汪志明版流体力学英文版chap

,汪志明版流体力学英文版chap6,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,汪志明版流体力学英文版chap6,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,*,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,汪志明版流体力学英文版chap,汪志明版流体力学英文版,chap6,Stress tensor,where is normal stress;is shear stress.,The subscript denote the outward normal line of the,surface on which stress apply.While denote the direction,of the stress.,2,汪志明版流体力学英文版,chap6,Stokes assume:1)fluid is isotropic;2)stress tensor is,continuous function of transformative velocity tensor ;,3)in static fluid,stress tensor infinitely approach to the,pressure function of static fluid.,On the above hypothesis,the general Newtonian internal,friction theorem can be written as:,here is called as the second viscosity.,The above equation is also called constitutional equation.,3,汪志明版流体力学英文版,chap6,Put the constitutional equation into Cauchy equation,then,regularize it with considering the incompressible fluid,continuity equation ,get,Above equation is known as Navier-Stokes equation,or,N-S equation.,4,汪志明版流体力学英文版,chap6,The N-S equation in Cartesian coordinates:,5,汪志明版流体力学英文版,chap6,The N-S equation in Cylindrical coordinates:,6,汪志明版流体力学英文版,chap6,Kinetic Energy Equation,The change of kinetic energy in unit time for specific,volume fluid is relative to the total change of works done by,outside force,pressure and viscous force,in unit time.,7,汪志明版流体力学英文版,chap6,Introduce viscous stress ,then,Substitute the viscous stress into kinetic energy expression,so,kinetic energy equation in integral form is:,It shows that the change of works done by outside force,pressure and viscous force is the sum of change of kinetic,and mechanical energy in unit time and in specific volume fluid.,The mechanical energy include reversible and irreversible,parts,which is used to balance the viscous force and dissipate in,thermal energy form.,8,汪志明版流体力学英文版,chap6,Dissipation function is defined to explain the change rate of irreversible mechanical energy:,In Cartesian coordinates,dissipation function is:,and in Cylindrical coordinates,it is,9,汪志明版流体力学英文版,chap6,Specially,if the compressibility can be ignored,is,Using divergence theorem,the differential kinetic energy,equation is given:,It is for incompressible fluid with linear viscosity:,10,汪志明版流体力学英文版,chap6,Internal Energy Equation,Consider internal energy ,thermal vector,and stress ,put them into internal energy,equation,gives,Its differential form is,11,汪志明版流体力学英文版,chap6,If gradient of temperature is zero,it become so called,Neumann equation:,It is again written as:,Above equation is also the first law of thermodynamics.,12,汪志明版流体力学英文版,chap6,Define enthalpy ,specific heat with constant,volume ,and let ,thermal,energy is got after putting them into Neumann equation:,13,汪志明版流体力学英文版,chap6,Define entropy ,the increase of enthalpy for,unit mass can be written as:,using,Neumann equation,then gives:,Since ,so,or ,this is the second law of thermodynamics.,14,汪志明版流体力学英文版,chap6,By make model we can study the fluid mechanics problem,about huge scale object in laboratory.,Consider two flow fields,including the model and the,prototype,respectively.If the ratio of corresponding forces,F,2,/F,1,in both flow fields is same,or there is a similarity of,forces,we call it as dynamic similitude.,Geometrically similar:a similarity of shape,Kinematic similitude:a similarity of streamline,15,The very basis of dimensional analysis is the principle of,dimensional homogeneity,which states every term in a,complete physical equation has the same measure formula.,For example:,The above equation is dimensionally homogeneity with,dimension being L4.A constant reference length l is used to,make dimensionless equation by dividing each term with L4:,汪志明版流体力学英文版,chap6,16,The Buckingham Pi ()theorem states that“any complete,physical relationship can be represented as one subsisting,between a set of independent nondimensional product com-,binations of the physical quantities concerned.,The least possible number of independent nondimensional,quantities which appear in the relationship is equal to the,number of related physical quantities less the number of,the fundamental units.,汪志明版流体力学英文版,chap6,17,To solve problems using,theorem,follow these steps:,1.Isolate the physical quantities from the given problem.,Identify the quantities Q,n,and the number n.,2.Select the M,L,T or F,L,T system.For each quantity Q,select appropriate dimensions and determine number m.,3.Construct the dimensional matrix and evaluate rank r.,4.Evaluate the(n-r)dimensionless coefficients.,5.Apply F(,1,2,n-r,)=0 to obtain the desired,empirical relationship governing the physical problem.,汪志明版流体力学英文版,chap6,18,Example 6.1 Consider a inclined cylindrical pipe with uniform,cross section,inner diameter D,axial length l,absolute rough-,ness,.Flow is stead and without end effects.If the average,velocity V,dynamic viscosity,and density,represent visco-,sity and inertia.Find the empirica