Acoustic diffraction by a half-plane in a viscous fluid medium

We consider the diffraction of a time-harmonic acoustic plane wave by a rigid half-plane in a viscous fluid medium. The linearized equations of viscous fluid flow and the no-slip condition on the half-plane are used to derive a pair of disjoint Wiener-Hopf equations for the fluid stresses and velocities. The Wiener-Hopf equations are solved in conjunction with a requirement that the stresses are integrable near the edge of the half-plane. Specific wave components of the scattered velocity field are given analytically. A Padé approximation to the Wiener-Hopf kernel function is used to derive numerical results that show the effect of viscosity on the velocity field in the immediate vicinity of the edge of the half-plane.     

Effects of chemical reactions on unsteady MHD free convection and mass transfer for flow past a hot vertical porous plate with heat generation/absorption through porous medium

The paper studies the effects of chemical reactions on unsteady MHD free convection and mass transfer flow of a viscous, incompressible, electrically-conducting fluid past an infinite hot vertical porous plate embedded in porous medium. Heat generation/absorption and viscous dissipation effects are included. The temperature of the plate is assumed to be spanwise cosinusoidally fluctuating with time. The governing equations are solved by perturbation technique. Numerical evaluation of the analytical results is performed. Graphical results for transient velocity and transient temperature profiles and tabulated results for skin-friction coefficient and Nusselt number are presented and discussed.     

Acoustic scattering by a rigid elliptic cylinder in a slightly viscous medium

A complete solution is obtained for the two-dimensional diffraction of a time-harmonic acoustic plane wave by an impenetrable elliptic cylinder in a viscous fluid. Arbitrary size, ellipticity, and angle of incidence are considered. The linearized equations of viscous flow are used to write down expressions for the dilatation and vorticity in terms of products of radially and angular dependent Mathieu functions. The no-slip condition on the rigid boundary then determines the coefficients. The resulting computations are facilitated by recently developed library routines for complex input parameters. The solution for the circular cylinder serves as a guide and a differently constructed solution for the strip is also given. Typical results in the "resonant" range of dimensionless wave number, displaying the surface vorticity and the far-field scattering pattern are included, with the latter allowing comparison with the inviscid case.     

Variable viscosity and thermal conductivity effects on MHD flow and heat transfer in viscoelastic fluid over a stretching sheet

The problem of flow and heat transfer of an electrically conducting viscoelastic fluid over a continuously stretching sheet in the presence of a uniform magnetic field is analyzed for the case of power-law variation in the sheet temperature. The fluid viscosity and thermal conductivity are assumed to vary as a function of temperature. The basic equations comprising the balance laws of mass, linear momentum, and energy modified to include the electromagnetic force effect, the viscous dissipation, internal heat generation or absorption and work due to deformation are solved numerically.     

Effect of the ion slip on the MHD flow of a dusty fluid with heat transfer under exponential decaying pressure gradient

In the present study, the unsteady Hartmann flow with heat transfer of a dusty viscous incompressible electrically conducting fluid under the influence of an exponentially decreasing pressure gradient is studied without neglecting the ion slip. The parallel plates are assumed to be porous and subjected to a uniform suction from above and injection from below while the fluid is acted upon by an external uniform magnetic field applied perpendicular to the plates. The equations of motion are solved analytically to yield the velocity distributions for both the fluid and dust particles. The energy equations for both the fluid and dust particles including the viscous and Joule dissipation terms, are solved numerically using finite differences to get the temperature distributions.     

Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions

This article investigates the natural convection flow of viscous incompressible fluid in a channel formed by two infinite vertical parallel plates. Fully developed laminar flow is considered in a vertical channel with steady-periodic temperature regime on the boundaries. The effect of internal heating by viscous dissipation is taken into consideration. Separating the velocity and temperature fields into steady and periodic parts, the resulting second order ordinary differential equations are solved to obtain the expressions for velocity, and temperature. The amplitudes and phases of temperature and velocity are also obtained as well as the rate of heat transfer and the skin friction on the plates. In presence of viscous dissipation, fluids of relatively small Prandtl number has higher temperature than the channel plates and as such, heat is being transferred from the fluid to the plate.     

Viscous scattering of a pressure wave: calculation of the fluid tractions on a biomimetic acoustic velocity sensor

In the paper we give a method for calculating the tractions (local forces) of the fluid motion determined by an incoming plane pressure wave on an artificial hair cell transducer structure. The sensing element of the transducer is a standing high aspect ratio cilium in the shape of a narrow thin curved beam (tape-like), which can be easily fabricated in micro-/nanotechnology. The method is based on considering the system of partial differential equations describing the motion of the compressible viscous fluid in an acoustic linearized approximation, and representation of the velocity field as a viscous acoustic single-layer potential. The boundary conditions, stating the cancellation of the velocity components on the solid beam, yield a two-dimensional (2-D) system of three integral equations over the beam's surface for the traction components. In the case of a narrow cilium, the system of integral equations furnishes a system of two 1-D integral equations over the symmetry curve of the structure for obtaining the tangential and normal components of the traction. This system is solved numerically by a finite (boundary) element method. The numerical code written for solving the problem was applied to some particular structures. The last structure is similar to the trichobothrium of a spider Cupiennius salei. The results obtained show that the curvature of the hair is enhancing sensitivity to flows directed normal to the main shaft of the hair confirming the assertion of Barth et al. [Philos. Trans. R. Soc. London, Ser. B 340, 445-461 (1993)].     

Behavior of nonlinear fluid viscous dampers for control of shock vibrations

Fluid viscous dampers have been widely used for suppression of high velocity shocks. While linear fluid viscous dampers have been used for a long time, nonlinear fluid viscous dampers show considerable promise due to their superior energy dissipation characteristics and significant reduction in the damper force compared to a linear fluid viscous damper for the same peak displacement. This paper presents results from experimental study to characterize fluid viscous dampers when subjected to half-cycle sine shock excitation. The mathematical formulation and a numerical study to evaluate the relative performance of structures with fluid viscous dampers subjected to short-duration shock (impulse) loading are also discussed. The influence of damper nonlinearity (α) and the supplemental damping ratio (ξ_sd) on response has been investigated. The supplemental damping ratio of nonlinear fluid viscous dampers when subjected to shock excitation is found by equivalent linearization using the concept of equal energy dissipation. The paper also presents some design charts, which can be used for preliminary decisions on parameters of nonlinear dampers to be used in design.     

Free convection effects and radiative heat transfer in MHD Stokes problem for the flow of dusty conducting fluid through porous medium

The present note deals with the effects of radiative heat transfer and free convection in MHD for a flow of an electrically conducting, incompressible, dusty viscous fluid past an impulsively started vertical non-conducting plate, under the influence of transversely applied magnetic field. The heat due to viscous dissipation and induced magnetic field is assumed to be negligible. The governing linear partial differential equations are solved by finite difference technique. The effects of various parameters (like radiation parameter N, Prandtl number Pr, porosity parameter K) entering into the MHD Stokes problem for flow of dusty conducting fluid have been examined on the temperature field and velocity profile for both the dusty fluid and dust particles.     

Efficiency of acoustic heating produced in the thermoviscous flow of a fluid with relaxation

Instantaneous acoustic heating of a fluid with thermodynamic relaxation is the subject of investigation. Among others, viscoelastic biological media described by the Maxwell model of the viscous stress tensor, belong to this type of fluid. The governing equation of acoustic heating is derived by means of the special linear combination of conservation equations in differential form, allowing the reduction of all acoustic terms in the linear part of the final equation, but preserving terms belonging to the thermal mode responsible for heating. The procedure of decomposition is valid for weakly nonlinear flows, resulting in the nonlinear terms responsible for the modes interaction. Nonlinear acoustic terms form a source of acoustic heating in the case of dominative sound, which reflects the thermoviscous and dispersive properties of a fluid. The method of deriving the governing equations does not need averaging over the sound period, and the final governing dynamic equation of the thermal mode is instantaneous. Some examples of acoustic heating are illustrated and discussed, conclusions about efficiency of heating caused by different sound impulses are made.     

Curle's equation and acoustic scattering by a sphere

Recent papers have initiated interesting comparisons between aeroacoustic theory and the results of acoustic scattering problems. In this paper, we consider some aspects of these comparisons for acoustic scattering by a sphere. We give a derivation of Curle's equation for a specific class of linear acoustic scattering problems, and, in response to previous claims to the contrary, give an explicit confirmation of Curle's equation for plane wave scattering by a stationary rigid sphere of arbitrary size in an inviscid fluid. We construct the complete solution for scattering by a rigid sphere in a viscous fluid, and show that the neglect of viscous terms in Curie's equation yields an incomplete prediction of the far field dipole pressure. We also consider the null field solution of the sphere scattering problem, and give its extension to the vorticity modes associated with viscosity. Finally, we construct a solution for an elastic sphere in a viscous fluid, and show that the rigid sphere/null field solution is recovered from the limit of infinite longitudinal and shear wave speeds in the elastic solid.     

Advanced Study of Unsteady Heat and Chemical Reaction with Ramped Wall and Slip Effect on a Viscous Fluid

This paper investigate the effect of slip boundary condition, thermal radiation, heat source, Dufour number,chemical reaction and viscous dissipation on heat and mass transfer of unsteady free convective MHD flow of a viscous fluid past through a vertical plate embedded in a porous media. Numerical results are obtained for solving the nonlinear governing momentum, energy and concentration equations with slip boundary condition, ramped wall temperature and ramped wall concentration on the surface of the vertical plate. The influence of emerging parameters on velocity,temperature and concentration fields are shown graphically.     

Joule heating and viscous dissipation effects on MHD flow past a semi-infinite inclined plate with variable surface temperature

A numerical investigation is performed to study the MHD free convection flow past a semi-infinite inclined plate subjected to a variable surface temperature. The Joule heating and viscous dissipation effects are taken into account in the energy equation. The governing equations of the flow are transformed into a nondimensional form using suitable dimensionless quantities. A fully developed implicit finite-difference scheme of Crank-Nicolson type is engaged to solve the dimensionless governing equations, which is more accurate, fast convergent, and unconditionally stable. The effects of the MHD, inclination angle, power law, Grashof number, Prandtl number, Joule heating, and viscous dissipation effects are studied on the velocity, temperature, shear stress, and heat transfer coefficients during transient periods. It is observed that the MHD has retarding effects on velocity.     

Fluid dynamics phenomena induced by power ultrasounds

Propagation of power ultrasound (from 20 to 800 kHz) through a liquid inside a cylindrical reactor initiates acoustic cavitation and also fluid dynamics phenomena such as free surface deformation, convection, acoustic streaming, etc. Mathematical modelling is performed as a new approach to predict where active bubbles are and how intense cavitation is. A calculation based on fluid dynamics equations is undertaken using computational fluid dynamics code; this is of great interest because such code provides not only the pressure field but also velocity and temperature fields. The link between the acoustic pressure and the cavitation field is clearly established. Moreover, the pressure profile near a free surface allows one to predict the shape of the acoustic fountain. The influence of the acoustic fountain on the wave propagation is shown to be important. The convective flow inside a reactor is numerically obtained and agrees well with particle image velocity measurements. Non-linearities arising from the dissipation of the acoustic wave are computed and lead to the calculation of the acoustic streaming. The superimposed velocity field (convective flow and acoustic streaming) succeeds in simulating the bubble behaviour at 500 kHz, for instance.     

Acoustic heating produced in the thermoviscous flow of a Bingham plastic

This study is devoted to the instantaneous acoustic heating of a Bingham plastic. The model of the Bingham plastic’s viscous stress tensor includes the yield stress along with the shear viscosity, which differentiates a Bingham plastic from a viscous Newtonian fluid. A special linear combination of the conservation equations in differential form makes it possible to reduce all acoustic terms in the linear part of of the final equation governing acoustic heating, and to retain those belonging to the thermal mode. The nonlinear terms of the final equation are a result of interaction between sounds and the thermal mode. In the field of intense sound, the resulting nonlinear acoustic terms form a driving force for the heating. The final governing dynamic equation of the thermal mode is valid in a weakly nonlinear flow. It is instantaneous, and does not imply that sounds be periodic. The equations governing the dynamics of both sounds and the thermal mode depend on sign of the shear rate. An example of the propagation of a bipolar initially acoustic pulse and the evolution of the heating induced by it is illustrated and discussed.     

Investigations of Viscous Dissipation in Stagnation Point Flow Past a Stretchable Riga Wall: Modern Analysis of Heat Transport

This article scrutinizes the features of viscous dissipation in the stagnation point flow past through a linearly stretched Riga wall by implementing Cattaneo-Christov heat flux model. Viscous dissipation is carried out in Cattaneo-Christov diffusion analysis for the first time in this letter. As a result of Cattaneo-Christov model, some extra terms of viscous dissipation are appeared in the energy equation. These extra terms of viscous dissipation are missing in the literature. On the utilization of suitable transformations, the equations governing the problem are reduced under the boundary layer approximation into the non-linear and dimensionless ordinary differential equations. Convergent approach is utilized to solve the dimensionless governing equations. The solution thus acquired is used to highlight the effects of emerging parameters on velocity distribution and fluid's temperature through the graphs. Features of the drag force (or skin friction co-efficient) are graphically interpreted. It is noticed that the presence of modified Hartman number helps to reduce the fluid's temperature but enhances the velocity profile. Further an enlargement in the value of thermal time relaxation parameter helps to decrease the temperature distribution.     

Energy velocity and quality factor of poroelastic waves in isotropic media

The energy velocity and Q factor of poroelastic acoustic waves in the context of classical isotropic Biot's theory are revisited. Special attention is paid to the high frequency regime when interphase interaction is viscoelastic. The analogy with viscoelastic behavior is emphasized in derivation of the energy balance equations which relate kinetic energy, potential energy, viscous power dissipation, and elastic energy stored associated with each wave. These lead to exact closed form expressions for the energy velocity and Q factor for both longitudinal and shear waves from energy principles. Most notably, the analysis of the resulting expressions reveals that the energy velocity of both longitudinal and shear waves equals (exceeds) the corresponding phase velocity in the case of the low (full) frequency range theory, and that the exact expression for the Q factor contains an additive correction due to viscoelastic interphase interaction.     

Mean force on a small sphere in a sound field in a viscous fluid

A mean force exerted on a small rigid sphere by a sound wave in a viscous fluid is calculated. The force is expressed as a sum of drag force coming from the external steady flow existing in the absence of the sphere and contributions that are cross products of velocity and velocity derivatives of the incident field. Because of the drag force and an acoustic streaming generated near the sphere, the mean force does not coincide with the acoustic radiation pressure, i.e., the mean momentum flux carried by the sound field through any surface enclosing the sphere. If the sphere radius R is considerably smaller than the viscous wave penetration depth delta, the drag force can give the leading-order contribution (in powers of delta/R) to the mean force and the latter can then be directed against the radiation pressure. In another limit, delta< or =R, the drag force and acoustic streaming play a minor role, and the mean force reduces to the radiation pressure, which can be expressed through source strengths of the scattered sound field. The effect of viscosity can then be significant only if the incident wave is locally plane traveling.     

Analytical and numerical calculations of optimum design frequency for focused ultrasound therapy and acoustic radiation force

Focused ultrasound therapy relies on acoustic power absorption by tissue. The stronger the absorption the higher the temperature increase is. However, strong acoustic absorption also means faster attenuation and limited penetration depth. Hence, there is a trade-off between heat generation efficacy and penetration depth. In this paper, we formulated the acoustic power absorption as a function of frequency and attenuation coefficient, and defined two figures of merit to measure the power absorption: spatial peak of the acoustic power absorption density, and the acoustic power absorbed within the focal area. Then, we derived “rule of thumb” expressions for the optimum frequencies that maximized these figures of merit given the target depth and homogeneous tissue type. We also formulated a method to calculate the optimum frequency for inhomogeneous tissue given the tissue composition for situations where the tissue structure can be assumed to be made of parallel layers of homogeneous tissue. We checked the validity of the rules using linear acoustic field simulations. For a one-dimensional array of 4 cm acoustic aperture, and for a two-dimensional array of 4 × 4 cm² acoustic aperture, we found that the power absorbed within the focal area is maximized at 0.86 MHz, and 0.79 MHz, respectively, when the target depth is 4 cm in muscle tissue. The rules on the other hand predicted the optimum frequencies for acoustic power absorption as 0.9 MHz and 0.86 MHz, respectively for the 1D and 2D array case, which are within 6% and 9% of the field simulation results. Because radiation force generated by an acoustic wave in a lossy propagation medium is approximately proportional to the acoustic power absorption, these rules can be used to maximize acoustic radiation force generated in tissue as well.     

Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation

Acoustic waves in tissues and weakly attenuative fluids often have an attenuation parameter, alpha(omega), satisfying alpha(omega)= alpha0omegay in which alpha0 is a constant, omega is the frequency, and y is between 1 and 2. This power law attenuation is not predicted by the classical thermoviscous wave equation and researchers have proposed different modified viscous wave equations in which the loss term is a convolution operator or a fractional spatial or temporal derivative. In this paper, acoustic waves undergoing power law attenuation are modeled by a modification to the thermoviscous wave equation in which the time derivative of the viscous term is replaced by a fractional time derivative. An explicit time domain, finite element formulation leads to a stable algorithm capable of simulating axisymmetric, broadband acoustic pulses propagating through attenuative and dispersive media. The algorithm does not depend on the Born approximation, long wavelength limit, or plane wave assumptions. The algorithm is validated for planar and focused transducers and results include radiation patterns from a viscous scatterer in a lossless background and signals reflected from a viscous layer. The program can be used to determine scattering parameters for large, strong, possibly viscous scatterers, in either a lossless or viscous background, for which analytic results are scarce.