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Water Resources Engineering
Published in P.K. Jayasree, K Balan, V Rani, Practical Civil Engineering, 2021
P.K. Jayasree, K Balan, V Rani
This is known in other fields as the diffusion equation or heat equation, it is a parabolic partial differential equation (PDE). This mathematical statement indicates that the change in hydraulic head with time (left-hand side) equals the negative divergence of the flux (q) and the source terms (G). This equation has both head and flux as unknowns, but Darcy’s law relates flux to hydraulic heads, so substituting it in for the flux (q) leads to Ss∂h∂t=−∇⋅(−K∇h)−G
Heat flux measurements: theory and applications
Published in Kaveh Azar, in Electronic Cooling, 2020
The analysis of both transient and periodic heat conduction problems is based on the diffusion equation, which is a parabolic partial differential equation. () α∂2T∂X2=∂T∂t
Heat Transfer Distributed Parameter Systems
Published in Nayef Ghasem, Modeling and Simulation of Chemical Process Systems, 2018
The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in each region over time. Use the general differential equation that describes the heat transfer in cylindrical coordinate: () ρCp∂T∂t+ρCp(vr∂T∂r+vθ1r∂T∂θ+vz∂T∂z)=k(1r∂∂r(r∂T∂r)+1r2∂2T∂θ2+∂2T∂z2)+ΦH
Determination of a spacewise-dependent heat source by a logarithmic-type regularization method
Published in Applicable Analysis, 2023
Bin Hu, Shuoping Xie, Zewen Wang
Let Ω be a bounded domain in with smooth boundary , and be the closure of Ω. Consider the second-order parabolic partial differential equation where is a linear elliptic operator of second order on , i.e. Moreover, is assumed to be symmetric and uniform with , which means that and with positive constants θ and η.
Inverse method identification of thermophysical properties based on solotone effect analysis for discontinuous Sturm–Liouville systems
Published in Inverse Problems in Science and Engineering, 2019
Joseph Fourier formulated and solved the transient heat conduction equation. This parabolic partial differential equation connects the thermophysical properties of a material to its temperature. Specifically, where β is the temperature, k the thermal conductivity and c the heat capacity.2 The temperature β is a function of time and any number of spatial dimensions. The thermophysical quantities k and c are positive, as is apparent from their definitions. In reality, the thermal conductivity and the heat capacity of a material varies with temperature, and hence with position and time. However, k and c are often assumed constant in any particular problem.
Effect of Atangana–Baleanu fractional derivative on a two-dimensional thermoviscoelastic problem for solid sphere under axisymmetric distribution
Published in Mechanics Based Design of Structures and Machines, 2023
Polymers are considered a significantly important industry material due to their physical properties and ease of deformation. So, this importance prompted Gross (1953) to find a mathematical model that describes this phenomenon. This model has been developed by many authors like Ferry (1961), Gurtin and Sternberg (1962), Stratonva (1971), and Pobedrya (1979). Il’yushin (1968) and Pobedrya (1969) introduced the coupled theory of thermoviscoelasticity. This theory has been discussed by Kovalenko and Karnaukhov (1972) and Medri (1988). Contrary to physical observations, the heat equation of this theory predicts an infinite speed of propagation for heatwaves because it is a parabolic partial differential equation.