The jet from a horizontal slot under gravity

1975 ◽  
Vol 344 (1639) ◽  
pp. 471-487 ◽  
Keyword(s):  
Integral Equation ◽  
Froude Number ◽  
Horizontal Plane ◽  
Nonlinear Integral Equation ◽  
Free Stream ◽  
Constructive Proof ◽  
Stream Line ◽  
Plane Flow ◽  
Steady Plane

The steady plane flow in a jet, falling under gravity from a slot in a horizontal plane, is studied. The fluid is incompressible and inviscid, the flow is irrotational, and the reciprocal e of the Froude number lies in the range 0 < ε ⩽ ½π A constructive proof of the existence of the solution of a nonlinear integral equation describing the free stream line is given. As ε → 0 this solution reduces to known formal approximations. The method is then applied to a class of related problems.

Sluice-gate problems with gravity

1977 ◽  
Vol 81 (1) ◽  
pp. 157-175 ◽  
Author(s):  
P. J. Budden ◽  
J. Norbury
Keyword(s):  
Integral Equation ◽  
Existence Of Solutions ◽  
Fluid Flows ◽  
Linear Integral Equation ◽  
Constructive Proof ◽  
Sluice Gate ◽  
Plane Flow ◽  
Steady Plane ◽  
Linear Integral ◽  
A Free Boundary Problem

AbstractIn this paper, a free-boundary problem of the steady plane flow of an ideal fluid through a slot is solved. The fluid flows under the effect of gravity between rigid boundaries, and then out of a slot as a jet which becomes horizontal at infinity downstream. A constructive proof of the existence of solutions to a non-linear integral equation is given for a parameter range 0 < ε < ε* ≃ 1·02 (where ε, the inverse Froude number, measures the effect of gravity). Approximations to the solution are then found for ε → 0.

Aerodynamics of slot-film cooling: theory and experiment

1985 ◽  
Vol 160 ◽  
pp. 15-27 ◽  
Author(s):  
A. D. Fitt ◽  
J. R. Ockendon ◽  
T. V. Jones
Keyword(s):  
Integral Equation ◽  
Film Cooling ◽  
Nonlinear Integral Equation ◽  
Free Stream ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Total Head ◽  
Slot Film Cooling ◽  
Range Of Values ◽  
Theory And Experiment

A simple model is proposed for the two-dimensional injection of irrotational inviscid fluid from a slot into a free stream. In a certain range of values of the ratio of free-stream to injection total heads, the film thickness satisfies a nonlinear integral equation whose solution enables the mass flow in the film to be found. Some experiments are described which both agree with this theory when it is relevant and indicate its deficiencies at other values of the total head ratio.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Analytical treatment of two-dimensional supersonic flow. II. Flow with weak shocks

1956 ◽  
Vol 248 (952) ◽  
pp. 499-515 ◽  
Keyword(s):  
Solution Process ◽  
Wave Interaction ◽  
Analytical Treatment ◽  
Plane Flow ◽  
Flow Equations ◽  
Interaction Regions ◽  
Steady Plane ◽  
Set Up ◽  
Downstream Flow ◽  
Weak Shocks

A scheme of approximate solution is presented for the treatment of shock waves in the steady, plane flow of a perfect gas. It is based on the neglect of any entropy variations produced by the shocks and hence is applicable only when the shocks are weak. The method provides an extension of Friedrichs’s (1948) results for simple waves to wave-interaction regions. By an examination of the solution of the continuous-flow equations in the neighbourhood of a known shock wave it is shown how the downstream flow may be calculated without reference to the particular shock shape (§2). There are certain cases in which this approach fails and they are discussed by means of a typical example in §3.3. Once the downstream flow has been calculated, it is possible to set up general equations for the determination of the shock (§ 2). Examples of the solution of these equations for typical problems are given in §3. In §4 there is a brief discussion of the validity of using homentropic theory and estimates of the errors involved in the solution process are obtained.

scholarly journals The Steady Profile of an Axisymmetric Ice Sheet

1981 ◽  
Vol 27 (95) ◽  
pp. 25-37 ◽  
Author(s):  
I. R. Johnson
Keyword(s):  
Aspect Ratio ◽  
Power Law ◽  
Viscous Incompressible Fluid ◽  
Ice Sheet ◽  
Plane Flow ◽  
The Third ◽  
Basal Sliding ◽  
Steady Plane ◽  
Third Dimension ◽  
Special Case

AbstractSteady plane flow under gravity of an axisymmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding is treated according to a power law between shear traction and velocity. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, and temperature variation through the ice sheet is neglected. Illustrations are presented for Glen’s power law (including the special case of a Newtonian fluid), and the polynomial law of Colbeck and Evans. The analysis follows that of Morland and Johnson (1980) where the analogous problem for an ice sheet deforming under plane flow was considered. Comparisons are made between the two models and it is found that the effect of the third dimension is to reduce (or leave unchanged) the aspect ratio for the cases considered, although no general formula can be obtained. This reduction is seen to depend on both the surface accumulation and the sliding law.

scholarly journals Fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces

2020 ◽  
Vol 21 (1) ◽  
pp. 135
Author(s):  
Godwin Amechi Okeke ◽  
Mujahid Abbas
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Nonlinear Integral Equation ◽  
Metric Spaces ◽  
Banach Algebras ◽  
Cone Metric Spaces ◽  
Nonlinear Operators ◽  
Complex Valued ◽  
Cone Metric

It is our purpose in this paper to prove some fixed point results and Fej´er monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type VolterraFredholm functional nonlinear integral equation in complex valued Banach spaces.

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