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The velocity of an unsteady flow of a viscous fluid of the second-grade MHD-type enclosed between two parallel side walls perpendicular to a plate was obtained by applying the integral transformation. The fluid is required to move by the plate, which over time t = 0 + subjected the fluid to shear stress. The solutions satisfy the given equation as well as the boundary and initial conditions, and they were separated into two types: steady state and transient state. Furthermore, through h → ∞ , we are able to recover the results found in the literature for motion across an infinite plate. Graphs depict the effect of the side walls and the time it takes to reach the steady state. The solutions are shown in graphs and discussed physically to examine the impact of different flow parameters. It is found that the fluid velocity decreases with an increasing fractional parameter β and second-grade parameter α. Also, it is noticed that the fluid velocity decreases with increasing values of Reynolds number and effective permeability. Numerous industrial products, including honey, paints, varnishes, coffee, chocolate and jelly, use this type of fluid flow concept. [ABSTRACT FROM AUTHOR]

The magnetohydrodynamic flow through a porous medium of a second-grade fluid between two side walls induced by an infinite plate that exerts an accelerated shear stress to the fluid over an infinite plate is examined. Expressions for velocity and shear stress are determined with the help of integral transforms. In the absence of side walls, all the solutions that have been obtained are reduced to those corresponding to the motion over an infinite flat plate. The Newtonian solutions are also obtained as limiting case of the general solution. Finally, influence of magnetic and porosity parameter is graphically highlighted. [ABSTRACT FROM AUTHOR]

Structured Silicon Carbide was proposed to be an ideal template for the production of arrays of edge specific graphene nanoribbons (GNRs), which could be used as a base material for graphene transistors. We prepared periodic arrays of nanoscaled stripe-mesas on SiC surfaces using electron beam lithography and reactive ion etching. Subsequent epitaxial graphene growth by annealing is differentiated between the basal-plane mesas and the faceting stripe walls as monitored by means of atomic force microscopy (AFM). Microscopic low energy electron diffraction (μ-LEED) revealed that the graphene ribbons on the facetted mesa side walls grow in epitaxial relation to the basal-plane graphene with an armchair orientation at the facet edges. The π-band system of the ribbons exhibits linear bands with a Dirac like shape corresponding to monolayer graphene as identified by angle-resolved photoemission spectroscopy (ARPES). [ABSTRACT FROM AUTHOR]

• A method for calculating the projected shade in solar greenhouse is proposed. • The distribution of solar radiation intensity in solar greenhouse are clarified. • The growth of tomato in different zones in solar greenhouse are tested. • The effects of azimuth and solar altitude angle on indoor shadow area were explored. The side walls and fixed insulation quilt of a solar greenhouse helped reduce heat loss from the room at night but produced shadows on the greenhouse floor during the day, affecting the growth and development of crops. In the present study, a method to calculate the projected area of the side walls and fixed insulation quilt in the greenhouse was derived. Then, a typical-shaped solar greenhouse and tomatoes were used as experimental research objects to investigate the spatial and temporal variation of indoor solar radiation intensity and tomato growth. The results showed that: 1) The average absolute error between the calculated and measured values of the ground shade of the solar greenhouse on four typical sunny days was 1.12%, 2.65%, and 2.02%, respectively. The area of selected greenhouse was 450m2, the maximum shaded area could account for 20.3% of the total area of the greenhouse, the shaded area projected by the insulation onto the greenhouse floor remained constant at different times, and that projected by the wall decreased and then increased parabolically with time. 2) The average solar radiation intensity in the greenhouse at the winter solstice tended to increase and then decrease with time, with a maximum value of 41,374.0 Lux. The uniformity of solar radiation distribution was reduced in the morning, evening and midday hours. 3) The tomato plant height, stem diameter, leaf area index and yield were 36.9%, 14.7%, 63.1% and 125.4% higher, respectively, in the high than in the low solar radiation zone. The results of the study can provide a theoretical basis for clarifying the dynamics of shaded areas and the spatial and temporal distribution of solar radiation intensity in greenhouses, and for optimizing the structural parameters of solar greenhouses. [ABSTRACT FROM AUTHOR]

Starting solutions corresponding to the oscillating motion of a Maxwell fluid between side walls perpendicular to a plate are established using integral transforms. Such solutions are scarce in the literature, the motion of the fluid being due to an oscillating shear on the boundary. The solutions corresponding to the motion over an infinite plate that applies an oscillating shear to the fluid are obtained as limiting cases of general solutions. All solutions are presented as a sum between steady-state and transient solutions and can easily be particularized to give the similar solutions for Newtonian fluids. They describe the motion of the fluid some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of initial conditions. However, they satisfy the boundary conditions and governing equations. Finally, the distance between walls for which the velocity of the fluid in the middle of the channel in unaffected by their presence and the required time to reach the steady-state are graphically determined. [ABSTRACT FROM AUTHOR]

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations. [ABSTRACT FROM AUTHOR]

Abstract: The velocity field and the adequate tangential stresses corresponding to the unsteady flow of an Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate are established by means of Fourier sine transforms. The solutions corresponding to Maxwell, second grade and Newtonian fluids, performing the same motion, appear as limiting cases of the solutions obtained here. In the absence of the side walls, namely when the distance between walls tends to infinity, all solutions that have been determined reduce to those corresponding to the flow over an infinite plate. Finally, for comparison, the velocity field at the middle of the channel as well as the shear stress on the bottom wall is plotted as a function of y for several values of t and of the material constants. The influence of the side walls on the motion of the fluid is also emphasized by graphical illustrations. [Copyright &y& Elsevier]

Abstract: The unsteady flow of an incompressible generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate has been studied using Fourier sine and Laplace transforms. The obtained solutions for the velocity field and shear stresses, written in terms of the generalized G and R functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. For α = β =1 and λ r → λ these solutions are going to the corresponding Newtonian solutions. Furthermore, the solutions for generalized Maxwell fluids as well as those for ordinary Oldroyd-B and Maxwell fluids, performing the same motion, are also obtained as limiting cases of our general solutions. In the absence of the side walls, namely when the distance between the two walls tends to infinity, the solutions corresponding to the motion over an infinite constantly accelerating plate are recovered. For λ r →0 and β →1, these solutions reduce to the known solutions from the literature. Finally, the effect of the material parameters on the velocity profile is spotlighted by means of the graphical illustrations. [Copyright &y& Elsevier]

Abstract: The unsteady flow of a Maxwell fluid caused by a suddenly moved plane wall between two side walls perpendicular to the plane is considered. Closed form solution is developed employing the Fourier sine transforms. The solution obtained for large values of time is similar to the steady solution of a Newtonian fluid. The unsteady solution for a Newtonian fluid have been recovered as a limiting case by choosing . The physical interpretation of the solution is presented through the graph and tables. [Copyright &y& Elsevier]

Abstract: Exact solutions for the unsteady flow of a second grade fluid induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane are established by means of the Fourier sine transforms. The similar solutions for Newtonian fluids, performing the same motions, are obtained as limiting cases for . The steady solutions, the same for Newtonian and non-Newtonian fluids, are also obtained as limiting cases for . In the absence of the side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Graphical illustrations show that the diagrams corresponding to the velocity field in the middle of channel and the shear stress at the bottom wall for a second grade fluid are going to be those for a Newtonian fluid if the normal stress module . [Copyright &y& Elsevier]

Abstract: The unsteady flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate is investigated. Exact solutions for the velocity field are established by means of the Fourier and Laplace transforms. The similar solutions for Maxwell and Newtonian fluids can be obtained as limiting cases of our results. In the absence of side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate. [Copyright &y& Elsevier]

In this article, unidirectional flows of fractional viscous fluids in a rectangular channel are studied. The flow is generated by the shear stress given on the bottom plate of the channel. The authors have developed a generalized model on the basis of constitutive equations described by the time-fractional Caputo–Fabrizio derivative. Many authors have published different results by applying the time-fractional derivative to the local part of acceleration in the momentum equation. This approach of the fractional models does not have sufficient physical background. By using fractional generalized constitutive equations, we have developed a proper model to investigate exact analytical solutions corresponding to the channel flow of a generalized viscous fluid. The exact solutions for velocity field and shear stress are obtained by using Laplace transform and Fourier integral transformation, for three different cases namely (i) constant shear, (ii) ramped type shear and (iii) oscillating shear. The results are plotted and discussed. [ABSTRACT FROM AUTHOR]