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Here the differentiation with respect to x$_{i}$ is taken for- mally as if x were a regular 3D vector not bound to a sphere of radius unity, and e$_{imn}$ is the Levi-Civita (unit anti-symmetric) tensor. The advantage of such a defini- tion is that not only do these spherical harmonics consti- tute an orthogonal set with respect to the integration of dot product over x , i.e. the quantity 

is zero unless j = j $^{′}$, k = k $^{′}$, l = l $^{′}$, but also operators G and K are diagonal in the chosen basis for a spherical vesicle, so that the integrals 

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are zero unless j = j $^{′}$, k = k $^{′}$, l = l $^{′}$. Given the centro- symmetry we impose, in our expansion of v ( x ) , u ( x ) , and F ( x ) , the absence of Y i 1 $_{,k,l}$( x ) , for even k, and Y i 2 $_{,k,l}$( x ) , and Y i 3 $_{,k,l}$( x ) , for odd k. In order to find the evolution equations we need to expand the velocity field to the order of O ( ε $^{2}$) . Coefficients for Y i $_{j,k,l}$( x ) for k > 6 are O ( ε $^{3}$) and thus can be neglected. It will be shown later that coefficients of 5th and 6th orders of vector spherical harmonics make corrections to the evolution equations for f$_{k,l}$ with k ≤ 4 which are O ( ε $^{3}$) . So only five first orders (starting from the zeroth one) of vector spherical harmonics should be taken into account in the expansion of any space-dependent quantity under consideration. 

In order to fulfil the condition of local inextensibility of the membrane we impose zero surface divergence of the velocity field[8]. 

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Here 

is the Jacobian matrix. We can take the derivatives as if x were a regular 3D vector, because the surface di- vergence is fully determined by the distribution of the velocity field on the membrane and does not depend on the continuation of the flow into the liquids. 

We expand equations (10,12) to order O ( ε $^{2}$) and the resulting integrals could be performed analytically. Pro- jecting the results of the integration on the space of vector spherical harmonics up to the fourth order, we obtain a set of equations satisfied by the coefficients entering the expansions of v ( x ) and Z ( x ) . Since the surface area is 

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evaluated up to order O ( ε $^{3}$) , while the velocity field in (12) is expanded only up to order O ( ε $^{2}$) , it is not appro- priate to use (12) in order to ensure the conservation of the whole surface area. Therefore, we only use the projec- tions of (12) on Y$_{k,l}$ ( x ) for k equal to 2 or 4 . Accordingly, we leave the isotropic part of Z ( x ) (denoted here as Z$_{0}$ ) undetermined. Once the final shape evolution equations are obtained we shall use the constraint that the time derivative of (9) is equal to zero in order to determine Z$_{0}$. This way of reasoning was used in previous studies [8–11]. Note that unlike other theories [8–11], since we expand the equations to higher order, it is not legitimate to replace dA ( x ) in (12) with d $^{2}$x prior to projection on the spherical harmonics sub-space. Such a substitu- tion would imply neglecting terms of order O ( ε $^{2}$) in final equations, and would be inconsistent with the spirit of the present theory. 

Having determined the velocity field thanks to the above expansions, we are in a position to obtain the fi- nal evolution equation by making use of the kinematic equation expressing the fact that the membrane velocity is equal to the fluid velocity at the membrane 

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Then the task is to substitute the expanded velocity field into this equation (in terms of coefficients of the velocity field) and project the resulting expression onto the space of spherical harmonics of interest. This then leads to the determination of the evolution equations that must be satisfied by the shape coefficients f$_{k,l}$. It can be shown that the following identities hold Y i 1 , 5 $_{,l}$( x ) x$_{i}$ = 0 , Y i 2 , 6 $_{,l}$( x ) x$_{i}$ = 7 Y$_{6}$$_{,l}$ ( x ) , and Y i 2 , 6 $_{,l}$( x ) x$_{i}$ = 6 Y$_{6}$$_{,l}$ ( x ) . There- fore the projection of the first part of the right hand side of (13) on the subspace of spherical harmonics of orders up to four does not depend on the coefficients of the vector spherical harmonics of orders five and six in the velocity field. Regarding the the second part, ∂$_{i}$f ( x ) is of order O ( ε ) and the coefficients for fifth and sixth orders of vector spherical harmonics in the velocity field expansion are of order O ( ε $^{2}$) so that their contribution in the shape evolution equation is of order O ( ε $^{3}$) , that is beyond our accuracy. 

It is convenient to decompose the applied shear flow into its elongational and rotational parts to simplify the final equations. The same decomposition can be applied to general linear flow: the quantity E$_{2}$ ( x ) = U$_{i}$ ( x ) x$_{i}$/ 2 defines the straining part, while the vector Ω$_{i}$ = e$_{ijk}$∂$_{j}$U$_{k}$ ( x ) / 2 represents the vorticity. Note, that u ( x ) in (10) is the velocity of the imposed flow evaluated at the point R ( x ) , which takes the following form 

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The shape evolution equations for the second and fourth order harmonics can be written in a compact form: 

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