Transport Phenomena in the Shock Initiation Problem

1971 ◽  
Vol 49 (20) ◽  
pp. 2532-2539
Author(s):  
James Parker Elliott
Keyword(s):  
Heat Flux ◽  
Transverse Energy ◽  
Constitutive Law ◽  
Navier Stokes ◽  
Axial Flux ◽  
Shock Initiation ◽  
Transverse Pressure ◽  
Special Cases ◽  
Small Times ◽  
Structure Problem

The collisionless solution of the shock initiation problem is considered. Expressions for the fluid mechanical variables are given for arbitrary initial equilibrium states on either side of the diaphragm. Whereas, in certain special cases, a constitutive law in Navier–Stokes form can be obtained for the stress, no such relationship exists in general. Departure from the Navier–Stokes law is found to be related to gradients in the transverse pressure that come about as a result of the mixing of molecules with different thermal energies. The heat flux is broken down as the sum of a flux of axial energy and a flux of transverse energy. The ratio of the axial flux to the total flux is examined in several cases. Comparison is made with the Chapman–Enskog value, and with the values assumed by other authors in studying the shock structure problem. Departure of this ratio from its near-equilibrium value of 1/3 is suggested as a means of assessing the validity of the collisionless solution. The possibility of extending the Navier–Stokes solution down to small times is discussed.

Analysis of flow and heat transfer over stretching/shrinking and porous surfaces

2021 ◽  
pp. 875608792110258
Author(s):  
Azhar Ali ◽  
Dil Nawaz Khan Marwat ◽  
Aamir Ali
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Current Model ◽  
Uniform Heat Flux ◽  
Similarity Transformations ◽  
Flow And Heat Transfer ◽  
Special Cases ◽  
Porous Surfaces

Flows and heat transfer over stretching/shrinking and porous surfaces are studied in this paper. Unusual and generalized similarity transformations are used for simplifying governing equations. Current model includes all previous cases of stretched/shrunk flows with thermal effects discussed so far. Moreover, we present three different cases of thermal behavior (i) prescribed surface temperature (ii) Variable/uniform convective heat transfer at plat surface and (iii) prescribed variable/uniform heat flux. Stretching/shrinking velocity Uw(x), porosity [Formula: see text], heat transfer [Formula: see text], heat flux [Formula: see text] and convective heat transfer at surface are axial coordinate dependent. Boundary layer equations and boundary conditions are transformed into nonlinear ODEs by introducing unusual and generalized similarity transformations for the variables. These simplified equations are solved numerically. Final ODEs represent suction/injection, stretching/shrinking, temperature, heat flux, convection effects and specific heat. This current problem encompasses all previous models as special cases which come under the scope of above statement (title). The results of classical models are scoped out as a special case by assigning proper values to the parameters. Numerical result shows that the dual solutions can be found for different possible values of the shrinking parameter. A stability analysis is accomplished and apprehended in order to establish a criterion for determining linearly stable and physically compatible solutions. The significant features and diversity of the modeled equations are scrutinized by recovering the previous problems of fluid flow and heat transfer from a uniformly heated sheet of variable (uniform) thickness with variable (uniform) stretching/shrinking and injection/suction velocities.

Application of generalized Fourier heat conduction law on MHD viscoinelastic fluid flow over stretching surface

2019 ◽  
Vol 30 (6) ◽  
pp. 3481-3496 ◽  
Author(s):  
Arif Hussain ◽  
Muhammad Yousaf Malik ◽  
Mair Khan ◽  
Taimoor Salahuddin
Keyword(s):  
Heat Flux ◽  
Fluid Flow ◽  
Friction Factor ◽  
Numerical Technique ◽  
Wall Heat Flux ◽  
Constitutive Law ◽  
Wall Friction ◽  
Theoretical Physics ◽  
Stretching Surface ◽  
Content Type

Purpose The purpose of current flow configuration is to spotlights the thermophysical aspects of magnetohydrodynamics (MHD) viscoinelastic fluid flow over a stretching surface. Design/methodology/approach The fluid momentum problem is mathematically formulated by using the Prandtl–Eyring constitutive law. Also, the non-Fourier heat flux model is considered to disclose the heat transfer characteristics. The governing problem contains the nonlinear partial differential equations with appropriate boundary conditions. To facilitate the computation process, the governing problem is transmuted into dimensionless form via appropriate group of scaling transforms. The numerical technique shooting method is used to solve dimensionless boundary value problem. Findings The expressions for dimensionless velocity and temperature are found and investigated under different parametric conditions. The important features of fluid flow near the wall, i.e. wall friction factor and wall heat flux, are deliberated by altering the pertinent parameters. The impacts of governing parameters are highlighted in graphical as well as tabular manner against focused physical quantities (velocity, temperature, wall friction factor and wall heat flux). A comparison is presented to justify the computed results, it can be noticed that present results have quite resemblance with previous literature which led to confidence on the present computations. Originality/value The computed results are quite useful for researchers working in theoretical physics. Additionally, computed results are very useful in industry and daily-use processes.

scholarly journals A Solution to Pressure Equation with Its Boundary Condition of Combining Tangential and Normal Pressure Relations

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1507
Author(s):  
Hui Xiao ◽  
Wei Liu
Keyword(s):  
Boundary Condition ◽  
Numerical Algorithm ◽  
Iteration Process ◽  
Normal Pressure ◽  
Constitutive Law ◽  
Navier Stokes ◽  
Numerical Verification ◽  
Pressure Equation ◽  
Navier Stokes Equation ◽  
Pressure Boundary Condition

Pressure is a physical quantity that is indispensable in the study of transport phenomena. Previous studies put forward a pressure constitutive law and constructed a partial differential equation on pressure to study the convection with or without heat and mass transfer. In this paper, a numerical algorithm was proposed to solve this pressure equation by coupling with the Navier-Stokes equation. To match the pressure equation, a method of dealing with pressure boundary condition was presented by combining the tangential and normal direction pressure relations, which should be updated dynamically in the iteration process. Then, a solution to this pressure equation was obtained to bridge the gap between the mathematical model and a practical numerical algorithm. Through numerical verification in a circular tube, it is found that the proposed boundary conditions are applicable. The results demonstrate that the present pressure equation well describes the transport characteristics of the fluid.

scholarly journals Diffusion of dust particles emitted from a fixed source

2015 ◽  
Vol 4 (4) ◽  
pp. 454
Author(s):  
Khaled Al-mashrafi
Keyword(s):  
Source Function ◽  
Strong Dependence ◽  
Vertical Direction ◽  
Dust Particles ◽  
Special Values ◽  
Diffusion Parameters ◽  
Special Cases ◽  
Small Times ◽  
The Mathematical Model ◽  
General Time

<p>In this paper, we investigate the mathematical model for the diffusion of dust particles emitted from a fixed source. Mathematically, the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time is solved. The solution in closed form for a source of general time dependence is obtained. A number of special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken are examined in detail. The numerical calculations show the strong dependence of the concentration of dust on the speed of the wind, the source, and its position in the vertical direction. It is also found that the diffusion parameters play an important role in the spread of the dust particles in the atmosphere. When diffusion is present only in the vertical direction, it is found that for small times the dust spreads with a front that travels with the speed of the wind.</p>

On the Generalized Navier-Stokes Equations

2003 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Ahmed Salem
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Hankel Transforms ◽  
Fourier Sine ◽  
Special Cases

In this paper, we present a general Inodel of the classical Navier-Stokes equations. With the help of Laplace, Fourier Sine transforms, finite Fourier Sine transforms, and finite Hankel transforms, an exact solutions for three different special cases have been obtained.

HYDRODYNAMIC CALCULATION OF CONTACTLESS SEALS WITH PLANE SLOTS IN DRIVES OF ELECTRIC POWER SYSTEMS

2021 ◽  
Vol 11 (2) ◽  
pp. 171-177
Author(s):  
Evgeny A. KRESTIN ◽  
Grigoriy V. SEREBRYAKOV
Keyword(s):  
Pressure Drop ◽  
Power Systems ◽  
Electric Power ◽  
Constant Pressure ◽  
Saturated Vapor ◽  
Electric Power Systems ◽  
Navier Stokes ◽  
Friction Forces ◽  
Continuity Equations ◽  
Special Cases

Non-contact seals with fl at slott ed gaps of drives of electric power systems used in switchgears of hydraulic units, as well as in pumps and hydraulic motors have been investigated. Calculation of seals based on average clearance results in an underestimation or overestimation of the leakage rate compared to the operational values. The regularity of the distribution of pressure and fl ow rate in the gap of a fl at conical slot is determined, and formulas for the fl ow rate (leakage) and friction forces acting on the walls of the conical slot are found. To solve the problem, the approximate Navier-Stokes and fl ow continuity equations are used. Several special cases of the fl ow of the working fl uid in diff erent gaps are considered: a plane-parallel gap with an oscillating wall and at a constant pressure gradient and a conical gap at diff erent ratios of the pressure drop and the frictional action of the moving channel wall. When the wall oscillates in a conical gap and constant pressure, the presence of an extremum is characteristic. In this case, an excess pressure appeared in the slott ed gap, creating a supporting force, and the pressure value became high enough. When the lower wall of the conical slot moves in the direction of the increasing gap, the pressure inside the slott ed channel, under certain conditions, can reach a complete vacuum, the value of which is limited by the bulk strength of the liquid and the pressure of saturated vapor at a given temperature. When the pressure drop and oscillations of the wall of the conical gap are additive, then at a suffi ciently high velocity of the wall movement, the pressure inside the slot can even increase and exceed the value of the supplied pressure.

scholarly journals Differential Sea-Ice Drift. II. Comparison of Mesoscale Strain Measurements to Linear Drift Theory Predictions

1974 ◽  
Vol 13 (69) ◽  
pp. 457-471 ◽  
Author(s):  
W. D. Hibler
Keyword(s):  
High Pressures ◽  
Constitutive Law ◽  
Elastic Behavior ◽  
Local Pressure ◽  
Strain Measurements ◽  
Linear Drift ◽  
Special Cases ◽  
Drift Theory ◽  
Sea Ice Drift ◽  
Frequency Behavior

A comparison of mesoscale strain measurements with the atmospheric pressure field and the wind velocity field indicate that the ice divergence rate and vorticity follow the local pressure and wind divergence with significant correlation. For low atmospheric pressures and converging winds the divergence rate was found to be negative with the vorticity being counter-clockwise. The inverse behavior was observed for high pressures and diverging winds. This behavior was shown to agree with predictions based upon the infinite boundary solution of a linearized drift theory in the absence of gradient current effects and using the constitutive law proposed by Glen (1970) for pack ice. The best least-squares values of the constitutive law parametersηandζwere found to be ≈ 1012kg/s. Using typical divergence rates these values yield compressive stresses of the magnitude of 105N/m which are similar to values suggested by the Parmerter and Coon (1972) ridge model. In general, the infinite boundary solution of the linear drift equation indicates that in a low-pressure region that is reasonably localized in space, the ice would be expected to converge for high compactness (winter) and diverge for low compactness (summer).Calculations were also carried out using a more general linear visco-elastic constitutive law that includes memory effects and which includes a generalized Hooke’s law as well as the Glen law as special cases. A best fit of this more general calculation with strain measurements indicates overall a better agreement with viscous behavior than with elastic behavior, with the frequency behavior of the estimated “viscosities” similar to the Glen law behavior at temporal frequencies less than ≈ 0.01 h−1.

scholarly journals The Heat Flux Vector(s) in a Two Component Fluid Mixture

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 77
Author(s):  
A. D. Kirwan ◽  
Mehrdad Massoudi
Keyword(s):  
Heat Flux ◽  
Interaction Force ◽  
Heat Fluxes ◽  
Fluid Mixture ◽  
Two Fluids ◽  
Heat Flux Vector ◽  
Special Cases ◽  
Different Temperatures ◽  
Phenomenological Coefficients ◽  
Kinematic Properties

Bulk kinematic properties of mixtures such as velocity are known to be the density weighed averages of the constituent velocities. No such paradigm exists for the heat flux of mixtures when the constituents have different temperatures. Using standard principles such as frame indifference, we address this topic by developing linear constitutive equations for the constituent heat fluxes, the interaction force between constituents, and the stresses for a mixture of two fluids. Although these equations contain 18 phenomenological coefficients, we are able to use the Clausius-Duhem inequality to obtain inequalities involving the principal and cross flux coefficients. The theory is applied to some special cases and shown to reduce to standard results when the constituents have the same temperature.

Numerical Investigation of a Transient Operation of a Heat Pipe With Experimental Validation

2004 ◽  
Author(s):  
Gustavo Gutierrez ◽  
Juan Catan˜o ◽  
Tien-Chien Jen
Keyword(s):  
Heat Flux ◽  
Heat Pipe ◽  
Stokes Equations ◽  
Solid Wall ◽  
Control Volume ◽  
Navier Stokes ◽  
Interface Temperature ◽  
Vapor Flow ◽  
Navier Stokes Equation ◽  
Wick Structure

In this paper, a full transient analysis of the performance of a heat pipe with a wick structure is performed. For the vapor flow, the conventional Navier-Stokes equations are used. For the liquid flow in the wick structure, which is modeled as a porous media, volume averaged Navier-Stokes equation are adopted. The energy equation is solved for the solid wall and wick structure of the heat pipe. The energy and momentum equations are coupled through the heat flux at the liquid-vapor interface that defines the suction and blowing velocities for the liquid and vapor flow. The evolution of the vapor-liquid interface temperature is coupled through the heat flux at this interface that defines the mass flux to the vapor and the new saturation conditions to maintain a fully saturation vapor all the time. A control volume approach is used in the development of the numerical scheme. A parametric study is conducted to study the effect of different parameters that affect the thermal performance of the heat pipe. And experimental setup is developed and numerical results are validated with experimental data. The results of this study will be useful for the heat pipe design and implementation in processes that are essentially transient and steady state conditions are not reached like for example drilling applications.

Axisymmetric Stagnation—Point Flow and Heat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent Angular Velocity and Uniform Transpiration

2006 ◽  
Vol 129 (1) ◽  
pp. 106-115 ◽  
Author(s):  
A. B. Rahimi ◽  
R. Saleh
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Reynolds Number ◽  
Angular Velocity ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow And Heat Transfer ◽  
Self Similar

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite rotating circular cylinder with transpiration U0 are investigated when the angular velocity and wall temperature or wall heat flux all vary arbitrarily with time. The free stream is steady and with a strain rate of Γ. An exact solution of the Navier-Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the angular velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may rotate with constant speed, with exponentially increasing/decreasing angular velocity, with harmonically varying rotation speed, or with accelerating/decelerating oscillatory angular speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semi-similar solutions of the unsteady Navier-Stokes equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent rotation velocity of the cylinder is, for example, a step-function. All the solutions above are presented for Reynolds numbers, Re=Γa2∕2υ, ranging from 0.1 to 1000 for different values of Prandtl number and for selected values of dimensionless transpiration rate, S=U0∕Γa, where a is cylinder radius and υ is kinematic viscosity of the fluid. Dimensionless shear stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder rotating with certain exponential angular velocity function and at particular value of Reynolds number is azimuthally stress-free. Heat transfer is independent of cylinder rotation and its coefficient increases with the increasing suction rate, Reynolds number, and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration.

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