Dumitru Vieru

Starting solutions for the flow of second grade fluids in a rectangular channel due to an oscillating shear stress

The unsteady motion of a second grade fluid between two parallel side walls perpendicular to a plate is studied by means of the Fourier sine and cosine transforms. Initially, the fluid is at rest and at time t = 0+, the plate applies an oscillating shear to the fluid. The solutions that have been obtained, presented under integral and series form and written as a sum between steady time-periodic and transient solutions can be easily reduced to the similar solutions for Newtonian fluids performing the same motion. They describe the motion of the fluid some time after its initiation. After that time, when the transient solutions disappear, the motion of the fluid is described by the steady time-periodic solutions that are independent of the initial conditions. In the absence of side walls, more exactly when the distance between walls tends to infinity, all solutions reduce to those corresponding to the motion over an infinite plate. As it was to be expected, the steady time-periodic solutions corresponding to sine and cosine oscillations of the shear stress on the boundary differ by a phase shift. Finally, the influence of side walls on the fluid motion, the required time to reach the steady periodic flow, as well as the distance between walls for which the velocity of the fluid in the middle of the channel is unaffected by their presence are established by numerical calculus and graphical illustrations. As expected, the time needed to reach the steady periodic flows is lower in the presence of side walls. It is lower for Newtonian fluids in comparison with second grade fluids and greater for sine oscillations in comparison to the cosine oscillations of the boundary shear.The unsteady motion of a second grade fluid between two parallel side walls perpendicular to a plate is studied by means of the Fourier sine and cosine transforms. Initially, the fluid is at rest and at time t = 0+, the plate applies an oscillating shear to the fluid. The solutions that have been obtained, presented under integral and series form and written as a sum between steady time-periodic and transient solutions can be easily reduced to the similar solutions for Newtonian fluids performing the same motion. They describe the motion of the fluid some time after its initiation. After that time, when the transient solutions disappear, the motion of the fluid is described by the steady time-periodic solutions that are independent of the initial conditions. In the absence of side walls, more exactly when the distance between walls tends to infinity, all solutions reduce to those corresponding to the motion over an infinite plate. As it was to be expected, the steady time-periodic solutions corresponding ...

Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source

Free convection flow of some water based fractional nanofluids over a moving infinite vertical plate with uniform heat flux and heat source is analytically and graphically studied. Exact solutions for dimensionless temperature and velocity fields, Nusselt numbers, and skin friction coefficients are established in integral form in terms of modified Bessel functions of the first kind. These solutions satisfy all imposed initial and boundary conditions and reduce to the similar solutions for ordinary nanofluids when the fractional parameters tend to one. Furthermore, they reduce to the known solutions from the literature when the plate is fixed and the heat source is absent. The influence of fractional parameters on heat transfer and fluid motion is graphically underlined and discussed. The enhancement of heat transfer in such flows is higher for fractional nanofluids in comparison with ordinary nanofluids. Moreover, the use of fractional models allows us to choose the fractional parameters in order to get a...

Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel

Time-nonlocal generalized model of the natural convection heat transfer and nanofluid flows through a rectangular vertical channel with wall conditions of the Robin type are studied. The generalized mathematical model with time-nonlocality is developed by considering the fractional constitutive equations for the shear stress and thermal flux defined with the time-fractional Caputo derivative. The Caputo power-law non-local kernel provides the damping to the velocity and temperature gradient; therefore, transport processes are influenced by the histories at all past and present times. Analytical solutions for dimensionless velocity and temperature fields are obtained by using the Laplace transform coupled with the finite sine-cosine Fourier transform which is suitable to problems with boundary conditions of the Robin type. Particularizing the fractional thermal and velocity parameters, solutions for three simplified models are obtained (classical linear momentum equation with damped thermal flux; fractional shear stress constitutive equation with classical Fourier’s law for thermal flux; classical shear stress and thermal flux constitutive equations). It is found that the thermal histories strongly influence the thermal transport for small values of time t. Also, the thermal transport can be enhanced if the thermal fractional parameter decreases or by increasing the nanoparticles’ volume fraction. The velocity field is influenced on the one hand by the temperature of the fluid and on the other by the damping of the velocity gradient introduced by the fractional derivative. Also, the transport motions of the channel walls influence the motion of the fluid layers located near them.Time-nonlocal generalized model of the natural convection heat transfer and nanofluid flows through a rectangular vertical channel with wall conditions of the Robin type are studied. The generalized mathematical model with time-nonlocality is developed by considering the fractional constitutive equations for the shear stress and thermal flux defined with the time-fractional Caputo derivative. The Caputo power-law non-local kernel provides the damping to the velocity and temperature gradient; therefore, transport processes are influenced by the histories at all past and present times. Analytical solutions for dimensionless velocity and temperature fields are obtained by using the Laplace transform coupled with the finite sine-cosine Fourier transform which is suitable to problems with boundary conditions of the Robin type. Particularizing the fractional thermal and velocity parameters, solutions for three simplified models are obtained (classical linear momentum equation with damped thermal flux; fractiona...

Natural convection flows and heat transfer with exponential memory of a Maxwell fluid with damped shear stress

Abstract Unsteady nonlinear boundary layer convection flows and heat transfer of a fractional Maxwell fluid near a vertical plate with constant thermal flux are studied. The fractional constitutive equations for the shear stress and thermal flux are formulated for the first time with the integral time-fractional operator of type Caputo–Fabrizio. These types of fractional relationships provide a weighted average exponential to the velocity gradient and temperature gradient. The studied model is constituted from a system of nonlinear partial differential equations coupled with fractional differential equations with initial and boundary conditions. The solutions for the shear stress, velocity, thermal flux and temperature are obtained by using the Laplace transform along with a suitable transformation of variables. In order to finding solutions of the studied model, two inverse Laplace transforms of exponential type are obtained. The particular cases when only the stress or thermal flux are damped, as well as the convection flows of the ordinary fluid are obtained as particular cases when the stress/thermal fractional parameter tends to 1.

On the Motion Induced by a Flat Plate That Applies Oscillating Shear Stresses to an Oldroyd-B Fluid: Applications

Starting solutions corresponding to the motion induced by an infinite flat plate that applies oscillating shear stresses to an Oldroyd-B fluid are developed using the Laplace and Fourier sine transforms. These solutions, presented as a sum between steady-state and transient solutions, describe the motion of the fluid some time after its initiation. After that time, when the transients disappear, the starting solutions tend to the steady-state solutions which are periodic in time and independent of the initial conditions. However, they satisfy the governing equation and boundary conditions. As a check of results, the known solution corresponding to the motion due to an infinite plate that applies a constant shear to the fluid is obtained as a limiting case of our cosine solution. Exact solutions for Maxwell and Newtonian fluids performing the same motions are obtained as special cases of general solutions. Furthermore, as an application, the solutions corresponding to the motion produced by an oscillating plate are provided.