Solutions of unsteady compressible boundary-layer equations

1966 ◽  
Vol 62 (3) ◽  
pp. 511-518 ◽  
Author(s):  
G. N. Sarma
Keyword(s):  
Prandtl Number ◽  
The Other ◽  
Stream Velocity ◽  
Arbitrary Parameter ◽  
Main Stream ◽  
Compressible Boundary Layer ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Compressible Boundary Layers ◽  
Small Times

AbstractThe unsteady two-dimensional compressible boundary layers have been studied by Sarma ((5)) assuming that the Prandtl number is unity and that the wall is in an arbitrary motion, the main stream being steady. In this paper the Prandtl number is taken to be an arbitrary parameter which need not be equal to unity, and it is assumed that the main stream velocity and temperature are perturbing about a steady mean, the wall being in an arbitrary motion. Solutions are obtained in two parts, one when the temperature gradient at the wall is perturbing about a zero mean and the other when the temperature of the wall is perturbing about a steady mean. Thus in this paper the theory given in Sarma ((5)) is made still more general. Following the work of Sarma ((4)–(6)) two types of solutions are developed in each part, one for large times and the other for small times.

Correlated incompressible and compressible boundary layers

1949 ◽  
Vol 200 (1060) ◽  
pp. 84-100 ◽  
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Boundary Layers ◽  
Absolute Temperature ◽  
Stream Velocity ◽  
Main Stream ◽  
Boundary Layer Equations ◽  
Compressible Boundary Layers ◽  
Incompressible Boundary ◽  
Similar Solutions

The boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the Prandtl number is unity. Various results in the theory of incompressible boundary layers are then taken over into the compressible theory. In particular, the existence of ‘similar’ solutions is proved, and Howarth’s method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity. A comparison with an exact solution is used to show that this method gives a closer approximation than does Pohlhausen’s.

Unsteady laminar boundary layers in a compressible stagnation flow

1975 ◽  
Vol 70 (3) ◽  
pp. 561-572 ◽  
Author(s):  
C. S. Vimala ◽  
G. Nath
Keyword(s):  
Free Stream ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Compressible Boundary Layer ◽  
Two Dimensional ◽  
Stagnation Flow ◽  
Laminar Boundary Layers ◽  
Implicit Finite Difference ◽  
Layer Flow ◽  
Heat Transfer Parameter

The unsteady laminar compressible boundary-layer flow in the immediate vicinity of a two-dimensional stagnation point due to an incident stream whose velocity varies arbitrarily with time is considered. The governing partial differential equations, involving both time and the independent similarity variable, are transformed into new co-ordinates with finite ranges by means of a transformation which maps an infinite interval into a finite one. The resulting equations are solved by converting them into a matrix equation through the application of implicit finite-difference formulae. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) a constantly accelerating stream and (ii) a fluctuating stream. The results show that in the former case both the skin-friction and the heat-transfer parameter increase steadily with time after a certain instant, while in the latter they oscillate, thus responding to the fluctuations in the free-stream velocity.

The heated laminar vertical jet

1967 ◽  
Vol 29 (2) ◽  
pp. 305-315 ◽  
Author(s):  
R. S. Brand ◽  
F. J. Lahey
Keyword(s):  
Boundary Layer ◽  
Temperature Distribution ◽  
Laminar Flow ◽  
Prandtl Number ◽  
Exact Solutions ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Prandtl Numbers ◽  
Vertical Jet ◽  
Steady Laminar

The boundary-layer equations for the steady laminar flow of a vertical jet, including a buoyancy term caused by temperature differences, are solved by similarity methods. Two-dimensional and axisymmetric jets are treated. Exact solutions in closed form are found for certain values of the Prandtl number, and the velocity and temperature distribution for other Prandtl numbers are found by numerical integration.

On the laminar compressible boundary layer induced by the passage of a plane shock wave over a flat wall

1967 ◽  
Vol 63 (3) ◽  
pp. 889-907 ◽  
Author(s):  
J. A. D. Ackroyd
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Prandtl Number ◽  
Plane Shock Wave ◽  
Plane Shock ◽  
Compressible Boundary Layer ◽  
Empirical Relationships ◽  
Flat Wall ◽  
Boundary Layer Equations ◽  
Semi Empirical

SummaryThe laminar compressible boundary layer induced by the passage of a plane shock wave over a flat wall is examined in detail. Use is made of the empirical viscosity-temperature relationship μ ∝ Tω. The boundary-layer equations are solved numerically for various values of the index ω, Prandtl number and shock strengths. The resulting solutions are then used to construct simple semi-empirical relationships for some of the more important boundary-layer parameters.

scholarly journals On the two-dimensional steady flow of a viscous fluid behind a solid body.―I

1933 ◽  
Vol 142 (847) ◽  
pp. 545-562 ◽  
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Velocity Distribution ◽  
Asymptotic Approximation ◽  
Constant Velocity ◽  
Boundary Layer Theory ◽  
The Other ◽  
Main Stream ◽  
Two Dimensional ◽  
Prandtl Equations

1.1. The purpose of this paper is to exhibit, for reasons given below, calculations of the velocity distribution some distance downstream behind any symmetrical obstacle in a stream of viscous fluid, but particularly behind an infinitely thin plate parallel to the stream, the motion being two-dimensional. For a slightly viscous fluid, Blasius worked out the velocity distribution in the boundary layer from the front to the downstream end of the plate; and in a previous paper, I calculated the velocity in the wake for a distance varying from 0.3645 to 0.5 of the length of the plate from its downstream end (according to distance from its plane). In these calculations the fluid was supposed unlimited, and the undisturbed velocity in front of the plate was taken as constant. The viscosity being assumed small, the work was carried out on the basis of Pranstl's boundary layer theory, with zero pressure gradient in the direction of the stream. The velocity is then constant everywhere expect within a thin layer near the plate, and in a wake which must gradually broaden out downstream. (The broadening of the wake just behind the plate is so gradual that it could not be shown by calculations of the accuracy obtained in I). Pressure variations in a direction at right angles to the stream are negligible, and so is the velocity in that direction. Later, Tollmien attcked the problem from the other end, and found a first asymptotic approximation for the velocity distribution in the wake at a considerable distance downstream. He simplified the Prandtl equations by assuming that the departure from the constant velocity, U 0 , of the main stream is small, and neglecting terms quadratic in this departure. In other words, he applied the notion of the Oseen approximation to the Prandtl equations. His result for the velocity is U = U 0 {1 - a X -½ exp (-U 0 Y 2 /4νX)}.

Steady flow in the laminar boundary layer of a gas

1949 ◽  
Vol 199 (1059) ◽  
pp. 533-558 ◽  
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Incompressible Fluid ◽  
Flat Plate ◽  
Horizontal Cylinder ◽  
Vertical Surface ◽  
Main Stream ◽  
Stream Flows ◽  
Boundary Layer Equations ◽  
Plane Jets

If the boundary-layer equations for a gas are transformed by Mises’s transformation, as was done by Kármán & Tsien for the flow along a flat plate of a gas with unit Prandtl number σ, the computation of solutions is simplified, and use may be made of previously computed solutions for an incompressible fluid. For any value of the Prandtl number, and any variation of the viscosity μ with the temperature T , after the method has been applied to flow along a flat plate (a problem otherwise treated by Crocco), the flow near the forward stagnation point of a cylinder is calculated with dissipation neglected, both with the effect of gravity on the flow neglected and with this effect retained for vertical flow past a horizontal cylinder. The approximations involved by the neglect of gravity are considered generally, and the cross-drift is calculated when a horizontal stream flows past a vertical surface. When σ =1, μ∞ T , and the boundary is heat-insulated, it is shown that the boundary-layer equations for a gas may be made identical, whatever be the main stream, with the boundary-layer equations for an incompressible fluid with a certain, determinable, main stream. The method is also applied to free convection at a flat plate (with the heat of dissipation and the variation with altitude of the state of the surrounding fluid neglected) and to laminar flow in plane wakes, but for plane jets the conditions σ =1, μ∞ T , previously imposed by Howarth,are also imposed here in order to obtain simple solutions.

The unsteady laminar boundary layer on a semi-infinite flat plate due to small fluctuations in the magnitude of the free-stream velocity

1972 ◽  
Vol 51 (1) ◽  
pp. 137-157 ◽  
Author(s):  
R. C. Ackerberg ◽  
J. H. Phillips
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Stream Velocity ◽  
Main Stream ◽  
Mean Flow ◽  
Boundary Layer Equations ◽  
Damped Oscillations ◽  
Asymptotic Values ◽  
The Mean

Asymptotic and numerical solutions of the unsteady boundary-layer equations are obtained for a main stream velocity given by equation (1.1). Far downstream the flow develops into a double boundary layer. The inside layer is a Stokes shear-wave motion, which oscillates with zero mean flow, while the outer layer is a modified Blasius motion, which convects the mean flow downstream. The numerical results indicate that most flow quantities approach their asymptotic values far downstream through damped oscillations. This behaviour is attributed to exponentially small oscillatory eigenfunctions, which account for different initial conditions upstream.

A general theory of unsteady compressible boundary layers with and without suction or injection

1965 ◽  
Vol 61 (3) ◽  
pp. 795-807 ◽  
Author(s):  
G. N. Sarma
Keyword(s):  
Boundary Layers ◽  
Incompressible Flows ◽  
Pressure Ratio ◽  
The Body ◽  
Main Stream ◽  
Compressible Boundary Layers ◽  
Small Times ◽  
Suction Or Injection ◽  
The Difference ◽  
Coefficient Of Viscosity

AbstractThe theory of unsteady two-dimensional compressible boundary layers is studied in two parts. Part I deals with the solutions when the main stream is steady and the body is in an arbitrary motion while Part II deals with the solutions when the main stream is steady and the body is at rest with an arbitrary unsteady suction or injection perturbing about a zero mean. General methods of obtaining solutions are given when the coefficient of viscosity is proportional to temperature and the Prandtl number is unity. The main feature of the work in this paper lies in subjecting the equations to a transformation similar to that of Howarth and in linearization similar to that of Lighthill. Following the work of Sarma in incompressible flows two types of solutions are developed, one for large times and the other for small times. General expressions for the skin friction, the pressure ratio, the heat transfer and the difference between the normal coordinates of compressible and incompressible flows are obtained.

Three-Dimensional Reconstruction of Two Forms of the Chaperonin GRO EL

1990 ◽  
Vol 48 (1) ◽  
pp. 258-259
Author(s):  
J.L. Carrascosa ◽  
G. Abella ◽  
S. Marco ◽  
M. Muyal ◽  
J.M. Carazo
Keyword(s):  
Three Dimensional ◽  
The Other ◽  
Two Dimensional ◽  
Series Method ◽  
Three Dimensional Reconstruction ◽  
Structural Components ◽  
E Coli ◽  
Dimensional Reconstruction ◽  
Morphogenetic Factors ◽  
Tilt Series

Chaperonins are a class of proteins characterized by their role as morphogenetic factors. They trantsiently interact with the structural components of certain biological aggregates (viruses, enzymes etc), promoting their correct folding, assembly and, eventually transport. The groEL factor from E. coli is a conspicuous member of the chaperonins, as it promotes the assembly and morphogenesis of bacterial oligomers and/viral structures.We have studied groEL-like factors from two different bacteria:E. coli and B.subtilis. These factors share common morphological features , showing two different views: one is 6-fold, while the other shows 7 morphological units. There is also a correlation between the presence of a dominant 6-fold view and the fact of both bacteria been grown at low temperature (32°C), while the 7-fold is the main view at higher temperatures (42°C). As the two-dimensional projections of groEL were difficult to interprete, we studied their three-dimensional reconstruction by the random conical tilt series method from negatively stained particles.

Polymer-Graphene Nanoassemblies and their Applications in Cancer Theranostics

2020 ◽  
Vol 20 (11) ◽  
pp. 1340-1351 ◽  
Author(s):  
Ponnurengam M. Sivakumar ◽  
Matin Islami ◽  
Ali Zarrabi ◽  
Arezoo Khosravi ◽  
Shohreh Peimanfard
Keyword(s):  
Mechanical Stability ◽  
Chemical Properties ◽  
The Other ◽  
Hydrophilic Polymer ◽  
Two Dimensional ◽  
Normal Tissues ◽  
Physical Chemical ◽  
Anti Cancer ◽  
Good Biocompatibility ◽  
Cancer Theranostics

Background and objective: Graphene-based nanomaterials have received increasing attention due to their unique physical-chemical properties including two-dimensional planar structure, large surface area, chemical and mechanical stability, superconductivity and good biocompatibility. On the other hand, graphene-based nanomaterials have been explored as theranostics agents, the combination of therapeutics and diagnostics. In recent years, grafting hydrophilic polymer moieties have been introduced as an efficient approach to improve the properties of graphene-based nanomaterials and obtain new nanoassemblies for cancer therapy. Methods and results: This review would illustrate biodistribution, cellular uptake and toxicity of polymergraphene nanoassemblies and summarize part of successes achieved in cancer treatment using such nanoassemblies. Conclusion: The observations showed successful targeting functionality of the polymer-GO conjugations and demonstrated a reduction of the side effects of anti-cancer drugs for normal tissues.

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