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The correct ANSWER is (d) \(\iint_V\rho \vec{V}.d\vec{S}\)

The explanation is: In general,

MASS FLOW rate=density × VELOCITY × area

For this CASE,

elemental mass flow rate = \(\rho \vec{V}.d \vec{S}\)

total mass flow rate=\(\iint_V\rho \vec{V}.d\vec{S}\)

Correct choice is (d) Incompressible flows at high Mach number

To explain I would say: EULERIAN equations are best suited for examining incompressible flows at high Mach number. They are used to study FLOW over the WHOLE AIRCRAFT.

The correct choice is (a) K

The best explanation: k represents thermal conductivity. As DIFFUSION in HEAT TRANSFER is heat conduction, diffusion COEFFICIENT becomes, thermal conductivity.

Right choice is (d) Eulerian

The best EXPLANATION: In diagram, fluid PARTICLES move and the field of observation remains in the same POSITION. This REPRESENTS Eulerian Approach.

Correct CHOICE is (b) Temperature

The explanation is: ENERGY conservation should be solved for a FLUID FLOW if we WANT the temperature distribution or if the system involves heat transfer.

Right option is (d) Control MASS, Control VOLUME, Control volume

Best explanation: Statement of Reynolds TRANSPORT Theorem: “The INSTANTANEOUS total change of B inside the control mass is equal to the instantaneous total change of B within the control volume plus the net flow of B into and out of the control volume”.

Right option is (B) The rate of CHANGE of element’s VOLUME

Easiest explanation: Applying mass conservation for this element,

time rate of change of mass = 0

\(\frac{D(\RHO \delta V)}{Dt}=0 \)

\(\delta V \frac{D(\rho)}{Dt}+\rho\frac{D(\delta V)}{Dt}=0 \)

\(\frac{D(\rho)}{Dt}+\rho\frac{1}{\delta V}\frac{D(\delta V)}{Dt}=0 \)

\(\frac{D(\rho)}{Dt}+\rho\nabla.\rho=0 \)

Thus, the term arises from the rate of change of element’s volume.

Right OPTION is (a) \(\VEC{V}.\nabla T\)

Easiest explanation: \(\vec{V}.\nabla T\) (dot product of VELOCITY vector gradient of T) is the convective derivative which is the time rate of change due to the MOVEMENT of the fluid element.

Right choice is (c) Eulerian

Explanation: REYNOLDS transport THEOREM can be used to convert the material volume form of the conservation EQUATIONS to Eulerian form.

Correct choice is (a) The rate of change of the total energy is EQUAL to the rate of heat addition and work EXTRACTION

To explain I would SAY: The first LAW of thermodynamics applied to a system states that “The rate of change of the total energy is equal to the rate of heat addition and work extraction”.

The CORRECT answer is (c) \(\rho(\NABLA.(p\vec{V})+\nabla.(\tau.\vec{V})) dx \,DY \,DZ\)

To elaborate: The RATE of work done is power which is the product of force and velocity. This can be represented by \((\nabla.(p\vec{V})+\nabla.(\tau.\vec{V})) dx \,dy \,dz\).

Right ANSWER is (a) \(\frac{\PARTIAL\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{ZY}}{\partial z}\)

Easy explanation: \(\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{zy}}{\partial z}\) are the DIFFUSION terms of the y-momentum equation. It involves all the shear STRESS tensors which are in the y-direction.

The correct option is (a) Newton’s SECOND LAW of motion

To elaborate: Momentum equation is derived using Newton’s second law of motion. This gives a relationship between force and acceleration. It ALSO gives the condition for momentum conservation.

The correct choice is (a) \(\nabla.(\rho \VEC{V})=0\)

To explain I WOULD say: Taking the continuity equation,

\(\frac{\PARTIAL\rho}{\partial t}+\nabla.(\rho \vec{V})=0\)

For steady flow, flow variables do not vary with time.

\(\frac{\partial\rho}{\partial t}=0\)

Thus, for steady flow \(\nabla.(\rho \vec{V})=0\).

The correct ANSWER is (b) \(div(\rho\Phi\vec{u})\)

Easy explanation: \(div(\rho\Phi\vec{u})\) is the convective TERM of the TRANSPORT EQUATION. CONVECTION is the flow of fluid into and out of the model of observation.

The correct choice is (c) \(\NABLA.\vec{V} = \frac{1}{\DELTA V}\frac{D(\delta V)}{DT} \)

To explain: For control volumes,

\(\frac{DV}{Dt}=\iiint_{V}(\nabla.\vec{V})dV\)

For INFINITESIMAL elements, V can be represented as δ V. Therefore,

\(\frac{D(\delta V)}{Dt}=\iiint_{\delta V}(\nabla.\vec{V})dV\)

As the elements are infinitesimally small, ∇.\(\vec{V}\) is the same for all elements. Hence,

\(\frac{D(\delta V)}{Dt}=(\nabla.\vec{V})\delta V\)

\(\nabla \vec{V}=\frac{1}{\delta V} \frac{D(\delta V)}{Dt}\).

Correct option is (a) Position coordinates are dependent on time

The best I can EXPLAIN: As the flow model is not stationary, its position coordinates vary ALONG with time. So, it RESULTS in a non-conservative equation.

Right choice is (d) Non-conservative integral

To explain: The given DIAGRAM represents a FINITE control VOLUME MOVING along with the flow. This will give a non-conservative integral EQUATION.

Right option is (d) CONTINUITY equation

For explanation: Continuity equation is used as given below.

\(\rho\FRAC{De}{Dt}=\rho\frac{\PARTIAL e}{\partial t}+\rho\VEC{V}.\nabla e \)

But,

\(\rho\frac{\partial e}{\partial t}=\frac{\partial(\rho e)}{\partial t}-e\frac{\partial \rho}{\partial t}\)

And

\(\rho\vec{V}.\nabla e=\nabla.(\rho e\vec{V})-e\nabla.(\rho \vec{V})\)

Therefore,

\(\rho\frac{De}{Dt}=\frac{\partial(\rho e)}{\partial t}-e\frac{\partial \rho}{\partial t}+\nabla.(\rho e\vec{V})-e\nabla.(\rho \vec{V})\)

\(\rho\frac{De}{Dt}=\frac{\partial(\rho e)}{\partial t}-e(\frac{\partial \rho}{\partial t}+\nabla.(\rho \vec{V}))+\nabla.(\rho e\vec{V})\)

APPLYING the continuity equation, \(\frac{\partial \rho}{\partial t}+\nabla.(\rho \vec{V})=0\), and hence

\(\rho\frac{De}{Dt}=\frac{\partial(\rho e)}{\partial t}+\nabla.(\rho e\vec{V})\).

Correct answer is (d) \(DIV(\rho u\vec{V})\)

For explanation I would SAY: : From general CONSERVATION EQUATION, the convection term is div(ρuΦ). For momentum equation, \(\Phi=\vec{V}\). So, the convection term in momentum equation is div(ρu\(\vec{V}\)).

The CORRECT choice is (B) div(ΓgradΦ)

The best explanation: div(ΓgradΦ) represents diffusion. Diffusion is the movement of FLUID from a high concentration to low concentration WITHIN the system.

Correct ANSWER is (B) Only moving models

To ELABORATE: Substantial derivatives arise as the coordinates MOVE and they VARY with time. So, they are applicable only to moving models.

The correct option is (B) Time RATE of change of the volume of a moving fluid element per unit volume

To explain I would say: DIVERGENCE of VELOCITY of a moving fluid model physically means that “time rate of change of the volume of a moving fluid element per unit volume”.

Correct option is (c) \(\rho\vec{f}.\vec{V}\)dx DY dz

To elaborate: MASS = density×volume

mass = ρdx dy dz

rate of WORK done = FORCE×velocity

rate of work done = \(\rho\vec{f}.\vec{V}\)dx dy dz

\(\rho\vec{f}.\vec{V}\)dx dy dz is the work done by the body force.

Correct choice is (d) x, y, Z, t

To explain I would say: There are four independent variables in Navier-Stokes EQUATIONS. Three spatial variables (x, y, z) and ONE time VARIABLE (t).

Correct answer is (a) 3

Easy explanation: Τxy indicate that the STRESS COMPONENT ACTS in the y-direction on a surface normal to the x-direction. REPRESENTING this in the diagram, 3 is the corresponding tensor.

The correct choice is (b) Directions of STRESS and NORMAL to the surface on which they are acting

The explanation: The two subscripts of stress TENSORS indicate the DIRECTION of the stress and that of the normal to the surface on which they act. So, stress tensors GIVE the location and direction of the stresses.

Correct option is (a) Control mass SYSTEM, Control VOLUME system

For explanation: EQUATIONS FORMED by considering the control mass system and control volume system are not the same even if the same physical law is used. A relation between these equations is established by Reynolds transport theorem.

Correct ANSWER is (d) Fourier’s law of heat CONDUCTION

To elaborate: Fourier’s law of heat conduction gives the relationship between heat ENERGY and temperature. This is used in the energy equation to convert heat TERMS to temperature terms.

The correct answer is (a) \(\rho\frac{Dv}{Dt}=-\nabla p+\rho g\)

The explanation: \(\frac{\rho Dv}{Dt}=-\nabla p+\rho g\) represents a Euler equation. All other EQUATIONS have this TERM μ∇^2v representing DIFFUSION.

Right answer is (c) z-momentum equation

To explain I would say: Mass DIFFUSION of the continuity equations are in general not included in the Navier-Stokes equations. This is because most of the fluid FLOW and thermodynamic processes do not include any CHANGE in concentration which is mass diffusion.

The correct option is (c) \(\frac{D\rho}{Dt}+\rho \nabla.\VEC{V}=0\)

To elaborate: \(\frac{\partial\rho}{\partial t}+\nabla.(\rho \vec{V})=0\) is the non-conservative DIFFERENTIAL EQUATION. Non-conservative differential equation is given by an infinitesimally small fluid element MOVING along with the flow.

Right option is (c) Total derivative

For EXPLANATION: SUBSTANTIAL derivative is the same as total derivative. However, total derivative is completely mathematical.

\(\FRAC{DT}{Dt}=\frac{\partial T}{\partial t}+U \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}+w \frac{\partial T}{\partial z}\)

\(\frac{DT}{Dt}=\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}+w \frac{\partial T}{\partial z}\).

The CORRECT choice is (a) \(\FRAC{D\vec{V}}{Dt}\)

For EXPLANATION: \(\frac{D\vec{V}}{Dt}\) is the substantial DERIVATIVE. \(\frac{d\vec{V}}{dt}\) is the local derivative. \(\frac{\partial \vec{V}}{\partial t}\) is the partial derivative.

Right choice is (d) Pathline

The explanation: Pathline is the one which represents the path of a FLUID element along its way. So, to DEFINE the positions of the parcels, pathline is the BEST.

The correct answer is (b) Stationary FINITE CONTROL volume

The EXPLANATION is: The diagram represents a finite control volume stationary in position. Fluid flows into and out of this model.

The CORRECT OPTION is (C) 5

Easiest explanation: There are 5 equations RELATED to solving a flow field.

Mass conservation equation

x-momentum equation

y-momentum equation

z-momentum equation

Energy equation

The correct ANSWER is (d) \(\tau=\mu\left\{(\nabla \vec{v})^T+(\nabla.\vec{v})^T\right\}+\lambda(\nabla.\vec{v})I\)

To explain: The shear stress tensor of a fluid element can be given by \(\tau=\mu\left\{(\nabla \vec{v})^T+(\nabla.\vec{v})^T\right\}+\lambda(\nabla.\vec{v})I\). This is not applicable for practical cases. HOWEVER, common fluids like AIR and WATER are assumed to be Newtonian for USING this relationship.

Correct option is (c) Net mass flow through the control SURFACE = RATE of change of mass inside the control volume

For EXPLANATION I WOULD say: Statement of mass CONSERVATION equation for a finite control volume fixed in space:

Net mass flow through the control surface is equal to the rate of change of mass inside the control volume.

Correct CHOICE is (a) Mass conservation

To EXPLAIN I would say: Continuity equation is derived from the mass conservation principle. It states that for an isolated system, the mass of the system must REMAIN CONSTANT.

The correct CHOICE is (a) Rate of change

Easy EXPLANATION: This term represents the rate of change of FLUID property INSIDE the model of flow. It does not involve any kind of flow of the property.