an incompressible flow, fluid density remains invariant along fluid element trajectories. This implies that the spatial distribution of non-interacting noninertial particles in such flows cannot develop density inhomogeneities beyond those that are already introduced in the initial condition. However, in certain practical situations, density is measured or accumulated on (hyper-) surfaces of dimensionality lower than the full dimensionality of the flow in which the particles move. An example is the observation of particle distributions sedimented on the floor of the ocean. In such cases, ev…
