Some Exact Solutions of Second Grade Fluid over the Plane moving with Constant Acceleration
Authors: Safdar Salman 
Published: September 2018

<span>Many materials such as clay coatings, drilling<br /><span>muds,suspensions, certain oils and greases, polymer<br /><span>melts,elastomers and many emulsions have been treated as <span>non–<br /><span>Newtonian fluids. It is difficult to suggest a singlemodel which<br /><span>exhibits all properties of non–Newtonianfluids as in caseof the<br /><span>Newtonian fluids. They cannotbe described in a simple model as<br /><span>for the Newtonian fluids and there has been much confusion overthe<br /><span>classification of non–Newtonian fluids. However,non–Newtonian<br /><span>fluids may be classified as: <span><strong>(i) </strong><span>fluidsfor which the shear stress<br /><span>depends on the shear rate;<span><strong>(ii) </strong><span>fluids for which the relation between<br /><span>the shearstress and shear rate depends on time; <span><strong>(iii) </strong><span>fluidswhich<br /><span>possess both elastic and viscous properties called visco elastic<br /><span>fluids or elastico–viscous fluids.Because of great diversity in the<br /><span>physical structure of non–Newtonian fluids, it does not seem<br /><span>possible to recommend a single constitutive equation for usein<br /><span>the cases described in <span><strong>(i)</strong><span>, <span><strong>(ii) </strong><span>and <span><strong>(iii)</strong><span>. Therefore,many constitutive<br /><span>equations for non–Newtonian fluids have been proposed. Most of<br /><span>them are empirical orsemi–empirical. For more general threedimensionalrepresentations the method of continuum mechanics<br /><span>is needed. Although many constitutive equationshave been<br /><span>suggested, many questions are still unsolved.Some of the<br /><span>continuum models do not givesatisfactory results in accordance</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span>with the availableexperimental data. Therefore, in many practical<br /><span>applications, empirical or semi–empirical equations havebeen used.<br /><span>A constitutive equation is a relation between stressand the<br /><span>local properties of the fluid. For a fluid at restthe stress is<br /><span>determined wholly by the static pressure.Although in the case of a<br /><span>fluid in relative motion therelation between stress and the local<br /><span>properties of thefluid is more complicated, some modifications<br /><span>maybe made such as the stress being dependent only onthe<br /><span>instantaneous distribution of fluid velocity in theneighborhood of<br /><span>the element. This distribution maybe expressed only in terms of the<br /><span>velocity gradientcomponents such as for a Newtonian fluid.<br /><span>However,non-Newtonian fluids cannot be described as simple as<br /><span>Newtonian fluids. One of the most popular modelsfor non-Newtonian<br /><span>fluids is the model that is called the second–order fluid or second<br /><span>grade fluid <span>[6]</span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></span>