Analytical Expressions For Velocity Research Articles

Numerical and analytical study of heat transfer in the wire coating process using Carreau–Yasuda fluid model

In this article, we present a non-isothermal study of the wire coating process using the Carreau–Yasuda fluid model. Our aim is to explore how the Carreau–Yasuda fluid model parameters are linked to key operational factors in the wire coating process. These factors encompass the radius of the coated wire, the volumetric flow rate, and the force needed to pull the wire. The equations that depict the flow and heat transfer within the die are solved: (a) analytically using the perturbation method to get the analytical expression for velocity, temperature distribution, and relevant engineering quantities; (b) numerically using shooting method with the “Runge–Kutta–Fehlberg algorithm.” The physical effects of the Weissenberg number (We), power law index (n), pressure gradient (A), and Brinkman number (Br) on the velocity profile, temperature distribution, and engineering quantities are discussed with the help of different graphs. Our findings reveal that, in the case of shear thinning (n < 1) with a constant pressure gradient, as compared to the Newtonian scenario, both the thickness of the coated wire and the force required to pull the wire decrease with an increase in the Weissenberg number (We). Moreover, the force required to pull the wire also decreases with an increase in the values of the pressure gradient parameter.

Configuration design and kinematics analysis of foldable composite space capture mechanism

To meet the requirements of high flexibility, high stiffness, and large working range of space debris cleaning technology, a foldable composite space capture mechanism is proposed based on the finite screw theory in this paper. The degree of freedom of the mechanism is analyzed by using the screw theory and its degree of freedom is obtained. The closed-loop vector method and DH method are used to analyze the kinematics of the mechanism, and the position solution of the mechanism is derived. The analytical expressions of velocity and acceleration solutions are obtained by differential calculation. According to the analysis of numerical example, the kinematic model of the mechanism is built in the simulation environment, and the simulation experiment is carried out for the motion trajectory of the mechanism. The correctness of theoretical analysis solution and method is proved by contrasting and analyzing motion trajectory, and it is verified that the space capture mechanism uses the foldable property to form large workspace to complete the capture of the target.

Heat transfer effects on the oscillatory MHD flow in a porous channel with two immiscible fluids

The MHD oscillatory flow of two immiscible, viscous liquids in a porous channel with heat transfer is the subject of this investigation. The two liquid layers with different viscosities flow in both regions. The analytical expressions for velocity and temperature distribution have been derived by solving the governing flow equations using the regular perturbation method. The effects of various parameters on the velocity, temperature, and Nusselt number have been shown graphically, and numerical values of skin friction and flow rate are presented in tabular form and discussed. According to our analysis, the mass flux reduces as the magnetic field strength rises. While the temperature of the liquid enhances with an increase in the Eckert number and the Prandtl number, the temperature distribution rises with a decrease in the thermal conductivity ratio. To validate the results, the analytical solutions are compared with the fourth-order numerical Runge–Kutta method coupled with the shooting approach, and the results are found to be in excellent agreement.

UNSTABLE MHD CASSON FLUID FLOW PHENOMENA THROUGH VERTICAL CHANNEL

Several studies have been made on unstable MHD Casson fluid flow past a perpendicular channel due to its significant executions in science and technology. For that reason, the major objective of the present article is to derive exact solutions for unstable free convection MHD flow of Casson fluid through a vertical channel. Employing suitable dimensionless variables, dimensional regulatory equations are transmuted into dimensionless form. Method of Laplace transform (MOLT) is applied to get analytical expressions for velocity and temperature of the flow. The acquired analytical results were plotted against different critical parameters to understand the flow phenomena. The obtained results are compared with earlier work with some limitations. The outcomes are in well concurrence with the existing ones.

Theoretical Analysis of Roll-over-web Coating of a Non-Newtonian Polymer Using Lubrication Approximation Theory

Roll-coating process plays an important role in many industries for its practical applications such as paint, PVC coated fabrics and plastic industries. There are several roll-to-roll coating methods including forward and reverse roll-coating. However, the roll-over-web coating study of the Rabinowitsch model is presented in this paper. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. We employ the regular perturbation technique to get analytical expressions for velocity, pressure, and pressure gradient. Engineering quantities such as power input function and roll-separating force are calculated by Runge-Kutta method. The dimensionless Rabinowitsch parameter effect on velocity, pressure, pressure gradient, load-carrying force, and power input are shown graphically. It is interesting to note that for the shear thickening case, the Rabinowitsch model predicts 35% higher pressure, while in the shear thinning case it predicts 29% less pressure in the nip region when compared to the Newtonian model. The force and power show a decreasing trend on increasing the dimensionless Rabinowitsch parameter a. Moreover, the separation point shifts right of its Newtonian value when fluid behaves like shear thickening and volumetric flow rate increases which causes the coating thickness to increase.

INFLUENCE OF MAGNETIC FIELD ON THE FLOW OF JEFFERY AND NEWTONIAN FLUIDS THROUGH COMPOSITE POROUS CHANNEL

This work concerns the flow of Jeffery and Newtonian fluids through a composite porous channel in the presence of a uniform magnetic field applied perpendicular to the direction of flow. The mathematical formulation of this problem represents the flow of Jeffery fluid through the porous channel sandwiched between two porous channels of the Newtonian fluid. The horizontal composite porous channel is divided into three porous channels of the same width and the fluid flow occurs due to a constant pressure gradient. The Jeffery equation has been used for the middle porous channel while the Brinkman equation has been employed for lower and upper porous channels. Continuity of velocities, continuity of stresses at interfaces, and impenetrability conditions at the outer surface of the composite porous channel have been used as boundary conditions. Analytical expressions for velocities, volumetric flow rate, and shear stresses are obtained for the respective channels. Effects of the viscosity ratio parameter (&amp;beta;), Hartmann number (&lt;i&gt;M&lt;/i&gt;), Jeffery parameters (&amp;lambda;&lt;sub&gt;1&lt;/sub&gt;), and permeability parameters (&amp;eta;) on the flow rate and fluid velocity are explained graphically and discussed. Numerical values of volumetric flow rate with respect to different flow parameters such as Hartmann number, viscosity ratio, and Jeffrey parameter are presented in tabular form.

Analytical investigation of transient free convection and heat transfer of a hybrid nanofluid between two vertical parallel plates

Flow and heat transfer of transient free convection of a hybrid nanofluid between two parallel plates are theoretically investigated. Effects of a magnetic field, thermal radiation, and a heat source or a sink are considered. Laplace transforms are used to solve the dimensionless governing equations. Analytical expressions of velocity and temperature profiles and shear stress and rate of heat transfer are presented. Moreover, the correlations between the parameters and the shear stress and the rate of heat transfer have been derived. Increasing alumina and copper nanoparticle volume fractions reduces the velocity and temperature. However, the converse is seen for increasing heat source or sink parameter. For increasing radiation parameter, the temperature decreases but the velocity first increases. When 5% and 10% alumina nanoparticles are mixed with 5% copper nanoparticle, the heat transfer through the left wall increases about 14% and 30% and that through the right wall enhances about 21% and 44% in comparison to the heat transfer of pure fluid. For 5% and 10% copper nanoparticles with 5% alumina nanoparticle, the corresponding values are 15% and 31% and 22% and 45%, respectively. The heat source or sink intensity also plays an important role in the thermal field and heat transfer.

Soret and heat generation effects on unsteady MHD Casson fluid flow in porous medium

This article studies motion with quadratic speed on the unsteady free convective MHD Casson fluid flow in a porous medium. The fluid in electrically conducting and bounding plate has ramped the surface concentration. The characteristics of heat and mass transfer are analyzed with thermal diffusion, heat generation, chemical reaction, and thermal radiation. The governing dimensionless equations are solved using the Laplace transform technique, and analytical expressions for velocity, temperature, and concentration are obtained. For both boundary conditions, the expression for Skin friction, Nusselt number, and Sherwood number are obtained and represented in tabular form. Effects of several pertinent parameters on velocity, temperature, and concentration distribution are presented graphically. It is seen that the mass buoyancy force increases with Soret number which leads to accelerated fluid flow. Chemical reaction and porosity of the fluid tend to reduced Skin friction coefficient, while Casson fluid and magnetic field tend to reverse effect on it.

Blood Flow with Multiple Stenoses in a Force Field

In the present manuscript, a mathematical model of steady and incompressible Casson fluid in a non-uniform tube having many stenoses in the presence of a force field is analyzed. Using mild stenosis approximation and appropriate boundary conditions, analytical expressions for velocity, pressure drop, impedance, and wall share stress have been computed due to their importance in the rheology of blood. The Casson fluid is used to depict the behavior of blood flow. The effects of different physical constraints on resistance to the flow and wall shear stress of the fluid are examined. The study ascertains that resistance to the flow and wall shear stress is maximum at duck of stenosis. It is also explored that an increase in the size of the stenosis in the artery affects the normal flow of the blood through vessels in the heart, body, and brain and this may lead to major cardiac disease problems like stroke, heart attack, etc.

Creeping flow of Jeffrey fluid through a swarm of porous cylindrical particles: Brinkman–Forchheimer model

The majority of the previous studies analyzed the flow of fluids with constant viscosity through membranes composed of porous cylindrical particles using the particle-in-cell approach with the Brinkman equation governing the flow through porous media. However, a slight variation in temperature affects the viscosity of the fluids and hence affects the filtration process of fluids through membranes. The motivation of this problem came from the fact that viscosity is concentration dependent due to presence of impurities and contaminants in the fluids and hence can be taken as function of position or temperature. The present work is a theoretical attempt to investigate the impact of temperature-dependent viscosity on the creeping flow of Jeffrey fluid through membrane consisting of the aggregates of the porous cylindrical particles. The flow pattern of the Jeffrey fluid is taken along the axial direction of the cylindrical particles, and the cell model approach is utilized to formulate the governing equations driven by a constant pressure gradient. The flow regime is divided into two-layer form, one is inside the porous cylindrical particle enclosing a solid core, which is governed by the Brinkman–Forchheimer equation, and another one is outside of the porous cylindrical particle, which is governed by the Stokes equation. Being a nonlinear equation, an analytical solution of the Brinkman–Forchheimer equation is intractable. To overcome this difficulty, the regular and singular perturbation methods have been employed to solve the Brinkman–Forchheimer equation under the assumption of temperature-dependent viscosity for small and large permeability of the porous medium, respectively; however, an analytical approach is utilized to solve the Stokes equation. The analytical expressions for velocity in different regions, hydrodynamic permeability of the membrane, and Kozeny constant are derived. The impact of various control parameters such as viscosity parameter, Forchheimer number, permeability of the porous medium, and Jeffrey fluid parameter on the above quantities are discussed and validated with previously published works on the Newtonian fluid in the limiting cases. The present work is in good agreement with the previously published work on Newtonian fluid under constant viscosity assumptions where the porous media flow was governed by the Brinkman equation. The remarkable observation of the present study is that higher viscosity and Jeffrey fluid parameters lead to enhanced velocity profile and hence the hydrodynamical permeability of the membranes. However, a decay in the Kozeny constant is observed with the increasing viscosity and Jeffrey fluid parameters. The coating of porous layer can be attributed to adsorption of polymers on the solid particles and further makes the present model to be more relevant in understanding the membrane filtration process.

Effects of Stenosis and Dilatation on Flow of Blood Mixed with Suspended Nanoparticles: A Study Using Homotopy Technique

The paper deals with a theoretical study on blood flow in a stenosed segment of an artery, when blood is mixed with nano-particles. Blood is treated here as a couple stress fluid. Stenosis is known to impede blood flow and to be the cause of different cardiac diseases. Since the arterial wall is weakened due to arterial stenosis, it may lead to dilatation /aneurysm. The homotopy perturbation technique is employed to determine the solution to the problem for the case of mild stenosis. Analytical expressions for velocity, shear stress at the wall, pressure drop, and flow resistance are derived. The impact of different physical constants on the wall shear stress and impedance of the fluid is examined by numerical simulation. Streamline patterns of the nanofluid are investigated for different situations.

Analytical Study on Couple Stress Fluid in an inclined Channel

Numerical and analytical solutions of Stokes theory of couple stress fluid under the effects of constant, space, and variable viscosity in the inclined channel are discussed here. The considered couple stress fluid is described mathematically with the definition of the stress tensor. The dimensional form of the boundary value problem is transformed into dimensionless form by defining dimensionless quantities and then solved with help of the perturbation technique. The analytical expressions of velocity and temperature of all cases based on the viscosity of the couple stress fluid are presented. For the validity of the perturbation solution, the Pseudo-Spectral collocation method is employed for each case of the viscosity model including constant, space, and Vogel’s models, respectively. The solution of the perturbation method and Pseudo-Spectral methods are shown together in the graphs. The effects of couple stress parameters on velocity and temperature distributions are also elaborated with physical reasoning in the results and discussion part. It is observed that velocity and temperature of fluid escalate via the pressure gradient parameter and Brinkman number while decelerating via couple stress parameter.

Oscillatory Viscoelastic Model of Blood Flow in Stenotic Artery

An analytic solution for the two-layered model of pulsatile blood flow in stenotic arteries is discussed. The blood flow in narrow arteries is assumed to obey the two-fluid model: non-Newtonian viscoelastic as Jeffrey fluid in core region, and Newtonian as plasma fluid in peripheral region. The present study is aimed to develop an oscillatory viscoelastic model of blood flow and obtain the analytic expressions for velocity of pulsatile blood flow in a stenotic artery. The velocity of pulsatile blood flow is used to discuss the wall shear stress and volume flow rate through stenotic artery and to relate relaxation time, retardation time, height of the stenosis with the velocity, volume flow rate and wall shear stress. The solution of the model boundary value problem is obtained by the method of special functions under the assumption of steady oscillations. It is found that the velocity is directly proportional to relaxation time and on the other hand retardation time is inversely proportional. For non-zero time period, the wall shear stress increases as relaxation time increases and decreases as retardation time increases. It reveals a significant influence of relaxation and retardation time on oscillatory blood flow in stenotic artery under the pathological state of cardiovascular disease.

Analytical Solutions for Unsteady Pipe Flows with Slip Boundary Condition

In this paper analytical expressions for time-dependent velocity profiles and pressure gradient are obtained for fully-developed laminar flows with given volume flow-rate conditions in circular pipe flows with slip boundary conditions. The governing equations are solved analytically using the traditional Laplace transform method together with Mellin’s inversion formula. The evolution of velocity profiles and pressure gradient for starting and pulsatile flow with slip boundary conditions are analyzed. New simplified expressions and perspectives on velocity and pressure gradient for no-slip and slip flows are obtained from the analytical results. New scalings in starting and pulsatile flows are proposed for pipe flows with no-slip and slip boundary conditions using non-dimensional numbers. Special attention is paid to the effect of slip factor and pulsatile flow frequency on the time-dependent skin-friction factor. Finally, by using the starting and pulsating flow results, analytical expressions of velocity and pressure for arbitrary inflow are obtained by approximating the arbitrary volume flow-rate by a Fourier series

Radiation and Diffusion Thermo Effects of Visco-Elastic Fluid Past a Porous Surface in the Presence of Magnetic Field and Chemical Reaction with Heat Source

An analysis of two dimensional free convective hydro-magnetic flow of an electrically conducting visco-elastic fluid past a vertical porous surface has been studied in the presence of magnetic field and chemical reaction of first order with heat source. Simultaneously effects of heat and mass transfer are also studied in combination of the physics of diffusion thermo (Dufour effect) and suction at the plate is assumed to be constant. The equations governing are solved by perturbation technique. Analytical expressions for velocity, temperature and concentration fields, non-dimensional skin friction coefficient and Nusselt number are obtained. The results are analyzed graphically for various values of visco-elastic parameter along with a combination of other flow parameters involved in the solution.

Thermal radiation effects on MHD flow with heat and mass transfer of micropolar fluid between two vertical walls

In the present study, the boundary layer free convective unsteady flow of an incompressible micropolar fluid under a uniform magnetic field is considered with thermal radiation in symmetric and asymmetric boundary conditions. The governing unsteady boundary layer momentum, angular momentum, energy and concentration equations with initial and boundary conditions are nondimensionalised and solved numerically. For validation of numerical solutions, analytical expression for velocity, micro-rotation, temperature and concentration profiles of steady state case have been obtained and compared through the tabular form. It is found that magnetic field , vortex viscosity and its steady state time has retarding effect on velocity and micro-rotation of micropolar fluid. For both cases (asymmetric and symmetric), thermal radiation parameter tends to improve velocity and micro-rotation profiles. Physically, due to increasing Nr, heat is generated in fluid flow, which leads to improvement in heat transfer as well as velocity throughout the fluid flow region.

Analysis of Peristaltic Transport of Non-Newtonian Fluids Through Nonuniform Tubes: Rabinowitsch Fluid Model

Abstract Peristaltic transport is an important mechanism of physiological phenomenon and peristaltic pumps. With the advancement of medical science, it has been established that the physiological fluids do not behave like Newtonian fluids. Therefore, in order to understand the behaviour and properties of physiological fluids during peristalsis, selection of appropriate fluid model is of great importance. In the present investigation, properties of peristaltic transport through nonuniform tube have been studied for non-Newtonian fluids using Rabinowitsch fluid model. Theoretical analysis has been presented for long wavelength and low Reynolds number approximation. To analyse various properties of the flow, analytical expressions for velocity, pressure gradient, pressure rise, friction force, and temperature have been obtained. The numerical results for the same have been obtained to present the effect of various physical and flow parameters on fluid velocity, pressure rise, friction force, and temperature. Significant variation of these properties has been observed in the analysis for non-Newtonian nature of the fluid and nonuniformity of the tube.

Magnetohydrodynamic Conjugate Flow of Casson Fluid Over a Vertical Plate Embedded in a Porous Medium with Arbitrary Wall Shear Stress

The influence of thermal radiation on unsteady magnetohydrodynamic (MHD) free convection flow of Casson fluid over a vertical plate embedded in porous medium with arbitrary wall shear stress has studied. Casson fluid is a non-Newtonian fluid. The partial differential equations that govern the flow, heat and mass transfer are transformed to non-dimensional by using suitable non-dimensional variables, which are then solved analytically by using the Laplace transform technique. Analytical expressions for velocity, temperature and concentration fields are determined. Additionally, the exact expressions of the Nusselt number and Sherwood number are also obtained. Numerical results of velocity, temperature and concentration profiles with various values of embedded flow parameters are graphically shown and the influences to the characteristic of the fluid flow are discussed in detail. The variations in Nusselt number and Sherwood number are displayed in tabular form.

Effects of temperature-dependent properties on wire-coating from a bath of FENE-P fluid

An analysis is carried out to study theoretically the coating of wire by drawing it from a pool of viscoelastic fluid in a pressure type-die under non-isothermal conditions. The constitutive equation of FENE-P model is used to characterize the rheology of the polymer melt. The equations describing the flow and heat transfer inside the die are solved (a) analytically for constant physical properties (b) numerically for temperature-dependent fluid properties. In the former case, the analytical expressions of velocity, volumetric flow rate, final radius of the coated wire and force required to pull the wire are reported. In the latter case, the fluid viscosity and relaxation time are considered as a function of temperature. The well-known Nahme–Griffith law is used to capture such dependence. Shooting method with Runge–Kutta–Fehlberg algorithm is employed to solve the coupled energy and momentum equations. The effects of different parameters on the velocity and the temperature distributions, volumetric flow rate, final thickness of the coated wire and force necessary to pull the wire are illustrated graphically for both the cases of constant and variable physical properties.

Heat and Mass Transfer in Three Dimensional Free Convective Oscillatory Flow in Porous Medium with Constant Heat and Mass Flux in Presence Radiation for an optically Thin Fluid

A theoretical analysis of free convective flow of viscous incompressible fluid through a porous medium bounded by a flat porous plate subject to periodic suction and constant heat and mass flux has been presented. The flow becomes three dimensional due to variation of suction velocity in transverse direction. Analytical expression for velocity, temperature, concentration and skin friction are obtained using perturbation technique. Numerical solutions are obtained for different values Grash of number (G r ), mass Grashof number(Gc), Prandtl number (Pr), Schmidt number (Sc) and Radiation (R). It is found that non-dimensional velocity increases with increase of Gr, Gc and Sc and decreases with increase of R, non-dimensional temperature decreases with increasing of R, Concentration decreases with increase of Sc. and skin friction co-efficient increases with increase of Gr and R but effect of Gr is more than R.