Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model

1989 ◽  
Vol 202 ◽  
pp. 339-366 ◽  
Author(s):  
S. A. Forth ◽  
A. A. Wheeler
Keyword(s):  
Boundary Layer ◽  
Simple Model ◽  
Binary Alloy ◽  
Numerical Solutions ◽  
Free Stream ◽  
Travelling Waves ◽  
Free Stream Velocity ◽  
Large Reynolds Number ◽  
Stream Velocity ◽  
Morphological Stability

In this paper we consider the effect of a model boundary-layer flow on the hydrodynamic and morphological stability of a simple model of the solidification of a binary alloy. We conduct a linear analysis and develop asymptotic solutions for large Schmidt number and large Reynolds number. We also present numerical solutions for data appropriate to a lead–tin alloy. We show that for modes parallel to the free-stream velocity the flow is responsible for the appearance of travelling waves and, for common values of the material parameters, may stabilize the morphological stability of the interface. However the morphological stability of modes perpendicular to the free-stream velocity is unaffected by the presence of the flow. The hydrodynamic stability of the boundary layer is very weakly affected by the presence of the interface, which we attribute to the large Schmidt numbers associated with real crystal growth situations.

scholarly journals Prediction of Heat Transfer From Laminar Boundary Layers, With Emphasis on Large Free-Stream Velocity Gradients and Highly Cooled Walls

1966 ◽  
Vol 88 (3) ◽  
pp. 249-256 ◽  
Author(s):  
L. H. Back ◽  
A. B. Witte
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Shear Stress ◽  
Velocity Gradient ◽  
Free Stream ◽  
Variable Density ◽  
Similarity Solutions ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Gradient Parameter

Laminar boundary-layer heat transfer and shear-stress predictions from existing similarity solutions are extended in an approximate way to perfect gas flows with a large free-stream velocity gradient parameter β and variable density-viscosity product ρμ across the boundary layer resulting from a highly cooled wall. The dimensionless enthalpy gradient at the wall gw′, to which the heat flux is related, is found not to vary appreciably with β. Thus the application of similarity solutions on a local basis to predict heat transfer from accelerated flows to an arbitrary surface may be a reasonable approximation involving a minimum amount of calculation time. Unlike gw′, the dimensionless velocity gradient at the wall fw″, to which the shear stress is related, is strongly dependent on β.

Effects of Partial Slip on Chemically Reactive Solute Distribution in MHD Boundary Layer Stagnation Point Flow Past a Stretching Permeable Sheet

2015 ◽  
Vol 13 (1) ◽  
pp. 29-36 ◽  
Author(s):  
Swati Mukhopadhyay
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Free Stream ◽  
Free Stream Velocity ◽  
Stagnation Point Flow ◽  
Partial Slip ◽  
Stream Velocity ◽  
Slip Parameter ◽  
Mhd Boundary Layer

Abstract This paper presents the magnetohydrodynamic (MHD) boundary layer stagnation point flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet in presence of partial slip. With the help of similarity transformations, the partial differential equations corresponding to momentum and the concentration equations are transformed into non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity increases with the increasing value of the ratio of the free stream velocity and the stretching velocity. Velocity decreases with the increasing magnetic parameter when the free-stream velocity is less than the stretching velocity but the opposite behavior is noted when the free-stream velocity is greater than the stretching velocity. Due to suction, fluid velocity decreases at a particular point of the surface. With increasing velocity slip parameter, velocity increases when the free-stream velocity is greater than the stretching velocity. But the concentration decreases in this case. Concentration decreases with increasing mass slip parameter.

Effects of Unsteady Free-Stream Velocity and Free-Stream Turbulence at a Stagnation Point

1983 ◽  
Vol 105 (1) ◽  
pp. 66-71 ◽  
Author(s):  
R. S. R. Gorla
Keyword(s):  
Stagnation Point ◽  
Numerical Solutions ◽  
Free Stream ◽  
Eddy Diffusivity ◽  
Descent Method ◽  
Free Stream Velocity ◽  
Steepest Descent Method ◽  
Wall Friction ◽  
Stream Velocity ◽  
Free Stream Turbulence

An analysis is presented to investigate the combined effects of transient free-stream velocity and free-stream turbulence at a stagnation point on a cylinder situated in a crossflow. A model has been successfully formulated for the eddy diffusivity induced by the free-stream turbulence. The governing momentum equation has been integrated by the steepest descent method. Numerical solutions are provided for the unsteady wall shear stress function for specific free-stream transients. The results are correlated by a new turbulence parameter. It has been found that the wall friction increases with increasing free-stream turbulence intensity. In the case of flows involving unsteady free-stream velocity, the friction factor increases with increasing values of the reduced frequency of oscillations.

Skin Friction and Heat Transfer for Laminar Boundary-Layer Flow With Variable Properties and Variable Free-Stream Velocity

1953 ◽  
Vol 20 (3) ◽  
pp. 415-421
Author(s):  
S. Levy ◽  
R. A. Seban
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flow ◽  
Free Stream ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Layer Flow

Abstract Numerical solutions of the momentum and energy equations are presented for particular types of laminar boundary-layer flow analogous to the Hartree “wedge flows.” Variation of the viscosity and of the thermal conductivity is considered under the circumstances of no dissipation, favorable pressure gradient, and the product of conductivity and density a constant. The solution is based on approximate representations of the velocity and temperature profiles in the boundary layer and these are of such character that the labor of calculation is minimized and the accuracy of the results preserved. The differential equations are reduced to two algebraic equations which rapidly yield the skin friction and the heat transfer in terms of the wall to free-stream temperature ratio for the desired value of Prandtl number. Numerical results are given for a range of wedge flows with gases of Prandtl number 0.70 and 1.0. These results reveal that when the free-stream velocity is variable the temperature difference between the wall and the free stream exerts a substantial effect on the velocity distribution in the boundary layer and on the skin-friction coefficient. Alternatively, the heat-transfer coefficient is not affected radically. A calculation method is presented for the determination of the heat transfer and skin friction for a flow with an arbitrary variation of velocity over an isothermal surface. This method utilizes the results of the present analysis for the variable property wedge flows.

The thermal boundary layer on a non-isothermal surface with non-uniform free stream velocity

1958 ◽  
Vol 4 (3) ◽  
pp. 321-329 ◽  
Author(s):  
E. M. Sparrow
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Wall Temperature ◽  
Thermal Boundary Layer ◽  
Free Stream ◽  
Temperature Data ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Universal Functions ◽  
Isothermal Surface

A formally exact solution for the thermal boundary layer on a non-isothermal surface subjected to non-uniform free stream velocity is presented in the form of a series. It is demonstrated that the solution can be recast in terms of universal functions, which are independent of the wall temperature data of particular problems, and which depend only on a single parameter characterizing the variation of the free stream velocity.

Darryl E. Metzger Memorial Session Paper: The Streamwise Development of Go¨rtler Vortices in a Favorable Pressure Gradient

1996 ◽  
Vol 118 (1) ◽  
pp. 162-171 ◽  
Author(s):  
M. V. Finnis ◽  
A. Brown
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Velocity Gradient ◽  
Free Stream ◽  
Free Stream Velocity ◽  
Velocity Variation ◽  
Radius Of Curvature ◽  
Stream Velocity ◽  
Mean Flow ◽  
Favorable Pressure Gradient

Measurements are presented of the streamwise velocity variation within a laminar boundary layer on a concave surface of 4 m radius of curvature for which the free-stream velocity gradient factor (ν/U02)dU0/dx was approximately 1 × 10−6. The stream velocity variation was consistent with the presence of counterrotating vortices resulting from the Go¨rtler instability. The vortices exhibited exponential growth over the streamwise extent of the measurements to a disturbance amplitude of approximately 13 percent of the local free-stream velocity. The vortex growth rates were found to be less than those for a zero velocity gradient factor, indicating that a favorable pressure gradient stabilizes the flow with respect to the Go¨rtler instability. Boundary layer profiles at local upwash and downwash positions are compared with the linear theory for which the mean flow was modeled using the Pohlhausen approximation to the solution of the boundary layer equations. The agreement between the measured and predicted profiles indicates that the linear stability theory can provide a fair approximation to the small amplitude growth of the Go¨rtler instability.

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