Conjugate equation of heat conduction for an isotropic linear viscoelastic body

1972 ◽  
Vol 23 (3) ◽  
pp. 1199-1201
Author(s):  
V. V. Kryuchkovskii ◽  
P. I. Khristichenko
Keyword(s):  
Heat Conduction ◽  
Viscoelastic Body ◽  
Conjugate Equation ◽  
Linear Viscoelastic ◽  
Equation Of Heat Conduction

scholarly journals Simplified Grinding Temperature Model Study

2019 ◽  
Vol 6 (2) ◽  
pp. a1-a7
Author(s):  
N. V. Lishchenko ◽  
V. P. Larshin ◽  
H. Krachunov
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Heat Source ◽  
Boundary Conditions ◽  
Grinding Temperature ◽  
Temperature Model ◽  
One Dimensional ◽  
Heating And Cooling ◽  
Equation Of Heat Conduction ◽  
The One

A study of a simplified mathematical model for determining the grinding temperature is performed. According to the obtained results, the equations of this model differ slightly from the corresponding more exact solution of the one-dimensional differential equation of heat conduction under the boundary conditions of the second kind. The model under study is represented by a system of two equations that describe the grinding temperature at the heating and cooling stages without the use of forced cooling. The scope of the studied model corresponds to the modern technological operations of grinding on CNC machines for conditions where the numerical value of the Peclet number is more than 4. This, in turn, corresponds to the Jaeger criterion for the so-called fast-moving heat source, for which the operation parameter of the workpiece velocity may be equivalently (in temperature) replaced by the action time of the heat source. This makes it possible to use a simpler solution of the one-dimensional differential equation of heat conduction at the boundary conditions of the second kind (one-dimensional analytical model) instead of a similar solution of the two-dimensional one with a slight deviation of the grinding temperature calculation result. It is established that the proposed simplified mathematical expression for determining the grinding temperature differs from the more accurate one-dimensional analytical solution by no more than 11 % and 15 % at the stages of heating and cooling, respectively. Comparison of the data on the grinding temperature change according to the conventional and developed equations has shown that these equations are close and have two points of coincidence: on the surface and at the depth of approximately threefold decrease in temperature. It is also established that the nature of the ratio between the scales of change of the Peclet number 0.09 and 9 and the grinding temperature depth 1 and 10 is of 100 to 10. Additionally, another unusual mechanism is revealed for both compared equations: a higher temperature at the surface is accompanied by a lower temperature at the depth. Keywords: grinding temperature, heating stage, cooling stage, dimensionless temperature, temperature model.

scholarly journals Influence of cheese ripening on the viscoelastic behaviour of edam cheese

2013 ◽  
Vol 19 (No. 1) ◽  
pp. 1-7 ◽  
Author(s):  
J. Buchar ◽  
I. Kubiš ◽  
S. Gajdůšek ◽  
I. Křivánek
Keyword(s):  
Viscoelastic Properties ◽  
Rheological Model ◽  
Deformation Behaviour ◽  
Viscoelastic Body ◽  
Original Method ◽  
Cheese Ripening ◽  
Linear Viscoelastic ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
Pressure Bar

The paper deals with the study of the effect of cheese ripening on parameters of a rheological model of cheese mechanical behaviour. The Edam cheese has been tested by the method of the Hopkinson Split Pressure Bar. The original method of the evaluation of viscoelastic properties has been used. The rheological model of the three element linear viscoelastic body, so called “standard linear solid” has been used. This model successfully describes the experimentally observed deformation behaviour of cheese specimens. The effect of the time of cheese ripening on the parameters of the rheological model has been demonstrated.

scholarly journals Mathematical Model of a Three-Dimensional Non-Isothermal Glacier

1976 ◽  
Vol 16 (74) ◽  
pp. 308-309
Author(s):  
S.S. Grigoryan ◽  
M.S. Krass ◽  
P.A. Shumskiy
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Small Parameter ◽  
Heat Generation ◽  
Curvilinear Coordinates ◽  
Shear Stresses ◽  
System Of Equations ◽  
Equation Of Heat Conduction

Abstract In the case of a non-isothermal glacier it is necessary to integrate the equations of dynamics together with the equation of heat conduction, heat transfer, and heat generation because of the interdependence (1) of strain-rate of ice on its temperature, and (2) of ice temperature on the rate of heat transfer by moving ice and on the intensity of heat generation in its strain. In view of the complexity of the whole system of equations, simplified mathematical models have been constructed for dynamically different glaciers. The present model concerns land glaciers with thicknesses much less than their horizontal dimensions and radii of curvature of large bottom irregularities, so that the method of a thin boundary layer may be used. The principal assumption is the validity of averaging over a distance of the order of magnitude of ice thickness. Two component shear stresses parallel to the bottom in glaciers of this type considerably exceed the normal stresses and the third shear stress, so the dynamics are described by a statically determined system of equations. For the general case, expressions for the stresses have been obtained in dimensionless affine orthogonal curvilinear coordinates, parallel and normal to the glacier bottom, and taking into account the geometry of the lower and upper surfaces. The statically undetermined problem for ice divides is solved using the equations of continuity and rheology, so the result for stresses depends considerably on temperature distribution. In the case of a flat bottom the dynamics of an ice divide is determined by the curvature of the upper surface. The calculation of the interrelating velocity and temperature distributions is made by means of the iteration of solutions (1) for the components of velocity from the stress expressions using the rheological equations (a power law or the more precise hyberbolic one) with the assigned temperature distribution, and (2) for the temperature with the assigned velocity distribution. The temperature distribution in the coordinate system used is determined by a parabolic equation with a small parameter at the principal derivative. Its solution is reduced to the solution of a system of recurrent non-uniform differential equations of the first order by means of a series expansion of the small parameter: the right part for the largest term of the expansion contains a function of the heat sources, and for the other terms it contains the second derivative along the vertical coordinate from the previous expansion term. Thus advection makes the main contribution to the heat transfer, and temperature in a glacier is distributed along the particle paths, changing simultaneously under the influence of heat generation. A relatively thin conducting boundary layer adjoins the upper and lower surfaces of a glacier, playing the role of a temperature damper in the ablation area. The equation of heat conduction (at the free surface) or of heat conduction and heat transfer (at the bottom) with the boundary conditions, and with the condition of the connection with the solution of the problem for the internal temperature distribution, is being solved for the boundary layer because of its small thickness. Beyond the limits of the boundary layer, heat conduction makes a small change in the temperature distribution, which can be calculated with any degree of accuracy.

Sign in / Sign up

Export Citation Format

Share Document